Paper Title: Substrate stabilization of Jahn--Teller distortion in a single molecule.





moritz.frankerl@dipc.org
Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany
Donostia International Physics Center (DIPC), E-20018 Donostia-San Sebastián, Spain

Institute for Experimental Physics, University of Regensburg, 93040 Regensburg, Germany
Department of Physical Chemistry, University of Innsbruck, 6020 Innsbruck, Austria

Donostia International Physics Center (DIPC), E-20018 Donostia-San Sebastián, Spain
IKERBASQUE, Basque Foundation for Science, E-48013, Bilbao, Spain

jascha.repp@ur.de
Institute for Experimental Physics, University of Regensburg, 93040 Regensburg, Germany

andrea.donarini@ur.de
Institute for Theoretical Physics, University of Regensburg, 93040 Regensburg, Germany


§ ABSTRACT 
    Charge-state transitions of a single Cu-phthalocyanine molecule adsorbed on an insulating layer of NaCl on Cu(111) are probed by means of alternate charging scanning tunneling microscopy. Real-space imaging of the electronic transitions reveals the Jahn–Teller distortion occurring upon formation of the first and second anionic charge states. The experimental findings are rationalized by a theoretical many-body model which highlights the crucial role played by the substrate. The latter enhances and stabilizes the intrinsic Jahn–Teller distortion of the negatively charged molecule hosting a degenerate pair of single-particle frontier orbitals. Consequently, two excess electrons are found to occupy, in the ground state, the same localized orbital, despite a larger Coulomb repulsion than the one for the competing delocalized electronic configuration. Control over the charging sequence by varying the applied bias voltage is also predicted.        


Substrate stabilization of Jahn–Teller distortion in a single molecule
    Andrea Donarini
    August 5, 2024
======================================================================



Introduction Scanning tunneling microscopy (STM) is an excellent tool for investigating the electronic structure of single molecules on surfaces. By combining atomic-scale imaging with spectroscopy of electronic states, STM enables a comprehensive understanding of molecular-scale systems. Ultrathin insulating films of 2-3 monolayers (ML) have been used to separate the molecule from the metallic substrate and thereby inhibit their electronic hybridization, providing access to the (near-) pristine electronic structure of the molecule <cit.>. This enabled the observation of vibrational side bands <cit.>, coherent electron-nuclear coupling <cit.>, many-body correlations <cit.>, charge-state lifetimes <cit.>, and coherent spin control <cit.> on a single molecule. Furthermore, this facilitated the demonstration of single-molecule fluorescence <cit.> permitting vibronic spectroscopy with submolecular resolution <cit.>, single-molecule electrofluorochromism <cit.>, energy transfer in molecular dimers <cit.>, Anti-Kasha emissions <cit.>, switching of the excitonic state of a single molecule <cit.>, and bistability between π-diradical open- and closed-shell states <cit.>.

By employing thicker insulating films (>20 ML), electron exchange with the substrate can be completely inhibited, enabling the stabilization of multiple charge states in  single-molecule investigations <cit.>. This approach requires atomic force microscopy (AFM) for the detection of electron exchange with the tip. It enabled probing and quantifying of single-electron transfer between molecules <cit.>, the reorganization energy upon single-molecule charging <cit.>, excited molecular states <cit.>, as well as their lifetime <cit.>.
Further, based on this approach the imaging of charge-state transitions in single molecules with angstrom resolution <cit.> has been developed. This technique, dubbed single-electron alternate charging scanning tunneling microscopy (AC-STM), is based on driving a single electron to tunnel between the conductive AFM tip and a single molecule deposited on a non-conductive substrate. Charging events are controlled by short voltage pulses (V_ac) of opposite polarity that are superimposed on the DC voltage (V_ dc) across the AFM junction, inducing alternating charging of the molecule. While several charge state transitions can be accessed by gating through V_ dc, the polarity of the voltage pulses of V_ac provides selectivity in probing a specific direction of the charge-state transition <cit.>.


 

Polar insulators strongly stabilize excess charges in adsorbed atoms and molecules <cit.>, facilitating orbital localization through self-trapping of polarons <cit.>, revealing that the insulator does not merely electronically isolate the adsorbed molecule, but plays an active role in determining its electronic properties.  Whereas the reorganization energy is a well studied manifestation of this effect, little is known about how the large polarizability of the environment influences other electronic properties, as, for example, the occupation sequence of the molecular orbitals.

Here we study the electronic states in a charged  Cu-phthalocyanine (CuPc) molecule as a model system that is known to exhibit a Jahn–Teller (JT) distortion. We employ AC-STM imaging to reveal the energetic sequence of orbital occupation for the JT distorted system. Injection of a second electron results in the occupation of the same formerly-degenerate orbital. Our theoretical analysis reveals that this sequence is not merely a molecular property but it is critically governed by the polarizable NaCl substrate, strongly enhancing the JT splitting. Moreover, we predict that the delocalized configuration is accessible at higher bias voltages where it dominates the electronic transition rate due to the state multiplicity.


Experiments We employ AC-STM [<ref>(a)], cf. SI for details, to image the spatial distribution of charge-state transitions for the neutral, anionic and dianionic CuPc species. The map of the 0 → 1^- transition [<ref>(c)] exhibits a 4-fold rotational symmetry, similar to the anionic resonance observed by STM on ultrathin NaCl layers (2 ML)  <cit.>. Toggling of the molecule yields, on average, the superposition of two images rotated of about 32 degrees with respect to each other <cit.>). As reported previously <cit.>, for the reverse transition (1^- → 0)[<ref>(d)] the symmetry reduces from 4- to 2-fold due to the JT distortion, which favors the occupation of one single LUMO <cit.>. 

In the dianionic ground state the second excess electron can either occupy the other (empty) LUMO or further populate the already occupied one. The actual preference results from a competition between JT distortion and Coulomb repulsion, the latter being stronger for two electrons occupying the same orbital.
The AC-STM images of the 1^- → 2^- and 2^- → 1^- transitions, displayed respectively in <ref>(e) and (f), reveal that the second excess electron is occupying the same orbital as the first one.

Model We model the molecule in the AC-STM with the Hamiltonian


    [] = [mol] + [tip] + [tun]+ [as] + [vib] + [e-ph] ,


where the minimal electronic description of the molecule (mol), metallic tip (tip) and tunnelling coupling (tun) is complemented by a small environmental asymmetry (as), the set of molecular and substrate vibrational modes (vib), and by their coupling to the electronic degrees of freedom (e-ph). We write the molecular Hamiltonian in the Fock space of the four frontier orbitals, which are the singly occupied molecular orbital (SOMO, S), the highest occupied molecular orbital (HOMO, H) and the two degenerate lowest unoccupied molecular orbitals (LUMOs, L_xz and L_yz) <cit.>.
The latter two are shown in <ref>(b).
The single-particle energies, ε_S = -12.0, ε_H = -11.7, ε_L_xz/L_yz = -10.7, are renormalized by a crystal field correction of Δ = 2.44 and all symmetry-allowed interaction matrix elements V_ijkl of the Coulomb interaction are retained. The tip is described as a non-interacting Fermi sea with harmonic lateral confinement <cit.> 
[tip] = ∑_kσ( ε_k + c V_b ) [kσ]c^†[kσ]c, where the tip states ε_k are shifted by the applied bias V_b subject to a lever arm c caused by the voltage drop in the insulating layer, [kσ]c destroys an electron in the tip with momentum k and spin σ. 
The tunneling is expressed as 
[tun] = ∑_ikσ t^tip_ikσ[kσ]c^†[iσ]d + H.c., only considering tunneling processes between tip and molecule since molecule-substrate tunneling is inhibited. The tunneling amplitudes t^tip_ikσ are modelled to be proportional to the overlap between the molecule and the tip wave functions and thereby contain the dependence on tip position. The tip wave functions are composed by a Gaussian component stemming from the lateral confinement and an evanescent part along the direction perpendicular to the molecular plane <cit.>.  We incorporate an environmental asymmetry in our model via [as] = -δ_ as ([xz]n-[yz]n), with δ_ as = 4k_BT = 2.




The vibronic Hamiltonian is a collection of harmonic oscillators [vib] = ∑_ζħω_ζ ([ζ]a^†[ζ]a + 1/2) containing molecular and salt modes. Based on symmetry considerations, the vibrational modes of the system fall into three categories, namely, (i) those that do not interact with charges in the molecule, (ii) those that affect both LUMOs the same way, and (iii) those that interact differently with each of the LUMOs. The C_4v symmetry of the combined salt and molecular system facilitates this distinction of vibronic modes. We do not consider the modes falling into the first category. The modes constituting the second category lead to a stabilization of charge states in the molecule. These are the ones transforming as the A_1 irreducible representation and we call them symmetric (S) modes from now on since they do not reduce the symmetry of the system. They are only sensitive to the global charge on the LUMOs, thus


    [e-ph]^ S = ∑_α∈{A_1} g_α ([xz]n  + [yz]n) ( [α]a^† + [α]a) .


The modes in the third category affect the charge distribution within a specific charge state. We refer to them as antisymmetric (AS) modes  since they lower the symmetry of the molecule.
They belong to the B_1 and B_2 irreducible representation. However, due to energetic considerations (cf. SI) we completely neglect the B_2 modes. The antisymmetric (B_1) modes couple to the imbalance between the populations of the  L_xz and L_yz orbitals, i.e.


    [e-ph]^ AS = ∑_β∈{B_1} g_β ([xz]n - [yz]n)( [β]a^† + [β]a) .


We calculated the molecular mode energies as well as the corresponding electron-phonon couplings g_α/β from first principles using  <cit.>, thus obtaining 14 symmetric (A_1)  and 14 antisymmetric (B_1) modes. The coupled salt modes can be combined to obtain just a single symmetric (A_1) and a single antisymmetric (B_1) mode. We take the corresponding electron-phonon couplings as free parameters in our theory and set them to g^salt_A_1 = 68 and g^salt_B_1 = 40, assuming for the energy the one of the transversal optical phonon mode ħω_TO = 20 <cit.>. A microscopic derivation of this minimal model yielding couplings constants of comparable strength is given in the SI. We show a comparison between one symmetric molecular mode and the symmetric salt mode in <ref>(a) and (c). The antisymmetric case is displayed in <ref>(b) and (d).

The influence of the symmetric modes on the many-body spectrum is readily obtained by a canonical Lang-Firsov transformation, leading to a renormalization of the single-particle energies ϵ̃_i = ϵ_i - ∑_αg_α^2/ħω_α as well as of the direct Coulomb interaction terms Ṽ_ijji = V_ijji - 2∑_αg_α^2/ħω_α and i,j ∈{ L_xz, L_yz}. Thus, the energy renormalization induced by the symmetric salt mode explains the stabilization of charged molecules and atoms adsorbed on polar dielectrics due to the Franck-Condon blockade <cit.>.     

For antisymmetric modes, the exchange and pair hopping between LUMOs shows an intricate interplay with the electron-phonon coupling. In this case, the Lang-Firsov transformation does not eliminate the coupling between the electronic and the mechanical degrees of freedom, which persists in the transformed exchange and pair-hopping terms. We identify, instead, a single JT active mode and resort for our analysis into a semi-classical adiabatic approximation. To this end, we introduce the canonical displacement operators

    [m]Q = ∑_β∈ B_1√(ħω_β) A_mβ( [β]a^†+[β]a),

where the first row of the transformation matrix reads A_1β = 1/g√(g_β^2/ħω_β) and yields the JT active mode [AS]Q:= [1]Q. The JT distortion is thus described by the operator 

    [JT](Q_ AS) = Q_ AS^2/4 +  g ([xz]n -  [yz]n) Q_ AS

being Q_ AS the classical displacement of the JT active mode, with the effective coupling g= √(∑_βg_β^2/ħω_β). 
On this basis, we simulate the experimentally investigated many-body eigenstates and energies by diagonalizing the effective Hamiltonian 


    [eff] = [ mol]H̃ + [as] + [JT] (Q_ AS) ,


along the coordinate Q_ AS of the JT-active antisymmetric mode, with [ mol]H̃ containing the Lang-Firsov renormalization due to the symmetric modes.




Electronic structure Some of the eigenenergies of [eff] are the adiabatic potential energy surfaces (APES) plotted in <ref>. The lowest parabola in <ref>(d) is the APES of the neutral state. This neutral ground state defines the reference configuration (Q_ AS = 0). The anionic states with an excess electron in one of the LUMO levels are orbitally degenerate for Q_ AS = 0. Their energy is lowered upon deforming along Q_ AS. Thus, the anionic ground state has one extra electron occupying the L_xz (L_yz) orbital for Q_ AS < 0 (Q_ AS > 0). The small environmental asymmetry δ_ as determines here that the global minimum has the excess electron in L_xz. The singlet-triplet splitting due to SOMO-LUMO exchange is not resolved on this energy scale. So far, the energy spectrum reflects the well-known JT effect. 
The richer structure of the dianionic APESs corresponds to two qualitatively different distributions of the excess electrons.
For one excess electron in each of the L_xz and L_yz orbitals, the coupling to the JT mode vanishes. Thus, the lowest energy of these states (with several different spin multiplicities) is found at Q_ AS = 0 in the upper part of <ref>(d). On the other hand, states with both excess electrons in the same former LUMO strongly couple to the JT-mode. Their ground state is found at Q_ AS≠ 0, analogously to the anionic configuration, but with an even larger JT deformation and reorganization energy. Pair hopping between the real-valued orbitals induces the avoided crossing at Q_ AS = 0, characteristic of the pseudo JT effect <cit.>.

Many-body transition rates A qualitative understanding of the measured charge-transitions is obtained by combining the electronic structure with few other constraints imposed by the experimental set-up. Specifically, the slow tip oscillations synchronized with the bias pulses allow for the system thermalization between consecutive tunnelling events and the large read-out tip-molecule distance yields single electron tunnelling. Moreover, the spectral broadening induced by the symmetric modes is much larger than the environmental asymmetry, which in turns exceeds the thermal broadening. Those premises imply, for example, that the neutral to anionic transition is 4-fold symmetric; it resembles the (incoherent) superposition of the L_xz and L_yz orbitals (with additional toggling), as it involves the spectrally unresolved states associated to both local minima with Q_ AS >0 and Q_ AS < 0 [<ref>(c) and <ref>(b)]. In contrast, environmental asymmetry ensures, for the thermalized anion, the predominant occupation of the state with Q_ AS<0; consequently the anionic to neutral transition acquires the 2-fold rotational symmetry of the  L_xz orbital [<ref>(d) and <ref>(c)]. Mere energetic arguments would predict also for the transition between the anion and the dianion the participation of the states corresponding to both the deformed local minima in the dianionic APES. Single electron tunnelling, though, forbids the contribution of the state with Q_ AS >0 and double occupation of the L_yz orbital, and ultimately yields a many-body transition resembling again L_xz [<ref>(e) and <ref>(g)]. The same appearance is expected for the transition from the dianionic to the anionic ground state [<ref>(h)] although quantum fluctuations in the configuration of the dianion do not completely exclude small contributions from the L_yz orbital, cf. <ref>(f) and the SI for the expression of the dianionic ground state as a function of the deformation coordinate.           

Electron tunnelling implies displacements of both the symmetric and antisymmetric modes. Their interplay is visualized in <ref>(e). The large electron-phonon coupling to the symmetric modes strongly suppresses the transitions between ground states along the effective symmetric coordinate Q_ S [black arrow in <ref>(e)]. On the contrary, w.r.t. the antisymmetric modes transitions between ground states are possible. This phenomenon, visualized for the neutral to anionic transition holds for all other considered transition between states with different charges.

The theoretical transition maps in <ref> and <ref> are calculated with Fermi's golden rule. The transition rate from an initial thermalized state towards a vibrationally excited state with neighbouring charge reads




    R^N→ N+1 = ∑_{n_α},f∑_i,j,σ∏_α∈{A_1}Γ_iσ,jσ(r_ tip)
    	P_λ_α({n_α})f^+(E_f - E_g - e  c V_b)
      ⟨ N+1,f  | [iσ]d^† | N,g ⟩⟨ N,g | [jσ]d | N+1,f ⟩ ,
    	
      
    	
       R^N+1→ N = ∑_{n_α},f∑_i,j,σ∏_α∈{A_1}Γ_iσ,jσ(r_ tip)
    	P_λ_α({n_α})f^-(E_g - E_f - e   c V_b)
      ⟨ N+1,g | [iσ]d^† | N,f ⟩⟨ N,f | [jσ]d | N+1,g ⟩ ,



where the single-particle rate matrix is given by Γ_iσ,jσ(r_tip) = 2π/ħ∑_k t_k,iσ^* t_k,jσδ(E_f - E_g - ϵ_k).
The symmetric modes' Franck-Condon factors yield the Poisson distributions P_λ_α({n_α}), which depend on the Huang-Rhys factors λ_α = g_α/ħω_α and {n_α} is the set of the final state's vibrational excitations w.r.t. all modes α∈{A_1}. Due to the large electron-phonon coupling λ_ salt≈ 5, the salt contribution converges to a Gaussian and it is the main source of spectral broadening <cit.>.   
The Fermi function f^+ (and related f^- = 1-f^+) with the tip temperature T = 6 and chemical potential μ=-4.65 is calculated at the difference between the molecular many-body energies shifted by the fraction c = 0.6 of sample bias V_b dropping between the molecule and the tip. The energy of the final state E_f = E_f^e + ∑_αħω_α n_α includes the electronic potential energy E_f^e (the local minima of the APES in <ref>) and the excitation energies of the symmetric modes, while E_g is the ground state energy of the initial state. Finally, <ref> includes the transition matrix elements between different charge states. To this end, we denote the initial state with |N,g⟩ (|N+1,g⟩) and the final state with |N+1,f⟩ (|N,f⟩). The sum over f extends to all possible states accessible from the designated initial state via a single electron tunnelling event. 
The tunneling rates in <ref> are depicted in <ref> as a function of the tip position and show a strong similarity to the experimental results, with also a remarkable agreement between all the experimental and theoretical bias voltages for the different transitions. 

We notice, in this respect, that the 1^- to 2^- transition denoted with a green arrow in <ref> is calculated, in accordance to the experiment, at the onset of the anion to dianion transition (V_b = 2.75). 




At higher biases we expect to open also transitions to the excited dianionic states, as sketched by the red arrows in <ref>(d) and (f). 
The bias dependence of the transition rate is  shown in <ref>. We evaluate <ref> in the tip positions visualized by the black and teal crosses in the insets. They select tunnelling events involving a single orbital, respectively L_xz for r_1 and L_yz for r_2. The form of the dianionic ground state determines the lower bias onset for the rate in position r_1. At a higher bias voltages both rates saturate, with the one at r_2 being twice as large as the one at r_1: the unoccupied orbital offers two spin channels as compered to the single one for an orbital already occupied by one electron. 
The predicted change of the full rate map is shown in the insets of <ref>.
The comparison to <ref>(g) and <ref> shows the emergence of the orthogonal LUMO (L_yz in our model) with an overall stronger signal.

Control over the landscape of the dianionic APES should also be obtained by changing the underlying salt. It is experimentally known that RbI, NaCl, and LiF have increasing influence onto the spectral broadening of molecular electronic states <cit.> and thus on the reorganization energy of the symmetric modes. We hypothesize that a similar trend also characterizes the antisymmetric modes, with a possible reversal of the orbital occupation sequence of CuPc on RbI with respect to the one on NaCl and LiF.


Conclusions We have shown how the electron-phonon coupling between a molecule and the underlying salt substrate determines the specific charge configuration within a molecule going beyond mere charge stabilization. We extend to this respect existing observations of charge localization in oligophenylene <cit.> and pentacene <cit.> to systems with JT distortion. The cooperative mechanism between salt substrate and molecule should generally appear in molecules and substrates sharing the same point group. This approach could pave the way to control charge configurations in molecules depending on the used substrate, with potential relevance in the study of intermolecular electron transfer, surface chemical reactivity and charge sensing. We also envisage the interplay between JT distortion of neighbouring molecules for the realization of cellular automata based on molecular arrays in which the local information is stored in the charge configuration of a single molecule.

Acknowledgement
M.F., L.L.P., J.R. and A.D. acknowledge financial support from the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) within Project-ID 314695032 - SFB 1277 (project B02). J.R. gratefully acknowledges funding from the ERC Synergy Grant MolDAM (no. 951519) and the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) through RE2669/6-2. T.F. and M.F. acknowledge support from the Spanish MCIN/AEI/ 10.13039/501100011033 (PID2020-115406GB-I00), the Basque Department of Education (PIBA-2023-1-0021), and the Euskampus Transnational Common Laboratory QuantumChemPhys.













































§ EXPERIMENTAL TECHNIQUE

Experiments were carried out with a low-temperature scanning tunneling and atomic force microscope equipped with a qPlus tuning fork <cit.> (resonance frequency f_0 ≈ 29.1 kHz, spring constant k = 1.8 kN m^-1, quality factor Q ≈ 3 × 10^4) in ultrahigh vacuum (p = 2 × 10^-10 mbar) and at a temperature of 6.2 K. As substrate we used Cu(111) single crystal covered with >20ML of NaCl. The molecules were deposited onto the sample kept at ≈ 7 K inside the microscope. Bias voltages are given as sample bias with respect to the tip. Positive constant-height offsets Δ z correspond to a distance decrease with respect to the AFM Δ f set point above the clean NaCl. For AC-STM experiments, the ac voltage pulses were produced by an arbitrary waveform generator (Agilent 33522A) and fed to the microscope head through semirigid coaxial high frequency cables.


§ CALCULATING MOLECULAR MODE PARAMETERS


The dependence of the electronic wave functions of the molecule on the nuclear positions is obtained by expanding the electronic Hamiltonian around the reference configuration R_0 up to first order in the displacement coordinate q. By definition, the first derivative of the ground state energy for the reference electronic state vanishes. This is not the case, though, for the each component of the electronic Hamiltonian. 
In terms of the normal modes coordinates, we thus obtain

    [e-ph] = ∑_l,α(∂[e]/∂q_l,α)_q_l,α=0∑_ζv^ζ_l,α√(ħ/2M_lω_ζ)( [ζ]a^† + [ζ]a),

with M_l the mass of the atom l and v^ζ the displacement vector of the normal mode ζ with frequency ω_ζ.
The molecular Hamiltonian  [mol] presented in the main text corresponds to [e](r,R_0). The matrix element describing the electron phonon coupling strength is given by

    g_ij^ζ = ∑_lα⟨ i|∂[e]/∂q_lα| j⟩v^ζ_lα√(ħ/2M_lω_ζ) .

These matrix elements can be evaluated by the INELASTICA python package <cit.>.
To evaluate them we first obtained the molecular geometry with the SIESTA DFT implementation <cit.>.
We employed a Single-Zeta-Pseudo (SZP) basis set and used the GGA-PBE exchange functional.
For details about the calculation of the mode energies ħω_ζ and electron phonon couplings g_ij^ζ we refer to the paper by Frederiksen et al. <cit.>.



§ SEPARATION OF SYMMETRIC AND ANTISYMMETRIC MODES


In this section, we aim to demonstrate the contrasting effects of symmetric and antisymmetric modes. We find that the symmetric modes solely lead to an energy renormalization depending on the total number of charges on the molecule. Conversely, the coupling to the antisymmetric modes varies across different orbital occupations, thereby exerting an influence on the charge distribution of the molecular ground state.

Our discussion begins with the component of the Hamiltonian that encompasses the vibronic degrees of freedom as well as the electron-phonon coupling

    [vib] + [e-ph] = ∑_ζħω_ζ([ζ]a^†[ζ]a + 1/2) + 
    		∑_ζσ∑_i,j = L± g^ζ_ij[iσ]d^†[jσ]d( [ζ]a^† + [ζ]a).

The sum is taken over all contributing modes ζ, the spin σ, and the degenerate LUMOs i and j, as we assume that any excess charge on the molecule to be confined solely to the LUMOs, thus restricting the electron-phonon coupling to these two orbitals.

The electron phonon coupling strength is proportional to the matrix elements with respect to the derivative of the electronic Hamiltonian, see <ref>,

    g_ij^ζ∝⟨ i|∂[e]/∂q_lα| j⟩.

The derivative ∂[e]/∂q_lα is an irreducible tensor operator which transforms in the same way as the respective normal mode ζ.
The wave functions i and j transform according to the E_g irreducible representation of the D_4h point group. Therefore, for the matrix element in <ref> to be nonzero we can only use modes ζ such that their irreducible representation Γ^ζ is contained in the symmetric product

    [ E_g⊗ E_g ]_+ = A_1g⊕ B_1g⊕ B_2g.

The modes transforming according to the A_1g irreducible representation couple to the global excess charge on the LUMOs. We call these the symmetric modes. The associated molecular distortion preserves the symmetry of the system. Therefore, they are not contributing to the JT effect. It is important to consider them, as they give a renormalization to the eigenenergies of the molecule depending on the charge state and they lead to a broadening in the transition rates.

On the other hand, the B_1g and B_2g modes couple to the unbalance between the occupation of the LUMOs. In particular, this imbalance concerns,  for the B_1g modes, the real valued basis L_xz and L_yz shown in Fig. <ref>. Contrary to the totally symmetric A_1g modes, these modes can lower the symmetry of the molecule. Thus, they are responsible for the JT distortion.



 §.§ Energy renormalization due to symmetric modes

In this section, we delve into the impact of the symmetric modes. All relevant parameters are shown in <ref>.
The symmetric modes couple diagonally to the LUMOs, i.e. i=j in <ref>,

    [vib]^A +  [e-ph]^A= ∑_α∈{A_1g}ħω_α([α]a^†[α]a + 1/2) + 
    		∑_α∈{A_1g} g_α∑_iσ[iσ]d^†[iσ]d( [α]a^† + [α]a).

All modes couple equally to the occupation of the L_xz and L_yz orbitals and thus we omit the subscript specifying the LUMO from the coupling strength g_α. The sum over α encompasses all modes transforming according to the A_1g irreducible representation and g_α is the coupling strength to the LUMOs of mode α. We apply the Lang-Firsov transformation to the full Hamiltonian 

    Ĥ̃ = 
      e^Ŝ( [mol] + [tip] + [tun]+ [as] + [vib] + [e-ph] ) e^-Ŝ ,

where the operator 

    Ŝ = ∑_α∈{A_1g}∑_iσλ_α[iσ]n( [α]a^†-[α]a) .

with λ_α = g_αħω_α. This transforms the annihilation and creation operators on the molecule as

    d̃̂̃_iσ = d̂_iσX,   d̃̂̃^†_iσ = d̂^†_iσX^†.

with X = e^-∑_αλ_α ([α]b^† - [α]b).
The phonon operators are shifted by

    ẫ_α = [α]a - λ_α∑_iσ[iσ]n,   ẫ^†_α = [α]a^† - λ_α∑_iσ[iσ]n.

The transformed molecular Hamiltonian thus obtains a renormalization of the single-particle LUMO energies ϵ̃_i  =  ϵ_i - ∑_α∈ A_1gg^2_α/ħω_α .
and of the direct Coulomb interaction between the LUMOs Ṽ_ijji = V_ijji - 2∑_α∈ A_1gg^2_α/ħω_α, with i,j = L_xz , L_yz.
Furthermore, the symmetric electron-phonon coupling term vanishes, leaving only the harmonic vibrations in the Hamiltonian which describes the effect of the symmetric modes.
The renormalization of the single-particle LUMO energies and the corresponding Coulomb interaction terms leads to a stabilization of charged states on the molecule.
The Lang-Firsov transformation of the tunneling Hamiltonian yields the X̂ operator introduced in Eq. (<ref>). Its matrix elements in the harmonic oscillator eigenbasis are the Franck-Condon factors giving the Poisson distribution in the rates of Eqs. (7) of the main text. Ultimately, the X̂ operator and thus the coupling to the A_1g modes is responsible for the broadening of the calculated transition rates.





 §.§ Treatment of antisymmetric modes

The antisymmetric modes are more sensitive to the actual distribution of the charge between the LUMOs, rather then to their global occupation. The electron vibron coupling Hamiltonian, written for the real basis depicted in Fig. <ref>, reads 

    [e-ph]^B = 
    	∑_β∈{B_1g} g_β( [xz]n - [yz]n)
        ( [β]a^† + [β]a)
    + 
    	∑_β∈{B_2g} g_β([xz]d^†[yz]d + [yz]d^†[xz]d)
        ( [β]a^† + [β]a),

where β encompasses all modes transforming according to either B_1g or B_2g.
This Hamiltonian gives rise to the Jahn-Teller (JT) effect, as each deformation favours the occupation of a determined LUMO thus lifting their degeneracy. The more complex form of the electron-phonon coupling combined with the structure of the exchange Hamiltonian makes the Lang-Firsov transformation unuseful. The latter does not decouple in this case  the electronic from the vibronic degrees of freedom. Therefore, we treat the antisymmetric displacements classically and estimate the ground state confirmation by searching for the minima of adiabatic potential energy surfaces (APES).
To this end, we introduce canonical displacements and momenta as follows

    x̂_β = √(ħω_β)( [β]a^†+[β]a) ,  p̂_β = iħ/2√(ħω_β)( [β]a^† - [β]a) .

This transformation allows us to express the Hamiltonian associated to the modes with the B_1g and B_2g symmetry as

    [vib]^B +  [e-ph]^B =  
      ∑_β∈{B_1/2 g}( ω_β^2 p̂^2_β + 1/4x̂_β^2 ) 
      + 
    	∑_β∈{B_1 g}g_β/√(ħω_β)( [xz]n - [yz]n)
        x̂_β
     +
     ∑_β∈{B_2 g}g_β/√(ħω_β)([xz]d^†[yz]d + [yz]d^†[xz]d)
        x̂_β.

We further simplify the problem by ignoring the kinetic terms and by turning x̂_β into the classical coordinate  x_β. At this point we should diagonalize the effective Hamiltonian 


    [eff]({x_β}) = Ĥ̃_ mol + [as] + ∑_βx_β^2/4 +  [e-ph]^B({x_β})

and look for the ground state energy and configuration for each charge sector. Given the presence of 14 B_1g and 14 B_2g modes, determining the energy minimum of this potentials proves impractical. A first screening on the modes can be performed using their reorganization energy Δ E_ reorg≡ħωλ^2. As can be seen from <ref>, the B_1g modes provide a higher reorganization energy than the B_2g modes. Therefore, we neglect the B_2g modes and focus our attention solely on the B_1g modes.



§ MANY-BODY EIGENSTATES




A crucial role for the understanding of the experiments is played by the effective Hamiltonian in Eq. (<ref>) from which the low energy many-body eigenstates of CuPc in its neutral, anionic and dianionic configuration can be extracted. We restrict ourselves to the Fock space generated from the single particle basis of four frontier orbitals: i.e. the SOMO (S), HOMO (H) and the two degenerate LUMOs. In particular, we will only consider, in the following, the neutral anionic and dianionic states with respectively 0,1 and 2 electrons in the LUMOs. This further simplification does not affect significantly the low energy eigenstates of the neutral and anionic molecule, as obtained by comparison with numerical diagonalization performed in the full Fock space associated to the four frontier orbitals. In the dianionic case this approximation excludes a pair of doublets with triple occupation of the LUMOs. The latter are though not relevant to explain the experimental results, as these states cannot be accessed with a single electron tunnelling event from the anionic ground state.  

As the LUMOs are degenerate, they allow for different representations. We show in <ref> the frontier orbitals with the LUMOs in their real valued representation (L_xz, L_yz). However, in the discussion of the many-body eigenstates, we will also use their rotationally invariant representation which is given by <cit.>

    |L+⟩ = 1/√(2)( |L_xz⟩ + i |L_yz⟩),   |L-⟩ = 1/√(2)( |L_xz⟩ - i |L_yz⟩).

The Hamiltonian we discuss is [eff](x_1) = Ĥ̃_ mol + x_1^2/4 + [e-ph]^B(x_1), where, to start, we restrict ourselves to the B_1g mode with the highest reorganization energy. The renormalization due to the symmetric modes only introduces the same shift to all eigenenergies of a given particle number. In practice, we have

    [eff](x_1) =  
    		∑_i (ε̃_i + Δ) [i]n + 1/2∑_ijkl∑_σσ'Ṽ_ijkl[iσ]d^†[kσ']d^†[lσ']d[jσ]d
    		+  x_1^2/4
    	+ ∑_σg_1/√(ħω_1)( [L+σ]d^†[L-σ]d + [L-σ]d^†[L+σ]d) x_1 .

The summation indices i, j, k and l run over all four frontier orbitals, i.e. i,j,k,l = S,H,L_+,L_-. The model comprises the renormalized single particle energies ε̃_i = ε_i - ∑_α g_α^2/ħω_α  with ε_S = -12.0, ε_H = -11.7, ε_L+/L- = -10.7, also shifted by a crystal field correction of Δ = 2.44. The Coulomb parameters concerning the direct interaction are also renormalized by the reorganization energy of the symmetric modes, i.e. Ṽ_ijkl = V_ijkl - 2∑_α g_α^2/ħω_α. Exchange and pair hopping contributions are instead not influenced by the Lang-Firsov transformation. The bare Coulomb parameters, calculated by direct integration from the frontier orbitals <cit.>, are given in <ref>.
The energy of the considered mode is ħω_1 = 185 and the coupling strength g_1 = 72.





 §.§ The neutral molecule


We start our discussion of the eigenstates form the neutral molecule. By diagonalizing <ref> in the reference position R_0, i.e. x_1 = 0, we find a spin degenerate doublet

    |3,D_0^↑⟩ = [S↑]d^†|Ω⟩,   |3, D_0^↓⟩ = [S↓]d^†|Ω⟩,

where |Ω⟩ is the reference state with doubly occupied HOMO but empty SOMO and LUMOs. We observe the characteristic unpaired spin in the SOMO, due to its localization and correspondingly high charging energy. The first-excited neutral energy level lies about 0.8 above the ground state <cit.> and thus we do not consider it in our calculations.  
The lowest eigenvalue of the Hamiltonian <ref> is  depicted in <ref> (a) as  a function of the displacement coordinate x_1 given in units of √(ħω_1) and represents the adiabatic potential energy surface (APES). As the LUMOs remain unoccupied in the neutral ground state, the electron-phonon coupling does not contribute to the potential energy and we only see the harmonic contribution due to the first antisymmetric mode.




 §.§ The anionic molecule

The effective Fock space for the CuPc anion comprises the eight states given by [Sσ]d^†[Liσ']d|Ω⟩ with σ, σ' = ↑, ↓ and i = ±, which are degenerate in the reference configuration as far as it concerns the direct Coulomb interaction. The exchange coupling between the electrons occupying the SOMO and the LUMO splits this eightfold degeneracy. The eigenstates in the reference configuration can thus be organized as two sets of triplets and singlets

    |4,T_0^+1,i⟩ = [S↑]d^†[Li↑]d^†|Ω⟩,
       |4,T_0^0,i⟩ = 1/√(2)(
        [S↑]d^†[Li↓]d^†
        -
        [Li↑]d^†[S↓]d^†)
        |Ω⟩,
       |4,T_0^-1,i⟩ = [S↓]d^†[Li↓]d^†|Ω⟩,
       |4,S_1,i⟩ = 1/√(2)(
        [S↑]d^†[Li↓]d^†
        +
        [Li↑]d^†[S↓]d^†)
        |Ω⟩,

with the triplet orbitally degenerate ground states separated from the singlets by 2 J_SL = 18 meV. These two sets, labeled by i = ±, arise due to the degenerate LUMOs and are mixed by the coupling to the B_1g vibrational mode. First, we discuss the two triplet states with S_z = +1. The eigenstates of the Hamiltonian <ref> when x_1≠ 0 read

    |4,T_0^+1,xz⟩ = 1/√(2)(|4,T_0^+1,+⟩ + |4,T_0^+1,-⟩),  |4,T_0^+1,yz⟩ = i/√(2)(|4,T_0^+1,+⟩ - |4,T_0^+1,-⟩).

We notice that these states are also eigenstates of <ref> if x_1 = 0, thus giving a single set of eigenstates at all deformations, although the corresponding eigenenergies vary as function of x_1, as shown in Fig. <ref>. From the definition of the rotationally invariant representation given in Eq. (<ref>) we can express these anionic eigenstates states in the real valued representation as 

    |4,T_0^+1,xz⟩ = [S↑]d^†[L_xz↑]d^†|Ω⟩,  |4,T_0^+1,yz⟩ = [S↑]d^†[L_yz↑]d^†|Ω⟩.

The same procedure can be applied to the other 4 triplet states. We obtain

    |4,T^0_0,iz⟩ = 1/√(2)(
      [S↑]d^†[L_iz↓]d^†
      -
      [L_iz↑]d^†[S↓]d^†)
      |Ω⟩,
       |4,T^-1_0,iz⟩ = [S↓]d^†[L_iz↓]d^†|Ω⟩,

with i = x/y.
Analogously, for the singlet states we obtain, in the real valued basis,

    |4,S_1,iz⟩ = 1/√(2)( 
     [S↑]d^†[L_iz↓]d^† 
     +
     [L_iz↑]d^†[S↓]d^†) |Ω⟩,

again with i =x/y. In each of the two dimensional subspaces with a given spin S and S_z the Jahn-Teller Hamiltonian in Eq. (<ref>) acquires the canonical form 

    H^ 1-_ JT = x_1^2/4 1_2 + g_1/√(ħω)_1x_1 σ_x

typical of a E ⊗ b_1 Jahn-Teller problem <cit.>.
The sign of the displacement along the x_1 coordinate determines whether the triplet-singlet set with the added electron in the L_xz or in the L_yz orbital experience a decrease or increase in energy. This leads to a preference of one set of triplet-singlet states over the other depending on the direction of distortion. The APES depicted in <ref> (b) illustrates this phenomenon. The plot shows four parabolas representing the potential energy of both triplet-singlet sets.



 §.§ The dianionic molecule



The effective Fock space for the dianionic molecule consists of twelve states, which, in the rotationally invariant basis, are represented as

    [Sσ]d^†[Liσ']d^†[Ljσ”]d^†|Ω⟩,

where i,j = ±, and σ' = σ̅” if i = j due to the Pauli exclusion principle. These states are degenerate for what concerns the direct Coulomb interaction (terms involving U in [mol]) with an energy E_5 = ε̃_S + 2(ε̃_H + ε̃_L) + 5Δ + Ũ_H + Ũ_L + 2(Ũ_SH + Ũ_SL)+ 4Ũ_HL. By considering the exchange and pair hopping interactions (terms involving J in [mol]), this degeneracy is lifted. We first discuss the spectrum and the eigenstates in the reference configuration x_1 = 0. The eigenenergies are given with respect to the degenerate energy.
The scheme of the energy levels is given in <ref> (a). The ground state is a spin quadruplet

    |5,Q_0^3/2⟩ = [S↑]d^†[L+↑]d^†[-↑]d^†|Ω⟩,
       |5,Q_0^1/2⟩ = 1/√(3)( 
      [L+↑]d^†[L-↑]d^†[S↓]d^† 
      -
      [S↑]d^†[L-↑]d^†[L+↓]d^† 
      +
     [S↑]d^†[L+↑]d^†[L-↓]d^†) 
     |Ω⟩,
       |5,Q^-1/2⟩ = 1/√(3)( 
      [S↑]d^†[L+↓]d^†[L-↓]d^†  
      -
      [L+↑]d^†[S↓]d^†[L-↓]d^† 
      +
      [L-↑]d^†[S↓]d^†[L+↓]d^†) 
      |Ω⟩,
       |5,Q_0^-3/2⟩ = [S↓]d^†[L+↓]d^†[L-↓]d^†|Ω⟩.

The energy of this quadruplet is E_Q_0 = E_5 -J^ex_+--2J^ex_ SL. The first excited states form a spin degenerate doublet, with energy  E_D_1 = E_5 - J^ex_+- + J^ex_SL. The corresponding eigenstates are given by

    |5,D_1^↑⟩ = 1/√(6)(
      2 [L+↑]d^†[L-↑]d^†[S↓]d^†
      +
      [S↑]d^†[L-↑]d^†[L+↓]d^†
      -
      [S↑]d^†[L+↑]d^†[L-↓]d^†)
      |Ω⟩ ,
       |5,D_1^↓⟩  =  1/√(6)(
      2 [S↑]d^†[L+↓]d^†[L-↓]d^†
      +
      [L+↑]d^†[S↑]d^†[L-↓]d^†
      -
      [L-↑]d^†[S↓]d^†[L+↓]d^†)
      |Ω⟩.

The following three energy levels are also spin doublets, with the following eigenenergies 

    E_D_2 = E_5 -J_+-^ p - J_ SL^ ex ,    E_D_3 = E_5  + J_+-^ p - J_ SL^ ex ,    E_D_4 = E_5  + J_+-^ ex - J_ SL^ ex .

The corresponding eigenstates are,

    |5,D_2/3^↑⟩  = 
      1/√(2)(
      [S↑]d^†[L+↑]d^†[L+↓]d^†∓[S↑]d^†[L-↑]d^†[L-↓]d^†) |Ω⟩, 
       |5,D_2/3^↓⟩ = 1/√(2)(
      [L+↑]d^†[S↓]d^†[L+↓]d^†∓[L-↑]d^†[S↓]d^†[L-↓]d^†)|Ω⟩,

and, respectively, 

    |5,D_4^↑⟩ = 1/√(2)(
                [S↑]d^†[L+↑]d^†[L-↓]d^† 
                +
                [S↑]d^†[L-↑]d^†[L+↓]d^†)
                |Ω⟩,
       |5,D_4^↓⟩ = 1/√(2)(
                [L+↑]d^†[S↓]d^†[L-↓]d^† 
                +
    		      [L-↑]d^†[S↓]d^†[L+↓]d^†)
                |Ω⟩.




The first eight states corresponding to the lower energies are not affected by the electron vibron coupling to the modes of symmetry B_1g. This is evident in <ref>, where we observe three parabolas - one is fourfold and the other two are twofold degenerate - with a minimum at x_1 = 0. This antisymmetric mode couples instead the eigenstates |5,D^σ_3⟩ and |5,D^σ_4⟩. For each of the two spin components, the Jahn-Teller Hamiltonian projected on the corresponding two dimensional basis reads 

    H^2-_ JT(x_1) = x_1^2/4 1_2 
    - J_+-^ ex - J_+-^ p/2σ_z 
    +2g_1/√(ħω_1)x_1σ_x .

Due to the finite energy splitting in the reference configuration (x_1 = 0) this Hamiltonian describes the pseudo-Jahn-Teller effect <cit.>. The existence of localized minima corresponding to deformed configurations is only guaranteed for a sufficiently large coupling, i.e. g > √(ħω_1 (J_+-^ ex - J_+-^ p))/4, as can be deduced by analysing the two highest eigenvalues of the dianionic effective Hamiltonian:

    E_D_3/4(x_1) = E_5 - J_ SL^ ex + J_+-^ ex + J_+-^ p/2 + x_1^2/4∓√((J_+-^ ex-J_+-^ p/2)^2 + 4g_1^2/ħω_1x_1^2) .

Differently from the anionic case, here also the eigenstates depend on the deformation coordinate x_1. In particular we obtain

    |5,D^σ_3(x_1)⟩   = cosθ(x_1) |5,D_3^σ⟩ - sgn  x_1 sinθ(x_1) |5,D_4^σ⟩ ,  
    |5,D^σ_4(x_1)⟩   =  sgn x_1 sinθ(x_1) |5,D_3^σ⟩+cosθ(x_1) |5,D_4^σ⟩ ,

where the mixing angle is defined as 

    θ(x_1) = 1/2arccos(1 + 16 g_1^2 x_1^2/ħω_1 (J_+-^ ex - J_+-^ p)^2)^-1/2 .

Within the single mode approximation considered here we obtain a mixing angle at equilibrium θ_ eq≈ 0.24π. The larger is the electron phonon coupling g with respect to the vibrational energy ħω_1 and the energy splitting J_+-^ ex - J_+-^ p of the reference configuration, the larger is the absolute value of the equilibrium configuration.  Thus, the mixing angle converges to θ = π/4. In view of the enhancement of the coupling introduced by the other molecular modes and, above all, by the insulating substrate, we expand the equilibrium eigenstate in the vicinity of this limiting case. 
The lowest energy state having the equilibrium configuration with x_1, eq < 0 reads

    |5,D^σ_3(x_1, eq)⟩   = 
    cos(θ_ eq - π4)1/√(2)(|5,D_3^σ⟩ + |5,D_4^σ⟩) 
    - 
    sin(θ_ eq - π4)1/√(2)(|5,D_3^σ⟩ - |5,D_4^σ⟩) 
       =  sgn σ [Sσ]d^†[
    cos(θ_ eq - π4) [L_xz↑]d^†[L_xz↓]d^† 
    +
    sin(θ_ eq - π4) [L_yz↑]d^†[L_yz↓]d^†]
    |Ω⟩
       ≡cos( θ_ eq - π4) |5,D^σ_xx⟩ + sin( θ_ eq - π4) |5,D^σ_yy⟩ .

This state shows a predominant double occupation of the real valued L_xz orbital (|5,D^σ_xx⟩) with a small admixture of the state with a doubly occupied L_yz orbital (|5,D^σ_yy⟩). The roles of L_xz and L_yz are exchanged if x_1, eq > 0. 

The experimental results we showed in Fig. 1 of the main text suggest that the dianionic ground state has both electrons in the same real valued LUMO. This is clearly not the case in our theoretical model if we consider one mode since the ground state is in the reference configuration where the lowest lying states are the quadruplets which have one electron in each LUMO. We can estimate the reorganization energy required to achieve the desired dianionic ground state by first considering that E_D_3 - E_Q_0 = J_+-^ ex + J_+-^ p + J_SL^ ex = 435 meV. The total reorganization energy scales with the square of the electron-phonon coupling. Thus, the one associated to the B_1g modes scales with the square of the electronic unbalance in the real valued LUMOs. It is thus necessary to achieve a reorganization energy per electron due to B_1g modes of least 109 for the state |5,D^σ_3(x_1, eq)⟩ to become the dianionic ground state.



§ COMBINATION OF MULTIPLE MODES WITH THE SAME SYMMETRY




In this section, our focus is on combining multiple modes that belong to the same irreducible representation into a single effective mode. This allows us to analyze the potential energy of the system with contributions from all modes of  the same irreducible representation. The combined system of molecule and salt has C_4v symmetry. The B_1g modes we calculated for the molecule are all in plane modes and thus they transform as B_1 modes with respect to the C_4v point group. Therefore, we can not only combine the molecular modes among themselves but also include salt contributions to them. We begin this procedure by introducing the transformations of the displacements and momenta given by

    Q̂_n = ∑_β∈{B_1}  A_nβx̂_β,   P̂_m = ∑_β∈{B_1}  B_mβp̂_β.

Imposing canonical commutation relations on these new coordinates yields

    [ Q̂_n, P̂_m] = ∑_ββ' A_nβB_mβ'[ x̂_β,p̂_β'] = 
    	i ħ∑_ββ' A_nβB_mβ'δ_ββ' = 
    	iħ∑_α A_nβB_mβ!= iħδ_nm.

We further assume A to be orthogonal and thus deduce B=A. Moreover, we fix the first row of the transformation matrix to be

    A_1β = 1/g√(g_β^2/ħω_β),

with g = √(∑_βg_β^2/ħω_β). This change of coordinate allows us rewrite the Jahn-Teller Hamiltonian as 

    Ĥ_JT({Q_n}) = 1/4∑_n Q_n^2 + g Q_1 (n̂_xz - n̂_yz) ,

in which the electron-phonon coupling only concerns one effective JT mode, with effective coupling strength g.
However, this transformation comes at the cost of a non-diagonal kinetic part in the Hamiltonian. We are still free to construct the matrix transformation A such that all the Jahn-Teller inactive modes remain orthogonal among themselves. We can thus consider them as phonon bath for the JT active mode Q_1 and we neglect them  in our further discussions. This is justified as far as we are not interested in the dynamics of the JT distortion, but rather on the new equilibrium configurations.

Only considering molecular contributions to the effective coupling strength yields a value of g=218√() corresponding to a reorganization energy of Δ E_B_1 = 48. From our previous discussion in <ref> we know that this reorganization energy is not enough for the states with two electrons in one LUMO to become the new ground states. We can also deduce this from the APES in <ref> (a), which is plotted along the effective JT coordinate Q_ AS := Q_1 in units of characteristic quantity

    L^ AS_c = √(1/g√(∑_βħω_β g_β^2)) ,

where β runs over all considered modes. We notice that an effective Huang-Rhys factor can be defined, starting from Eq. (<ref>), for the JT active mode Q_ AS  as half of the ratio between the deformed equilibrium configuration Q_ AS, min of the anion and the characteristic scaling constant L^ AS_c. In other terms, it reads 

    λ^ AS_ eff = g/L^ AS_c = [ (∑_βg_β^2/ħω_β)^3/∑_β'ħω_β' g_β'^2]^1/4 .

 The latter reduces to the common expression λ = g_β/(ħω_β) when the sums are restricted to a single mode. 
 The deformed minima in the dianionic configuration are closer to the dianionic minimum energy but this still is not enough to explain the double charging of the same LUMO.
 However, if we consider also the contributions from the salt modes with the parameters given in the main text, we obtain g = 359√(), yielding a reorganization energy of Δ E_B_1 = 129. We show the corresponding APES along the Q_ AS displacement coordinate in <ref> (b). For the neutral and anionic charge states we observe no qualitative changes. For the dianionic molecule we see instead a qualitative change, as |5,D^σ_xx⟩ becomes the new global ground state.
The next highest lying states are the |5,D^σ_yy⟩ states. These two pairs of states are split by 2δ_ as, due to the  asymmetry which we included in our Hamiltonian. Otherwise, analogously to the anionic pairs of triplet and singlet states, they would be degenerate.
The energy splitting between the dianionic ground states |5,D^σ_xx⟩ and the quadruplet states is, with our choice of parameters, 70. This analysis confirms that the salt substrate in this system has an effect on the charge distribution within a single molecule going beyond mere stabilization of extra charges. We notice, moreover that, even with the contribution of the salt, λ_ eff≈ 1 thus allowing for direct transitions between vibrational ground state with different charges. The classical limit of non-adiabatic vertical transition does not apply to the JT active mode. Analogous considerations can be reserved also to the symmetric (A_1) modes, yielding a significant difference in the effective Huang-Rhys factors
 
    λ_ eff^ S = 2.36 ,  λ_ eff^ AS = 1.11 .






§ MICROSCOPIC MODEL OF THE SALT ELECTRON-PHONON COUPLING



We present in this section a simple microscopic model to estimate the electron-phonon coupling between an excess charge on the molecule and surface phonons of the underlying salt. We consider the bare Coulomb interaction between the excess electrons in the molecule and the salt ions. Comparison with experimental results indicate that this method seems to overestimate the values for the coupling.


 §.§ Linear vibronic coupling values

We consider vibration of the ions in all three space directions, thus accounting for  both the longitudinal and the transversal phonon modes in the salt. For simplicity, we limit ourselves to an Einstein model of independent harmonic oscillators with the energies ħω_L = 32 for the vibrations parallel to the surface and ħω_T = 20 for the ones perpendicular to it. The linear electron-phonon  couplings are calculated from the change of the Coulomb interaction between the LUMOs and the salt ions upon displacing of the latter. Specifically we evaluate

    [ g_l^α]_ij = q_l e^2/4πϵ_0*∫  drϕ^*_i(r)ϕ_j(r) Δ^α_0l/|r-R^0_l-u_l|u_lα_{u_lα}=0.

Here l labels an atom in the salt, i,j = L± specify one of the degenerate LUMOs and α = x,y or z represents the spatial direction of the displacement. Furthermore, we have q_i = ± 1 depending on the atom being a chloride (+1) or sodium (-1) atom, e is the elementary charge and ϵ_0 the vacuum permittivity.
The atom's equilibrium position is specified by R^0_l, u_l is the displacement coordinate of the considered atom and we have introduced the zero point fluctuations Δ^x/y_0l = √(ħ/2m_lω_L) and Δ^z_0l = √(ħ/2m_lω_T).
The central copper atom of the molecule, which we define to be at the origin, is centered on a chloride ion and we take 20 layers of salt into account. We checked that the integral in Eq. (<ref>) is converged in the z direction (perpendicular to the salt surface) when taking into account this number of layers. In the x and y direction we consider a square with an area of 400^2, which again suffices for the convergence of the electron phonon couplings. Moreover, we choose a distance of 2.8 between salt and molecule, which is consistent with values found in the literature <cit.>. By calculating the derivative in Eq. (<ref>), we obtain the expression

    [ g_l^α]_ij = q_l e^2/4πϵ_0∫ dxdydz
        ϕ^*_i(x,y,z)ϕ_j(x,y,z)(α-R^0_lα)Δ^α_0l/[(x-R^0_lx) ^2+(y-R^0_ly)^2+(z-R^0_lz)^2]^3/2 ,

which represents the starting point for the numerical calculation of the coupling constants.


 §.§ Numerical evaluation of the integral

The numerical integration of <ref> faces the problem of a divergent integrand when the integration variable coincides with one of the equilibrium positions of the atoms in the salt r = R_l. The integral is though not diverging. To better understand this issue, we expand the product of the wave functions around the diverging point as

    ϕ^*_i(r)ϕ_j(r) =     ϕ^*_i(R_l^0)ϕ_j(R_l^0) + 
    	∑_α(∂/∂ r_αϕ^*_i(r)ϕ_j(r))_ r =  R^0_l( r_α - R^0_lα) + 
       1/2∑_α∑_β(∂^2/∂ r_α∂ r_βϕ^*_i(r)ϕ_j(r))_ r =  R^0_l( r_α - R^0_lα) ( r_β - R^0_lβ) + ...,

where α,β = x,y or z. We now evaluate the integral <ref> for the first few terms of this expansion in the vicinity of a diverging point r =  R_l^0.
Furthermore, we drop all constants since we are only interested in the behavior of the integral when the integrand is diverging and without loss of generality set α = z.
By shifting to cylindrical coordinates with ζ = z-R_l z and ρ = √((x-R_l x)^2 + (y-R_l y)^2) and integrating over a small volume around the diverging point, we obtain, for the constant term in Eq. (<ref>)

    ∫_0^2π  dϕ∫_0^R   dρ∫_-ϵ^ϵ  dζρζ/( ρ^2+ζ^2 )^3/2 = 0.

For the linear component of <ref>, we distinguish two cases. The first case is for α = z, in which it is convenient to choose spherical coordinates with r = √(ρ^2 +ζ^2) and θ = arctan( ρ/ζ). We thus express the integral in the vicinity of the diverging point in these coordinates and obtain 

    -∫_0^2π  dϕ∫_0^π  dθ∫_0^R   dr r cos^2(θ)sin(θ)	,

which is not diverging and vanishes in the limit R → 0. The second case we need to discuss is when α≠ z. We can choose α=x as the case with α = y is completely analogous.
Now, we rotate our coordinate system and obtain ζ' = y-R_l y and ρ' = √((x-R_l x)^2 + (z-R_l z)^2).
With this transformation, the local integral is proportional to

    ∫_0^2π  dϕ∫_0^R   dρ'∫_-ϵ^ϵ  dζ'ρ'^3 cos(ϕ)sin(ϕ)/( ρ'^2+ζ'^2 )^3/2 = 0 .

Analogously, one can prove that also the integrals involving higher orders expansions will be finite and vanish in the limit of infinitesimal integration volume around the point with diverging integrand. Therefore, in the numerical evaluation we can set the integrand at the diverging point to 0 and can just use standard numerical integration schemes. 
The electron-phonon coupling relative to each vibrational mode of a salt ion is a Hermitian 2x2 matrix which can be expanded in terms of the Pauli matrices. We obtain 

    g_l^α = g^α_l,A_1 1_2 + g^α_l,B_1σ^x+g^α_l,B_2σ^y.

as, due to the form of the LUMO orbitals, the σ^z component vanishes exactly. We labelled the different electron-vibron coupling components according to the representations of the corresponding irreducible tensor operator with respect to the C_4v point symmetry group. In fact,


    ∑_σ∑_ij[Liσ]d^†δ_ij[Ljσ]d∈ A_1 , 
      ∑_σ∑_ij[Liσ]d^†σ^x_ij[Ljσ]d∈ B_1 , 
      and  ∑_σ∑_ij[Liσ]d^†σ^y_ij[Ljσ]d∈ B_2 .


So far we have shown how to calculate the electron-phonon couplings for each individual atom in the salt. We are, however, more interested in the collective behavior of the substrate. Thus, despite of the simplified Einstein model, we look for the modes with finite coupling to the excess charge in the molecular LUMOS. Analogously to the scheme proposed in Sec. <ref> we thus combine all modes with the same symmetry. We start with collective transversal modes, whose coordinates read

    x_Γ^T = ∑_l g^z_l,Γ/√(∑_m(g^z_m,Γ)^2)ũ_l z,

being Γ = A_1, B_1, and B_2 the corresponding irreducible representation and ũ_l z = u_l z/Δ^z_0l the displacement of each individual atom in z direction normalized by the respective zero point fluctuation. Analogously, we define for the longitudinal modes the collective mode coordinates

    x_Γ^L = ∑_l g^x_l,Γũ_l x + g^y_l,Γũ_l y/√(∑_m (g_m,Γ^x)^2 + (g_m,Γ^y)^2) .

These coordinate transformations identify six collective salt modes with a finite electron phonon coupling. All the other modes are not influenced by an excess charge on the CuPc molecule and are treated as a dissipative phonon bath.
The coupled modes exhibit respectively the transversal and longitudinal coupling strengths g^T_Γ = √(∑_l (g^z_l,Γ)^2) and g^L_Γ = √(∑_l (g^x_l,Γ)^2+(g^y_l,Γ)^2), with Γ = A_1, B_1 and B_2. Starting from the numerical calculation of the integrals in (<ref>), one obtains the following coupling strengths for the collective modes

    g^T_A_1   = 116,         g^T_B_1 = 20.6,         g^T_B_2 = 2.5
    
    		g^L_A_1   = 92.7,         g^L_B_1 = 11.1,         g^L_B_2 = 11.7.

Similarly to what we observed with the molecular modes, also in the salt the contribution to the electron-phonon coupling of the B_1 mode dominates over the one of the B_2 modes. Therefore, we concentrate our discussion solely on the A_1 and B_1 modes. They yield a total reorganization energy

    Δ  E_reorg^salt = (g^T_A_1)^2+(g^T_B_1)^2/ħω_T+
    	(g^L_A_1)^2+(g^L_B_1)^2/ħω_L = 966.

These values of the reorganization energy due to the salt deformation can be taken as an estimate for their order of magnitude. Their quantitative accuracy is though problematic.

On the one hand, by following the conclusions from the experiments of Fatayer et al. <cit.> we expect that the total reorganization energy upon charging should be in the range of 300≤Δ E_reorg^tot≤500. As the molecular component is Δ E_reorg^mol = 187, a salt reorganization energy in the range 113≤Δ E_reorg^salt≤313 is to be expected. 
This leads us to the conclusion that our model overestimates the total reorganization energy stemming from the salt.

On the other hand, in the molecular reference condition the energy difference between the dianionic ground state (both electrons in different LUMOs) and the localized configuration is ≈200.
Thus the minimum reorganization energy stemming from the antisymmetric modes, in order to obtain the localized configuration as the dianionic state with lowest energy, must be ≈100.

We assume that these problems can be solved by a more realistic treatment of the salt and salt-molecule interaction which includes the dispersion relation of the phonons and a screening of the bare Coulomb interaction between the molecule and the underlying ions. While keeping the form of the Hamiltonian Eq. (<ref>), we have considered the effective couplings as free parameters and explored their effects on the many body transitions. 
We have chosen them to be Δ E_A_1^salt = 231.2 and Δ E_B_1^salt = 80.
Combined with the molecular contributions this choice of parameters gives a total reorganization energy of Δ E_A_1 + Δ E_B_1 = 500.
Thus not overestimating the total reorganization energy while having the localized dianionic configuration as the ground state.






 Paper Title: Describe, Transform, Machine Learning: Feature Engineering for Grain Boundaries and Other Variable-Sized Atom Clusters.

























































[cor1]Corresponding author

BYU_CS]C. Braxton Owens
LANL1]Nithin Mathew
BYU_Phys]Tyce W. Olaveson
NASA]Jacob P. Tavenner
LANL1]Edward M. Kober
Baylor]Garritt J. Tucker
BYU_Phys]Gus L. W. Hartcor1
gus.hart@gmail.com
BYU_ME]Eric R. Homercor1
eric.homer@byu.edu

[BYU_CS]organization=Department of Computer Science Brigham Young University,
            
            city=Provo,
            postcode=84602, 
            state=UT,
            country=USA
            


[LANL1]organization=Group T-1, Theoretical Division, Los Alamos National Laboratory,
            
            city=Los Alamos,
            postcode=87544, 
            state=NM,
            country=USA
            
            
[LANL2]organization=Center for Non-Linear Studies, Theoretical Division, Los Alamos National Laboratory,
            
            city=Los Alamos,
            postcode=87544, 
            state=NM,
            country=USA
            
            
[NASA]organization=KBR, Inc., Intelligent Systems Division, NASA Ames Research Center,
            
            city=Moffett Field,
            postcode=94035, 
            state=CA,
            country=USA
            
            
[Baylor]organization=Department of Physics, Baylor University,
            
            city=Waco,
            postcode=76798, 
            state=TX,
            country=USA
            

[BYU_Phys]organization=Department of Physics and Astronomy Brigham Young University,
            
            city=Provo,
            postcode=84602, 
            state=UT,
            country=USA
            

[BYU_ME]organization=Department of Mechanical Engineering Brigham Young University,
            
            city=Provo,
            postcode=84602, 
            state=UT,
            country=USA
            

§ ABSTRACT 

Obtaining microscopic structure-property relationships for grain boundaries are challenging because of the complex atomic structures that underlie their behavior. This has led to recent efforts to obtain these relationships with machine learning, but representing a grain boundary structure in a manner suitable for machine learning is not a trivial task. There are three key steps common to property prediction in grain boundaries and other variable-sized atom clustered structures. These are: (1) describe the atomic structure as a feature matrix, (2) transform the variable-sized feature matrices of different structures to a fixed length common to all structures, and (3) apply machine learning to predict properties from the transformed feature matrices. We examine these feature engineering steps to understand how they impact the accuracy of grain boundary energy predictions. A database of over 7000 grain boundaries serves to evaluate the different feature engineering combinations. We also examine how these combination of engineered features provide interpretability, or the ability to extract insightful physics from the obtained structure-property relationships.






Grain Boundaries Atomic Structure Structure Descriptor Machine learning Feature Engineering Structure-Property Relationships














§ INTRODUCTION


Due to the impact of grain boundaries (GBs) on material properties <cit.>, there is a need to better understand the relationship between the structure of a GB and its corresponding properties. With expanding computing power, increasingly large amounts of data, and advances in data-driven approaches, there has been a push for suitable representations of GBs in order to predict their properties <cit.>. However, accurate property prediction is not the only measure of success. Models and representations that provide insight into structure-property relationships are key to advance our understanding.
 
Representing a GB starts with defining its structure, since it has both macroscopic and microscopic characteristics. Macroscopically, five degrees of freedom are used to define the GB character: Three to define a misorientation between two crystals, (often given by a normalized rotation axis [uvw] and angle θ), and two to define a boundary plane (hkl). Microscopically, the positions of the atoms result in 3n degrees of freedom. Additionally, a GB can assume various metastable configurations under any given set of macroscopic constraints.<cit.>. 

While there have been impressive developments in macroscopic representations for better understanding GBs <cit.>, the atomic structure is what defines a GB's properties. This article concentrates on the microscopic structure-property relationships since the macroscopic structure acts as a constraint on the microscopic structure.   

One microscopic method for defining a GB is the structural unit model <cit.>. This model describes the atomic structure of a quasi two-dimensional GB as a series of repeating atomic “structural units" that are characteristic of the boundary's local atomic arrangement. This simplifies analysis of its geometry and properties as long as it is quasi two-dimensional. In recent years, the structural unit model has been modified to more accurately represent a GB by considering the effects of metastable structures <cit.>. Other early methods for defining local atomic environments for characterizing GBs include: the centrosymmetry parameter (CSP) <cit.>, Voronoi index <cit.>, excess volume <cit.>, common neighbor analysis (CNA) <cit.>, the Polyhedral Unit Model <cit.>, and local entropy <cit.>. 





Although the early methods have been used mainly to classify GB atoms <cit.> they have also been used for machine learning predictions of atomic level properties <cit.>. Advanced methods, some of which are described below, have also been used to predict atomic level properties in GBs <cit.>. 


However, when using machine learning to predict properties of a GB as a whole, the descriptor must be transformed in some way to achieve a consistent feature size. This notion of transformation is an important one for variable-sized atomic structures such as GBs because different GBs will have different numbers of atoms (features) in their structure. Transformation is one step in a three step feature engineering process common to almost all machine learning predictions for variable-sized atom structures, such as GBs. 
These steps are illustrated in Figure <ref> and are described as follows:
1. Describe the atomic structure with an encoding algorithm, descriptor, or fingerprint of some kind, which is often represented as a matrix or vector. 2. Transform the mapping of the variable length descriptor for each structure to a fixed length descriptor common across all structures in a dataset. 3. Apply Machine Learning models or regression algorithms, to learn and then predict the property of a given atomic structure. 

We illustrate the consistency of these steps, occasionally combined in different orders or sequences, in numerous applications of machine learning to predict properties based on microscopic GB structure in the paragraphs below. These examples will also serve to highlight the variety of descriptors, transforms, and machine learning algorithms employed by the community.

Snow et al. <cit.> utilized the graphs underlying CNA to describe the atoms in each GB, categorizing them into 2205 distinct environments. They then performed principal component analysis to reduce these environments into 84 principal components, which constitutes a second description step. To standardize the representation size, a transform was applied, representing each GB by the proportions of the 84 components present. This transformed representation was used as input to a linear regression machine learning model to predict GB energy.

Guziewiski et al. <cit.> also explore this concept of proportions, utilizing the diamond-structure identification and the polyhedral template matching algorithms to describe GB atoms. This was transformed into a fixed-length density metric for each GB by counting the number of atoms within each polyhedral template class and normalizing this by the GB area. Random forest machine learning models were then used to predict both the GB energy and the tensile strength of the GB. 

Gomberg et al. <cit.> utilized a specialized pair correlation function <cit.> to describe their GB structures. This function is unique due to its use of a probability distribution function, allowing equal sampling for each GB and simultaneously transforming the descriptor into a fixed length. This representation was further refined to the first two principal components, constituting a second describe step, which is then used as input into a regression machine learning model. Dang and Yu extended Gomberg's method by incorporating the standard deviation of the pair correlation function through a weight parameter <cit.>.

More recently, the GB community has used atomic structure descriptors developed by the machine learned interatomic potential community. These descriptors are attractive because they are inspired by the symmetries and physical response inherent to the atoms, as described by Musil et al.<cit.>.
Rosenbrock et al. <cit.> implemented one of these physics inspired descriptors, called the smooth overlap of atomic positions (SOAP), to describe the GB and then transformed the SOAP descriptor into a fixed-length vector by averaging over the atom environments. These are then used to predict GB energy using a support vector machine learning model. Later, Fujii et al.<cit.> used SOAP to calculate a local distortion factor, describing how similar a GB atom's environment is to a bulk atom's environment. This was then transformed into a fixed-length using complete-linkage clustering. The GB thermal conductivity was then predicted by a ridge regression machine learning algorithm.

 
In this paper, we examine the impact of using different descriptors, transforms, and machine learning models for feature engineering for predicting GB energy, as illustrated in Figure <ref>. The exact descriptors, transforms, and machine learning algorithms we employ are listed in Figure <ref> and described in detail in the Methods Section. The feature engineering is tested on a dataset comprising 7304 aluminum GBs, which provides comprehensive coverage of the 5-dimensional macroscopic space of crystallographic character <cit.>. The interplay of various descriptors, transforms, and machine learning algorithms are analyzed for their effect on the accuracy of their predictions. In addition, we examine how feature reduction impacts the results for select cases. Finally, we assess the interpretative ability of some key descriptors to establish meaningful connections to the inherent structure of the GB. Prioritizing interpretability is imperative to increase our understanding of atomic GB structure-property relationships.






§ RESULTS





 §.§ Accuracy of Predictions


As illustrated in Figure <ref>, there are essentially 3 methods or “knobs” that can be adjusted to improve accuracy of predictions: descriptor, transform, and machine learning algorithm. As noted above, the descriptors, transforms, and machine learning algorithms we employ are listed in Figure <ref> and described in detail in the Methods Section. 



We start by examining Figure <ref>, which shows parity plots comparing machine learning predictions of GB energy against the GB energy values calculated in LAMMPS, as reported in <cit.>. There is one plot for each of the 7 descriptors examined in this work: The Atomic Cluster Expansion (ACE), SOAP, Atom Centered Symmetry Functions (ACSF), Strain Functional (SF) descriptors, a Graph description, CSP, and CNA. For each of these 7 descriptors, the results are reported for the combination of transform and machine learning algorithm that resulted in the highest overall accuracy predicted for GB energy using this dataset. The accuracy of predictions is measured by the mean absolute error (MAE) and R-squared (R^2) values. These metrics are used in tandem to best illustrate the performance of a given model. Finally, these 7 descriptors are accompanied by a parity plot labeled as “Random SOAP” where the SOAP descriptor is used as input, but with the GB energy values shuffled so that every SOAP vector points to a random GB energy value in the set. This serves as a worst-case reference value since a shuffled dataset would be expected to have no correlations.

In examining Figure <ref>, the SOAP descriptor combined with Linear Regression achieves the highest accuracy, with a low MAE of 3.89 mJ/m2 and a high R^2 of 0.99, indicating near-perfect correlation between predicted and actual values. In contrast, the Random SOAP model, where GB energies are shuffled, has a high MAE of 46.96 mJ/m2 and a negative R^2 of -0.23, confirming no predictive capability. While ACE and SF descriptors also achieve high accuracy, ACSF exhibit intermediate performance and descriptors like graph (graph2vec), CNA, and CSP exhibit significantly higher MAE and lower R^2, indicating poorer predictive performance. This suggests that the higher complexity descriptors do indeed capture more relevent information for predicting grain boundary energy.

Figure <ref> illustrates key aspects of feature engineering used to predict GB energy. First, it can be seen that due to the nature of this dataset, with many GBs concentrated about the mean GB energy value of 497 mJ/m2, it is possible to obtain a relatively low MAE, even in the case of the “Random SOAP” model. It is for this reason that we report both the MAE and the R^2 values. Caution must be exercised in assuming a model is good just because the MAE is low. One must also see a high R^2 value to show that the model results in correlated predictions; a negative value for the R^2 metric indicates that it would have been better to simply predict the mean.

Second, one can see that the `average' transform is selected as the transform providing the most accurate predictions in four of seven cases. Third, in three of seven cases, the machine learning algorithm that provides the highest accuracy is linear regression. In the other four of seven cases, MLPRegression is the most accurate. But the three cases with linear regression have much better predictions than those with MLPRegression. Fourth, one can see that the combination of MAE and R^2 values provide a nice summary of the accuracy that can be visibly seen in the parity plots. Fifth, the stark contrasts of the MAE and R^2 values between the SOAP and “Random SOAP” models illustrates that there is valuable information in the features of the averaged SOAP that is correlated with the GB energy of a given structure. This is strictly true when comparing SOAP and “Random SOAP” and likely true when comparing “Random SOAP” with the other descriptors which categorize the atomic information of the GBs in distinct manners.





While important insights can be gained from these comparisons in Figure <ref>, great care must be exercised because they are not direct comparisons; the models are different. Better comparisons can be made by holding as many variables constant between the models as possible. The three key steps (describe, transform, and machine learning) represent a 3-dimensional space of combinations, as illustrated in Figure <ref>. However, since not all combinations were evaluated, we choose to examine 2-dimensional subsets, holding constant one of the three methods, as illustrated by the yellow bands in Figure <ref>. The methods that are held constant are chosen because they typically preform better than their counterparts for the GB energy predictions examined in this work. 

The first subset we analyze compares different descriptors and machine learning algorithms, keeping the averaging transform constant since it performes well for 4 of 7 descriptors in Figure <ref>. The accuracy comparison for this subset is given in Figure <ref>a. The ACE, SOAP, and SF descriptors performed exceptionally well across multiple machine learning algorithms. The R^2 values were consistently above 0.95 in the majority of cases, indicating strong predictive capabilities, while the MAE values were comparatively low, <10 mJ/m2 in the majority of cases, reflecting accuracy in the energy predictions. ACSF had higher errors (MAE just less than 20 mJ/m2 and R^2 around 0.8). CSP, CNA, and the graph descriptor had errors around 25 mJ/m2 and R^2 < 0.67.


One might be tempted to make quick judgements about the quality of the different descriptors, but caution must be exercised because the number of features for each descriptor vary drastically, as shown by Table <ref> in the Methods section. Each descriptor forms a unique basis to represent an environment. For the SOAP, ACE, and SF techniques, the user specifies the desired order of radial basis functions, the degree of spherical harmonics, or the polynomial order to be included in the representation. Similarly, for the ASCFs, the user selects sets of 2- and 3-body functions. The graph used in combination with graph2vec creates a nearest-neighbor connected graph where the user defines a cutoff distance for the nearest neighbors. In contrast, the CNA and CSP descriptors are both singular-valued quantities for each atom, though it is noted that CNA is built on top of local graphs that could be used instead of the integer classification, as in <cit.>.


In the cases where the user picks the level of expansion, the user can determine what cutoff can be used to obtain a desired level of accuracy; this trades computation time and memory required for a larger basis that hopefully captures more of the physics. For example, when we increased the number of ACSF features from 8 to 37, the R^2 value increased from 0.65 to 0.81 but it required significantly more computation time.
One could also compare the fact that ACSFs and SFs have approximately the same number of terms, 37 and 36 respectively, but the ACSFs perform worse than SFs.  This suggests that more or different interaction pairs are probably required to detect the structural features that determine GB energy with ACSF. In fact, with a greater understanding of the ACSF implementation, it is possible that one could obtain higher accuracy even with the same number of terms. 
Although we see that this method of “covering your bases” by increasing the number of features does quite well in terms of accuracy, there is power in a pre-training choice of basis based on physical beliefs about the material, which ideas are discussed in the interpretability section of the results.

What is remarkable about the two singular-valued descriptors, CSP and CNA, is that, despite being less accurate than the SOAP, ACE, SF, and ACSF descriptors, they still achieve respectable accuracy (c.f. Figure <ref>a).  
In fact, it is simple enough that we show the equation for the LR model, 
γ = 280.26 × x_CSP + 173.14, 
where γ is the GB energy and x_CSP is the average of the scalar CSP values for a given boundary. Similarly for CNA, the single coefficient linear function is 
γ = 505.88 × x_CNA -370.71, 
where x_CNA is the average of the integer CNA values that refer to different structure types.

Finally, in Figure <ref>a, we include the graph descriptor, despite its use of a different transform, graph2vec. This descriptor performs worse than the CSP and CNA descriptors, even though it encodes more information than the singular-valued descriptors. However, as with the other descriptors, several parameters could be optimized for better predictions, including the cutoff distance of the graph, the weighting of the graph, and the selection of subgraphs in the graph2vec transform.

In summary, the effect of descriptor on accuracy in Figure <ref>a shows that in general, more features is better. The density-based descriptors, many of which are created for machine learned interatomic potentials, appear to be better at capturing the complex and intricate nature of the local atomic environments. 

In examining the role of machine learning algorithms for a given descriptor, one can see that the linear models (LR, LASSO, RidgeCV) generally outperform other types of machine learning models for nearly all descriptors when using the average transform (c.f. Figure <ref>a). The SVM (SVR) and Nearest Neighbor (KNN) models frequently rank next in performance, followed closely by Neural Network (MLPRegression) and ensemble (AdaBoostRegressor) methods. All these models perform better than the "Random SOAP" input, demonstrating their ability to identify meaningful correlations between the features and GB energy. 

The second subset we examine in the 3-D space holds the SOAP descriptor constant and varies the transform and machine learning methods, as illustrated in Figure <ref>. The results for this subset are illustrated in Figure <ref>b, where it should be noted that the average column in Figure <ref>b is the same as the SOAP column in Figure <ref>a due to the intersection of the two 2-D cross-sections. This analysis shows that in all cases in Figure <ref>b, the averaging transform significantly outperforms the other methods; MAE is less than 10 mJ/m2 for the average transform in most cases and greater than 29 mJ/m2 for the other transforms in all cases. 

All the transforms assume some prior on the important features to transform. The average transform assumes that an average environment is the most important information to preserve. KMeans clustering assumes that the clustering in the dataset and the locations of those clusters is the most important information to preserve. Largest simplex assumes that it is important to represent the data with subsections of the data that are far apart and maximize the simplex volume. CUR assumes that specific subsets, actual rows and columns of the original data, are critical and obtains these through matrix decomposition.

Given that GB energy is calculated from the sum of the energy of all the atoms divided by the area of the boundary, it is not surprising that the best tranform is an average of the atomic environments. In other words, the assumption behind the average transform aligns closely with the calculation method for GB energy, making it a suitable representation. Conversely, the assumptions underlying other transforms do not as accurately capture the relationship between the atomic collection and GB energy. However, there may be other cases where a different transform better matches with the property of interest. For example, in some cases extreme values of a distribution control the behavior, such as in fracture, and a different transform may better capture that relationship. Therefore, we hypothesize that the best transform for accurate predictions is one that preserves the relationship between the way a collection of atoms relates to the property of interest.

However, it is also possible that the choices made in this work about the transform hyperparameters resulted in poor predictions we did not seek to optimize these hyperparameters. For example, the target rank used in the CUR transform is 20 and the largest simplex transform employed 10 dimensions; it is not clear that these values are sufficient or insufficient. Similarly, the KMeans transform employed 100 clusters and it is not clear that this is representative of the number of clusters in any of the descriptors. 

It is also worth noting that the SOAP descriptor appears to fall victim to the curse of high dimensionality when transformed with KMeans clustering. On average, the KMeans clustering transform performs worse when applied to SOAP. This is likely because the high dimensionality of SOAP descriptors leads to a phenomenon known as “distance uniformity". In high-dimensional spaces, feature values tend to be equidistant from each other, making it difficult for clustering algorithms to distinguish between similar and dissimilar data points. 

The third and fourth 2-D subsets 
we examine from the 3-D space illustrated in Figure <ref>
hold a different machine learning algorithm constant. Specifically, the linear regression model was picked because of its high accuracy and MLPRegression was also picked because of the popularity of deep learning models. The results for these subsets are presented in Figures <ref>c and <ref>d where the effect of different descriptors and transforms can be seen. First, in comparing the two tables, the average transform is better with linear regression in all but the case of ACSF. The higher accuracy between linear regression and MLPRegression is evenly divided for the largest simplex transform. MLPRegression is better in two of three cases for the CUR transform and in one of four cases for the KMeans transform. However, in many of these cases, the accuracy values approach or exceed predictions by “Random SOAP”, making it difficult to judge the value of the improvements. Furthermore, these all perform worse than the average transform.  

In the case of CSP and CNA, the GaussianKDE transform performs better than the average transform for both linear regression and MLPRegression, with the exception of CNA by linear regression. Also for these, MLPRegression performs better than linear regression for three of four cases considered. In these singular-valued descriptors, the more sophisticated GaussianKDE transform and MLPRegression allow it to obtain slightly better predictions. 

This examination of the effect of all three key steps (describe, transform, machine learning) shows that the descriptor plays an outsize role in the quality of the predictions. However, the transform of the features also plays an important role and some important information can be lost at this step if care is not exercised. Finally, the machine learning algorithm appears to play more of a secondary role; if the features are correlated with the property of interest, multiple algorithms can often extract the relationship (though some methods appear to perform better than others depending on the circumstance).

   



 §.§ Feature Selection


Although SOAP achieves the highest accuracy when predicting GB energy, in our implementation SOAP also uses the most features of any of the descriptors (c.f. Table <ref>). ACE and SF achieve comparable accuracy but only use 121 and 36 terms, respectively. In fact, these two occasionally outperform SOAP. It should be noted that any of these could be adjusted to use more or fewer terms to achieve higher or lower levels of accuracy. One can also use feature selection methods to remove redundant or irrelevant information in the machine learned structure-property models. Feature selection is an important step towards interpretable machine learning models because of the challenge of interpreting the meaning of high-dimensional descriptors <cit.>.

To identify lower dimensional representations we implement a feature selection method that uses the least absolute shrinkage and selection operator (LASSO) to identify what terms are most important for retaining high accuracy. LASSO is formulated as a minimization of a least squares error plus an L1 norm regularization term scaled by the parameter λ which controls the trade-off between fitting the model accurately to the training data and keeping the model coefficients (parameters) small and sparse. This LASSO model is defined by


    min_β{1/Ny - Xβ^2_2 + λβ_1 }


For this feature selection analysis, we use SOAP due to its prominence in GB predictions and SF for its accuracy and interpretability, as detailed in the following section. Figure <ref> plots the number of non-zero terms(blue) and the model accuracy(red) of LASSO as the size of λ increases for both SOAP (Figure <ref>a) and SF (Figure <ref>b). The sparsity of the parameters increases with increased λ values. This model is trained using the average transform and illustrates just how many parameters can be neglected while maintaining a high R^2 value. The `elbow' of the R^2 curve marks where the model begins to significantly drop in accuracy.


Figure <ref> illustrates that at low λ values SF retains all 36 terms and achieves an R^2 value of 0.95. SOAP starts with more terms but can be reduced to 209 terms with a comparable R^2 value. At the high end of λ values, SOAP and SF achieve an R^2 value of 0.43 using 4 and 2 terms, respectively,. 

This illustrates that although the accuracy decreases, both of these descriptors can be reduced to a very small representation space while preserving the most important information. It is noteworthy that averaged values of both CSP and CNA achieve higher accuracy predictions of GB energy with a single scalar value with R^2 values of 0.67 for both. Thus, these two singular-valued descriptors are quite expressive and are better than just a few terms of the other descriptors. But, perhaps this is not surprising since these descriptors were designed to easily identify defects and other changes in structure with a single value, while the other descriptors were created to provide a more nuanced description of an environment with a much larger number of terms.  Consequently, while feature selection can be helpful in removing redundant and irrelevant information, the descriptors selected have a big impact on identifying important features that contribute to the machine-learned structure-property model.





 §.§ Interpretability




The goal in this work is to obtain structure-property models that are both accurate and interpretable. Interpretability is of critical interest for the advancement of science since machine learning models could easily become black box models that we can't understand. We have examined the impacts that the descriptor, transform, and machine learning model have on accuracy and methods to select the features that have the biggest impact on the models. We now focus on extracting interpretable information from these models. 



  
Descriptors
ACE, SOAP, SF, ACSF, CSP, and CNA all represent the local environment around each atom. These representations often involve sums over neighboring atoms, where each term in the sum depends on the distance and/or angular relationship between the central atom and its neighbors. ACE, SOAP, and SF all represent the angular relationships in terms of spherical harmonic expansions, while ACSF uses a more explicit n body expansion of the angular terms. SOAP utilizes a truncated Gaussian function for the radial distribution and then recasts its basis function in a bispectrum approach. This results in the mixing of angular and radial nodes in the invariant basis functions. ACE utilizes multiple radial functions within a specified cutoff radius to capture the radial environments. In SF, the radial and angular nodes of the expansion are kept separate resulting in descriptors that are analogous to an atomic orbital expansion. SF then represents this information in the minimal set of invariants required to characterise deformations up to the 4th order (i.e., second derivatives or curvature of the strain). ACSF uses multiple Gaussian radial terms which are then convolved with the angular terms. All of the methods carry out their expansions to a level of accuracy that can be defined by the user.

In contrast to the other methods, the graph description (used in the graph2vec transform) characterizes the GB in a periodic, weighted graph. In this framework, the nodes of the graph correspond to the spatial positions of atoms and the edges are weighted by the distance between neighboring atoms within a cutoff distance. Thus, this method captures the complex arrangements of the atoms in a GB as a whole.

All the descriptors transform the 3D spatial coordinates of atoms into a high-dimensional feature space. This transformation is designed to capture complex interactions and symmetries, but it also means that the resulting descriptors are often far removed from the intuitive, three-dimensional space in which atoms actually exist. The use of physical descriptors (such as bond bending terms or spherical harmonics) leaves open the possibility that terms could be interpretable. However, the derivation of the terms often renders this difficult, and most approaches do not offer any interpretations of their descriptors. This is not a criticism of these descriptors as they were not designed for the purpose of interpretability. 

Many of these descriptors were designed for machine learned potentials or for classification of atom neighborhoods. In other words, many of these descriptors were meant for forward modeling only, but to extract interpretable information, we need to go backwards through forward models. Therefore, pointing out weaknesses in backwards applications to extract specific terms is not a completely fair criticism. Nevertheless, because we wish to understand which descriptors provide this backwards path to interpretability we examine them anyway.

In a backwards application, ACE and SOAP are less than ideal because they do not retain knowledge of how their rotationally invariant descriptors are oriented in space. In contrast, the SF approach explicitly retains such terms. Additionally, for ACE and SOAP, one can know which degree of the spherical harmonics are important as illustrated in Supplemental Figure S2, 
but finer structural detail within that degree is lost. It may be possible to preserve some of that information if interpretability is desired. In a backwards application of ACSF, it is not clear to the authors how one would interpret the values of the 2- and 3-body functions. At a minimum the summation over neighbors makes it difficult to connect with specific atomic structures. However, it is possible that improved application of ACSF descriptors could be useful in extracting the local symmetries and deviations therefrom that could result in interpretability. 

Graph2vec's interpretability suffers due to its abstract representation of graph embeddings. In its current implementation, there is no method by which to go backwards and make use of the subgraphs that were extracted, but this could be added for improved interepretability. CSP and CNA are non-unique descriptors where multiple atomic configurations can produce identical values, making it challenging to pinpoint specific structural features responsible for observed properties.


As a result, the inherent complexity of these descriptors makes it challenging to reverse-engineer the key features of the structure-property relationships. Most of these descriptors do not support easy interpretability because they were not designed for this purpose. However, the SF approach is distinct in characterizing the invariants of physical deformations up to the 4th order. This descriptor is examined in additional detail in the case study below.




  
Transforms
Just like descriptors, the ability to go backwards through transforms is crucial for interpretability. Averaging, the best transform for the most accurate descriptors, can not be applied backwards. One can only make conclusions about the average atomic environments. As explained in the Accuracy subsection, there are assumptions inherent to each of the transforms. Averaging best matches the physics of how GB energy is calculated. 

On the other hand, the largest simplex and CUR methods actually preserve specific features from the input matrix. By selecting the most important rows, these techniques effectively fix the size of the matrix that describes the GB. In other words, they choose the atomic environments that, even before training on data, are likely to preserve valuable information based on the assumptions of these transforms listed earlier. This deliberate selection ensures that the matrix representation of each GB remains concise and meaningful. Therefore, both CUR and largest simplex methods identify original atomic environments that can be analyzed for their interpretability and impact on the predictions.

The KMeans clustering transform, while fixing the matrix representation, does not preserve the original rows of the matrix but rather a number of cluster centers. Thus it is like averaging where information is lost in the transform application.  In the case of KMeans clustering the original rows can be identified by finding the nearest neighbors to the cluster centers. Although this can be difficult in higher dimensions. 

Finally, it is important to note however that other properties of interest might have different relationships with the atoms involved and therefore a different transform might work best, as discussed in the Accuracy subsection above. 



  
Machine Learning Models
Finally, the ability to extract interpretable information from the machine learning models depends heavily on the model used. When linear models are used, one can readily identify the features that have the greatest impact on the predictions based on values of the coefficients. The relationships are easily defined and understood. One can even reduce the features using a feature selection or regularization approach <cit.>, as discussed above, to more easily identify the important features. However, in non-linear models identifying the most important features is not as simple. 

If a KNN performs well, that may be indicative of clustering in the input features, which would suggest that the clustering is relevant to the property of interest. Ensemble methods like AdaBoostRegressor fit the data multiple times, focusing more on difficult cases with additional iterations. While decision trees provide inherent interpretability, ensembles of them make that more difficult. However, these algorithms can export importance scores to learn about which features are of greatest interest.

In our Support Vector Regression (SVR) model, we utilized a linear kernel to fit the data with a hyperplane. The choice of a linear kernel allowed us to maintain a straightforward relationship between the features and the output. This approach ensured that the model remained easily interpretable, as each feature's effect on the output could be independently assessed through the corresponding coefficients. By employing a linear kernel, we avoided the complexities associated with high-dimensional transformations, which are common in nonlinear kernel methods.

A neural network model uses a series of layers whose connectedness and construction can be highly variable. In the neural network each layer typically has a non-linear activation function to learn non-linear relationships. While this may improve predictions, the overall structure of the neural network makes it difficult to extract interpretable information.

However, in any of these cases, one can use additional tools, such as those that fit into the category of Explainable AI <cit.>, to extract the features of greatest impact or importance.

Having just discussed how the method selected for each step in the process of predicting structure-property relationships impacts interpretability, the discussion has remained theoretical. In the following subsection, we examine a case study where we can be more specific about the ability to extract meaningful information from machine learned models.




 §.§ Case Study of Interpretability





    
    
    
    
    
    
    
    
    
    
    
    
    
    
    

Here we examine how a combination of one descriptor, one transform, and features selection in two different machine learning models provide interpretability. We employ the SF descriptor because, as noted above, it was defined with the express purpose of retaining a physical meaning. We employ the average transform because, as noted above, it retains a connection to how our property of interest, GB energy, is calculated. Finally, we examine feature selection in two different machine learning methods to illustrate the differences related to interpretability. 


As discussed above, regularized linear models allow easy identification of the most important features in a model. We revisit the results of the LASSO application to the SF illustrated in Figure <ref> and described in the Feature Selection subsection above. As the model complexity is reduced, the terms that remain can be considered the most important for GB energy prediction and interpretability. The last five SF terms to be removed by LASSO, and therefore the top five terms for predicting GB energy, are listed in Table <ref>. Next to each term in Table <ref> is the sign of the correlation of that term in the model with its effect on the GB energy prediction.


To understand the impact of a non-linear model, we employ Extra Trees regression. For interpretability, this is used in conjunction with SHapley Additive exPlanations (SHAP) analysis <cit.>. SHAP values explain the prediction of an instance by computing the contribution of each feature to the prediction. This method is based on game theory and helps in attributing the prediction output to individual features, offering a fair and consistent way to understand the model behavior.





Figure <ref> plots the SHAP analysis of an Extra Trees regression, showcasing the top 10 SF features sorted by their average SHAP value. 
A SHAP value indicates how much a given feature changes the output of the model compared to the baseline prediction; positive and negative SHAP values correspond to a positive and negative effect on the predicted property, respectively. The colors represent the values of the features for each data point; with red and blue values corresponding to high and low values of a particular feature, respectively. For example, if a dot is blue and located on the right side (positive SHAP value), the low value of that feature increases the value predicted by the model. 
The top five features from the SHAP analysis are also listed in Table <ref>, along with the sign of the correlation between the term and its effect on the predicted GB energy in that model.




There is significant alignment between the LASSO and SHAP analysis, as illustrated in Table <ref>. Four of the top five SF terms are the same in both analyses. Furthermore, the sign on the correlation of these four terms is the same. These four terms are P4I8, P1I0, P2I0, and P4I9, and are accompanied by the P3I4 and O2I0 terms that only appear in one model. 

To further confirm the expected correlation of these top terms from the LASSO and SHAP analysis, we plot the average SF values for each GB against both excess volume and energy in Figure <ref>. Excess volume is included as it is known to have a positive correlation with GB energy <cit.>. It is noted that correlation plots for all SF terms are plotted in Supplemental Figures S8-S10. 
The top 5 features from these supplmental plots with the highest correlation with GB energy are listed in Table <ref> along with the sign of the correlation. It is worth noting that five of the six correlations plotted in Figure <ref> have the same sign as that of the models listed in Table <ref>. The exception is the O2I0 term, which has an opposite sign in the model but only has a weak correlation with energy.

As noted earlier, SF comes with an added interpretability benefit since each SF descriptor classifies something unique about the deformation. Each SF term can be classified into one of five categories: density, deformation magnitude, deformation type, and internal and external orientation[Supplemental Figures S8-S10 
identify the categories for all 36 terms. Also, note that some of the categories from <cit.> have been renamed: deformation magnitude was net deformation, deformation type was directionality, and the two orientation types were not listed as a categories.] <cit.>.

The P4I8 term is one of three density metrics, and is an r^4 average for all the atoms in the neighborhood of a given atom. Figure <ref> shows that there are reasonable correlations with excess volume and GB energy, which haveR^2 values of 0.53 and 0.72, respectively.

The P1I0, P2I0, and P3I4 terms are all categorized as deformation magnitude.
P1I0 characterizes the gradient in the density. 
P2I0 characterizes the net deviatoric strain, akin to the von Mises strain invariant of the neighborhood. There should be substantial deformations of the fcc structures in the lattice neighboring the GB and shearing will be one of the primary means for this. The P3I4 term measures the extensional contribution of the strain gradient. The importance of this term can be attributed to the presence of strain gradients at GBs, especially in the case of GBs that can be represented as dislocation arrays. It is similar to the P1I0 term, where it is a gradient term, but it is weighted by r^3 rather than r^1. As illustrated in Figure <ref>, the P1I0 and P3I4 have strong correlations with excess volume and energy, while P2I0's correlations are weak.


The P4I9 metric is an internal orientation metric that defines the orientation between the local lattice and the P4I6 measure of the shear. The latter is similar to the net shear metric P2I0, except that it is weighted by r^4 rather than r^2.  
Finally, the O2I0 is an external orientation metric that is also highlighted by the LASSO method; this terms defines the orientation of the lattice shearing (measured by P2I0) with respect to the normal of the GB. Thus, it has some similarity to P4I9. Given that aluminum is not isotropic (albeit with a relatively small Zener ratio), it is not surprising that the amount of shear necessary to accommodate the mismatch at a GB will be related to the direction of that shear i.e., the crystal orientation. 

Interestingly, the P4I9 and O2I0 terms both have little to no correlation with excess volume and energy on their own. But in conjunction with the other terms in the models, they are deemed more important for the prediction of GB energy than other terms that have strong correlation with energy. For example, the P2I2 term (a density metric based on r^2 weighting) has much stronger correlation with energy than the P4I9 term, but the models are not predicting based on any single term alone, but the combined effect of multiple terms. Thus, some of these terms may provide more of a secondary effect that can distinguish nuanced variations of the GB energy and such effects may not be apparent in 2-D cross-sections examining single-value correlations.

The consistency of the top features and their identifiable correlations with energy in all but one case illustrate that this combination of SF descriptor, average transform, and both linear and non-linear machine learning models is capturing useful trends. The average transform tells about general trends of the atoms in the GB but not about specific local atomic environments. The linear model provides detailed insight into the influence of each parameter. The non-linear model provides this insight through the SHAP analysis. Most importantly, the SF descriptor connects the features with physical, interpretable attributes of the GB structure-property relationship.

However, the analysis also uncovers the fact that different models will extract different features. While there are only minor differences in the top five features listed in Table <ref>, the difference becomes more dramatic as more terms are compared. In addition, other models will extract entirely different features; a Bayesian linear model applied to the data showed correlations with many of the SF metrics listed above, but sometime with the same effect and sometimes with an opposite effect on the GB energy. However, the Bayesian analysis, which is described in the supplemental material, required an additional step in describing the data (  subtracting the mean and dividing by the standard deviation). The model input is slightly different and leads to different correlations as a result. Care must be taken since the machine learning is only finding correlations in the data that has been through feature engineering, it is not finding the causal relationships for the predictions.




§ DISCUSSION

Prediction in variable-sized atom-clustered structures consistently require three steps: describe, transform, machine learning. Each of these steps, illustrated in Figure <ref>, play a role in the resulting model accuracy and interpretation. Such feature engineering is frequently employed to provide more accurate predictions or better interpretation of the results. In the fast growing environment of machine learning and artificial intelligence models, the diversity in steps taken by different groups makes it challenging to know which methods lead to improved accuracy and interpretability. We have examined this challenge in GBs, which, like all variable-sized atom clusters, require a transform to obtain consistent feature sizes. By attempting to standardize the various steps of the feature engineering process, we have aimed to understand how each step in the process affects the accuracy and interpretability of the resulting model predictions.


Descriptors play an essential role in taking atom structure information, most often represented by Cartesian coordinates, and mapping that to a feature vector that encodes the most critical information.
Our findings underscore the robustness of physics-inspired structural representations <cit.> in capturing the intricate behaviors of GBs. Notably, descriptors such as the SOAP, ACE, and SF demonstrated superior predictive accuracy, underscoring their potential in advancing computational materials science. SF stands out in this group of accurate predictors because it has a low feature count and each feature has a physical meaning in terms of the strain in the neighborhood of each atom. Perhaps one of the interesting conclusions is that higher order deformations (i.e., strain gradients and higher) should be considered for accurate predictions of GB energy. This is likely the main reason behind higher predictive capability of SOAP and ACE compared to SF, as the current implementation of SF considers only up to 4th order terms, whereas spherical harmonics up to 12th order were considered for both SOAP and ACE. This is also corroborated by LASSO analysis for SOAP descriptors (Supplemental Figures S1 and S2), 
which shows persistence of l=8,10 terms. 


In short, it appears that better accuracy can always be achieved with additional descriptor information. For example, there are numerous cases where concatenation of one or more descriptor improves the learning <cit.>. But, longer feature vectors complicate interpretability. Additionally, some descriptors that can encode a lot of information cannot be processed in reverse to provide physical insight from model predictions. Therefore, those descriptors that encode physics that can be readily extracted from a model prediction are likely to provide the greatest insight into the resulting structure-property models. 


The transforms applied to obtain consistent feature sizes from the variable-sized input data impact both the accuracy and interpretability of the resulting model. As hypothesized in this work, the average transform provides the best accuracy because it is the most similar to the procedure used to calculate GB energy from the atomic structure; the excess energy, relative to the bulk energy, for all the atoms is summed and divided by the area of the boundary. We further hypothesize that other properties of interest may benefit from descriptors that capture the important features of that property. For example, properties controlled by extreme values in a distribution may benefit from a transform that captures the structural aspects of those extreme values. However, while some transforms might provide more accurate predictions, they may make interpretation difficult. The average transform is one of these and it can not be processed in reverse to tell us how the distribution of values that were averaged might be critical to a certain structure-property relationship.


The machine learning models and algorithms also play a significant role in both accuracy and interpretability. As discussed above, simpler models are preferred to complex models and the linear models provide high accuracy in many cases. The linear models are easier to interpret because the contribution of any given feature can be easily discerned. However, non-linear models can make use of Explainable AI tools <cit.>, such as SHAP analysis discussed above, to extract interpretable meaning from the resulting model. Models can sometimes be overly complex; we suggest opting for simpler linear models whenever feasible.

From an accuracy standpoint, we recognize that we have not considered all combinations of descriptors, transforms, and machine learning models. Neither have we done an exhaustive search of the hyperparameters for each of the descriptors, transforms and machine learning models beyond a simple gridsearch of hyperparameters for the machine learning models. We cannot guarantee that higher accuracy couldn't be achieved with adjustments to the models we examined. But, by providing a systematic approach, and a consistent dataset across all the models evaluated in this work, we provide a framework and benchmark against which new and improved models can be tested, like the MNIST dataset has served for benchmarking machine learning efforts in optical character recognition <cit.>. Furthermore, by providing standard steps and language for the comparison of models along with a deliberate attempt to employ principles of the machine learning community, we hope the grain boundary community can identify the best methods to obtain structure-property relationships that will drive innovation.

 
Feature selection is a tool that can be used in conjunction with machine learning models to reduce the feature vector to those items that are the most critical for the accuracy in the model. The assumption from an interpretability standpoint is that these selected features are the most important for the model and can therefore be used to obtain insight into the structure-property models.


The final case study illustrated how the SF descriptor provides insight into the density and deformations that correlate with GB energy. Four of the top five features were shared between a linear and non-linear model.
This nuanced view, where some features are consistently highlighted across methods while others are unique to specific approaches, offers a richer understanding of the predictive landscape. It suggests that while some attributes of GBs are universally recognized by various predictive models, others may be more method-dependent, possibly due to underlying assumptions or mathematical formulations inherent to each technique. This insight not only enriches our understanding of feature selection dynamics but also guides further investigation into the specific roles these features play in material structure-property relationships.

The implications of these results offer a pathway towards more precise and efficient predictive models that can be instrumental in materials design and engineering. By enhancing our ability to predict GB properties, these findings could facilitate the development of materials with optimized mechanical properties, thereby having a profound impact on various industrial applications.







§ METHODS


In the application of machine learning grain boundaries (GBs), feature engineering <cit.> is crucial for enhancing model performance by tailoring input data to more accurately reflect the underlying problem and prepare for predictive modeling. Figure <ref> illustrates the three key steps taken for predicting GB properties: employing descriptors to encode or describe the structure (step 1), transforms to standardize the data size across different GBs (step 2), and  machine learning or prediction algorithms to predict a GB property from the input data (step 3). The methods and techniques employed in this work for the three key steps are described below.



 §.§ Descriptors


The descriptors employed in this work are inspired by a phylogenetic tree of structural representations shown in Figure <ref>. The tree is reproduced and adapted from a paper titled “Physics-Inspired Structural Representations for Molecules and Materials,” by Musil et al.  <cit.>. They emphasized the requirements for developing descriptors that map the atomic positions in Cartesian coordinates to a new metric space commonly called the feature space, the representation <cit.>, descriptor <cit.>, or fingerprint <cit.>. This mapping is required for predictive modeling because atom locations represented as Cartesian coordinates cannot uniquely characterise materials <cit.>. Specifically, atom locations do not preserve symmetries and any translation or rotation of a material will change the Cartesian coordinates.
Thus, a descriptor should address requirements of completeness, symmetry, smoothness, and additivity. These requirements and the general process is discussed in additional detail in the supplemental materials. 





In this work, we considered descriptors from all the branches. A more complete description of all the methods considered is provided in the supplemental materials. In the end, we selected the following methods listed in the tree by Musil et al.  <cit.>: smooth overlap of atomic positions (SOAP), atomic cluster expansion (ACE) <cit.>, and Behler-Parinello's atom centered symmetry functions (ACSF) <cit.>. These three methods are boxed in green in Figure <ref>. We also include several other methods. First, we include a recent Strain Functional (SF) Descriptor, which define the minimal set of invariants required to characterise deformations up to the 4th order (i.e., second derivatives or curvature of the strain) <cit.>. This method is similar to SOAP in having smoothly declining neighborhood function, compared to the sharper neighborhood cutoffs of ACE, MTP and SNAP. Second, we include two methods commonly used to identify GB atoms: Centrosymmetry Parameter (CSP) and Common Neighbor Analysis (CNA). Finally, we include a periodic graph description of a boundary where every atom in the GB is defined by a node, which is connected to edges within a certain cutoff distance. Each of these additional methods is listed in Figure <ref> in green under the branch we identified as most relevant. Techniques considered but not selected from other branches are discussed in the supplemental materials, with reasons being that in some cases they were not appropriate for characterization of single species large GB structures and in other cases, code was not readily available for implementation.

 
 



 
The SOAP, ACE, and ACSF implementations all required user defined parameters that impact how many terms is used to represent an atom environment. SOAP and ASCF are implemented using the `DScribe' python library <cit.>, a powerful tool designed for creating descriptors in atomistic systems. For SOAP, all default parameters were used excluding , , , and . ACSF are unique in the sense that the user defines a list of interactions i.e. pairs, triplets for the various functions used by ACSF. Since these lists can not be described briefly here, they are included in the supplemental materials. The implementation of the ACE algorithm was achieved using the `ace.jl' package within the Julia programming environment <cit.>. A basis was created using the following parameters: , , , , , and . The SF descriptors are currently 4th order, but can be extended to higher orders. SF used the following parameters:  and . The graph description has the nodes of the periodic graph listed as their spatial coordinates. The edges are weighted by the distance to their neighbors that are within  Å of the atom (node) of interest. CNA <cit.> and CSP <cit.> both represent each atom's environment with a single value, which is calculated during the creation of the dataset using LAMMPS <cit.>. Table <ref> shows the feature lengths of each of our descriptors denoted by M.






 §.§ Transforms

Since GBs and other variable-sized atom-clustered structures can have variable numbers of features, the feature size must be standardized through some sort of transform. This will transform the N_i × M feature representation from the descriptor to a P × M feature representation that is identical for all atomic structures. 


In this work we examine 6 possible transforms :  average, CUR or skeleton matrices <cit.>, KMeans clustering <cit.>, the largest simplex <cit.>, kernel density estimation (GaussianKDE) <cit.>, and graph2vec <cit.>.  The motivation behind this set of transforms, along with a more detailed explanation of each is provided in the supplemental materials. 


The Average, largest simplex, CUR and Kmeans transforms are used with almost all descriptors. GaussianKDE is used only with CSP and CNA and graph2vec is used only with the graph descriptor. 
Table <ref> lists the theoretical input and output size of each transform along with the P values used in this work. It is noted that no attempt was made to find the optimal P value for each implementation. Explanations for why each P value was chosen is given below along with the descriptions of the transforms.




 §.§ Machine Learning

Machine learning is oftentimes yet another mapping to a different feature space. To understand the impact of this step, the research employed a comprehensive and systematic approach to compare a diverse set of machine learning algorithms. This set includes three linear models: LinearRegession, Lasso, and RidgeCV; one support vector machine: SVR; two ensemble methods: Extra Trees and AdaBoost; one nearest neighbor method: KNN; and one neural network (deep learning) model: MLPRegresssion. All of these methods are available for implementation via the sklearn python library <cit.>. This set of algorithms were selected using the sklearn documentation where various supervised methods are grouped by methodology. 


Central to this methodology was the implementation of grid search for hyperparameter optimization across all algorithms. This was implemented in an effort to provide a uniform and equitable basis for comparison between models. 

We disclose that beyond the grid search, which required user input, little additional effort was taken for the optimization of the hyperparameter values. Typically convergence was costly from a time perspective for the non-averaging transforms due to the increased complexity, so the range of hyperparameters defined in the grid search was customised according to convergence time and not accuracy. While we recognize that this may limit the accuracy of these predictions, our data consisted of only a train and test set, so this also minimized the chances of overfitting.

A 5-fold cross-validation strategy was applied to all models.  
This approach assesses the models' performance on unseen data, providing a dependable estimate of their generalization capabilities. For performance assessment, both the Mean Absolute Error (MAE) and the Coefficient of Determination (R^2) were recorded. The MAE provided an understanding of the average absolute error made by the models, while the R^2 offered insights into the proportion of variance in the dependent variable that could be explained by the independent variables, acting as another method for measuring the accuracy of the model.

Comprehensive documentation of all aspects of the model training process, including hyperparameter values, cross-validation details, and performance metrics, was maintained for the reproducibility of the research. A code base with a sample dataset is included to ensure that the experiments can be replicated and validated by others in the scientific community.



 §.§ Dataset

To compare the diverse descriptors, transforms, and machine learning techniques, we employ a recently published atomistic dataset of aluminum GBs  <cit.>. This dataset provides a comprehensive sampling of the five degrees of crystallographic character. The datasest includes 7304 pure Aluminum GB structures and their corresponding energies. The construction and process of obtaining the minimum energy structure for each GB is described in detail <cit.> and the structures are available for download <cit.>.
These GB structures were created with the emperical embedded atom model (EAM) potential created by Mishin et al. <cit.>. Thus, the physics inherent to these structures is limited to the accuracy of this emperical potential and the methods used to construct the GBs.
For purposes of limiting data storage, each GB includes ± 15 Å of atoms relative to the expected location of the GB. Because the size of some of the GBs in the original dataset were too large for some of the descriptor implementations, we use a subset of 7174 GBs, which are those GB structures that contain less than 35,000 atoms. It is expected that this dataset contains sufficient diversity both in crystallographic character and atomic structure to serve as a robust basis for the comparisons and interpretations provided in this work.




§ ACKNOWLEDGEMENTS

This work was primarily supported by the U.S. National Science Foundation (NSF) under Award #DMR-1817321. EMK and NM were supported by internal LANL LDRD funding (XX9A, XXG0).



§ COMPETING INTERESTS


All authors declare no financial or non-financial competing interests. 



§ DATA AVAILABILITY


The datasets generated and/or analysed during the current study are available in the Mendeley Data repository, https://doi.org/10.17632/4ykjz4ngw <cit.>.



§ CODE AVAILABILITY


The underlying code for this study is available at 

https://github.com/braxtonowens/gbcomparehttps://github.com/braxtonowens/gbcompare. 



§ CREDIT AUTHOR STATEMENT


C. Braxton Owens: Conceptualization, Methodology, Software, Formal analysis, Data Curation, Writing - Original Draft, Writing - Review & Editing, Visualization;
Gus L. W. Hart, Eric R. Homer - Conceptualization, Methodology, Formal analysis, Writing - Original Draft, Writing - Review & Editing, Visualization, Supervision, Funding acquisition
Tyce W. Olaveson: Conceptualization, Software, Formal analysis;
Nithin Mathew, Jacob P. Tavenner, Edward M. Kober, Garritt J. Tucker - all performed the following related to the Strain Functional analysis: Formal analysis, Writing - Review & Editing;


 elsarticle-num 
 



































 Paper Title: Unsteady slip pulses under spatially-varying prestress.






eran.bouchbinder@weizmann.ac.il
Chemical and Biological Physics Department, Weizmann Institute of Science, Rehovot 7610001, Israel

§ ABSTRACT 
It has been recently established that self-healing slip pulses under uniform background/ambient stress (prestress) τ_ b are intrinsically unstable frictional rupture modes, i.e., they either slowly expand or decay with time t. Furthermore, their spatiotemporal dynamics have been shown to follow a reduced-dimensionality description corresponding to a special L(c) line in a plane defined by the pulse propagation velocity c(t) and size L(t). Yet, uniform prestress is rather the exception than the rule in natural faults. Here, we study the effects of a spatially-varying prestress τ_ b(x) (in the fault direction x) on 2D slip pulses, initially generated under a uniform τ_ b along a rate-and-state friction fault. We consider both periodic and constant-gradient prestress distributions τ_ b(x) around the reference uniform τ_ b. For a periodic τ_ b(x), pulses either sustain and form quasi-limit cycles in the L-c plane or decay predominantly monotonically along the L(c) line, depending on the instability index of the initial pulse and the properties of the periodic τ_ b(x). For a constant-gradient τ_ b(x), expanding and decaying pulses closely follow the L(c) line, with small systematic shifts determined by the sign and magnitude of the gradient. We also find that a spatially-varying τ_ b(x) can revert the expanding/decaying nature of the initial reference pulse. Finally, we show that a constant-gradient τ_ b(x), of sufficient magnitude and specific sign, can lead to the nucleation of a back-propagating rupture at the healing tail of the initial pulse, generating a bilateral crack-like rupture. This pulse-to-crack transition, along with the above-described effects, demonstrate that rather rich rupture dynamics can emerge from a simple, spatially-varying prestress. Furthermore, our results show that as long as pulses exist, their spatiotemporal dynamics are related to the special L(c) line, providing an effective, reduced-dimensionality description of unsteady slip pulses under spatially-varying prestress.

Unsteady slip pulses under spatially-varying prestress
    Eran Bouchbinder
    Received 2 August 2023 / Accepted 7 May 2024
======================================================





§ BACKGROUND AND MOTIVATION



Classical models of earthquakes, which consider a single frictional rupture that propagates along a homogeneous and planar fault separating unbounded linear elastic blocks in the presence of a uniform background/ambient stress (prestress) τ_ b, offer insight into the physics and mechanics of faulting <cit.>. Single frictional rupture is classified into crack-like modes and self-healing slip pulses <cit.>, where the latter feature a characteristic size L that implies a finite slip duration at points along the fault. Yet, our understanding of even these relatively simple models is incomplete.

Major recent progress has been made in understanding the spatiotemporal dynamics of self-healing slip pulses propagating along rate-and-state friction faults under a uniform prestress τ_ b <cit.>. It has been shown that slip pulses are intrinsically unstable objects that cannot propagate steadily, i.e., they either expand (L̇(t)>0) or decay (L̇(t)<0) with increasing time t. Yet, it has been shown that the unsteady pulses dynamics are slow and follow a special L(c) line in a plane defined by the pulse propagation velocity c(t) and size L(t). The special L(c) line corresponds to steady pulse solutions parameterized by τ_ b, which is composed of unstable points that nevertheless form a dynamic attractor at a given τ_ b. As such, special L(c) line should be regarded as an equation of motion for unsteady pulses under uniform prestress. This reduced-dimensionality description remains valid for other pairs of pulse quantities, e.g., the peak slip velocity v_ p(t) and L(t) <cit.>.

Real faults in the earth crust generally feature spatially-varying prestress <cit.>. Such nonuniform prestress distributions may emerge from previous inhomogeneous slip histories, or to effectively mimic either spatial variations in the frictional strength or geometrical fault irregularities. The latter include barriers <cit.>, bends <cit.>, double fault bends <cit.>, branched fault systems <cit.> and fault surface roughness <cit.>. Consequently, it has been recognized quite a long time ago that earthquake models should be extended to include spatially-varying prestress distributions <cit.>. It is particularly interesting to understand whether and to what extent a nonuniform prestress may contribute to the emergence of slip complexity, including phenomena such as non-self-healing slip pulses <cit.>, rupture arrest, back-propagating rupture <cit.> and pulse-to-crack transitions <cit.>.

Indeed, some works have studied the effects of nonuniform prestress on frictional rupture dynamics in various spatial dimensionalities, including in three dimensions (3D) <cit.>, in two dimensions (2D) <cit.> and in one dimensional (1D) space <cit.>, to list a few among others. The PhD thesis of <cit.> (see chapter 5 therein) focused on 2D pulse propagation in nonuniform prestress fields, arguing that the interaction of pulses with inhomogeneous prestress distributions is of particular interest as the finite size of pulses implies larger sensitivity to prestress spatial variations compared to crack-like rupture, which may tend to smooth out their effect.

Our goal in this paper is to study 2D unsteady pulse dynamics under spatially-varying prestress distributions τ_ b(x) of two classes. One is a periodic prestress and the other is a constant-gradient prestress. We will employ the above-mentioned reduced-dimensionality description of unsteady slip pulses, originally developed under uniform prestress <cit.>, to quantify pulse dynamics under nonuniform τ_ b(x) and to test the degree by which this theoretical framework extends to a broader class of problems. Our analysis also employs well defined initial pulse states, corresponding to controllably perturbed steady-state pulse solutions.



§ PROBLEM SETUP: INITIAL PULSES AND THE SPATIALLY-VARYING PRESTRESS Τ_ B(X)



We consider slip dynamics along a rate-and-state friction fault separating two infinite linear elastic blocks under anti-plane shear. The displacement field is u_z(x,y,t), where y=0 defines the fault and z is perpendicular to the xy plane. The frictional strength τ(v,ϕ) is a functional of the slip velocity field v(x,t)≡2∂_t u_z(x,y=0^+,t) and of the state field ϕ(x,t). The latter follows the “aging law” ∂_tϕ(x,t)=1-v(x,t) ϕ(x,t)/D, where D is a characteristic slip displacement <cit.>. τ(v,ϕ), whose explicit expression is given in the Supplementary Material (SM, appended below), features an N shape under steady-state sliding (for which ϕ=D/v), with a transition from rate-weakening to rate-strengthening friction occurring at a local minimum (v_ min,τ_ min) of the steady-state friction strength (see SM).

Spatiotemporal slip dynamics for t>0 are described by the boundary integral equation <cit.>

    τ[v(x,t),ϕ(x,t)] = τ_ b(x) -μ/2c_ sv(x,t) + s(x,t)  ,

balancing the frictional strength on the left-hand-side with the interfacial shear stress on the right-hand-side. The latter features three contributions; the first is the spatially-varying prestress τ_ b(x), the second is the so-called radiation-damping term (where μ is the shear modulus and c_ s is the shear wave-speed) and the third, s(x,t), is a spatiotemporal convolutional integral that accounts for the long-range interaction of different parts of the fault, mediated by bulk elastodynamic deformation. Equation (<ref>), along with the ϕ evolution equation, are solved in the spectral domain <cit.> using the open-source library cRacklet <cit.> in a periodic domain of size W, i.e., x∈[-12W, 12W] with periodic boundary conditions (see SM). The problem formulation is completed once the initial conditions and τ_ b(x) are specified.




In setting up the initial conditions, our goal is to employ well defined and controlled states that will allow us to address our main question regarding the interaction of slip pulses with nonuniform prestress distributions. Consequently, we are not interested in spontaneous frictional rupture nucleation under nonuniform prestress, but rather in generating propagating self-healing slip pulses with known properties. To this end, we invoke the steady-state pulse solutions of <cit.>, obtained under a uniform prestress, to be denoted as τ^_ b. Each such solution is characterized by time-independent slip velocity v^(x) and state ϕ^(x) fields, which propagate steadily at a velocity c^ and feature a characteristic size L^, see Fig. <ref>a. Such solutions were shown to correspond to “saddle configurations” separating expanding from decaying pulses. This implies that small positive perturbations of the steady-state pulse, which we quantify through an instability index Δϵ>0 (see SM), lead to a growing pulse, while small negative perturbations of the steady-state pulse with Δϵ<0 (see SM), lead to a decaying pulse, where |Δϵ| determines the initial rate of instability.

Yet another major advantage of using controllably perturbed steady-state pulses, obtained under uniform prestress τ^_ b, as initial conditions for Eq. (<ref>), is that they set a dynamical reference for the analysis under spatially-varying prestress τ_ b(x) to follow. Specifically, as already stated, it has been recently demonstrated that the unsteady pulse motion under uniform prestress τ^_ b, e.g., as quantified by the time evolution of L(t) and c(t), closely follows the special L(c) line in the L-c plane, whether the pulse grows/expands (L̇(t)>0 for Δϵ>0) or decays (L̇(t)<0 for Δϵ<0), see Fig. <ref>c. The special L(c) line, which corresponds to steady-state pulse solutions parameterized by τ^_ b <cit.>, has been shown to be a dynamic attractor (also for pulses nucleated away from it) and hence to provide a reduced-dimensionality description of unsteady slip pulses under uniform prestress <cit.>. The degree by which this reduced-dimensionality description extends to unsteady pulse dynamics under spatially-varying prestress is a central question addressed in this work.

To complete the problem formulation, we need to specify the spatially-varying prestress distributions τ_ b(x) in Eq. (<ref>). A first class of prestress distributions corresponds to a periodic (sinusoidal) variation of the form

    τ_ b(x) = τ^_ b[1+δsin(2π n x/W) ]  .

Here, the prestress distribution is characterized by two parameters: a dimensionless amplitude δ and a wavelength λ=W/n determined by n (expressed in terms of the domain size W). A second class of prestress distributions corresponds to a constant spatial gradient dτ_ b(x)/dx that is characterized by a tilt angle θ according to

    τ_ b(x) = τ^_ b[1+2tan(θ) x/W ]  ,

where θ>0 corresponds to a counterclockwise tilt. Examples of members of the two τ_ b(x) classes are shown in Fig. <ref>b.

The fault average of both distributions in Eqs. (<ref>)-(<ref>) satisfies ⟨τ_ b(x)⟩_x=τ^_ b, where the latter characterizes the uniform prestress for which the perturbed steady-state (serving as the t=0 initial condition) was obtained. This setup mimics physical situations in which an unsteady slip pulse obtained under a uniform prestress τ^_ b enters at t=0^+ a region of nonuniform prestress τ_ b(x) (with a fault average that equals τ^_ b), and its subsequent t>0 dynamics are tracked. Note also that we are interested in prestress distributions that vary rather slowly on the characteristic scale L^ (the size of the initial slip pulse). In the other limit, where the characteristic spatial scale of variation of τ_ b(x) is smaller than L^, we expect the pulse to smooth out the spatial variability of the prestress (see Sect. <ref> in the SM).






§ PULSE DYNAMICS UNDER PERIODIC PRESTRESS



We first consider the periodic prestress distributions in Eq. (<ref>). We set ⟨τ_ b(x)⟩_x/τ_ min=1.03 and study pulse dynamics in a system with W=480 m, for various Δϵ, δ and n. In Fig. <ref>a, we present the pulse dynamics in the L-c plane for Δϵ=-0.1, i.e., a pulse that decays under a uniform prestress. For δ=0.02 and n=3, it is observed that the pulse sustains, instead of decaying, and forms a limit cycle in the L-c plane (orange circles). The limit cycle, corresponding to a pulse whose L(t) and c(t) vary nearly periodically as it propagates (see Fig. <ref> in the SM), remains close to the special L(c) line (solid black line) and encircles the corresponding uniform prestress fixed point (square). In this case, the periodic prestress transforms a decaying pulse (obtained under a uniform prestress corresponding to the fault average of the periodic one) into a long-lived, sustained pulse. Changing the overall sign of the periodic prestress (i.e., setting δ=-0.02), leads to a decaying pulse, which closely follows the special L(c) line (green diamonds). These results indicate that the gradient of τ_ b(x) near x=0, in particular its sign, plays an important role in determining the fate of the pulse (i.e., whether it decays or sustains and follows a limit cycle in the L-c plane).

In Fig. <ref>b, we set Δϵ=-0.1 and δ=0.02, and vary the wavelength of the periodic prestress (by setting n=2,3,4, see legend). It is observed that sustained pulses, forming quasi-limit cycles in the L-c plane close to the special L(c) line, emerge. The quasi-limit cycles are pushed to larger L and c values with increasing wavelength (decreasing n). We use the term “quasi-limit cycle” as in some cases the limit cycle is not exact and since we did not always follow the dynamics over very long times (and system sizes W), due to computational constraints; consequently, we cannot always determine whether the limit cycle is exact or approximate (e.g., drifts with increasing time, see Fig. <ref> in the SM).

In Fig. <ref>c, we fix the periodic prestress parameters δ=0.01 and n=4, and vary the instability index Δϵ of the initial pulse. For Δϵ=0.1 (brown cycles), a quasi-limit cycle close to the special L(c) line emerges (the cycle in the L-c plane seems to be drifting upwards with time). Upon decreasing Δϵ, setting it to Δϵ=-0.1 (orange squares), a quasi-limit cycle also appears to emerge (recall that δ>0). Upon further decreasing Δϵ, setting it to Δϵ=-0.4 (green triangles), the pulse initially attempts to form a limit cycle in the L-c plane, but appears to be drifting to smaller L and c values, and to decay. Finally, in Fig. <ref>d, we fix Δϵ=-0.1 and n=4, and vary the amplitude δ>0 of the periodic prestress (see legend). It is observed that the emerging quasi-limit cycle broadens with increasing δ>0. Overall, the results presented in Fig. <ref> indicate that a periodic prestress distribution can reverse the fate of the original pulse (under uniform prestress), leading to either decaying pulses or to quasi-limit cycles in the L-c plane. In both cases, for the parameters considered, the unsteady pulse dynamics remain rather close to the special L(c) line, which corresponds to steady-state pulse solutions parameterized by a uniform pretress τ^_ b <cit.>.









§ PULSE DYNAMICS UNDER CONSTANT-GRADIENT PRESTRESS



We next consider the constant-gradient prestress distributions in Eq. (<ref>). We set ⟨τ_ b(x)⟩_x/τ_ min=1.05 and study pulse dynamics in a system with W=480 m, for various instability indices Δϵ and tilt angles θ. In Fig. <ref>, we present the pulse dynamics in the L-c plane for Δϵ=-0.295, i.e., a pulse that decays under a uniform prestress (corresponding to θ=0^∘ in the figure, see green squares), and various values of θ. Upon imposing a negative prestress gradient, corresponding to θ=-10^∘ (green circles), the pulse decays with a small down shift compared to the reference θ=0^∘ results, yet it still follows rather closely the special L(c) line (solid black line). Upon imposing a positive prestress gradient of θ=5^∘ (green diamonds), the pulse decays with a small up shift compared to the reference θ=0^∘ results.

The dynamics qualitatively change upon setting θ=6^∘ (yellow diamonds), where the pulse changes its character and becomes a growing/expanding pulse, which closely follows the special L(c) line. A similar growing/expanding pulse is observed for θ=10^∘ (brown circles), featuring a small up shift compared to the θ=6^∘ results. These results indicate that for a given instability index Δϵ<0, there exists a threshold tilt angle θ_ c>0 that can reverse the nature/fate of the pulse, from decaying to growing. For Δϵ=-0.295, θ_ c is between 5^∘ and 6^∘. In the inset of Fig. <ref>, we plot θ_ c against Δϵ<0, demonstrating that there is an interplay between the instability index of the initial pulse and the threshold tilt angle θ_ c>0 needed to induce a transition from decaying to growing/expanding pulses.

Our analysis above focused on initial self-healing slip pulses that remain pulses under various spatially-varying prestress conditions. Yet, as noted in Sect. <ref>, earthquakes can take the form of either pulse-like or crack-like rupture, with significant implications for the duration of slip, earthquake energy budget, rupture scaling laws and seismic radiation. The dynamic selection of these rupture modes/styles and the spontaneous transition between them are important topics under active investigation, e.g., see <cit.>. It is generally established <cit.> that crack-like rupture is more prevalent under larger average prestress levels. Consequently, it would be interesting to see whether for ⟨τ_ b(x)⟩_x/τ_ min=1.05 (a higher average prestress compared to the one considered in Sect. <ref>), a spatially-varying prestress distribution can spontaneously induce a pulse-to-crack rupture transition.

To address this question, we present in Fig. <ref> (top row) a series of equal-time snapshots of v(x,t) (Fig. <ref>a) and τ(x,t) (Fig. <ref>b) obtained for ⟨τ_ b(x)⟩_x/τ_ min=1.05 and Δϵ=0.005, under a constant-gradient prestress with θ=-9^∘. It is observed that the nonuniform prestress transforms a growing pulse (Δϵ>0) into a decaying pulse, which propagates a certain distance from left to right (in the positive x direction), until it arrests/dies off. Note that pulse arrest leaves behind it a heterogeneous stress distribution (brightest, rightmost snapshot in panel (b)). Interestingly, it is observed that near the healing tail of the initial pulse, around x/W≃-0.03 (also marked by an arrow), there seems to be a spontaneous attempt to nucleate secondary rupture. The latter is evident in τ(x,t) (Fig. <ref>b), but requires a logarithmic scale to be discerned in v(x,t), as done in the inset of Fig. <ref>a.

In Fig. <ref> (bottom row), we present the spatiotemporal dynamics for a slightly stronger negative gradient, corresponding to θ=-10^∘ (everything else being the same as in the top row). It is observed that this nonuniform prestress leads to a qualitative change in the dynamics. First, the attempted nucleation near the healing tail of the initial pulse, around x/W≃-0.03, becomes a successful nucleation event, giving rise to an asymmetric bilateral crack-like rupture. The rupture edge that propagates from right to left (in the negative x direction) corresponds to a back-propagating rupture. The rupture edge propagating in the forward (positive x) direction, catches up with the initial pulse that also propagates in the forward direction and merges with it. Second, upon merging of the two forward-propagating rupture fronts, a persistent asymmetric bilateral crack-like rupture emerges. Overall, it is observed that a spatially-varying prestress can induce a pulse-to-crack transition.




The crack-like nature of the two persistent, oppositely propagating rupture edges is evident in the inset of Fig. <ref>c (note the logarithmic y-axis), where a persistent finite slip velocity is left in the central part, while the two oppositely propagating rupture edges move further away. Similarly, the crack-like nature of the emerging rupture is also manifested in the residual stress observed in Fig. <ref>d. These results, which are presented again in a space-time plot in Fig. <ref>, demonstrate that rich spatiotemporal rupture dynamics may emerge from rather simple spatially-varying prestress distributions, revealing phenomena such as back-propagating rupture and pulse-to-crack transitions, which are of interest in various geophysical contexts, e.g., in relation to earthquake source models and seismological inversion analyses <cit.>. Finally, we note — as highlighted in <cit.> — that once pulse-like rupture transforms into a crack-like rupture, the dynamics of latter can no longer be meaningfully represented in the L-c plane and an intrinsic relation to the special L(c) line — which is essentially an equation of motion for self-healing slip pulses under uniform prestress — does not exist anymore.



§ SUMMARY AND CONCLUDING REMARKS



In this paper, we studied the interaction of well-defined initial self-healing slip pulses with two classes of spatially-varying prestress distributions τ_ b(x), one corresponding to a periodic variation and one to a constant gradient, see Eqs. (<ref>)-(<ref>). Our findings can be summarized along two major axes. The first concerns the richness (or to some extent the complexity) of rupture dynamics emerging from initial slip pulses interacting with a spatially-varying prestress (compared to their uniform prestress counterparts). The second concerns the theoretical description of unsteady slip pulse dynamics under spatially-varying prestress, particularly in relation to the recently established reduced-dimensionality description of pulse dynamics <cit.>.

We have demonstrated that simple nonuniform prestress distributions can give rise to rather rich rupture dynamics. In particular, we showed that a nonuniform prestress can reverse the nature/fate of a slip pulse, from decaying/shrinking to growing/expanding and vice versa, for a fixed fault-averaged prestress, see Figs. <ref>-<ref>. We also showed that pulses under periodic prestress can settle into propagating states in which their properties, e.g., pulse size L(t), peak slip velocity v_ p(t) and propagation velocity c(t), vary periodically or quasi-periodically as they propagate. Finally, we demonstrated that prestress spatial gradients can lead to the spontaneous nucleation of secondary rupture near the healing tail of a pulse, to back-propagating rupture and to pulse-to-crack transitions, see Figs. <ref>-<ref>.

The above mentioned rupture phenomena are geophysically documented phenomena of importance, which continue to pose challenges to our understanding. For example, it is known that while most seismic studies typically only estimate the average rupture velocity, seismic radiation is predominantly affected by the spatiotemporal variation of the rupture velocity <cit.>. Our finding, e.g., in relation to the emergence of quasi-periodically propagating slip pulses, can contribute in this context. Moreover, as noted in <cit.>, slip pulse propagation and arrest give rise to heterogeneous after-event stresses (cf., top row of Fig. <ref>) and hence may offer a mechanism for sustaining prestress heterogeneity over many earthquake cycles.

On the more theoretical side, our results demonstrate that the reduced-dimensionality description of unsteady pulses, originally established under uniform prestress conditions <cit.>, is still relevant under some nonuniform prestress conditions. Specifically, it has been shown that such pulse dynamics presented in a plane defined by a pair of pulse properties — e.g., its size L(t) and propagation velocity c(t) —, remain close to a special, previously derived line L(c). For example, periodically (or quasi-periodically) propagating slip pulses under nonuniform prestress correspond to limit cycles (or quasi-limit cycles) in the L-c plane that remain rather close to the special L(c) line at all times, see Figs. <ref>-<ref> and SM.

Consequently, while the special L(c) line is not a strict equation of motion for unsteady pulses under nonuniform prestress, in many cases it does provide a good approximation for the dynamics, in the sense that the knowledge of one pulse property, e.g., c(t), can be used to infer another property, e.g., L(t). Finally, we also demonstrated that similarly to the uniform prestress case <cit.>, the emerging theoretical picture remains valid upon using other pairs of pulse properties, e.g., L(t) and the peak slip velocity v_ p(t) (see SM).


Acknowledgements. We acknowledge support by the Israel Science Foundation (ISF grant no. 1085/20), the Minerva Foundation (with funding from the Federal German Ministry for Education and Research), the Ben May Center for Chemical Theory and Computation, and the Harold Perlman Family. We are grateful to Prof. Eric Dunham for posing to us the basic question, to Dr. Fabian Barras for his help with setting up the initial conditions in cRacklet and to Dr. Efim Brener for useful discussions.








              Supplementary material

























The goal of this document is to provide some technical details regarding the results presented in the manuscript and to offer some additional supporting data.





§ THE INTERFACIAL CONSTITUTIVE RELATION (FRICTION LAW)



The left-hand-side of Eq. (<ref>) corresponds to the frictional strength τ[v,ϕ], where v(x,t) is the slip velocity field and ϕ(x,t) is the internal state field. Exactly as in <cit.>, we have

    τ(v,ϕ)/σ = [1+blog(1+ϕ/ϕ^⋆)] [f_0/√(1+(v^⋆/v)^2)+α log(1+  v/v^⋆)] ,

where the frictional parameters appearing in the expression are given in the table in Fig. <ref>(left). This table also reports on the linear elastic bulk parameters of Eq. (<ref>) (the spatiotemporal convolutional integral s(x,t) therein features in addition to μ and c_ s, which appear in the radiation-damping term, also Poisson's ratio ν) and the value of the applied normal stress σ. As explained in the manuscript, the internal state field ϕ(x,t) evolves according the “aging law”, taking the form ∂_tϕ(x,t)=1-v(x,t) ϕ(x,t)√(1+v^*/v(x,t))/D, which includes the characteristic slip distance D, whose value is also given in the table in Fig. <ref>(left). Note that the regularization factor √(1+v^*/v(x,t)) was not included in the “aging law” expression in the manuscript, as it is relevant only at extremely small slip velocities (note the value of v^* in the table in Fig. <ref>). Under spatially homogeneous steady sliding, we have ∂_tϕ=0, which allows to express ϕ_ ss (the steady sliding value of the internal state field) in terms of v_ ss (the steady sliding value of the slip velocity field). Once done, one can obtain the steady sliding friction curve τ(v_ ss), plotted in Fig. <ref>(right). τ(v_ ss) follows a generic N shape, featuring a local minimum at (v_ min, τ_ min) <cit.> (marked on the figure). The existence of a minimum, though, does not play a crucial role in the problem we solved; yet, the characteristic slip velocity v_ min=0.005971 m/s and characteristic frictional strength τ_ min=0.3406679 MPa can be used to nondimensionalize various quantities, as done in the manuscript. We also note that using the previously-calculated effective fracture energy G_ c=0.65 J/m^2 of slip pulses <cit.> for our frictional parameters, one can construct a characteristic pulse length as L_ n=(c_ s/v_ min)^2 (G_ c/μ)=15.2 m. The latter is used to normalize the size of slip pulses L(t) in the manuscript.



§ PERTURBED STEADY-STATE PULSES AS INITIAL CONDITIONS



Our main goal in the manuscript is to systematically and controllably study the interaction of self-healing slip pulses with spatially-varying prestress distributions. To that aim, we need well-defined and controlled initial conditions that correspond to slip pulses. Here, we explain how this is achieved, providing additional details on top of the discussion offered in the manuscript.



 §.§ Steady-state self-healing pulses in finite periodic domains



Similarly to <cit.>, we employ the steady-state slip pulse solutions recently obtained under a uniform prestress in <cit.>. The latter solutions take the form v__ ss(ξ) and ϕ__ ss(ξ), with ξ=x-c^t being a co-moving space-time coordinate -∞<ξ<∞ and c^ is the steady pulse propagation velocity, see Fig. <ref>a, where the finite size of the steady-state pulse, L^, is also marked. Specifically, for a given spatially-varying prestress distribution τ_ b(x), we use a steady-state pulse solution (or a slightly perturbed version of it quantified by the instability index Δϵ, see Sect. <ref>) corresponding to a uniform prestress of magnitude ⟨τ_ b(x)⟩_x, i.e., the spatial fault average of τ_ b(x), at t=0. The latter is used as an initial condition for the t>0 spatiotemporal dynamics described by the boundary integral equation (Eq. (<ref>)), which is solved using the open-source library cRacklet <cit.>. cRacklet is a boundary element library based on a spectral formulation of linear elastodynamics <cit.>, and as such is applicable in finite domains of size W in the fault direction x (the domain is infinite in the perpendicular y direction, the fault is located at y=0), where periodic boundary conditions are employed.

The periodic domain of size W is clearly in conflict with steady-state pulses that propagated steadily at a velocity c^ indefinitely in the past, -∞<t<0. That is, generating slip history (past propagation) that extends beyond the domain boundary implies that the past spuriously affects future dynamics. The way to resolve this conflict has been discussed in detail in <cit.>, see in particular section S-3A of the Supplemental Materials file therein. In a nutshell, we require that the past propagation distance of the steady-state pulse, denoted by w, is much smaller than the spatial periodicity domain W and much larger than the steady-state pulse size L^, i.e., L^≪w≪W (see Fig. S1 in <cit.>). Effectively, this means that instead of indefinite past propagation (i.e., for -∞<t<0), we account for finite time past propagation over -t_0<t<0, where t_0>0 is determined by the spatial position in which the past propagation is initiated. Once a given slip history is generated, it is used to calculate the spatiotemporal convolutional integral and the resulting field s(x,t=0) is used in Eq. (<ref>) as an initial condition. The latter can be supplemented with a perturbation in the initial internal state field, quantified by the instability index Δϵ, as will be discussed in Sect. <ref>.





The above-described procedure for mimicking an initial condition corresponding to a propagating steady-state pulse in a finite, periodic system has one implication that has not been discussed in <cit.>. Specifically, the fact that the past indefinite propagation over -∞<t<0 is replaced by a finite past propagation over -t_0<t<0 gives rise to a frozen-in and localized artificial stress concentration due to the unaccounted for slip accumulated over -∞<t<-t_0. To demonstrate the point, we present in Fig. <ref>a the spatiotemporal evolution of the shear stress field τ(x,t), corresponding to one of the solutions discussed in <cit.> with a uniform prestress. The desired slip pulse initial stress distribution appears in the thick black curve. It is observed that the latter is indeed realized, except for a localized deviation near x/W≃-0.25. The latter persists as time progresses, upon which the pulse propagates to the right, further away from the artificial stress concentration. The corresponding dynamics of the slip velocity field v(x,t) are presented in Fig. <ref>b. In <cit.>, we made sure that the frozen-in, artificial stress concentration was located sufficiently away from the central part of the slip pulse such that the dynamics were not affected. The artifact can be readily removed by applying a static slip that exactly relaxes this artificial stress concentration at t=0, as done in Fig. <ref>c-d. Comparing the top and bottom rows of Fig. <ref>, we indeed observe that the frozen-in, artificial stress concentration does not affect the pulse dynamics. Nevertheless, in the present work, we eliminate it in generating the initial condition, at t=0, under uniform prestress. The spatially-varying prestress distribution τ_ b(x) is introduced at t=0^+, as explained in the manuscript.




The steady-state slip pulse solutions we employ as initial conditions have relatively long, small-amplitude tails at the healing front. An additional consequence of the computational domain periodicity in the presence of such initial conditions is that once the pulse propagates through the periodic boundary, it interacts with its own tail. Such interactions affect the dynamics of the pulse and are hence avoided, i.e., the simulations are stopped before this happens. Moreover, the initial pulse is placed such that its edge (e.g., defined by the location of its peak slip velocity) coincides with the center of the domain at x=0, where we also have τ_ b(x=0)=τ^_ b. Whenever a longer evolution time is required, the domain size W is increased (e.g., see results in Figs. <ref>-<ref>), but in general these computations are very demanding.

Finally, we note that for the constant-gradient prestress distributions τ_ b(x) there is a jump between τ_ b(x=-12W) and τ_ b(x=12W), which is formally inconsistent with the periodic boundary conditions. Yet, we verified that this jump does not affect the pulse evolution as long as its edge does not reach the domain boundary, which is indeed avoided, as explained above. Moreover, we verified that smoothly connecting τ_ b(x) across the domain boundary, to conform with the periodic boundary conditions, does not affect our results.



 §.§ Introducing initial pulse perturbations and the instability index



As shown in <cit.> and discussed in the manuscript, steady-state pulses under uniform prestress correspond to “saddle configurations” separating expanding from decaying pulses. That is, slightly perturbed steady-state pulses under uniform prestress either grow (expand) or decay (shrink), depending on the sign of the perturbation. Let us schematically denote the perturbation by ϵ, to be formally defined below. In an infinite system, steady-state pulses are expected to correspond to the marginal state, ϵ=0. However, fitting an infinite-system steady-state pulse into a finite, periodic domain of size W, as explained in Sect. <ref>, inevitably introduces perturbations (as also discussed in <cit.>). Consequently, initial pulses generated as explained in Sect. <ref> will either grow or decay under uniform prestress, i.e., they feature ϵ0, where the sign of ϵ determines whether the pulse grows or decays and its magnitude determines the initial growth/decay rate.

To quantify the latter, we adopt a similar perturbation scheme to that used in <cit.>. Let us denote by ϕ̃(x,t=0) the internal state field resulting from fitting an infinite-system steady-state pulse into a finite, periodic domain of size W, as discussed in Sect. <ref>. Next, we define ϵ through the following amplitude perturbation ϕ(x,t=0)=(1+ϵ) ϕ̃(x,t=0) and ask at what value of ϵ, the subsequent t>0 dynamics of ϕ(x,t=0) under uniform prestress (in the periodic, finite W domain) correspond to the marginal state (where decaying pulse become growing, or vice versa). We denote this value of ϵ by ϵ_ c, which obviously depends on W and on the level of uniform prestress τ^_ b. Specifically, for W=480 m, we find ϵ_ c=-0.4 for τ^_ b=1.03τ_ min and ϵ_ c=-0.295 for τ^_ b=1.05τ_ min.

We take advantage of the above results in order to generate initial conditions using well-controlled, perturbed initial steady-state pulses pulse by varying ϵ. For each uniform prestress τ^_ b, we define the instability index as Δϵ≡ϵ_ c-ϵ such that Δϵ>0 leads to an growing/expanding pulse, while Δϵ<0 leads to a decaying/shrinking pulse. Furthermore, |Δϵ| determines the initial rate of instability. For example, in Fig. <ref>, we set Δϵ=0.005>0 under uniform prestress and indeed the t>0 dynamics correspond to a very slowly growing/expanding pulse. In the manuscript, we used the above procedure to generate initial pulse states characterized by τ^_ b and Δϵ. These are used as initial conditions for the spatially-varying prestess problem, where a nonuniform τ_ b(x) is introduced at t=0^+ and the subsequent t>0^+ dynamics are studied, as discussed extensively in the manuscript.



§ ADDITIONAL SUPPORTING RESULTS


Here, we present a few additional results that further support the results discussed in the manuscript.



 §.§ Dynamics in the L-v_ p plane


In Figs. <ref>c, <ref>, and <ref>, we explored the reduced-dimensionality description of unsteady slip pulse dynamics in the L-c plane. Specifically, we tested the degree by which the pulse dynamics under various conditions follow the special L(c) line, corresponding to a family of steady-state pulse solutions parameterized by the value of the uniform prestress. In <cit.>, it was established that unsteady pulse dynamics under a given uniform prestress also follow the line corresponding to a family of steady-state pulse solutions when presented in a plane defined by other pairs of pulse properties/observables, e.g., the pulse peak slip velocity v_ p(t) and its size L(t).




In Fig. <ref>a, we plot the very same pulse dynamics under periodic prestress τ_ b(x) as in Fig. <ref>c, but in the L-v_ p plane (same color and symbol code as in Fig. <ref>c, see legend therein), where the solid line corresponds to the special L(v_ p) line. It is observed that the relations between the unsteady pulse dynamics and the special L(v_ p) line (and the proximity to it) are very similar to the corresponding relations to the special L(c) in Fig. <ref>c. Likewise, we present in Fig. <ref>b the same results for two cases presented in Fig. <ref> under constant-gradient prestress τ^_ b(x) (quantified by the tilt angle θ), but in the L-v_ p plane. Yet again, the results — in particular the relations to the special L(v_ p) line — are similar for the two reduced-dimensionality descriptions.


 

In Fig. <ref> and in Figs. <ref>c, <ref>, and <ref>, we focused on the reduced-dimensionality description of unsteady pulse dynamics. In Fig. <ref>, for completeness, we also present spatial snapshots of the pulse dynamics shown in Fig. <ref>a and in Fig. <ref>c. We use the very same parameters for the 3 pulses presented therein (see legend of Fig. <ref>c), but employ a system size twice as large, W=964 m. The latter allows us to probe pulse dynamics over larger propagation distances (and hence times), and to address the nature of the emerging limit cycles for growing pulses (e.g., whether the limit cycle is approximate or exact). Presenting pulse dynamics for the larger W in the reduced-dimensionality representation is, of course, possible but would be visually less clear due to the appearance of more overlapping cycles in the plane.

In Fig. <ref>a, we present equal-time snapshots corresponding to the pulse dynamics presented by the green triangles in Fig. <ref>a and in Fig. <ref>c. The spatial snapshots clearly demonstrate the decaying nature of this pulse, where we superimposed the envelope of the peak slip velocity (thicker line) for clarity. In Fig. <ref>b, we present equal-time snapshots corresponding to the pulse dynamics presented in the orange squares in Fig. <ref>a and in Fig. <ref>c. It is observed that the pulses seem to settle into a rather robust limit cycle, though there appears to be some increase in the pulse amplitude during the fourth cycle. Finally, we present in Fig. <ref>c equal-time snapshots corresponding to the pulse dynamics presented in the brown circles in Fig. <ref>a and in Fig. <ref>c. As is already clear from these two figures, the pulse amplitude shown in Fig. <ref>c indeed systematically increases with the cycle number.



 §.§ Pulse dynamics under prestress distributions featuring smaller scale spatial variations



Our analysis in this work focused on prestress distributions τ_ b(x) that vary over lengthscales that are larger than the size of the initial pulse L^. Here, we briefly discuss pulse dynamics when this scale separation does not hold. Similarly to Fig. <ref>, we consider larger systems with W=964 m. Since this value is different from the one used in the manuscript, to avoid any possible confusion we do not quantify the wavelength λ of the periodic prestress τ_ b(x) through n, according to n=W/λ, but rather use λ itself. That is, we rewrite Eq. (<ref>) as τ_ b(x)=τ^_ b[1+δsin(2π x/λ) ], making no reference to W. We set τ^_ b=1.03 τ_ min, for which L^=14.6 m. In Fig. <ref>a, we first consider λ=120 m, still featuring λ≫L^, with δ=0.01 and Δϵ=-0.1, which is identical to the case considered in Fig. <ref>b. As observed, and similarly to other examples presented in Fig. <ref>, in this case the periodic prestress reverses the fate of the pulse, from decaying under uniform prestress (Δϵ<0) to a sustained pulse, nearly forming a limit cycle in the L-c plane. Next, also in Fig. <ref>a, we reduced the periodic prestress wavelength to be λ=15 m, essentially identical to the initial pulse size L^=14.6 m. It is observed that the lack of scale separation between the prestress spatial variation and the initial pulse size leads to a qualitative change in the pulse dynamics in this case. That is, the pulse in this case follows a spiral decay trajectory close to the special L(c) line, see green circles in Fig. <ref>a.

In Fig. <ref>b, we again start with a reference case featuring λ≫L^, this time with with δ=0.01 and Δϵ=0.1>0. The latter is identical to the case considered in Fig. <ref>c, corresponding to a growing pulse, following upward drifting cycles in the L-c plane. Next, also in Fig. <ref>b, we reduced the periodic prestress wavelength to be λ=5 m, i.e., smaller than the initial pulse size L^=14.6 m. It is observed that under these conditions the pulse grows (while performing oscillations, which are not easily discerned in the figure), remaining close to the special L(c) line. This behavior, up to the short wavelength oscillations, is similar to the pulse dynamics under uniform prestress (recall that Δϵ=0.1>0). In fact, in the limit in which the prestress varies on a spatial scale much smaller than L^, we expect the pulse to smooth out the spatial variability of the prestress such that its dynamics are essentially the same for the average uniform prestress. To demonstrate this point, we set τ_ b(x)=τ^_ b[1+ζ(x) ], where ζ(x) is a random variable corresponding to a Gaussian distribution with zero mean and a standard deviation of 0.01. ζ(x) features no spatial correlations, i.e., its value at each numerical grid point x is independent of its values elsewhere. The results presented in Fig. <ref>c indeed demonstrate that the pulse dynamics, whether growing or decaying (see legend), smooth/average out the noise ζ(x) and identify with the corresponding uniform prestress results.















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@bib@innerbibempty

[Rice(1980)]Rice1980a
  author author J. R. Rice, title title The mechanics of earthquake
  rupture, in https://doi.org/10.1.1.161.3251 booktitle Phys. Earth's Inter. (year 1980) pp. pages 555–649NoStop
[Freund(1998)]Freund1998
  author author L. B. Freund, @noop  title Dynamic Fracture
  Mechanics (publisher Cambridge university press, address Cambridge, year 1998)NoStop
[Scholz(2002)]Scholz2002
  author author C. H. Scholz, @noop  title The mechanics of
  earthquakes and faulting (publisher Cambridge university
  press, year 2002)NoStop
[Heaton(1990)]Heaton1990
  author author T. H. Heaton, title title Evidence for and
  implications of self-healing pulses of slip in earthquake rupture, https://doi.org/10.1016/0031-9201(90)90002-F journal
  journal Phys. Earth Planet. Inter. volume 64, pages 1 (year 1990)NoStop
[Perrin et al.(1995)Perrin,
  Rice, and Zheng]Perrin1995
  author author G. Perrin, author J. R. Rice,  and author G. Zheng, title title Self-healing slip pulse on a frictional
  surface, https://doi.org/10.1016/0022-5096(95)00036-I journal journal J. Mech. Phys. Solids volume 43, pages 1461 (year
  1995)NoStop
[Lu et al.(2007)Lu,
  Lapusta, and Rosakis]lu2007pulse
  author author X. Lu, author N. Lapusta, and  author A. J. Rosakis,  title title Pulse-like and crack-like ruptures in
  experiments mimicking crustal earthquakes, @noop  journal journal Proceedings of the National Academy of
  Sciences volume 104, pages 18931
  (year 2007)NoStop
[Gabriel et al.(2012)Gabriel, Ampuero, Dalguer, and  Mai]Gabriel2012
  author author A.-A. Gabriel, author J.-P. Ampuero, author L. A. Dalguer, and author P. M. Mai, title title The transition of dynamic
  rupture styles in elastic media under velocity-weakening friction, https://doi.org/10.1029/2012JB009468 journal journal J. Geophys. Res. Solid Earth volume
  117, pages B09311 (year 2012)NoStop
[Brener et al.(2018)Brener,
  Aldam, Barras, Molinari,  and Bouchbinder]Brener2018
  author author E. A. Brener, author M. Aldam,
  author F. Barras, author J.-F. Molinari, and author E. Bouchbinder, title title Unstable Slip Pulses and Earthquake Nucleation as
  a Nonequilibrium First-Order Phase Transition, https://doi.org/10.1103/PhysRevLett.121.234302 journal
  journal Phys. Rev. Lett. volume 121,  pages 234302 (year 2018)NoStop
[Pomyalov et al.(2023a)Pomyalov, Barras,
  Roch, Brener, and Bouchbinder]pomyalov2023dynamics
  author author A. Pomyalov, author F. Barras,
  author T. Roch, author
  E. A. Brener, and author
  E. Bouchbinder, title
  title The dynamics of unsteady frictional slip pulses, https://doi.org/10.1073/pnas.2309374120 journal journal Proceedings of the National Academy of Sciences volume 120, pages e2309374120 (year
  2023a)NoStop
[Hanks(1974)]hanks1974faulting
  author author T. C. Hanks, title title The faulting mechanism of
  the san fernando earthquake, @noop  journal journal Journal of Geophysical Research volume
  79, pages 1215 (year 1974)NoStop
[Kanamori and Stewart(1978)]kanamori1978seismological
  author author H. Kanamori and author G. S. Stewart, title title Seismological aspects of
  the guatemala earthquake of february 4, 1976, @noop  journal journal Journal of Geophysical Research: Solid
  Earth volume 83, pages 3427 (year 1978)NoStop
[Hartzell and Brune(1979)]hartzell1979horse
  author author S. Hartzell and author J. N. Brune, title title The Horse Canyon earthquake of August 2,
  1975—Two-stage stress-release process in a strike-slip
  earthquake, @noop  journal journal
  Bulletin of the Seismological Society of America volume 69, pages 1161 (year
  1979)NoStop
[Aki(1979)]aki1979characterization
  author author K. Aki, title title Characterization of barriers
  on an earthquake fault, @noop  journal journal Journal of Geophysical Research: Solid Earth volume 84, pages 6140 (year
  1979)NoStop
[Bache et al.(1980)Bache,
  Lambert, and Barker]bache1980source
  author author T. C. Bache, author D. G. Lambert,  and author T. G. Barker,  title title A source model for the March 28, 1975, Pocatello
  Valley earthquake from time-domain modeling of teleseismic P waves, @noop  journal journal Bulletin
  of the Seismological Society of America volume 70,  pages 405 (year 1980)NoStop
[Beroza and Mikumo(1996)]beroza1996short
  author author G. C. Beroza and author T. Mikumo, title title Short slip duration in
  dynamic rupture in the presence of heterogeneous fault properties,  @noop  journal journal Journal of
  Geophysical Research: Solid Earth volume 101,  pages 22449 (year 1996)NoStop
[Matsumoto et al.(2018)Matsumoto, Yamashita, Nakamoto,
  Miyazaki, Sakai, Iio,
  Shimizu, Goto, Okada,
  Ohzono et al.]matsumoto2018prestate
  author author S. Matsumoto, author Y. Yamashita, author M. Nakamoto,
  author M. Miyazaki, author S. Sakai, author
  Y. Iio, author H. Shimizu, author K. Goto, author T. Okada, author M. Ohzono, et al., title title Prestate of stress and
  fault behavior during the 2016 Kumamoto earthquake (M7.3), @noop 
  journal journal Geophysical Research Letters  volume 45, pages 637 (year
  2018)NoStop
[Mildon et al.(2019)Mildon,
  Roberts, Faure Walker, and Toda]mildon2019coulomb
  author author Z. Mildon, author G. P. Roberts,
  author J. Faure Walker, and  author S. Toda, title title Coulomb pre-stress and fault bends are ignored yet
  vital factors for earthquake triggering and hazard, @noop 
  journal journal Nature Communications volume 10, pages 2744 (year
  2019)NoStop
[Lozos et al.(2011)Lozos,
  Oglesby, Duan, and Wesnousky]lozos2011effects
  author author J. C. Lozos, author D. D. Oglesby,
  author B. Duan, and author S. G. Wesnousky, title title The effects of double fault bends on rupture
  propagation: A geometrical parameter study, @noop  journal journal Bulletin of the Seismological Society of
  America volume 101, pages 385
  (year 2011)NoStop
[Duan and Oglesby(2007)]duan2007nonuniform
  author author B. Duan and author D. D. Oglesby, title title Nonuniform prestress from
  prior earthquakes and the effect on dynamics of branched fault systems,  @noop  journal journal Journal of
  Geophysical Research: Solid Earth volume 112
  (year 2007)NoStop
[Fang and Dunham(2013)]Fang2013
  author author Z. Fang and author E. M. Dunham, title title Additional shear
  resistance from fault roughness and stress levels on geometrically complex
  faults, https://doi.org/10.1002/jgrb.50262 journal
  journal J. Geophys. Res. Solid Earth volume 118, pages 3642 (year
  2013)NoStop
[Heimisson(2020)]heimisson2020crack
  author author E. R. Heimisson, title title Crack to pulse
  transition and magnitude statistics during earthquake cycles on a
  self-similar rough fault, @noop  journal journal Earth and Planetary Science Letters volume
  537, pages 116202 (year 2020)NoStop
[Day(1982)]day1982three
  author author S. M. Day, title title Three-dimensional simulation
  of spontaneous rupture: the effect of nonuniform prestress, @noop 
  journal journal Bulletin of the Seismological
  Society of America volume 72, pages
  1881 (year 1982)NoStop
[Idini and Ampuero(2020)]idini2020fault
  author author B. Idini and author J.-P. Ampuero, title title Fault-zone damage
  promotes pulse-like rupture and back-propagating fronts via quasi-static
  effects, @noop  journal journal
  Geophysical Research Letters volume 47, pages e2020GL090736 (year 2020)NoStop
[Ding et al.(2024)Ding,
  Xu, Fukuyama, and Yamashita]ding2024back
  author author X. Ding, author S. Xu, author E. Fukuyama, and author F. Yamashita, title
  title Back-propagating rupture: Nature, excitation, and
  implications, @noop  journal journal
  arXiv preprint arXiv:2406.01286  (year 2024)NoStop
[Brantut et al.(2019)Brantut, Garagash, and Noda]brantut2019stability
  author author N. Brantut, author D. I. Garagash, and author H. Noda, title title Stability of pulse-like
  earthquake ruptures, https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1029/2019JB017926
  journal journal J. Geophys. Res. Solid Earth  volume 124, pages 8998 (year
  2019)NoStop
[Johnson(1990)]johnson1990initiation
  author author E. Johnson, title title On the initiation of
  unidirectional slip, @noop  journal journal Geophysical Journal International volume
  101, pages 125 (year 1990)NoStop
[Elbanna(2011)]elbanna2011pulselike
  author author A. E. Elbanna, title Pulselike ruptures on strong
  velocity-weakening frictional interfaces: dynamics and implications,  @noop  Ph.D. thesis, school California Institute of
  Technology (year 2011)NoStop
[Barras et al.(2023)Barras,
  Thøgersen, Aharonov, and Renard]barras2023earthquakes
  author author F. Barras, author K. Thøgersen, author E. Aharonov, and author F. Renard, title title How do earthquakes stop?
  Insights from a minimal model of frictional rupture, @noop 
  journal journal Journal of Geophysical Research:
  Solid Earth volume 128, pages
  e2022JB026070 (year 2023)NoStop
[Dieterich(1979)]Dieterich1979a
  author author J. H. Dieterich, title title Modeling of rock
  friction: 2. Simulation of preseismic slip, https://doi.org/10.1029/JB084iB05p02169 journal journal J. Geophys. Res. volume 84, pages 2169 (year 1979)NoStop
[Ruina(1983)]Ruina1983
  author author A. L. Ruina, title title Slip instability and state
  variable friction laws, https://doi.org/10.1029/JB088iB12p10359
  journal journal J. Geophys. Res. volume 88, pages 10359 (year
  1983)NoStop
[Nakatani(2001)]Nakatani2001
  author author M. Nakatani, title title Conceptual and physical
  clarification of rate and state friction: Frictional sliding as a thermally
  activated rheology, https://doi.org/10.1029/2000JB900453
  journal journal J. Geophys. Res. Solid Earth  volume 106, pages 13347 (year 2001)NoStop
[Marone(1998)]Marone1998a
  author author C. Marone, title title Laboratory-derived
  friction laws and their application to seismic faulting, https://doi.org/10.1146/annurev.earth.26.1.643 journal
  journal Annu. Rev. Earth Planet. Sci. volume 26, pages 643 (year 1998)NoStop
[Baumberger and Caroli(2006)]Baumberger2006Solid
  author author T. Baumberger and author C. Caroli, title title Solid friction from
  stick-slip down to pinning and aging, https://doi.org/https://doi.org/10.1080/00018730600732186 journal journal Advances in Physics volume 55, pages 279 (year 2006)NoStop
[Das(1980)]das1980numerical
  author author S. Das, title title A numerical method for
  determination of source time functions for general three-dimensional rupture
  propagation, https://doi.org/10.1111/j.1365-246X.1980.tb02593.x
  journal journal Geophysical Journal
  International volume 62, pages 591
  (year 1980)NoStop
[Breitenfeld and Geubelle(1998)]Breitenfeld1998
  author author M. S. Breitenfeld and author P. H. Geubelle, title title Numerical analysis of
  dynamic debonding under 2D in-plane and 3D loading, https://doi.org/10.1023/A:1007535703095 journal journal Int. J. Fract. volume 93, pages 13 (year 1998)NoStop
[Roch et al.(2022)Roch,
  Barras, Geubelle, and Molinari]roch2022cracklet
  author author T. Roch, author F. Barras,
  author P. H. Geubelle, and  author J.-F. Molinari,  title title cRacklet: a spectral
  boundary integral method library for interfacial rupture simulation, https://doi.org/10.21105/joss.03724 journal journal Journal of Open Source Software volume
  7, pages 3724 (year 2022)NoStop
[Pomyalov et al.(2023b)Pomyalov, Lubomirsky, Braverman, Brener, and  Bouchbinder]pomyalov2023self
  author author A. Pomyalov, author Y. Lubomirsky, author L. Braverman, author E. A. Brener, and author E. Bouchbinder, title title Self-healing
  solitonic slip pulses in frictional systems, https://doi.org/10.1103/PhysRevE.107.L013001 journal
  journal Physical Review E volume
  107, pages L013001 (year
  2023b)NoStop
[Madariaga(1977)]Madariaga1977
  author author R. Madariaga, title title High-frequency
  radiation from crack (stress drop) models of earthquake faulting, https://doi.org/10.1111/j.1365-246X.1977.tb04211.x journal
  journal Geophys. J. Int. volume 51,  pages 625 (year 1977)NoStop
[Bar-Sinai et al.(2014)Bar-Sinai, Spatschek, Brener, and  Bouchbinder]Bar-Sinai2014
  author author Y. Bar-Sinai, author R. Spatschek, author E. A. Brener, and author E. Bouchbinder, title title On the
  velocity-strengthening behavior of dry friction, https://doi.org/10.1002/2013JB010586 journal journal J. Geophys. Res. Solid Earth volume
  119, pages 1738 (year 2014)NoStop








 Paper Title: Thermal Conductivity Predictions with Foundation Atomistic Models.






Theory of Condensed Matter Group of the Cavendish Laboratory, University of Cambridge, Cambridge, UK

Theory of Condensed Matter Group of the Cavendish Laboratory, University of Cambridge, Cambridge, UK

Engineering Laboratory, University of Cambridge, Cambridge, UK 

ms2855@cam.ac.uk
Theory of Condensed Matter Group of the Cavendish Laboratory, University of Cambridge, Cambridge, UK 

§ ABSTRACT 
Recent advances in machine learning have led to foundation models for atomistic materials chemistry, potentially enabling quantum-accurate descriptions of interatomic forces at  reduced computational cost. 
These models are benchmarked by predicting materials’ properties over large databases; however, these computationally intensive tests have been limited to basic quantities related to harmonic phonons, leaving uncertainty about the reliability for complex, technologically and experimentally relevant anharmonic heat-conduction properties.
Here we present an automated framework that relies on foundation models to compute microscopic vibrational properties, and employs them within the Wigner formulation of heat transport to predict the macroscopic thermal conductivity in solids with arbitrary composition and structure.
We apply this framework with the foundation models M3GNet, CHGNet, MACE-MP-0, and SevenNET to 103 diverse compounds, comparing predictions against first-principles references and introducing a benchmark metric based on conductivity.
This framework paves the way for physics-aware, accurate predictions of vibrational and thermal properties, and for uncovering materials that violate semiclassical Boltzmann transport and feature exceptional  heat-shielding or thermoelectric performance.



Thermal Conductivity Predictions with Foundation Atomistic Models

    Michele Simoncelli
    August 5, 2024
===================================================================

Over the past decades, several research groups have tackled the challenging computational task of fitting the Born-Oppenheimer potential energy surface as a function of atomic coordinates <cit.>. These efforts resulted in the development of so-called machine-learning potentials (MLPs), which allow us to reproduce first-principles energies and microscopic interatomic forces with nearly the same accuracy and orders-of-magnitude lower computational cost.
These developments enable the prediction of macroscopic observables from the integration of atomistic properties at a computational cost much reduced compared to first-principles methods, effectively opening avenues to design materials for target applications from theory. For example, engineering the magnitude of the macroscopic thermal conductivity of a material through changes in its atomistic composition and structure is crucial for neuromorphic computing <cit.>, thermoelectric energy harvesting <cit.>, and aerospace <cit.> technologies. 
Major drawbacks of MLPs-based methods are the significant work required to generate a first-principles database used to train and validate MPLs <cit.>, 
as well as the applicability limited to specific materials' compositions or structural phases.

Past works attempted to bypass these limitations by employing end-to-end machine-learning methods, which predict microscopic <cit.> or macroscopic <cit.> materials' properties very efficiently, but with the compromise of not rigorously resolving the fundamental physics that underlies them.

Fittingly, recent work <cit.> has formally demonstrated the possibility to obtain a complete description (in the mathematical sense) of atomic environments (and thus of forces) by employing Message Passing Neural Networks (MPNNs) models <cit.> with many-body messages <cit.> (to be precise, with the MACE architecture <cit.> this can be achieved using 4-body terms and one message pass).

This breakthrough has enabled the development of foundation machine-learning potentials (fMLPs) <cit.>, which are trained across nearly all chemical elements and can be directly combined to describe materials with diverse structures and compositions. 

Therefore, fMLPs could potentially overcome the problems related to complex training and the very limited transferability of conventional MLPs, ultimately enabling physics-aware predictions of macroscopic observables from the integration of atomistic quantities. 
Recent research efforts have assessed the accuracy of fMLPs
in predicting, e.g., the structural stability of solids <cit.> or  the harmonic vibrational frequencies <cit.>. 

However, fMLPs have not yet been tested for predicting vibrational-anharmonic and heat-conduction properties, due to the complexity of the computational framework that relates the interatomic forces to the macroscopic conductivity <cit.>.


Here we present an automated framework that uses fMLPs to compute anharmonic vibrational and thermal properties. We employ this to benchmark against first-principles reference data <cit.> the predictions obtained from state-of-the-art (SOTA) fMLPs M3GNet <cit.>, CHGNet <cit.>, MACE-MP-0 <cit.>, and SevenNET <cit.> (all trained on the Materials Project DFT-PBE database <cit.> and hereafter collectively referred to as `mp-fMLPs').
In particular, after detailing the methods to calculate harmonic and anharmonic vibrational properties of solids using fMLPs, we discuss how these quantities determine the thermal conductivity within the recently developed Wigner heat-Transport Equation (WTE) <cit.>, which generalizes the Boltzmann Transport Equation (BTE) <cit.> accounting not only for heat carried by particle-like propagation of phonons, but also for conduction arising from phonons' wave-like tunneling.
Thus, this framework is employed to predict from DFT-PBE or mp-fMLPs the thermal conductivity of 103 compounds made up of 34 different chemical species and having wurtzite, zincblende, or rocksalt structure.
We introduce descriptors to quantify the accuracy of fMLPs in predicting anharmonic and thermal properties, and show that they could potentially be used as a metric to benchmark or fine-tune fMLPs.
Finally, we show how the framework introduced paves the way for rigorous, physics-aware predictions of vibrational and thermal properties in materials with arbitrary composition and structure, highlighting its potential to find materials that violate semiclassical Boltzmann conduction relevant for  thermal-insulation or thermoelectric applications.



Results

From vibrational energy to thermal conductivity

The temperature-dependent thermal conductivity (κ(T)) of a solid is a macroscopic, experimentally and technologically relevant quantity that describes the capability to conduct heat.
To predict such quantity, we start from the Born-Oppenheimer Hamiltonian for atomic vibrations <cit.> expanded up to anharmonic third order in displacements from equilibrium, and we account for kinetic-energy perturbations due to isotopes <cit.>, 
 
    Ĥ=∑_R,b,αp̂^2_Rbα/2 M_b + 1/2∑_R,b,α
    R',b',α'∂^2 V/∂u_Rbα∂u_R'b'α'|_ equ_Rbαu_R'b'α'
       +1/3!∑_R,b,α
    R',b',α'∂^3 V/∂u_Rbα∂u_R'b'α'∂u_R”b”α”|_ equ_Rbαu_R'b'α'u_R”b”α” 
       +∑_R,b,α(m_b/M_b-1)p̂^2_Rbα/2 M_b,   
      6mm

where p̂_Rbα and û_Rbα are the momentum and positions-displacement operators along the Cartesian direction α for the atom b having isotope-averaged mass M_b and position R+τ_b (here, R is the Bravais-lattice vector and τ_b the position in the crystal's primitive cell); the last term in Eq. (<ref>) describes kinetic-energy perturbations induced by isotopes <cit.> (m_b is the exact mass of the atom at position  τ_b, 

which can deviate from the isotopically-averaged mass M_b).
The leading (harmonic) term in such an equation determines the vibrational frequencies; in  particular,  the Fourier transform of the mass-rescaled second-order derivative of the interatomic potential yields the dynamical matrix at wavevector q,
D(q)_bα,b'α'=∑_R∂^2 V/∂u_Rbα∂u_R'b'α'|_ eqe^-iq·(R+τ_b-τ_b')/√(M_bM_b').
By diagonalizing the dynamical matrix,

    ∑_b'α'D(q)_bα,b'α'ℰ(q)_s,b'α'=ω^2(q)_sℰ(q)_s,bα,

one obtains from the eigenvalues the phonon energies ħω(q)_s of the solid (s is a  band index ranging from 1 to 3N_ at, where N_ at is the number of atoms in the primitive cell), and from the eigenvectors ℰ(q)_s,bα the displacement patterns of atom b in direction α for the normal mode s.

The third derivative in Eq. (<ref>), instead, determines the anharmonic linewidth ħΓ_ a(q)_s (energy broadening due to three-phonon interactions <cit.>) of the phonon (q)_s:

    ħΓ_ a(q)_s=π/N_c^2∑_q'q''
    s',s”{2[N̅(q')_s'-N̅(q'')_s'']δ(ω(q)_s+ω(q')_s'-ω(q'')_s'') 
       + [1+N̅(q')_s'+N̅(q'')_s'']δ[ω(q)_s-ω(q')_s'-ω(q'')_s'']}×
       |
    ∑_α,α',α''
    b,b',b'',R'R''∂^3 V/∂ u_0bα∂ u_R'b'α'∂ u_R''b''α''ℰ(q)_s,bαℰ(q')_s',b'α'ℰ(q'')_s'',b''α''
       √(ħ^3/8)Δ(q+q'+q'')e^-i[q·τ_b+q'· (R'+τ_b')+q''· (R''+τ_b'')]/√(M_b M_b' M_b''ω(q)_sω(q')_s'ω(q'')_s'')|^2,
      6mm

where N̅(q)_s=[exp(ħω(q)_s/k_ BT)-1]^-1 is the Bose-Einstein distribution at temperature T, Δ(q+q'+q'') is the Kronecker delta (equal to 1 if q+q'+q'' is a reciprocal lattice vector, zero otherwise), δ is the Dirac delta. 
The last line in Eq. (<ref>) accounts for the presence of isotopic-mass disorder and yields the following linewidth <cit.>

    ħΓ_i(q)_s=   ħπ/2 N_c[ω(q)_s]^2
    ∑_q',s'δ[ω(q)_s-ω(q')_s']
       ×∑_b g_2^b |∑_αℰ(q)^⋆_s,bαℰ(q')_s',bα|^2,

where g_2^b=∑_i f_i,b(M_b -m_i,b/M_b)^2 describes the variance the isotopic masses of atom b (f_i,b and m_i,b are the mole fraction and mass, respectively, of the ith isotope of atom b; M_b=∑_i f_i,b m_i,b is the weighted average mass).

 
The recently developed WTE  <cit.> allows to predict the thermal conductivity of solids accounting for the interplay between structural disorder, anharmonicity, and Bose-Einstein statistics of vibrations. This offers a comprehensive approach to describe ordered `simple crystals' having phonon interband spacings much larger than the linewidths  <cit.>, completely disordered glasses  <cit.>, as well as the intermediate regime of `complex crystals' with interband spacings  smaller than the linewidths <cit.>. 
To assess how the accuracy in the prediction of the conductivity is affected by the precision with which fMLPs describe harmonic and anharmonic (third-order) force constants in Eq. (<ref>), it is sufficient to consider the conductivity obtained from the WTE solved in the single-mode relaxation-time approximation (SMA)  


    κ(T)=1/𝒱 N_c ∑_q,sC(q)_s ||v(q)_s,s||^2 /3 [Γ(q)_s]^-1
       +1/𝒱N_ c∑_q,s≠ s'ω(q)_s+ω(q)_s'/4[C(q)_s/ω(q)_s+C(q)_s'/ω(q)_s'] ||v(q)_s,s'||^2 /3
       ×1/2[Γ(q)_s+Γ(q)_s']/[ω(q)_s-ω(q)_s']^2+1/4[Γ(q)_s+Γ(q)_s']^2 ;
    32mm

where C(q)_s=ħ^2ω^2(q)_s /k_ B T^2N̅(q)_s[N̅(q)_s+1] is the specific heat of the vibration with energy ω(q)_s and total linewidth Γ(q)_s=Γ_ a(q)_s+Γ_ i(q)_s,
v^β(q)_s,s'=∑_b,α,b',α'ℰ^⋆(q)_s,bα[∇^β_q√(D(q))_bα,b'α' ]ℰ(q)_s',b'α' is the velocity operator coupling eigenstates s and s' at the same wavevector q (its diagonal elements s=s' are the usual phonon group velocities)  <cit.>, N_ c is the number of q-points used to sample the Brillouin zone and 𝒱 is the crystal's primitive-cell volume.
The first line on the right-hand side of Eq (<ref>) describes a conduction mechanism in which vibrations carry the heat C(q)_s by propagating particle-like with velocity v(q)_s,s over the lifetime [Γ(q)_s]^-1. It can be rigorously shown <cit.> that it coincides with the conductivity emerging from the Peierls-Boltzmann equation <cit.>, and will be henceforth referred to as κ_ P.
The term on the second and third lines of Eq. (<ref>) accounts for conduction through a wave-like tunneling mechanism between pairs of vibrational eigenstates. This term arises from the coherence between two different phonon modes s,s' at the same wavevector q (i.e., it becomes more signicant as their frequency difference ω(q)_s-ω(q)_s' becomes smaller) and is therefore referred to as `coherences conductivity', κ_ C. It has been shown in Refs. <cit.> that in simple crystals κ_ P≫κ_ C because particle-like propagation dominates over the wave-like tunneling. In contrast, in complex crystals both these mechanisms co-exist and can have comparable magnitude, implying κ__ P∼κ__ C. Finally, Refs. <cit.> have shown that in strongly disordered oxide glasses κ_ P≪κ_ C.

Vibrational and thermal properties from fMLPs

In general, κ(T) is highly sensitive to both harmonic and anharmonic vibrational properties <cit.>; 
therefore, the accuracy of fMLPs in describing these properties can be quantified by comparing their predictions for κ(T) with those obtained from reference first principles data.
To accomplish this goal, we begin by calculating the harmonic and third-order anharmonic force constants that determine the solid's vibrational energy (Eq. (<ref>)). We do this by using either: (i) Density-Functional Theory (DFT) with the Perdew, Burke, and Ernzerhof (PBE) functional (see Refs. <cit.> for details); or (ii) SOTA non-proprietary mp-fMLPs trained on DFT-PBE.

Then, we employ these force constants to determine harmonic vibrational properties (frequencies, velocity operators, and isotopic linewidths, see Eq. (<ref>) and Eq. (<ref>)) and anharmonic linewidths (Eq. (<ref>)). Subsequently, these atomistic vibrational properties are used in Eq. (<ref>) 
to calculate the thermal conductivity of the 103 diverse compounds in the phononDB-PBE database. The database contains rocksalt, zincblende, and wurtzite binary compunds and involves 34 chemical species, including alkali metals (Cs, K, Li, Na, Rb), alkaline earth metals (Be, Mg, Ca, Sr, Ba), transition metals (Ag, Cu, Zn, Cd), post-transition metals (Al, Ga, In, Pb), metalloids (As, Sb, Si, Te, B), nonmetals (H, C, N, O, P, S, Se), and halogens (F, Cl, Br, I).
Computational details are  provided in the Methods. 






In Fig. <ref> we compare the thermal conductivity predicted from first principles (DFT-PBE) or from the mp-fMLP MACE-MP-0 (trained on DFT-PBE data) <cit.>. We highlight how predictions from DFT-PBE or MACE-MP-0 are generally consistent within a factor of 2 across most materials,  see inset. 
We note that in about 20% of the compounds studied the semiclassical particle-like BTE fails to fully describe heat transport, and it is crucial to employ the more general WTE, see Fig. <ref> in the Methods.
These cases generally have ultra-low conductivity (κ_ TOT≲2 W/mK), and are characterized by having wave-like tunneling conductivity (κ_C, accounted for by the WTE but missing from the BTE) with magnitude comparable to the particle-like propagation conductivity (κ_P, accounted for by both the WTE and BTE).

To understand the microscopic origin of the discrepancies in the macroscopic DFT-PBE and MACE-MP-0 conductivities, we select three representative materials — wurtzite BeO, zincblende BeTe, and rocksalt RbH — and show in Fig. <ref> their phonon band structures, specific heat at constant volume, and the macroscopic thermal conductivity resolved in terms of contributions from microscopic phonon modes.
Examining the phonon band structures in the first column, we observe that MACE-MP-0 tends to underestimate the high-frequency optical vibrational modes compared to DFT-PBE. To further investigate these  differences, we plot the DFT-PBE phonons considering the non-analytical correction term <cit.> (NAC, red lines or scatter points), or not (dotted green line). This long-range interaction is responsible for the energy splitting between the longitudinal-optical and transversal-optical modes in polar dielectric materials <cit.>, and is not fully considered in fMLPs trained using a radial force  cutoff (e.g., for MACE-MP-0 such cutoff is 6 Å, while it is 5 Å  for SevenNet<cit.>, CHGNet<cit.>, and M3GNet<cit.>). 
This explains why the DFT-PBE phonons without NAC are in closer agreement with the MACE-MP-0 phonons (blue lines or scatter points). 
Importantly, it is noticeable that even without NAC, DFT-PBE frequencies tend to be higher than MACE-MP-0 frequencies, confirming the general tendency of MACE-MP-0 to underestimate vibrational frequencies discussed in Ref. <cit.>.
Nevertheless, Fig. <ref> shows that considering or not the NAC term has a negligible impact on the specific heat at constant volume and on the temperature-dependent conductivity (κ(T)) of BeO, BeTe and RbH.
This can be understood from the third column of Fig. <ref>, where we show that thermal transport in these materials is dominated by low-energy (acoustic) phonons, and these are negligibly affected by NAC. Specifically, these plots report the frequency-linewidth distributions, resolving both the anharmonic and isotopic contributions to the linewidths (Eq. <ref> and Eq. <ref>, respectively), and also quantifying how much a single phonon mode contributes to the total conductivity with the following expression:

    𝒦(q)_s=C(q)_s || v(q)_s,s||^2/3 [Γ(q)_s]^-1
       +∑_s'≠ sC(q)_s/C(q)_s+C(q)_s'ω(q)_s+ω(q)_s'/2[C(q_s/ω(q)_s+C(q_s'/ω(q)_s']
       ×|| v(q)_s,s'||^2/31/2[Γ(q)_s+Γ(q)_s']/[ω(q)_s'-ω(q)_s]^2+1/4[Γ(q)_s+Γ(q)_s']^2.

This equation describes how much a single phonon mode (q)_s contributes to heat conduction. Specifically, the term on the first line accounts for the aforementioned particle-like conduction mechanism; the term on the second and third lines, instead, describes the wave-like conduction that originates from tunneling between two non-degenerate phonons (q)_s and (q)_s' — here the single-phonon contributions from (q)_s is resolved using the ratio between specific heats C(q)_sC(q)_s+C(q)_s' as weight, as discussed by  Eq. (E3) in <cit.>. 

The frequency-linewidth distributions in Fig. <ref>a show overall agreement between DFT-PBE and MACE-MP-0 for the phonon modes that are mainly contributing to conduction — the acoustic modes and the optical modes with ħω(q)_s<k_B T — and we see that this implies very similar values for the corresponding macroscopic κ(T) (with a relative difference <10%).
Importantly, Fig. <ref>b shows 
compatibility between κ(T) predicted from DFT-PBE or MACE-MP-0 can also result from cancellation of errors; specifically, in BeTe there are visible differences in the frequency-linewidth distributions, and these largely cancel out when integrated to determine the conductivity. 
Finally, Fig. <ref>c illustrates that the presence of systematic, non-compensating differences in the microscopic frequency-linewidth distributions can also directly translate into significant discrepancies on the macroscopic conductivities (difference of a factor of 2.7). 
Additionally, we note that depending on the the chemical composition, the anharmonic linewidths at room temperature can either dominate over the isotopic linewidth (e.g., in BeO) or not (e.g., in BeTe). In the former case, the thermal conductivity is practically unaffected by considering or not the isotopic contributions to the linewidth, while in the latter case considering isotopic scattering yields a reduction of κ(300 K) of about ∼ 75 % (see Table. <ref>). 


The results above motivate us to investigate when good agreement between DFT and fMLP conductivities is obtained from accurately described microscopic harmonic and anharmonic vibrational properties, or because of compensation of errors.
To achieve this goal, we resolve the discrepancies in the total macroscopic thermal conductivities using the Symmetric Relative Error (SRE), 

    SRE[κ]=2|κ_fMLP-κ_DFT| /κ_fMLP+κ_DFT .

Then, to  determine whether a low value for SRE[κ]  originates from accurately described microscopic vibrational properties, or because of compensation of errors, we introduce the 
Symmetric Relative Mean Error (SRME) on the single-phonon conductivity contribution 𝒦(q)_s:

    SRME[{𝒦(q)_s}]=2/N_c 𝒱∑_qs|𝒦_fMLP(q)_s-𝒦_DFT(q)_s| /κ_fMLP+κ_DFT ,  6mm

where 
𝒦_DFT(q)_s refers to Eq. (<ref>)  evaluated using DFT, and 
𝒦_fMLP(q)_s refers to the same equation evaluated using fMLP.

Fig. <ref> illustrates that a large SRME generally implies large SRE.  Importantly, knowing both SRE and SRME enables us to identify when microscopic error compensation occurs — this is indicated by a large SRME but small SRE. Three representative cases are highlighted  in Fig. <ref>: wurtzite BeO shows low SRME and thus correspondingly low SRE;  rocksalt RbH exhibits high error in both SRME and SRE; zincblende BeTe displays high SRME but low SRE due to compensation of microscopic errors.
We note that the SRME[𝒦(q)_s] error can stem from discrepancies in the harmonic (second-order) or anharmonic (third-order) force constants. Therefore, in Fig. <ref> in the Methods we discuss how to disentangle the SRME and SRE into errors on harmonic vibrational properties and anharmonic vibrational properties.

Overall, this analysis highlights the importance of benchmarking the accuracy of fMLPs not only at the macroscopic but also at the microscopic level. In particular the macroscopic SRE[κ] alone is not a reliable descriptor for the fMLPs' ability to capture the harmonic and anharmonic physics underlying heat conduction. However, achieving both a small macroscopic SRE[κ] and a small microscopic SRME[κ] is a sufficient condition for accurately describing both the microscopic physics and the macroscopic thermal conductivity.




Accuracy of various SOTA foundation models

The previous section demonstrated that the SRE and SRME descriptors can effectively measure the accuracy of fMLPs in describing vibrational and thermal properties. Therefore, here we utilize these descriptors to compare the accuracy of different non-proprietary mp-fMLPs on the 103 diverse  structures contained in the phononDB-PBE database<cit.>.

Fig. <ref> compares the accuracy of the four foundation models M3GNet<cit.>, CHGNet<cit.>, MACE-MP-0<cit.>, and SevenNet<cit.> mp-fMLPs. 
To determine if a certain mp-fMLP tend to systematically overestimate or underestimate the conductivity, it is useful to rely on the Symmetric Relative Difference (SRD) in the total Wigner conductivity from DFT-PBE or mp-fMLP,

    SRD[κ]=2κ^ fMLP-κ^ DFT/κ^ fMLP+κ^ DFT,

which ranges from -2 to +2, resolving both overestimation and underestimation of the conductivity. 
Figure  <ref>a summarizes the SRD for all 103 binary compounds in the phononDB-PBE database, as depicted in the violin plot <cit.>. In this plot, the width of each violin shape is related to the percentage of materials with SRD values indicated on the y-axis. The median SRD value is marked by a white scatter point, while the black boxes illustrate the interquartile range, and the whiskers show 1.5 times the interquartile range below and above the first and third quartile points.  
Unstable structures with negative phonon frequencies were included considering κ^ fMLP=0, i.e., SRD=-2.
This analysis reveals that all the mp-fMLPs assessed in this study tend to  underestimate the thermal conductivity. This finding is consistent with the observed systematic underestimation of vibrational frequencies noted in Ref. <cit.>, and might also derived from overestimation of anharmonic linewidths. 




To examine whether the SRD shown in Fig. <ref>a is influenced by error compensation, we analyze in Fig. <ref>b the violin plots for SRME[{𝒦(q)_s}]. 
We see that the SRME distribution for MACE-MP-0 is the closest to zero, followed by SevenNet, M3GNet, and CHGNet. 
In addition, to quantify the overall accuracy of a fMLPs in predicting the macroscopic conductivity over a materials' database, without resolving the possible compensation of microscopic errors, it is informative to consider the mean of the modulus of the deviations, i.e., the mean of the distribution of SRE (<ref>) — we prefer this over the mean of the SRD distribution, as the latter can be close to zero in the presence of very broad but symmetric distribution. 
Importantly, we note that the mean for SRE[κ] and mean for SRME[𝒦(q)_s] are expected to be comparable in the absence of compensation of microscopic errors. In contrast, in the presence of compensation of microscopic errors, the mean SRE[κ] is significantly lower than mean SRME[𝒦(q)_s].
The mean values for SRME[𝒦(q)_s] and SRE[κ] are reported in Table <ref>.





In summary, our findings reveal that the mean SRE[κ] and the mean in SRME[𝒦(q)_s] are effective metrics for assessing the accuracy of mp-fMLPs in predicting macroscopic thermal conductivity. These metrics not only indicate the precision in predicting the macroscopic conductivity (low SRE[κ]), but can also assess whether accurate conductivity predictions follow from accurate estimates of microscopic vibrational and thermal properties (low SRE[κ] and low SRME[𝒦(q)_s]) or result from compensation of microscopic errors (low SRE[κ] and large SRME[𝒦(q)_s]).


Discussion

We have introduced a robust and automated computational framework for directly predicting the thermal conductivity of solids using foundation Machine Learning Potentials. This framework has been employed to benchmark the out-of-the-box accuracy of SOTA fMLPs 
trained on the Materials Project DFT-PBE database <cit.> (mp-fMLPs) — SevenNet<cit.>, MACE-MP-0<cit.>, CHGNet<cit.>, and M3GNet<cit.> — in predicting both microscopic harmonic and anharmonic vibrational properties, and the macroscopic thermal conductivity. 
In particular, we compared predictions from mp-fMLP against those obtained from the 
first-principles phononDB-PBE databasem, which includes 103 diverse compounds, made up of 34 different chemical species and having wurtzite, zincblende, or rocksalt structure <cit.>.  
Our results show that the overall most accurate mp-fMLPs, MACE-MP-0<cit.>, yields conductivities compatible within a factor of two of the DFT-PBE values for 69% of the materials in the phononDB-PBE database<cit.>. 
We have shown that non-proprietary SOTA mp-fMLPs yield thermal conductivity predictions that tend to underestimate the corresponding first-principles DFT-PBE values, quantifying these discrepancies with the parameter SRD[κ].
Most importantly, we have introduced a descriptor quantifies the performance of mp-fMLPs in predicting both microscopic harmonic and anharmonic vibrational properties, and the macroscopic, experimentally measurable  thermal conductivity — the Symmetric Relative Mean Error on the conductivity SRME[𝒦(q)_s]. This descriptor provides a metric that could be useful to training or fine-tune fMLPs, and its mean computed over a database of materials can be used to benchmark the accuracy of different families of mp-fMLPs. 
We have also provided computational recipes for incorporating the Mean in SRME[𝒦(q)_s] into standard benchmark frameworks used to assess the accuracy of fMLPs <cit.>.
Finally, we have discussed how our automated framework to compute the Wigner conductivity can accurately describe materials that violate semiclassical Boltzmann transport. These often feature very low conductivity (≲2 W/mK) and are potentially relevant for e.g., heat-shielding <cit.> and thermoelectric <cit.> technologies.
Ultimately, the computational framework and analyses that we presented pave the way for quantum-accurate, physics-aware predictions of thermal conductivity of materials with arbitrary composition and structure.






Methods
 
Failures of semiclassical Boltzmann transport in binary crystals with low conductivity.

Our automated framework can be used to identify materials in which the semiclassical BTE fails.

In particular, the BTE <cit.> describes heat propagation exclusively in terms of particle-like phonons that propagate and scatter akin to the particles of a classical gas. While the BTE is accurate in weakly anharmonic and ordered `simple crystal' (characterized by phonon interband spacings much larger than the linewidths<cit.>), it fails in strongly anharmonic or disordered materials (with phonon interband spacings smaller than the linewidths). It has been recently shown <cit.> that in the latter cases heat conduction originates not only from particle-like propagation but also from wave-like interband tunneling of phonons. The recently developed WTE generalizes the BTE by accounting for both  particle-like propagation and wave-like tunneling contributions to the conductivity (κ_P and κ_C, respectively; the total conductivity is given by κ_TOT=κ_P+κ_C), providing a comprehensive approach that overcomes the failures of the BTE.

Our automated framework computes the Wigner thermal conductivity<cit.>, and therefore can be used to find materials that have very low thermal conductivity, relevant for thermoelectric or heat-shielding technologies.
In Fig. <ref> we show that among the 103 compounds analyzed, the BTE fails in materials with low thermal conductivity (κ_ TOT≲ 2 W/mK), such as AgBr and AgCl, since the wave-like tunneling conductivity becomes comparable to the particle-like propagation  conductivity. We also note that in simple crystals with large conductivity, the BTE is accurate as the population conductivity dominates over the wave-like tunneling conductivity.


Symmetric Relative Mean Error on harmonic and anharmonic vibrational properties.
The analyses reported in the main text discusses SRME[𝒦(q)_s] as a descriptor that is informative of the accuracy of fMLPs in describing microscopic harmonic and anharmonic properties. In this section we provide the tools to resolve whether the SRME[𝒦(q)_s] originates from errors in microscopic harmonic properties, anharmonic properties, or both.
To this aim, we employ the expression for the microscopic, single-phonon contributions to the thermal conductivity (<ref>) to define descriptors that quantity the accuracy with which harmonic and anharmonic properties are predicted.
In particular, we note that the conductivity contribution of a phonon (q)_s is a function of: (i) all the phonon frequencies {ω(q_s'} at a fixed wavevector q and variable mode s'=1,…,N_ s (N_ s is the number of phonon bands, equal to 3 times the number of atoms in the crystal's unit cell <cit.>); (ii) all velocity-operator elements {v(q)_s,s'} at fixed q, s and variable s'; (iii) all isotopic linewidths at fixed q and variable s', {Γ_ i(q)_s'}, which depend solely on harmonic properties, see Eq. <ref>; (iv) the anharmonic linewidths {Γ_ a(q)_s'} at fixed q and variable s'. This can be summarized using the following notation: 
𝒦(q)_s=𝒦(q)_s[{ω(q)_s'},{v(q)_s,s'},{Γ_ i(q)_s'},{Γ_ a(q)_s'}]
where the curly brackets denote the set of values of a certain quantity over all the bands at fixed q (e.g., {ω(q)_s'}={ω(q)_1,ω(q)_2,…,ω(q)_N_s}). 

To quantify the impact of errors on the harmonic (har) and anharmonic (anh) properties on the single-phonon  conductivity contributions, we define the Symmetric Relative Mean Error (SRME) on these properties as follows:  

    
    SRME[har]=2∑_qs|
        𝒦(q)_s[{ω_ fMLP(q)_s'},{v_ fMLP(q)_s,s'},{Γ_ i, fMLP(q)_s'},{Γ_ a, DFT(q)_s'}]-
        𝒦_ DFT(q)_s
        | /∑_qs(
        𝒦(q)_s[{ω_ fMLP(q)_s'},{v_ fMLP(q)_s,s'},{Γ_ i, fMLP(q)_s'},{Γ_ a, DFT(q)_s'}]
        +
        𝒦_ DFT(q)_s
        ) ,


    SRME[anh]=2∑_qs|
        𝒦(q)_s[{ω_ DFT(q)_s'},{v_ DFT(q)_s,s'},{Γ_ i, DFT(q)_s'},{Γ_ a, fMLP(q)_s'}]-
        𝒦_ DFT(q)_s
        | /∑_qs(
        𝒦(q)_s[{ω_ DFT(q)_s'},{v_ DFT(q)_s,s'},{Γ_ i, DFT(q)_s'},{Γ_ a, fMLP(q)_s'}]
        +
        𝒦_ DFT(q)_s
        ),


where we have used the shorthand notation
𝒦_ DFT(q)_s=
𝒦__ DFT(q)_s[{ω__ DFT(q)_s'},{v__ DFT(q)_s,s'},{Γ_ i, DFT(q)_s'},{Γ_ a, DFT(q)_s'}].
Intuitively, SRME[har] (<ref>) is large when harmonic vibrational properties 
(ω(q)_s, or v(q)_s,s', or Γ_ i(q)_s) differ between DFT and fMLP, while SRME[anh] (<ref>) is large when the anharmonic linewidths 
(Γ_ a(q)_s) differ between DFT and fMLP. 
We show in Fig. <ref> that small SRME[har] and small SRME[anh] (e.g., as in BeO), imply a small SRE[κ]

Fig. <ref> also shows that often large SRME[har] and large SRME[anh] (e.g. RbH) translate into large  SRE[κ]; however there also cases (e.g. BeTe) in which large SRME[har] and large SRME[anh] can also compensate each other and result in a small SRE[κ]. This confirms the statements made in the main text on  SRE[κ]  not being a reliable descriptor for the capability of fMLPs to capture the harmonic and anharmonic physics underlying heat conduction. In contrast, having both small SRME[har] and small SRME[anh] is a sufficient condition to accurately capture harmonic and anharmonic vibrational properties, as well as  thermal conductivity.

Overall, these tests highlight the importance of benchmarking for the accuracy of fMLPs in predicting both microscopic harmonic and anharmonic vibrational properties, motivating the introduction of the SRME[{𝒦(q)_s}].



Influence of isotopic scattering on conductivity.



To analyze the influence of isotope scattering on the conductivity, we compare thermal conductivity values with and without considering the linewidths from isotope mass-disorder, as described in Eq. (<ref>). The results are summarized in Table <ref>. In wurtzite BeO and rocksalt RbH, the impact of isotope scattering is minimal or small, respectively. However, in zincblende BeTe, as shown in Fig. <ref>b, isotopic and anharmonic linewidths have comparable values, both affecting  the thermal conductivity. In this material, MACE-MP-0 significantly overestimates some linewidths compared to DFT-PBE, and this compensates for the overestimation in other regions. Consequently, MACE-MP-0 shows a significantly larger error in thermal conductivity when isotope scattering is not considered.



Automated Wigner Conductivity Workflow.
The workflow, outlined in Fig. <ref>, includes structure relaxation, force constant computation, and thermal conductivity analysis. 
To automate the calculation of interatomic forces, we developed an interface between the <cit.> (version 3.23.0) and the <cit.> (version 3.2.0) packages.
Atomic positions and cell parameters were simultaneously relaxed to minimize stresses and ensure positive phonon frequencies. Since the determination of irreducible q points during thermal conductivity calculations depends on the crystal symmetry, it is essential to consider symmetries. During the initial relaxation stage, symmetries are explicitly enforced as constraints while simultaneously relaxing atomic positions and cell parameters. After each step of atomic position relaxation, cell parameters are adjusted if the total forces exceed a a certain threshold (values are fMLP-specific and discussed later). In the second stage, the symmetry constraint is removed to allow for finer atomic adjustments and a complete relaxation of the structure. The joint relaxation of atomic positions and cell parameters is applied once more. If the symmetry is preserved, the final relaxed structure is used. If symmetry is broken, the structure from the first stage, with enforced symmetry, is retained. Finally, an additional relaxation of atomic positions with fixed cell parameters is performed using a stricter threshold to ensure the structure reaches a stable energy minimum at a fixed volume for accurate phonon calculations.




To ensure positive phonons, strict thresholds were established, tailored to the performance of each individual model. M3GNet required thresholds of 2.0× 10^-3 eV/Å and 8.0× 10^-4 eV/Å. For CHGNet, MACE-MP-0, and SevenNet, the thresholds were set to 10^-4 eV/Å and 5.0× 10^-5 eV/Å for initial and final values, respectively. In the supercell force-constant calculations, we used the same parameters used in the DFT reference data<cit.>.

Thermal conductivities were computed using  following Ref. <cit.>. A 19×19×19 q-mesh was used for rocksalt and zincblende structures, and a 19×19×15 q-mesh for wurtzite structures, with collision operator computed using the tetrahedron method. For Fig. <ref>, anharmonic linewidths were calculated on a 9×9×9 q-mesh to ensure clarity. We utilized version 3 of  for anharmonic linewidth calculations and employed the Wang method <cit.> for the NAC term, as implemented in the <cit.> package (version 2.26.6). When structures with negative phonon frequencies were found with mp-fMLPs, these were discarded from subsequent thermal-conductivity analysis.


DFT and mp-fMLP calculations.
Structures were taken from the Materials Project (MP)<cit.>. The Perdew, Burke, and Ernzerhof (PBE) exchange-correlation functional <cit.> was employed for DFT calculations, consistent with the approach used for generating MP dataset. The computational details for the DFT calculations can be found in Refs. <cit.>. As discussed in Refs. <cit.>, mp-fMLP were trained on the MP data using the PBE functional. M3GNet was used through the <cit.> (version 1.1.2) package with the M3GNet-MP-2021.2.8-PES model. The CHGNet version 0.3.8 was used. The MACE-MP-0 2024-01-07-mace-128-L2 model was run through LAMMPS<cit.>. SevenNet was used through the  package (version 0.9.2) with the SevenNet-0_11July2024 model.


Computational cost. 
The SevenNet, MACE, and CHGNet calculations were performed on an Nvidia Tesla A100 SXM4 80GB GPU, with the relaxation and force calculations for all 103 structures costing approximately ∼ 3 GPU hours per model. M3GNet was executed on AMD EPYC 7702 CPUs, with relaxation and force calculations costing around ∼ 30 CPU hours. Thermal conductivity calculations for all 103 structures required about ∼ 50 CPU hours per model, and similar resources were needed for evaluating thermal conductivity from the DFT force constants. Overall, the analysis of these four models required approximately ∼ 9 GPU hours and ∼ 280 CPU hours.



Data availability.
The phononDB-PBE dataset including the displacements generated by  and the corresponding force sets are available <https://github.com/atztogo/phonondb> <cit.>. The remaining dataset will be made available in the future.


Code availability.
The phono3py and  packages are available at <https://github.com/phonopy/>; the  package is available at <https://gitlab.com/ase/ase>.  containing the M3GNet model is available at <https://github.com/materialsvirtuallab/matgl>; the CHGNet model and package is available at <https://github.com/CederGroupHub/chgnet>; the SevenNet model and package is available at <https://github.com/MDIL-SNU/SevenNet>. The MACE used through LAMMPS<cit.> is available at <https://github.com/ACEsuit/lammps>, and the MACE-MP-0 model is available at <https://github.com/ACEsuit/mace-mp>. The automatic framework and the analysis code will be made available soon.




Competing interests.
The authors declare no competing interests.


Author contributions.
M.S. conceived and supervised the project. B.P. developed the automated computational framework and performed the numerical calculations with inputs from P.A., G.C., and M.S. B.P. and M.S. analyzed and organized the data, with inputs from P.A. All authors contributed to discussing the data, writing and editing the manuscript.



§ ACKNOWLEDGEMENTS
 

M. S. acknowledges support from: (i) Gonville and Caius College; (ii) the Swiss National Science Foundation (SNSF) project P500PT_203178; (iii) the Sulis Tier 2 HPC platform (funded by EPSRC Grant EP/T022108/1 and the HPC Midlands+consortium); (iv) the Kelvin2 HPC platform at the NI-HPC Centre (funded by EPSRC and jointly managed by Queen’s University Belfast and Ulster University). P.A. acknowledges support from SNSF through Post.Doc mobility fellowship P500PN_206693. 
We thank William Chuck Witt for the useful discussions, and we gratefully acknowledge Atsushi Togo for having provided harmonic and anharmonic force constants computed using density functional theory (PBE functional) from the phononDB-PBE database <cit.>.












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[Blank et al.(1995)Blank,
  Brown, Calhoun, and Doren]blank_neural_1995
  author author T. B. Blank, author S. D. Brown,
  author A. W. Calhoun, and  author D. J. Doren, https://doi.org/10.1063/1.469597 journal journal
  The Journal of Chemical Physics volume 103,  pages 4129 (year 1995)NoStop
[Bartók et al.(2010)Bartók, Payne, Kondor, and Csányi]bartok_gaussian_2010
  author author A. P. Bartók, author M. C. Payne,
  author R. Kondor, and author G. Csányi, https://doi.org/10.1103/PhysRevLett.104.136403 journal
  journal Physical Review Letters volume
  104, pages 136403 (year 2010)NoStop
[Drautz(2019)]drautz_atomic_2019
  author author R. Drautz, https://doi.org/10.1103/PhysRevB.99.014104 journal journal Physical Review B volume 99, pages 014104 (year 2019),  note publisher: American Physical SocietyNoStop
[Seko et al.(2019)Seko,
  Togo, and Tanaka]seko_group-theoretical_2019
  author author A. Seko, author A. Togo, and  author I. Tanaka, https://doi.org/10.1103/PhysRevB.99.214108 journal journal Physical Review B volume 99, pages 214108 (year 2019)NoStop
[Kocer et al.(2022)Kocer,
  Ko, and Behler]kocer_neural_2022
  author author E. Kocer, author T. W. Ko, and  author J. Behler, https://doi.org/10.1146/annurev-physchem-082720-034254 journal journal Annual Review of Physical Chemistry  volume 73, pages 163 (year
  2022)NoStop
[Batzner et al.(2022)Batzner, Musaelian, Sun, Geiger, Mailoa, Kornbluth, Molinari, Smidt, and Kozinsky]batzner_e3-equivariant_2022
  author author S. Batzner, author A. Musaelian,
  author L. Sun, author
  M. Geiger, author J. P. Mailoa, author M. Kornbluth, author N. Molinari, author T. E.  Smidt, and author B. Kozinsky, https://doi.org/10.1038/s41467-022-29939-5 journal journal Nature Communications volume 13,  pages 2453 (year 2022), note
  publisher: Nature Publishing GroupNoStop
[Togo and Seko(2024)]togo_–fly_2024
  author author A. Togo and author A. Seko,  https://doi.org/10.1063/5.0211296 journal journal The Journal of Chemical Physics volume
  160, pages 211001 (year 2024)NoStop
[Nataf et al.(2024)Nataf,
  Volz, Ordonez-Miranda, Íñiguez González, Rurali, and Dkhil]nataf_using_2024
  author author G. F. Nataf, author S. Volz,
  author J. Ordonez-Miranda,
  author J. Íñiguez González,
  author R. Rurali, and author B. Dkhil, https://doi.org/10.1038/s41578-024-00690-1 journal journal Nature Reviews Materials , pages 1 (year 2024)NoStop
[Kimber et al.(2023)Kimber,
  Zhang, Liang, Guzmán-Verri,
  Littlewood, Cheng, Abernathy,
  Hudspeth, Luo, Kanatzidis,
  Chatterji, Ramirez-Cuesta, and  Billinge]kimber_dynamic_2023
  author author S. A. J.  Kimber, author J. Zhang, author C. H. Liang,
  author G. G. Guzmán-Verri,
  author P. B. Littlewood,
  author Y. Cheng, author D. L. Abernathy, author
  J. M. Hudspeth, author
  Z.-Z. Luo, author M. G. Kanatzidis, author T. Chatterji, author A. J.  Ramirez-Cuesta, and author S. J. L. Billinge, https://doi.org/10.1038/s41563-023-01483-7 journal journal Nature Materials volume 22, pages 311 (year 2023)NoStop
[Pazhedath et al.(2024)Pazhedath, Bastonero, Marzari, and  Simoncelli]pazhedath_first-principles_2023
  author author A. Pazhedath, author L. Bastonero, author N. Marzari,  and author M. Simoncelli,  @noop  journal journal accepted in Phys.
  Rev. Applied  (year 2024), https://arxiv.org/abs/2309.10789 arXiv:2309.10789 [cond-mat.mtrl-sci]
  NoStop
[Witt et al.(2023)Witt,
  van der Oord, Gelžinytė, Järvinen, Ross, Darby, Ho,
  Baldwin, Sachs, Kermode,
  Bernstein, Csányi, and Ortner]witt_acepotentialsjl_2023
  author author W. C. Witt, author C. van der Oord,
  author E. Gelžinytė, author T. Järvinen, author
  A. Ross, author J. P. Darby, author C. H.  Ho, author W. J. Baldwin, author M. Sachs,
  author J. Kermode, author N. Bernstein, author
  G. Csányi, and author
  C. Ortner, https://doi.org/10.1063/5.0158783 journal journal The Journal of Chemical Physics volume
  159, pages 164101 (year 2023)NoStop
[Okabe et al.(2024)Okabe,
  Chotrattanapituk, Boonkird, Andrejevic, Fu, Jaakkola, Song, Nguyen, Drucker, Mu,
  Wang, Liao, Cheng, and  Li]okabe_virtual_2024
  author author R. Okabe, author A. Chotrattanapituk, author A. Boonkird, author N. Andrejevic, author X. Fu,
  author T. S. Jaakkola, author Q. Song, author
  T. Nguyen, author N. Drucker, author S. Mu, author Y. Wang, author B. Liao, author Y. Cheng, and author
  M. Li, journal journal Nature Computational Science https://doi.org/10.1038/s43588-024-00661-0 10.1038/s43588-024-00661-0
  (year 2024)NoStop
[Ojih et al.(2024)Ojih,
  Al-Fahdi, Yao, Hu, and  Hu]ojih2024graph
  author author J. Ojih, author M. Al-Fahdi,
  author Y. Yao, author
  J. Hu, and author
  M. Hu, https://doi.org/10.1039/D3TA06190F journal journal Journal of Materials Chemistry A volume
  12, pages 8502 (year 2024)NoStop
[Rodriguez et al.(2023)Rodriguez, Lin, Yang, Al-Fahdi, Shen, Choudhary, Zhao, Hu, Cao, Zhang et al.]rodriguez2023million
  author author A. Rodriguez, author C. Lin,
  author H. Yang, author
  M. Al-Fahdi, author
  C. Shen, author K. Choudhary, author Y. Zhao, author J. Hu, author B. Cao, author H. Zhang, et al., https://doi.org/10.1038/s41524-023-00974-0 journal journal npj Computational Materials volume 9,  pages 20 (year 2023)NoStop
[Rose et al.(2023)Rose,
  Kozachinskiy, Rojas, Petrache, and Barceló]rose_three_2023
  author author V. D. Rose, author A. Kozachinskiy,
  author C. Rojas, author M. Petrache, and author P. Barceló, http://arxiv.org/abs/2303.12853 title Three iterations of
  $(d-1)$-WL test distinguish non isometric clouds of $d$-dimensional
  points (year 2023), note arXiv:2303.12853
  [cs]NoStop
[Gilmer et al.(2017)Gilmer,
  Schoenholz, Riley, Vinyals,  and Dahl]gilmer_neural_2017
  author author J. Gilmer, author S. S. Schoenholz, author P. F. Riley, author O. Vinyals,  and author G. E. Dahl, in  https://proceedings.mlr.press/v70/gilmer17a.html booktitle Proceedings of the 34th International Conference on
  Machine Learning (publisher PMLR, year
  2017) pp. pages 1263–1272, note iSSN:
  2640-3498NoStop
[Batatia et al.(2023)Batatia, Kovács, Simm, Ortner, and Csányi]batatia_mace_2023
  author author I. Batatia, author D. P. Kovács, author G. N. C. Simm, author C. Ortner, and  author G. Csányi, http://arxiv.org/abs/2206.07697 title MACE: Higher Order
  Equivariant Message Passing Neural Networks for Fast and
  Accurate Force Fields (year 2023), note
  arXiv:2206.07697 [cond-mat, physics:physics, stat]NoStop
[Chen and Ong(2022)]chen_universal_2022
  author author C. Chen and author S. P. Ong,  https://doi.org/10.1038/s43588-022-00349-3 journal
  journal Nature Computational Science volume 2, pages 718 (year 2022), note publisher: Nature Publishing GroupNoStop
[Deng et al.(2023)Deng,
  Zhong, Jun, Riebesell,
  Han, Bartel, and Ceder]deng_chgnet_2023
  author author B. Deng, author P. Zhong,
  author K. Jun, author
  J. Riebesell, author
  K. Han, author C. J. Bartel, and author G. Ceder, https://doi.org/10.1038/s42256-023-00716-3 journal journal Nature Machine Intelligence volume 5,  pages 1031 (year 2023), note
  publisher: Nature Publishing GroupNoStop
[Batatia et al.(2024)Batatia, Benner, Chiang, Elena, Kovács, Riebesell, Advincula, Asta, Avaylon, Baldwin, Berger, Bernstein, Bhowmik, Blau, Cărare, Darby, De, Della Pia, Deringer, Elijošius, El-Machachi,
  Falcioni, Fako, Ferrari,
  Genreith-Schriever, George, Goodall, Grey, Grigorev, Han, Handley, Heenen, Hermansson, Holm, Jaafar, Hofmann, Jakob, Jung, Kapil, Kaplan, Karimitari, Kermode, Kroupa, Kullgren, Kuner, Kuryla, Liepuoniute, Margraf, Magdău, Michaelides,
  Moore, Naik, Niblett,
  Norwood, O'Neill, Ortner,
  Persson, Reuter, Rosen,
  Schaaf, Schran, Shi,
  Sivonxay, Stenczel, Svahn,
  Sutton, Swinburne, Tilly,
  van der Oord, Varga-Umbrich, Vegge, Vondrák, Wang, Witt, Zills, and Csányi]batatia_foundation_2024
  author author I. Batatia, author P. Benner,
  author Y. Chiang, author A. M. Elena, author
  D. P. Kovács, author
  J. Riebesell, author
  X. R. Advincula, author
  M. Asta, author M. Avaylon, author W. J.  Baldwin, author F. Berger, author N. Bernstein,
  author A. Bhowmik, author S. M. Blau, author
  V. Cărare, author J. P. Darby, author S. De, author F. Della Pia, author V. L. Deringer, author R. Elijošius, author Z. El-Machachi, author F. Falcioni, author E. Fako,
  author A. C. Ferrari, author A. Genreith-Schriever, author J. George, author
  R. E. A. Goodall, author
  C. P. Grey, author
  P. Grigorev, author
  S. Han, author W. Handley, author H. H.  Heenen, author K. Hermansson, author C. Holm,
  author J. Jaafar, author S. Hofmann, author
  K. S. Jakob, author
  H. Jung, author V. Kapil, author A. D.  Kaplan, author N. Karimitari, author J. R. Kermode, author N. Kroupa,
  author J. Kullgren, author M. C. Kuner, author
  D. Kuryla, author G. Liepuoniute, author J. T.  Margraf, author I.-B.  Magdău, author A. Michaelides, author J. H. Moore, author A. A. Naik,
  author S. P. Niblett, author S. W. Norwood, author
  N. O'Neill, author C. Ortner, author K. A.  Persson, author K. Reuter, author A. S. Rosen,
  author L. L. Schaaf, author C. Schran, author
  B. X. Shi, author E. Sivonxay, author T. K.  Stenczel, author V. Svahn, author C. Sutton,
  author T. D. Swinburne, author J. Tilly, author
  C. van der Oord, author
  E. Varga-Umbrich, author
  T. Vegge, author M. Vondrák, author Y. Wang, author W. C.  Witt, author F. Zills, and author G. Csányi, https://doi.org/10.48550/arXiv.2401.00096 title A foundation model for atomistic materials chemistry (year 2024), note arXiv:2401.00096 [cond-mat,
  physics:physics]NoStop
[Park et al.(2024)Park,
  Kim, Hwang, and Han]park_scalable_2024
  author author Y. Park, author J. Kim, author S. Hwang, and author
  S. Han, https://doi.org/10.1021/acs.jctc.4c00190 journal journal Journal of Chemical Theory and Computation volume 20, pages 4857 (year 2024), note publisher: American Chemical SocietyNoStop
[Yang et al.(2024)Yang,
  Hu, Zhou, Liu, Shi, Li, Li, Chen,
  Chen, Zeni, Horton,
  Pinsler, Fowler, Zügner,
  Xie, Smith, Sun,
  Wang, Kong, Liu,
  Hao, and Lu]yang2024mattersimdeeplearningatomistic
  author author H. Yang, author C. Hu, author Y. Zhou, author
  X. Liu, author Y. Shi, author J. Li, author G. Li, author Z. Chen, author
  S. Chen, author C. Zeni, author M. Horton, author R. Pinsler,
  author A. Fowler, author D. Zügner, author
  T. Xie, author J. Smith, author L. Sun, author Q. Wang, author L. Kong, author C. Liu, author
  H. Hao, and author
  Z. Lu, https://arxiv.org/abs/2405.04967 title Mattersim: A deep
  learning atomistic model across elements, temperatures and pressures
  (year 2024), https://arxiv.org/abs/2405.04967
  arXiv:2405.04967 [cond-mat.mtrl-sci] NoStop
[Dunn et al.(2020a)Dunn, Wang,
  Ganose, Dopp, and Jain]Dunn2020_gnome
  author author A. Dunn, author Q. Wang, author A. Ganose, author
  D. Dopp, and author
  A. Jain, https://doi.org/10.1038/s41524-020-00406-3 journal journal npj Computational Materials volume 6,  pages 138 (year 2020a)NoStop
[Riebesell et al.(2024)Riebesell, Goodall, Benner, Chiang, Deng, Lee, Jain,  and Persson]riebesell_matbench_2024
  author author J. Riebesell, author R. E. A. Goodall, author P. Benner,
  author Y. Chiang, author B. Deng, author
  A. A. Lee, author A. Jain, and author K. A.  Persson, https://doi.org/10.48550/arXiv.2308.14920
  title Matbench Discovery – A framework to evaluate machine
  learning crystal stability predictions (year 2024), note arXiv:2308.14920 [cond-mat]NoStop
[Dunn et al.(2020b)Dunn, Wang,
  Ganose, Dopp, and Jain]Dunn_matbench2020
  author author A. Dunn, author Q. Wang, author A. Ganose, author
  D. Dopp, and author
  A. Jain, https://doi.org/10.1038/s41524-020-00406-3 journal journal npj Computational Materials volume 6,  pages 138 (year 2020b)NoStop
[Yu et al.(2024)Yu,
  Giantomassi, Materzanini, Wang, and Rignanese]yu_systematic_2024
  author author H. Yu, author M. Giantomassi,
  author G. Materzanini, author J. Wang, and author
  G.-M. Rignanese, https://doi.org/10.48550/arXiv.2403.05729 title Systematic
  assessment of various universal machine-learning interatomic potentials
  (year 2024), note arXiv:2403.05729
  [cond-mat]NoStop
[Deng et al.(2024)Deng,
  Choi, Zhong, Riebesell,
  Li, Jun, Persson, and  Ceder]deng_overcoming_nodate
  author author B. Deng, author Y. Choi, author P. Zhong, author
  J. Riebesell, author
  Z. Li, author K. Jun, author K. A.  Persson, and author G. Ceder, https://doi.org/10.48550/arXiv.2405.07105 title Overcoming
  systematic softening in universal machine learning interatomic potentials by
  fine-tuning (year 2024), note
  arXiv:2405.07105NoStop
[Lee et al.(2024)Lee,
  Hegde, Wolverton, and Xia]lee_accelerating_2024
  author author H. Lee, author V. I. Hegde,
  author C. Wolverton, and  author Y. Xia, https://doi.org/10.48550/arXiv.2407.09674 title Accelerating
  High-Throughput Phonon Calculations via Machine Learning
  Universal Potentials (year 2024), note
  arXiv:2407.09674 [cond-mat]NoStop
[Fugallo et al.(2013)Fugallo, Lazzeri, Paulatto, and  Mauri]fugallo2013ab
  author author G. Fugallo, author M. Lazzeri,
  author L. Paulatto, and  author F. Mauri, https://doi.org/10.1103/PhysRevB.88.045430 journal journal Phys. Rev. B volume 88, pages 045430 (year 2013)NoStop
[Togo et al.(2015)Togo,
  Chaput, and Tanaka]phono3py
  author author A. Togo, author L. Chaput, and  author I. Tanaka, https://doi.org/10.1103/PhysRevB.91.094306 journal journal Phys. Rev. B volume 91, pages 094306 (year 2015)NoStop
[Seko et al.(2015)Seko,
  Togo, Hayashi, Tsuda,
  Chaput, and Tanaka]seko_prediction_2015
  author author A. Seko, author A. Togo, author H. Hayashi, author
  K. Tsuda, author L. Chaput, and author I. Tanaka, https://doi.org/10.1103/PhysRevLett.115.205901 journal
  journal Physical Review Letters volume
  115, pages 205901 (year 2015)NoStop
[Tadano et al.(2014)Tadano,
  Gohda, and Tsuneyuki]tadano_anharmonic_2014
  author author T. Tadano, author Y. Gohda, and  author S. Tsuneyuki, https://doi.org/10.1088/0953-8984/26/22/225402 journal
  journal Journal of Physics: Condensed Matter volume 26, pages 225402 (year
  2014)NoStop
[Carrete et al.(2017)Carrete, Vermeersch, Katre, van Roekeghem, Wang, Madsen, and  Mingo]carrete_almabte_2017
  author author J. Carrete, author B. Vermeersch,
  author A. Katre, author A. van Roekeghem, author
  T. Wang, author G. K. Madsen, and author N. Mingo, https://doi.org/10.1016/j.cpc.2017.06.023 journal journal Computer Physics Communications volume
  220, pages 351 (year 2017)NoStop
[Simoncelli et al.(2019)Simoncelli, Marzari, and Mauri]simoncelli_unified_2019
  author author M. Simoncelli, author N. Marzari, and author F. Mauri, https://doi.org/10.1038/s41567-019-0520-x journal journal Nature Physics volume
  15, pages 809 (year 2019)NoStop
[Simoncelli et al.(2022)Simoncelli, Marzari, and Mauri]simoncelli_wigner_2022
  author author M. Simoncelli, author N. Marzari, and author F. Mauri, https://doi.org/10.1103/PhysRevX.12.041011 journal journal Physical Review X volume 12, pages 041011 (year
  2022)NoStop
[Simoncelli et al.(2023)Simoncelli, Mauri, and Marzari]simoncelli_thermal_2023
  author author M. Simoncelli, author F. Mauri,  and author N. Marzari, https://doi.org/10.1038/s41524-023-01033-4 journal journal npj Computational Materials volume 9,  pages 1 (year 2023)NoStop
[Knoop et al.(2023)Knoop,
  Purcell, Scheffler, and Carbogno]knoop_anharmonicity_2023
  author author F. Knoop, author T. A. Purcell,
  author M. Scheffler, and  author C. Carbogno, https://doi.org/10.1103/PhysRevLett.130.236301 journal
  journal Physical Review Letters volume
  130, pages 236301 (year 2023)NoStop
[Jain et al.(2013)Jain,
  Ong, Hautier, Chen,
  Richards, Dacek, Cholia,
  Gunter, Skinner, Ceder, and  Persson]materialsproject
  author author A. Jain, author S. P. Ong,
  author G. Hautier, author W. Chen, author
  W. D. Richards, author
  S. Dacek, author S. Cholia, author D. Gunter, author D. Skinner, author G. Ceder, and author K. A.  Persson, https://doi.org/10.1063/1.4812323
  journal journal APL Materials volume 1, pages 011002 (year
  2013)NoStop
[Peierls(1929)]peierls_zur_1929
  author author R. Peierls, https://doi.org/10.1002/andp.19293950803 journal journal Annalen der Physik volume 395, pages 1055 (year
  1929)NoStop
[Tamura(1983)]tamura_isotope
  author author S.-i. Tamura, https://doi.org/10.1103/PhysRevB.27.858 journal journal Phys. Rev. B volume
  27, pages 858 (year 1983)NoStop
[Simoncelli et al.(2020)Simoncelli, Marzari, and Cepellotti]PhysRevX.10.011019
  author author M. Simoncelli, author N. Marzari, and author A. Cepellotti, https://doi.org/10.1103/PhysRevX.10.011019
  journal journal Phys. Rev. X volume 10, pages 011019 (year
  2020)NoStop
[Dragašević and Simoncelli(2023)]dragasevic_viscous_2023
  author author J. Dragašević and author M. Simoncelli, http://arxiv.org/abs/2303.12777 journal journal arXiv:2303.12777  (year
  2023)NoStop
[Liu et al.(2023)Liu,
  Liang, Yang, Yang,
  Yang, Song, Mei,
  Csányi, and Cao]liu_unraveling_nodate
  author author Y. Liu, author H. Liang, author L. Yang, author
  G. Yang, author H. Yang, author S. Song, author Z. Mei, author G. Csányi, and author
  B. Cao, https://doi.org/10.1002/adma.202210873 journal journal Advanced Materials volume 35, pages 2210873 (year 2023)NoStop
[Harper et al.(2024)Harper,
  Iwanowski, Witt, Payne, and  Simoncelli]harper_vibrational_2023
  author author A. F. Harper, author K. Iwanowski,
  author W. C. Witt, author M. C. Payne, and author M. Simoncelli, https://doi.org/10.1103/PhysRevMaterials.8.043601 journal
  journal Physical Review Materials volume
  8, pages 043601 (year 2024), note
  publisher: American Physical SocietyNoStop
[Di Lucente et al.(2023)Di Lucente, Simoncelli, and Marzari]di_lucente_crossover_2023
  author author E. Di Lucente, author M. Simoncelli, and author N. Marzari, https://doi.org/10.1103/PhysRevResearch.5.033125
  journal journal Physical Review Research  volume 5, pages 033125 (year
  2023)NoStop
[Jain(2020)]jain_multichannel_2020
  author author A. Jain, https://doi.org/10.1103/PhysRevB.102.201201 journal journal Phys. Rev. B volume
  102, pages 201201 (year 2020)NoStop
[Simoncelli et al.(2024)Simoncelli, Fournier, Marangolo,
  Balan, Béneut, Baptiste,
  Doisneau, Marzari, and Mauri]simoncelli_temperature-invariant_2024
  author author M. Simoncelli, author D. Fournier, author M. Marangolo,
  author E. Balan, author K. Béneut, author
  B. Baptiste, author
  B. Doisneau, author
  N. Marzari, and author
  F. Mauri, https://doi.org/10.48550/arXiv.2405.13161 title
  Temperature-invariant heat conductivity from compensating crystalline and
  glassy transport: from the Steinbach meteorite to furnace bricks
  (year 2024), note arXiv:2405.13161
  [cond-mat]NoStop
[Caldarelli et al.(2022)Caldarelli, Simoncelli, Marzari,
  Mauri, and Benfatto]caldarelli_many-body_2022
  author author G. Caldarelli, author M. Simoncelli, author N. Marzari,
  author F. Mauri, and author L. Benfatto, https://doi.org/10.1103/PhysRevB.106.024312 journal journal Physical Review B volume 106, pages 024312 (year 2022)NoStop
[Fiorentino and Baroni(2023)]fiorentino_green-kubo_2023
  author author A. Fiorentino and author S. Baroni, https://doi.org/10.1103/PhysRevB.107.054311 journal journal Physical Review B volume 107, pages 054311 (year
  2023)NoStop
[Bandi et al.(2024)Bandi,
  Jiang, and Marianetti]bandi_benchmarking_2024
  author author S. Bandi, author C. Jiang, and  author C. A. Marianetti, https://iopscience.iop.org/article/10.1088/2632-2153/ad674a journal journal Machine Learning: Science and Technology 
  (year 2024)NoStop
[Gonze and Lee(1997a)]gonze_dynamical_1997
  author author X. Gonze and author C. Lee,  https://doi.org/10.1103/PhysRevB.55.10355 journal
  journal Physical Review B volume 55,  pages 10355 (year 1997a)NoStop
[Hintze and Nelson(1998)]hintze_violin_plots_1998
  author author J. L. Hintze and author R. D. Nelson, https://doi.org/10.1080/00031305.1998.10480559
  journal journal The American Statistician  volume 52, pages 181 (year
  1998)NoStop
[Ziman(1960)]ziman1960electrons
  author author J. M. Ziman, https://global.oup.com/academic/product/electrons-and-phonons-9780198507796?cc=gb   lang=en   #
  title Electrons and phonons: the theory of transport
  phenomena in solids (publisher Oxford university press,  year 1960)NoStop
[Larsen et al.(2017)Larsen,
  Mortensen, Blomqvist, Castelli, Christensen, Dułak,
  Friis, Groves, Hammer,
  Hargus, Hermes, Jennings,
  Jensen, Kermode, Kitchin,
  Kolsbjerg, Kubal, Kaasbjerg,
  Lysgaard, Maronsson, Maxson,
  Olsen, Pastewka, Peterson,
  Rostgaard, Schiøtz, Schütt,
  Strange, Thygesen, Vegge,
  Vilhelmsen, Walter, Zeng,  and Jacobsen]ase-paper
  author author A. H. Larsen, author J. J. Mortensen, author J. Blomqvist, author I. E. Castelli, author R. Christensen, author M. Dułak,
  author J. Friis, author M. N. Groves, author
  B. Hammer, author C. Hargus, author E. D.  Hermes, author P. C.  Jennings, author P. B.  Jensen, author J. Kermode, author J. R. Kitchin, author E. L. Kolsbjerg, author J. Kubal,
  author K. Kaasbjerg, author S. Lysgaard, author
  J. B. Maronsson, author
  T. Maxson, author T. Olsen, author L. Pastewka, author A. Peterson, author C. Rostgaard, author J. Schiøtz, author O. Schütt, author M. Strange, author K. S.  Thygesen, author T. Vegge, author L. Vilhelmsen,
  author M. Walter, author Z. Zeng, and author
  K. W. Jacobsen, http://stacks.iop.org/0953-8984/29/i=27/a=273002 journal
  journal Journal of Physics: Condensed Matter volume 29, pages 273002 (year
  2017)NoStop
[Togo et al.(2023)Togo,
  Chaput, Tadano, and Tanaka]togo_implementation_2023
  author author A. Togo, author L. Chaput,
  author T. Tadano, and author I. Tanaka, https://doi.org/10.1088/1361-648X/acd831 journal journal Journal of Physics: Condensed Matter volume
  35, pages 353001 (year 2023), note
  publisher: IOP PublishingNoStop
[Wang et al.(2010)Wang,
  Wang, Wang, Mei,
  Shang, Chen, and Liu]Wang_2010
  author author Y. Wang, author J. J. Wang,
  author W. Y. Wang, author Z. G. Mei, author
  S. L. Shang, author
  L. Q. Chen, and author
  Z. K. Liu, https://doi.org/10.1088/0953-8984/22/20/202201 journal
  journal Journal of Physics: Condensed Matter volume 22, pages 202201 (year
  2010)NoStop
[Gonze and Lee(1997b)]Gonze1997
  author author X. Gonze and author C. Lee,  https://doi.org/10.1103/physrevb.55.10355 journal
  journal Physical Review B volume 55,  pages 10355 (year 1997b)NoStop
[Togo(2023)]togo_first-principles_2023
  author author A. Togo, https://doi.org/10.7566/JPSJ.92.012001 journal journal Journal of the Physical Society of Japan  volume 92, pages 012001 (year 2023)NoStop
[Perdew et al.(1996)Perdew,
  Burke, and Ernzerhof]perdew_generalized_1996
  author author J. P. Perdew, author K. Burke, and  author M. Ernzerhof, https://doi.org/10.1103/PhysRevLett.77.3865 journal journal Physical Review Letters volume 77,  pages 3865 (year 1996)NoStop
[Ko et al.(2023)Ko,
  Nassar, Miret, Liu, and  Ong]ko_2023_8025189_matgl
  author author T. W. Ko, author M. Nassar, author S. Miret, author
  E. Liu, and author
  S. P. Ong, https://doi.org/10.5281/zenodo.8025189 title Materials graph
  library (year 2023)NoStop
[Plimpton(1995)]plimpton_lammps_1995
  author author S. Plimpton, https://doi.org/https://doi.org/10.1006/jcph.1995.1039
  journal journal Journal of Computational
  Physics volume 117, pages 1 (year 1995)NoStop




 Paper Title: Dzyaloshinskii-Moriya interaction torques and domain wall dynamics in van der Waals heterostructures.




These authors contributed equally to this work

These authors contributed equally to this work

Email: sdong@seu.edu.cn
Key Laboratory of Quantum Materials and Devices of Ministry of Education, School of Physics, Southeast University, Nanjing 211189, China


§ ABSTRACT 
Since the discovery of two-dimensional ferroelectric and ferromagnetic materials, the van der Waals (vdW) heterostructures constructed by ferroelectric and ferromagnetic monolayers have soon become the ideal platforms to achieve converse magnetoelectric functions at the nanoscale, namely to use electric field to control magnetization. In this Letter, by employing density functional theory calculations and dynamic simulations of atomic spin model, we study the key role of interfacial Dzyaloshinshii-Moriya interaction (DMI) in CrI_3-In_2Se_3 vdW heterostructures. Our work demonstrates feasible DMI torques pumped by ferroelectric switching, which can drive current-free and low-energy consumption domain wall motion. Moreover, such interfacial DMI can also significantly enlarge the Walker field in magnetic field-driven domain wall technique.  


Dzyaloshinskii-Moriya interaction torques and domain wall dynamics in van der Waals heterostructures
    Shuai Dong
    August 5, 2024
====================================================================================================



Originated from spin-orbit coupling (SOC), Dzyaloshinshii-Moriya interaction (DMI) plays a vital role in modern spintronics. By coupling neighbor spins in the antisymmetric form H_D=D· (S_i×S_j)  <cit.>, DMI can lead to many noncollinear spin textures, e.g. canting antiferromagnetism, chiral domain walls, as well as skyrmions  <cit.>. It is well known that noncollinear spin textures usually produce nontrivial spin-electron scattering for transportation, which can contribute to topological (quantum) Hall effects and motion of quasi-particles <cit.>, facilitating functional manipulations at the nanoscale. Because the presence of DMI relies on the lack of space-inversion symmetry, DMI widely exists in those non-centrosymmetric magnetic materials.  In particular, helimagnets such as Mn_xFe_1-xSi, Fe_xCo_1-xSi, and FeGe, in which the DMI causes  spiral orders and/or skyrmions, have attracted lots of attentions <cit.>. Besides, the asymmetric interfaces in artificial heterostructures or superlattices, can also produce the interfacial DMI which leads to various emergent phenomena <cit.>.

Reversibility of DMI vector provides a promising route to control the chirality of magnetism. Multiferroics allows the coexistence of more than one ferroic order within a single phase material, which provide a platform to control DMI vector via electric field or mechanical methods <cit.>. For some specific multiferroic systems, e.g. BiFeO_3, due to the locking between oxygen octahedral distortions and dipole moments, the DMI vector can be reversed by ferroelectric switching, leading to the fipping of canting moment <cit.>. By reversing the DMI vector, the topological spin textures or solitons can be manipulated <cit.>. Following this mechanism, Yu et al demonstrated the chirality-dependent skyrmion-skyrmion interactions <cit.>.  

The magnetic chirality also widely exists in domain walls, and thus DMI also plays an essential role in domain wall dynamics <cit.>. For example, the reversal of DMI vector can produce a dissipative transverse motion during the chirality switching, as demonstrated in perovskite multiferroics  <cit.>. Comparing with the scenario based on current-dependent torques <cit.>, such a pure electric field control has the advantage for its lower energy consumption, which has been coined as a new class of “DMI torques” <cit.>. 





Van der Waals (vdW) heterostructures are ideal platforms to design electronic devices with controllable DMI, benefiting from their high-usage interfaces and easy control of stacking modes. As a widely studied system, α-In_2Se_3 monolayer was theoretically predicted to be ferroelectric  <cit.>, which was then fabricated and confirmed in experiments <cit.>. After that, In_2Se_3-based heterostructures have been theoretically studied and designed for nonvolatile electric field manipulations of various physical properties, including magnetism <cit.>, topological states or spin textures <cit.>, band alignments and charge transfer <cit.>. 
Although domain walls have been attracting more and more attentions in two-dimensional systems <cit.>, the DMI control of domain wall dynamics remains less touched.

In this Letter, the DMI torques and domain wall dynamics of CrI_3-In_2Se_3 vdW heterostructures are studied. Our first-principles calculations indicate the interfacial DMI in CrI_3 monolayer can be induced in these heterostructures and the full reversal of DMI vector can be achieved in the In_2Se_3/CrI_3/In_2Se_3 sandwich-like heterostructure, which can generate DMI torques to drive the motion of domain walls. Furthermore, the existing of interfacial DMI can enlarge the critical field of Walker breakdown in magnetic field-driven domain wall motion. Such magnetoelectric vdW heterostructures shed lights on the low energy-consuming domain wall nanoelectronic applications.

CrI_3 monolayer is a well-recognized two-dimensional ferromagnetic sheet with a magnetic easy axis perpendicular to its plane  <cit.>. The effective spin Hamiltonian for CrI_3's honeycomb lattice can be written as:

    H=-J∑_<i,j>S_i·S_j-∑_<i,j>D_ij·(S_i×S_j)-A∑_i(S_i^z)^2,

where S is the normalized spin vector, i/j are site indexs, and <> denotes nearest-neighbors. As depicted in Fig. <ref>(a). The first term is the ferromagnetic exchange (J>0), and the second term describes the interfacial DMI with the coefficient vector D. Due to the inversion symmetry, D=0  for isolated CrI_3 monolayer. However, the symmetry can be broken from the original D_3d to C_3v for CrI_3 monolayer on substrate, producing nonzero DMI vectors satisfy D_μ=e_z×e_μ  (μ=1, 2, 3) as depicted in Fig. <ref>(a). The last term is the magnetocrystalline anisotropy with the easy axis along the z-axis (A>0). 

Due to the presence of DMI, the magnetic domain walls should belong to the Néel-type with specific chirality (i.e. clockwise (CW) or counter-clockwise (CCW)) to reduce the energy cost from DMI, as sketched in Fig. <ref>(b).  In such a honeycomb lattice, although domain walls are possibly propagating along the zig-zag or armchair directions, i.e. the shadow stripes in Fig. <ref>(a), they can be proved being equivalent in the magnetic dynamics <cit.>. Then in the continuous limit, the model Hamiltonian for Néel-type domain walls propagating along the x-axis can be expressed as:

    H=∫[J/2(∇S)^2-De_y·(S×∇S)-4A/3(S^z)^2]dx,

where D is the amplitude of DMI vector. Therefore, the domain wall can be described as S^x=sech(x/Δ) with the characteristic width Δ=√(3J/8A)  <cit.>. It's worthy to note that the effective width of domain walls in the honeycomb lattice should be πΔ a_0 (a_0 is the Cr-Cr distance as defined in Fig. <ref>(a)).



To obtain the effective coefficients from real heterostructures, the first-principles calculations based on density functional theory (DFT) are performed, and technical details of calculations can be found in Supplementary Materials (SM) <cit.>. Both CrI_3 and In_2Se_3 monolayers are hexagonal lattices, as shown in Fig. <ref>(a-b), and their lattice constants are well-matched by following the (1×1)/(√(3)×√(3)) stacking mode with less than 0.5% mismatch. The CrI_3/In_2Se_3 bilayer and In_2Se_3/CrI_3/In_2Se_3 trilayer are considered. Here, the energetically most favorable stacking modes are displayed in Fig. <ref>(c-f), while other stacking modes are discussed in SM <cit.>. The CrI_3/In_2Se_3 bilayer is polar despite the polarization direction of In_2Se_3. The In_2Se_3/CrI_3/In_2Se_3 trilayer can be polar or nonpolar, depending on the combination of polarizations of two In_2Se_3 layers, as shown in Fig. <ref>(e-f). For bilayer heterostructures CrI_3/↑ and CrI_3/↓ (here ↑ or ↓ denotes the polarization direction in In_2Se_3 layer), the energy is calculated as the function of interlayer distance L. As shown in Fig. <ref>(g), their optimal distances L_↑ and L_↓ are only slightly different. Similar conclusion is also applicable to the trilayer heterostructures, in which the optimal L's slightly depend on the polarization of In_2Se_3 layers.

As aforementioned, the DMI can be induced by the proximity effects at interfaces. Microscopically, DMI can originate from multiple factors, including the lattice distortion, orbital hybridization, electrostatic field, and charge transfer. Here, although the lattice distortions between Cr-I-Cr bonds are rather tiny after optimization, the charge density is significantly different between the upper I and lower I surfaces, due to the electrostatic field from In_2Se_3, as shown in Fig. <ref>(a). To calculate the DMI vector, two noncollinear magnetic orders with identical exchange energy but opposite chiralities as depicted in Fig. <ref>(b), are calculated with SOC. The energy difference between these two states is only contributed by the DMI according to D=(E_CCW-E_CW)/4.  


All calculated coefficients including J, D, and A are summarized in Fig. <ref>(c-d). On one hand, comparing with monolayer CrI_3, the ferromagnetic exchange J is strengthened by 15%-28% in these heterostructures, while the magnetocrystalline anisotropy A is reduced by 0-26%. Thus the ferromagnetism of CrI_3 should remain robust. On the other hand, nonzero D's appear only in those polar heterostructures. For bilayers, the amplitude of DMI's are estimated as 0.16 meV and 0.3 meV for CrI_3/↑ and CrI_3/↓ respectively. Interestingly, the sign of DMI does not change upon the ferroelectric switching, implying that the single interface itself contributes to 0.23 meV while the reversable DMI contributes to 0.07 meV. For trilayer, the DMI is completely reversed from 0.08 meV in ↑/CrI_3/↑ to -0.08 meV in ↓/CrI_3/↓, while it is zero in ↑/CrI_3/↓ or ↓/CrI_3/↑ as expected from the inversion symmetry.

To study the spin dynamics of domain walls in the CrI_3 layer, we employed the atomic simulations by solving Landau-Lifshitz-Gilbert (LLG) equation <cit.>: 

    ∂S/∂ t=γ/μ_s(S×∂ H/∂S)+α(S×∂S/∂ t),

where γ=gμ_ B/ħ is the gyromagnetic ratio. μ_s=3 μ_ B is the atomic magnetic moment for Cr^3+. The last term is for Gilbert damping with coefficient α, which is typically determined by electron spin resonance in experiments. A stripy honeycomb lattice N=600×4 is adopted, with periodic boundary conditions along the y-direction and open boundary conditions along the x-direction. The two x-ends are fixed as spin-up and spin-down. The fourth-order Runge-Kutta method is used to solve the LLG equation to obtain the time-dependent evolution of spins.  



In the trilayer heterostructure, the DMI vector can be reversed by flipping the polarization, i.e. from +0.08 meV to -0.08 meV, which is anticipated to pump the DMI torques for spin dynamics <cit.>. Here, we study an isolated domain wall at the honeycomb lattice, starting from a sharp boundary between the spin-up domain and spin-down domain. As shown in Fig. <ref>(a), due to the DMI (D=0.08 meV) in ↑/CrI_3/↑, the final stable state is a CW Néel-type domain wall. After the polarization flipping to ↓/CrI_3/↓, the DMI energy will produce an effective field acting on the domain wall, and thus pumps a DMI torque as:

    Γ_D=-2Dγ/μ_s(S·e_y)∇S,

which will reverse the chirality of domain wall and induce a transverse motion as well. As shown in Fig. <ref>(b), with a large damping coefficient α=0.2, the domain wall moves by ∼15 nm along the x-direction within 0.8 ns, accompanying its chirality reversal to CCW. This dynamic process with a smaller damping coefficient α=0.1 is also calculated for comparison. As shown in Fig. <ref>(c), the transverse motion distance d reaches ∼30 nm within ∼1.5 ns. The motion of domain wall converges to stop gradually, thus it is not precise to define the motion time and the final distance   d_s. In fact, for the honeycomb lattice, the DMI torque-driven domain wall motion can be well described by:

    d=a_0Δ/αarctan[sinh(t/Q)],

where Q=8Δ(α+α^-1)μ_s/(3πγ D) denotes the characteristic time and the maximal moving distance is d_s=a_0Δπ/α. More details about the derivatives can be found in Supporting Information, and this kind of domain wall motion is similar to the "rolling-downhill"-type motion discovered in multiferroic perovskites <cit.>. All the simulation results are in perfect agreement with the theory, in which the moving distance is proportional to α^-1 and independent on the strength of DMI. Here, we can define the effective time for the motion as ∼2π Q, thus the average velocity can be calculated by v̅=3a_0πγ D/[16(α^2+1)μ_s]. The average velocity v̅ increases with decreasing α and its maximal limitation is v_m=3a_0πγ D/16μ_s when α→0, in consistent with the simulation results as shown in Fig. <ref>(d). For the In_2Se_3/CrI_3/In_2Se_3 heterostructure, the maximal average velocity is estimated as ∼20 m/s, which is close to the typical velocity of magnetic field-driven ferromagnetic domain walls below 50 mT <cit.>. 



For completeness of domain wall motion, we also investigate the magnetic field-driven domain wall motion with different polarizations. Considering a magnetic field h along the z-direction, the Zeeman energy term can be expressed as H=-hμ_s∑_ie_z·S_i, which gives rise to a magnetic field torque as:

    Γ_h=γ he_z×S.

The snapshots of domain wall dynamics in CrI_3/↓ are displayed in Fig. <ref>(a-b), where the magnetic field torque together with the Gilbert damping produce a dissipative torque to shift the domain wall and reduce the Zeeman energy. In this process, there is a critical field, called Walker field h_W <cit.>. Below h_W, the dynamic process is mainly the steady motion, while it becomes a processional motion above h_W (Fig. <ref>(d)). Here, the time-dependent domain wall motions under various magnetic fields are plotted in Fig. <ref>(c-d), which indicates h_W∼152 mT.

It is known that this h_W orignates from the in-plane magnetic anisotropy. However, for CrI_3 monolayer itself, the fully occupied t_ 2g orbitals at Cr ion together with the D_3d point symmetry of honeycomb lattice, preserve the in-plane symmetry and thus the in-plane magnetic anisotropy is almost zero <cit.>, as confirmed in our calculations. Therefore, the magnetic in-plane anisotropy comes from the interfacial DMI. With the coarse-graining approximation, the Walker field can be derived as:

    h_W=3πα D/8Δμ_s,

which agrees with above numerical value very well as shown in Fig. <ref>(d).

Without this interfacial DMI, e.g. in CrI_3 monolayer, the domain wall motion can only be the processional motion, which leads to a velocity v_p=a_0αγΔ h/(1+α^2). In heterostructures, below h_W, the domain wall motion reaches a final steady velocity v_s after a very short speeding duration (∼0.04 ns). This v_s=a_0γΔ h/α is much higher than the v_p. The numerical results of average velocity v̅ are plotted in Fig. <ref>(d) as a function of magnetic field for different heterostructures. Compared with those D=0 cases (CrI_3 monolayer and non-polar ↑/CrI_3/↓), all polar heterostructures own larger h_W and higher v̅, as expected. In particular, below h_W, the velocity of domain wall can be up to ∼140 m/s for CrI_3/↓, which is considerably large and similar to the gate-voltage effect in those magnetic film experiments <cit.>. 

In conclusion, by employing first-principles calculations and atomic spin dynamics simulations, we have studied the mechanism of interfacial DMI and domain wall dynamics in CrI_3-In_2Se_3 heterostructures. We have demonstrated the DMI torques can be pumped into In_2Se_3/CrI_3/In_2Se_3 trilayer heterostructures by polarization flipping, which contributes an efficient domain wall motion without either magnetic field or current. Furthermore, the existing and tunable DMI within heterostructures also plays an important role in improving the critical Walker field for magnetic field-driven domain wall motion, exhibiting the tunable and nonvolatile bias effect. In addition, our results are completely portable to other vdW heterostructures, which can be described by the effective spin model as well. Our results have performed rich manipulations of magnetic domain walls, which can be achieved in magnetoelectric vdW heterostructures.



This work was supported by the National Natural Science Foundation of China (Grant Nos. 12325401 & 12274069), China Scholarship Council, Postgraduate Research & Practice Innovation Program of Jiangsu Province (Grant No. KYCX21_0079), and the Big Data Computing Center of Southeast University. 






 Paper Title: Bimodal Scaling Law and Size Effect In Superelastic Nanopillars.






Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong
Department of Materials Science and Engineering, Sharif University of Technology, Tehran, Iran

xianchen@ust.hk
Department of Mechanical and Aerospace Engineering, Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong

§ ABSTRACT 

Shape memory alloys that can deform and then spring back to their original shape, have found a wide range of applications in the medical field, from heart valves to stents. As we push the boundaries of technology creating smaller, more precise tools for delicate surgery treatments, the behavior of these alloys at tiny scales becomes increasingly crucial. In this study, we discover that the size effect of critical stress required for stress-induced phase transformation is not universal. We propose an orientation-dependent power decay law, indicating a specific increase in critical stress for pillars smaller than 1 micron meter for the nominally soft [001] and hard [111] orientations. Additionally, we observe high transformability with 11% recoverable strain under high stress (2 GPa) through lattice frustration at 200nm scale. This research opens new avenues for exploring the superior elastic behavior of shape memory alloys for nanodevices.



Bimodal Scaling Law and Size Effect In Superelastic Nanopillars
    Xian Chen
    August 5, 2024
===============================================================





Shape memory alloys, characterized by their large deformability under moderate stress and the ability to revert to their original shapes upon stress removal, have demonstrated their success in many biomedical applications such as artificial heart valves, vascular stents and other biocompatible implants <cit.>. As the demand escalates for miniaturized implants tailored for the treatment of neurological diseases and microsurgical tools<cit.>, the superelastic properties of shape memory alloys exhibited at micro-to-nanoscales become increasingly pertinent. The size-dependent functionality and reversibility of these materials are garnering considerable attention in the realm of nanotechnology.

Nanomechanics experiments have empirically demonstrated that the critical stress required for stress-induced transformation in shape memory alloys increases as the sample size diminishes. <cit.> Concurrently, however, the prominence of superelastic strain gradually recedes as dimensions reduce from micron to nano meters.  For instance, in NiTi compression pillars, the superelasticity becomes less distinct below 400 nm, and disappears entirely around 200 nm.<cit.> A similar trend was observed in single crystal Ni_54Fe_19Ga_27 nanopillars with diameters less than 400 nm.<cit.> Furthermore,  many nanocrystalline NiTi-based alloys exhibit a weakened superelastic feature accompanied by a significantly narrowed hysteresis loop during stress-induced phase transformation.<cit.> The size effect in these alloys was typically observed in grains measuring around 20 nm, which is considerably smaller than the size effect seen in single-crystal nanocompression experiments. In some case, small grains under plastic deformation may induce nano-sized amorphous clusters, still showing magnificent recoverable strain upon 2 GPa compressive stress.<cit.>, which makes the NiTi alloy attractive under extreme loading conditions.  Unlike NiTi alloys, the single crystal [001] oriented CuAlNi nanopillars exhibit definitive superelasticity even around 260 nm corresponding to a wide open stress hysteresis under moderate stress-induced phase transformations. <cit.> Superelastic transformability at a high stress has never been reported in CuAl-based shape memory micro/nanopillars. Moreover, the amorphization process and orientation dependent size effect are rarely reported for CuAl-based alloys. Grasping these effects is pivotal in revolutionizing the development of Cu-based polycrystals for small-scale applications.    

For most shape memory alloys, the scaling law associated with the critical stress required for phase transformation has been heuristically related to the plasticity model<cit.> used in non-transforming metals <cit.>. It has been postulated that the critical stress for phase transformation bears a resemblance to the yield strength of ductile metals through a universal power law that scales as σ_c ∝ d^n with the exponent n < 0 for a wide dimension variation from 10 μm to 200 nm <cit.>. Empirically, the exponent of [210] oriented single crystal NiTi nanopillars was reported to be -0.23 <cit.>, however no significant size correlation was observed for [001] and [111] oriented nanopillars. For CuAlNi, the exponent was fitted to be -2 for [001] oriented nanopillars  <cit.>, showing magnificently higher growth of critical stress with size decreasing. It is important to verify whether the scaling law of the critical stress with respect to the micro/nano pillar's size is generally applicable across different crystallographic orientations. Whether the universal power law is valid for the critical stress of shape memory alloys remains an open question.

To address these questions, in this paper, we will study the nanopillars from 5 μm to 185 nm for different orientations by in situ nanomechanical compression experiments. The target alloy is counted as a relatively ductile shape memory alloy in the Cu-based alloy family. During both thermal-driven and stress-induced phase transformations, it forms typical laminated twin martensite from austenite through an elastic transition layer. From microstructural compatibility's point of view <cit.>, this alloy is very similar to equal-atomic NiTi. With the microstructural affinity to the nitinol alloys, this alloy is a good candidate to be explored for high stress-induced transformability.   




The alloy shows substantial orientation dependency of superelasticity under both tensile and compressive stresses <cit.>. Besides, it exhibits a strong elastic anisotropy of austenite lattice. It allows for the possibilities of various deformation mechanisms co-activated during the stress-induced phase transformations. Methods for modulus determination is included in the Supplemental Materials <cit.>. The maximum elastic modulus appears along [111] orientation, while the softest modulus is given by [001] orientation. According to the orientation map of the modulus in Fig. S1 <cit.>, we chose the grains close to the softest (∼ [001] pillar) and hardest (∼ [111] pillar) orientations as illustrated in Fig. <ref>. The uniaxial microcompression tests were conducted on the nominal [001] pillar with modulus 9 GPa and the nominal [111] pillar with modulus 65 GPa. Unlike traditional nanopillar used in the nanocompression experiments, we fabricated a series of zero-tapered cuboidal nanopillars with sizes varying from 5 μm to below 200 nm (Fig. S2 <cit.>). The fabrication process is detailed in Methods <cit.>. In such a way, the true stress-strain curves can be obtained without the influence of misalignment between the loading cell and tested pillars, and the true stress-stain behaviors can be accurately captured, especially for the submicron sized pillars. 



The uniaxial compressive stress-strain curves of softer [001] and stiffer [111] oriented pillars at sizes varying from about 200 nm to 3000 nm are presented in Figs. <ref>a and b. Each pillar was loaded until the stress-induced phase transformation from austenite to martensite was completed, then fully unloaded to recover the reverse transformation. More loading curves are shown in the Supplemental Materials, Fig. S4 and S5 <cit.>.
The superelastic behavior, associated with the increasing critical stress of phase transformation, is observed in both orientations. This behavior is evident from the micron scale down to approximately 500 nm. However, when the size falls below 500 nm, the superelasticity begins to weaken in the case of the stiffer [111] oriented pillars, and the residual strain starts to accumulate.
As the critical stress for the martensitic transformation approaches to the flow stress, slips are observed to occur accompanied by phase transformation along [111] compression, captured in 1060 nm, 790 nm, and 270 nm pillars (Fig. <ref>b). Among them, the superelasticity is measurable from the stress-strain curves, characterized as the open hysteresis loop with a partially recoverable plateau strain. The sides of some cuboidal pillars show clear steps due to dislocation slip, which suggests the co-activation of martensitic transformation and plasticity. 

On the other hand, in the softer [001] orientation, superelasticity is preserved down to 200 nm, with gradually increasing critical stresses. The superelastic transformability and reversibility are captured for all tested sizes despite some residual strains. From the SEM images of the pillars at the maximum load, none of the nanopillars show any sign of plastic deformation. The martensitic transformation still dominates the deformation mechanism for the whole range of sizes.  

A quantitative analysis of the critical stress is presented in Fig.<ref>a, which obviously reveals a bimodal scaling law over the entire size range for both tested orientations. Different scaling laws are separated at the critical size around 1 μm. In the case of the [001] orientation, micron-sized pillars exhibit a critical stress of approximately 200 MPa, which escalates to 500 MPa for nanopillars of about 200 nm. By contrast, the [111] orientation demonstrates a critical stress of around 600 MPa for micron-sized pillars, which progressively increases and ultimately approaches an asymptote at 1.1 GPa. The stiffer [111] orientation presents a significantly higher critical stress than the softer [001] orientation, accompanied by a distinct scaling trend. This underscores a pronounced size effect dependent on orientation.



Within the blow-up region in Fig. <ref>b, the scaling of critical stress fits an orientation dependent power decay model as 

    σ_c = σ_[uvw] - λ_[uvw]d^n

for the cuboidal pillar side length 0< d < 1000 nm with positive parameters σ_[uvw], λ_[uvw] and n > 0. Theoretically, the value of σ_[uvw] refers to the upper bound of the critical stress as the sample size asymptotically reduces to zero. The geometric interpretation of factor λ_[uvw] depends on the exponent. By our experiments, the power decay exponent n=2 for [111] orientation and n=1.05 for [001] orientation. As the critical stress of [111] nanopillars scales quadratically and [001] nanopillars scales linearly, with λ_[111] = 0.00033 as the area scaling factor and λ_[001] = 0.215 as the length scaling factor.

Compared to the yield strength of alloy: 867MPa for the soft orientation and 1128 MPa for the hard orientation<cit.>, it is natural that both martensitic transformation and plasticity can be co-activated in the nanopillars along [111] orientation as σ_[111] = 1010 MPa. When the loading direction is perfectly aligned with the pillar, the martensitic transformation may initiate first. However, it is quickly followed by a slip because the flow stress is sufficiently close to the critical stress required for phase transformation. As a result, we observe slips and martensite twins in some [111] nanopillars (Fig. <ref>b). Additional evidence supporting the combined deformation by slips and martensite transformation can be found in Fig. S4 <cit.>. Even in the case of larger pillars, such as those with a side length of 3 μm, the slip may occur after the phase transformation, as documented in Supplemental Video S1 <cit.>. At the same length scale (i.e. 3μm), the in situ observation in Video S2 <cit.> demonstrates that the deformation mechanism can be primarily governed by martensitic transformation, devoid of slips, which was achieved through meticulous control of loading alignment. When it comes to miniature-sized pillars (270 nm), the pillar exhibited slip prior to the phase transformation at stresses exceeding 1 GPa during loading. This in situ deformation was recorded in Supplemental Video S3 <cit.>.

For the [001] nanopillars, the critical stress has an upper limit of σ_[001] = 550 MPa, a value significantly lower than the flow stress. As a result, the martensitic transformation can be detected even in pillars smaller than 200 nm. These experimental findings elucidate the size-dependent deformation mechanisms observed in various types of shape memory alloys. For instance, the martensitic transformation in CuAlNi [001] nanopillars can be characterized due to the critical stress limit being substantially lower than its flow stress. Conversely, in NiTi nanopillars, the critical stress is inherently much larger. As the size diminishes, the critical stress of NiTi readily approaches the flow stress. Consequently, the pillar may yield prior to phase transformation due to stress localization during the loading process.



The theoretical scaling model in (<ref>) for the soft orientation [001] suggests a big potential for achieving high transformability under high stress. We select several nanopillars around 200 nm to investigate the ultimate elasticity that can be achieved by further increasing the maximum load. Fig <ref>a shows the stress-strain curves of a 220 nm cuboidal pillar under the uniaxial nanocompression in two consecutive loading cycles. During the first cycle, the nanopillar surprisingly withstands up to 2.6 GPa while recovering 9% strain upon unloading. The strength and elastic deformation become more pronounced during the second cycle. The nanopillar recovers 11% strain from 3.1 GPa compressive stress. The reference configuration as well as the deformed configurations after two magnificent stress cycles are shown in Figs. <ref>b-d. Approximately 2% and 4% residual strains were accumulated after the first and second cycles respectively. The high transformability is achieved at a high stress over 2 GPa in 220 nm pillar without showing noticeable superelastic plateau. In contrast, Fig. <ref>e exhibits the normal superelastic behavior captured in a 890 nm cuboidal pillar that forms normal martensite microstructures during loading, as seen in Figs. <ref>f-h.      

The deformation mechanisms, as seen in Supplemental Videos S4 and S5 <cit.> for two other nanopillars measuring 205 nm and 185 nm, are sensitive to both the loading condition and alignment. The 205 nm pillar exhibited a high recoverable strain at a high stress level (approximately 2 GPa), while the 185 nm pillar demonstrated a superelastic plateau strain at around 500 MPa. More loading curves of nanopillars around 200 nm were given in Supplemental Materials, Fig. S5 <cit.>.

To explore the deformation mechanism at the nanoscale for the soft orientation [001], we utilized high-resolution transmission electron microscopy (HRTEM) to examine the local lattice distortion of the post-deformed nanopillars with approximately 200 nm in size. For comparison, we also examined the 890 nm pillar, which exhibits normal superelasticity under moderate stress.



Figs. <ref>a and d provide a low magnification view of the microstructure. The central portion of each pillar was magnified under TEM, as shown in Fig. <ref>b for the 200 nm nanopillar and Fig. <ref>e for the 890 nm pillar. In the 200 nm nanopillar, amorphization was identified in the high-resolution TEM images. In contrast, the selected area electron diffraction in Figure <ref>e indicates that the 890 nm pillar retains its crystalline status. The diffraction pattern can be indexed by the space group Fm3m. 

Fig. <ref>c demonstrates that local lattice frustration leads to the emergence of amorphous clusters within the 200 nm nanopillar. These clusters, which are amorphous in nature, are embedded in the crystalline matrix. The amorphous-crystalline heterogeneous structure has the ability to relax elastically even after significant stress levels. This characteristic is reminiscent of the behavior observed in nanocrystalline nitinol alloys <cit.>. A distinguishing feature here is the initiation of the amorphization process by the single crystal lattice frustration. Notably, the size of this nanopillar significantly exceeds the grain sizes commonly reported in the literature. Such an interaction expands the potential applications of Cu-based shape memory alloys, particularly in the realm of miniaturized devices.

To summarize, we conducted an in situ nanomechanical compression experiment to explore the size effect and scaling law of in two primary orientations: the nominal stiff direction [111] and the nominal soft direction [001]. Our findings revealed that the size effect on the critical stress for the martensitic phase transformation does not satisfy a universal power law from micron to nanometer sizes. An orientation-dependent power decay law suggests a specific increase in critical stress for pillars smaller than 1 micron meter. Most notably, we discovered high deformability with exceptional recovery capability under high stress (2 GPa) along the elastically soft direction [001] when the nanopillar sizes are reduced to 200 nm. This research paves the way for revealing the superior superelastic behavior of shape memory alloys for nanotechnology applications.



M. K and X.C. thank the financial support under GRF Grants No. 16203021, 16204022 and No. 16203023 by Research Grants Council, Hong Kong.






 Paper Title: Phonon-Assisted Auger Decay of Excitons in Doped Transition Metal Dichalcogenide Monolayers.






Institute for Theoretical Physics and Astrophysics and Würzburg-Dresden Cluster of Excellence ct.qmats, University of Würzburg, Am Hubland, 97074 Würzburg, Germany


vasilipe@buffalo.edu
Department of Electrical Engineering, University at Buffalo, Buffalo, NY 14228, USA



§ ABSTRACT 
The competition between the radiative and nonradiative lifetimes determines the optical quantum yield and plays a crucial role in the potential optoelectronic applications of transition metal dichalcogenides (TMDC). Here, we show that, in the presence of free carriers, an additional nonradiative decay channel opens for excitons in TMDC monolayers. Although the usual Auger decay channel is suppressed at low doping levels by the simultaneous momentum and energy conservation laws, exciton-phonon coupling relaxes this suppression. By solving a Bethe-Salpeter Equation, we calculate the phonon-assisted Auger decay rates in four typical TMDCs as a function of doping,  temperature, and dielectric environment.  We find that even for a relatively low doping of 10^12 cm^-2, the nonradiative lifetime ranges from 16 ps to 165 ps in different TMDCs, offering competition to the radiative decay channel.   


Phonon-Assisted Auger Decay of Excitons in Doped Transition Metal Dichalcogenide Monolayers
    Vasili Perebeinos
    August 5, 2024
===========================================================================================






§ INTRODUCTION

The exceptional properties exhibited by two-dimensional (2D) materials have revolutionized the landscape of materials science and nanotechnology.<cit.> The interplay between radiative and nonradiative recombination processes in transition metal dichalcogenides (TMDCs) is fundamental to their optoelectronic properties.<cit.> These two-dimensional semiconductors exhibit strong light-matter interactions,<cit.> making them promising candidates for various photonic and optoelectronic applications<cit.>. The optical quantum yield,<cit.> a critical parameter for such applications, is directly influenced by the competition between the radiative<cit.> and nonradiative<cit.> exciton lifetimes.<cit.>

In TMDC monolayers, excitons, or bound electron-hole pairs, play a crucial role in determining the material's optical properties.<cit.> The radiative recombination of these excitons results in photon emission, which contributes to the material's luminescence. However, nonradiative recombination pathways can significantly impact the overall quantum efficiency of these materials. Therefore, understanding these dynamics is crucial for optimizing TMDC-based devices for various optoelectronic applications.<cit.>

In this work, we demonstrate that the nonradiative exciton phonon-assisted Auger decay channel can effectively compete with the radiative decay even for moderately low doping levels, especially in smaller bandgap TMDCs. The conventional Auger decay mechanism, a nonradiative process often observed in semiconductors,<cit.> is suppressed in TMDC monolayers at low doping concentrations. This suppression is attributed to the simultaneous conservation of momentum and energy in these 2D systems. However, phonon emission or absorption can relax momentum conservation, enabling phonon-assisted Auger decay of excitons available even at low doping levels.<cit.> The Auger decay with phonon assistance has been considered in bulk semiconductors.<cit.> In this work, we extend the phonon-assisted Auger decay mechanism to strongly bound excitons in 2D van der Waals materials. Strong exciton-phonon coupling in TMDCs<cit.> is responsible, for example, for the temperature-dependent width of photoluminescent spectra.<cit.> as well as magneto-optical spectroscopy<cit.> and provides new possibilities for ultrafast intralayer-to-interlayer exciton transitions and interlayer charge transfer.<cit.>



§ METHODS AND MODEL

We use standard notations throughout this paper, where e>0 stands for the elementary charge, ε_0 is the dielectric constant, and k_B is the Boltzmann constant.



 §.§ Excitons

To compute the phonon-assisted Auger decay rate for excitons in TMDCs we first obtain the single-particle spectrum from the model tight-binding Hamiltonian derived in Ref. Fang2015:PRB. Within the tight-binding model, a given single-particle state |nk⟩ with wave vector k in band n and energy ϵ_nk can be written as the Bloch sum

    |nk⟩=1/√(N)∑_ν,R^ık·Ra_nkν|ν,R⟩,

where |ν,R⟩ is the orbital ν centered at the Bravais lattice point R and N is the total number of primitive unit cells considered. The coefficients a_nkν in Eq. (<ref>) are determined from the tight-binding Hamiltonian via

    ∑_ν'ℋ_νν'(k)a_nkν'=ϵ_nka_nkν,

where the 11-band tight-binding Hamiltonian ℋ_νν' as well as its corresponding hopping parameters are taken from Ref. Fang2015:PRB. For simplicity, we use the 11-band model without spin-orbit coupling in the remainder of this work.

The tight-binding states determined from Eqs. (<ref>) and (<ref>) are then used to compute the exciton dispersion via a Bethe-Salpeter equation (BSE). For an exciton S with momentum q, this BSE can be conveniently written as<cit.>

    [Ω_S(q)-ϵ_c(k+q)+ϵ_v(k)] 𝒜^Sq_vck
    
    =∑_v'c'k'[𝒦^d_vck,v'c'k'(q)+𝒦^x_vck,v'c'k'(q)]𝒜^Sq_v'c'k',

where Ω_S(q) denotes the exciton energy and 𝒜^Sq_vck the coefficients of the linear combination forming the excitonic state

    |Ψ_Sq⟩=∑_vck𝒜^Sq_vckĉ^†_ck+qĉ_vk|GS⟩≡X̂^†_Sq|GS⟩.

Here ĉ^†_ck (ĉ_vk) is the creation (annihilation) operator of an electron with momentum k in a conduction band c (valence band v) and |GS⟩ is the ground state with fully occupied valence bands and unoccupied conduction bands, while X̂^†_Sq is the creation operator of an exciton in state S with momentum q. Equation (<ref>) contains the direct and exchange terms

    𝒦^d_vck,v'c'k'(q)=-∑_νν'a^*_ck+qνa_c'k'+qνa_vkν'a^*_v'k'ν'
    ×[1/N∑_R^-ı(k-k')·R W(|R|)]

and

    𝒦^x_vck,v'c'k'(q)=∑_νν'a^*_ck+qνa_vkνa_c'k'+qν'a^*_v'k'ν'
    ×[1/N∑_R^-ıq·R V(|R|)],

respectively. Equations (<ref>) and (<ref>) have been derived assuming point-like orbitals. The direct and exchange terms are computed in the real space from the screened interaction and the bare Coulomb potentials W(|R|) and V(|R|), respectively. If we do not account for doping in the BSE, W(|R|)=V(|R|). In 2D materials this potential can be well modeled by the Keldysh potential<cit.>

    W(|R|)=e^2/8ε_0 r_0[H_0(ε|R|/r_0)-Y_0(ε|R|/r_0)],

where H_0 is the Struve function, Y_0 the Bessel function of the second kind, ε the average dielectric background constant and r_0 the monolayer polarizability parameter.





 §.§ Phonon-assisted Auger Decay Rate

Having obtained Ω_S(q) and 𝒜^Sq_vck from Eq. (<ref>), we are now in a position to compute the phonon-assisted Auger scattering rate.

In the second-order processes we consider (see Fig. <ref>), the initial state is given by a zero-momentum exciton and an electron with momentum k' in the lowest conduction band c_0. By emitting a phonon with momentum -q [see Fig. <ref>(a)] or absorbing a phonon with momentum q [see Fig. <ref>(b)], the exciton acquires finite momentum q. This intermediate state, in turn, collapses to the final state by Auger decay of the exciton to an electron-hole pair. The energy conservation then requires Ω_S(0)+ϵ_c_0k'=ħω_-q+ϵ_c'k'+q for phonon emission and Ω_S(0)+ϵ_c_0k'+ħω_q=ϵ_c'k'+q for phonon absorption, where ħω_±q is the phonon dispersion and ϵ_ck the electron dispersion in conduction band c. During the transition to the final state, we allow for excitations to an arbitrary conduction band c' (not only within the lowest conduction band c_0).


We model the interaction mediating the collapse from the initial state to the final state by the Hamiltonian Ĥ=Ĥ_e-ph+Ĥ_e-e with the exciton-phonon interaction

    Ĥ_e-ph=g/√(N)∑_SkqX̂^†_Sk+qX̂_Sk(b̂^†_-q+b̂_q)

and the electron-electron interaction

    Ĥ_e-e=1/N∑_cc'vkk'qU^c',c_0,c,v_k'+q,k',k+q,kĉ^†_c'k'+qĉ_c_0k'ĉ_ck+qĉ^†_vk.

Equation (<ref>) contains the exciton creation (annihilation) operator X̂^†_Sk (X̂_Sk) defined in Eq. (<ref>). In our model the exciton-phonon coupling strength in Eq. (<ref>) is given by a constant g, while the Auger Coulomb matrix element in Eq. (<ref>) is given by

    U^c',c_0,c,v_k'+q,k',k+q,k=∑_νν'a^*_c'k'+qνa_ck+qνa_c_0k'ν'a^*_vkν'
    ×[1/N∑_R^-ı(k'-k)·R V(|R|)].

Here Eq. (<ref>) has been calculated employing the same approximations as those used to arrive at Eqs. (<ref>) and (<ref>).



Using second order perturbation theory and Fermi's golden rule, we can determine the Auger scattering rate as

    ħ/τ=2π g^2/N^3      ∑_c',k',S,q|∑_cvk𝒜^Sq_vckU^c',c_0,c,v_k'+q,k',k+q,k|^2/[Ω_S(q)-Ω_S(0)+ħω_-q]^2
          × f(ϵ_c_0k')[1-f(ϵ_c'k'+q)][1+n_-q]
          ×δ[ϵ_c'k'+q-ϵ_c_0k'+ħω_-q-Ω_S(0)]
    
    +2π g^2/N^3      ∑_c',k',S,q|∑_cvk𝒜^Sq_vckU^c',c_0,c,v_k'+q,k',k+q,k|^2/[Ω_S(q)-Ω_S(0)-ħω_q]^2
          × f(ϵ_c_0k')[1-f(ϵ_c'k'+q)]n_q
          ×δ[ϵ_c'k'+q-ϵ_c_0k'-ħω_q-Ω_S(0)],

where the first and second terms are the contributions due to phonon emission and absorption, respectively, while

    f(ϵ)=1/exp[(ϵ-μ)/k_BT]+1;  n_q=1/exp(ħω_q/k_BT)-1

denote the Fermi-Dirac and Bose-Einstein distribution functions with temperature T and chemical potential μ. In order to obtain Eq. (<ref>), we have assumed that only the lowest conduction band c_0 s occupied by excess electrons[Strictly speaking, occupied conduction bands also lead to additional screening of W(|R|), resulting in W(|R|)≠ V(|R|). For simplicity, we neglect this screening in W(|R|). Additional screening due to doping leads to smaller exciton binding energies as well as to a band gap renormalization, which mostly compensates for the reduction in binding energy. Since we only consider relatively small doping densities, we assume that these effects will not significantly affect the exciton wave function 𝒜^Sq_vck.] and that phonon-assisted transitions can occur only between this band and any other conduction band.





 §.§ Exciton-Phonon Coupling Strength

To estimate the strength of the exciton-phonon coupling, we use the experimentally measured full-width fall-maximum γ from temperature-dependent photoluminescence spectra (shown in Fig. <ref>) according to Fermi's Golden rule:

    γ(T)=γ_0+2π g^2 DOS_exc n_ph,

where γ_0 is a temperature-independent broadening due to defects and sample inhomogeneities and the temeprature dependence arises from the phonon population n_ph according to Eq. (<ref>). The exciton density of states is evaluated according to DOS_exc=m A/(2πħ^2), where A is the area of the primitive unit cell and the exciton mass m_exc=m_c+m_v is given by the sum of conduction and valance band effective masses from Table <ref>. The values of g from the best fits are given in Table <ref>. Note that we only consider phonon absorption in Eq. (<ref>), since there are no states to scatter to as a result of the phonon emission for the lowest-energy excitons and we do not differentiate between acoustic and optical phonon contributions to the full-width half maximum by using a fixed value of the phonon energy of ħω_ph=20 meV for all TMDCs considered in Eqs. (<ref>) and (<ref>).





§ RESULTS

Equations (<ref>)-(<ref>) can then be used to compute the phonon-assisted Auger decay rate. Here, we present the results for four different TMDCs, namely MoS_2, MoSe_2, WS_2 and WSe_2. As shown in Fig. <ref>, for all four TMDCs the scattering rates for the process considered in this work increase with doping densities and correspond to lifetimes in the sub ns regime at densities up to 10^13 cm^-2.



In systems with strong dielectric screening, on the other hand, the phonon-assisted Auger scattering rate is suppressed, as illustrated in Fig. <ref>: The stronger the dielectric screening, the smaller the Auger scattering rate. In the limit of an infinitely large dielectric constant, the phonon-assisted exciton Auger decay rate vanishes as shown in Fig. <ref> containing data computed from Eq. (<ref>) along with an empirical fit. 

The temperature dependence follows from the probabilities of phonon emission and absorption, i.e. 1+2n_ph. The overall rate depends sensitively on the bandgap of the TMDCs, that is, the smaller the bandgap, the stronger the phonon-assisted exciton Auger decay rate. To probe the proposed phonon-assisted exciton Auger decay mechanism, we want the experimental sample to be free of defects to limit other nonradiative defect-assisted decay channels. The gate can supply the free carrier in a TMDC in a controllable way, such that the measured phonon-assisted exciton Auger decay rate could be analyzed according to the following empirical formula: 

    1/τ=1+2n_ph/τ_0ρ/ρ_0ϵ_0+1/ϵ+1,

where τ_0 is the phonon-assisted exciton Auger decay lifetime measured at typical experimentally relevant conditions, that is, ρ_0=10^12 cm^-2, ϵ_0=3, and T=4 K. The value of τ_0 falls in the range of 16 ps to 165 ps depending on the bandgap of the TMDC material. 



§ CONCLUSIONS

We have demonstrated that the phonon-assisted exciton Auger decay mechanism can effectively compete with the radiative decay channel in defect-free samples, even for moderately low doping levels. The predicted values of the phonon-assisted Auger lifetime for excitons under typical experimental conditions are predicted to be in the range of 16 ps to 165 ps depending on the bandgap of the TMDC material. We believe that our work will stimulate further experimental work.  


We gratefully acknowledge the support by the National Science Foundation under Grant No. 2230727 and the computational facilities at the Center for Computational Research at the University at Buffalo (<http://hdl.handle.net/10477/79221>).






 Paper Title: Brittleness of metallic glasses dictated by their state at the fragile-to-strong transition temperature.







achraf.atila@uni-saarland.de



Department of Material Science and Engineering, Saarland University, Saarbrücken, 66123, Germany



§ ABSTRACT  
The effect of cooling on the brittleness of glasses in general, and bulk metallic glasses (BMGs) in particular, is usually studied with continuously varying cooling rates; slower cooling rates lead to stiffer, harder, and more brittle glasses than higher cooling rates. These protocols obscure any potential discontinuity that a glass might experience depending on whether its microstructure resembles that of a fragile or a strong glass-forming liquid. Here, we use large-scale molecular dynamics to simulate the nanoindentation behavior of model BMGs (Zr_0.6Cu_0.3Al_0.1) obtained by rapidly quenching equilibrium melts from temperatures above and below the fragile-to-strong transition temperature T_fst, leading to fragile and strong glasses, respectively. 
While the contact modulus deduced from the indentation simulation evolves smoothly with the temperature T_q from which the melt is quenched, the plastic response changes quasi-discontinuously as T_q passes through T_fst. In particular, strong glasses develop highly asymmetric flow profiles with mature shear bands, in contrast to fragile glasses. Quantitative differences reveal themselves not only through a formal von Mises localization parameter analysis but also through image analysis of flow patterns using pre-trained artificial intelligence models. Moreover, seemingly erratic flow profiles for our indentation geometry produced surprisingly reproducible and, thus, deterministic features.
It remains to be determined to what extent other classes of glass formers follow our observation that the degree of brittleness is significantly influenced by whether the melt is fragile or strong when it falls out of equilibrium at the glass transition temperature.




Brittleness of metallic glasses dictated by their state at the fragile-to-strong transition temperature
    Martin H. Müser
    August 5, 2024
=======================================================================================================





§ INTRODUCTION

Bulk metallic glasses (BMGs) have elastic strain limits of up to 5% <cit.> and high tensile <cit.> as well as yield <cit.> strengths of order 1 GPa exceeding that of crystalline and poly-crystalline metals by an order of magnitude.
At the same time, BMGs have defied the rule that materials cannot be simultaneously strong and tough <cit.>.
In fact, BMGs are known to have the highest known damage tolerance <cit.>, which is the product of fracture yield strength and toughness. 
On the downside, BMGs generally strain to soften, making them prone to localized deformations in the form of shear bands <cit.>, whereby undesirable surface markings can be produced, which ultimately evolve into cracks. This contrasts conventional metals, which usually strain hardens as the dislocations are the main carriers of plasticity <cit.>. This behavior limits the moldability of most BMGs and their widespread technological applicability.

One subject of ongoing research is how the thermal history of a BMG affects its atomic structure and, thereby, its deformation mechanisms, most notably shear bands <cit.>. 
It is established both experimentally <cit.> and from simulations <cit.> that (bulk-metallic) glasses produced with a small cooling rate are stronger but also more prone to shear banding than those quenched more quickly. 
In contrast, silica-based oxide glasses are always brittle.
A distinguishing feature of silica-based glasses is the ratio of the fragile-to-strong transition (FST) temperature T_fst and the glass transition temperature T_g, i.e., typical values of  T_fst/T_g clearly exceed unity for silicates but are of order unity for BMGs. 
This prompts us to hypothesize that the brittleness of a glass is significantly influenced by the degree to which its atomic structure resembles that of a strong or fragile melt.
In particular, the question arises whether the quasi-discontinuity of both dynamic and static properties of the melt at the FST translates into discontinuities in the mechanical properties of the corresponding glass.

A few notes on the FST, which has been argued to occur in any glass-forming melt <cit.>, and the glass transition itself may be in place. 
The (equilibrium) viscosity and atomic diffusion coefficients cross over from a non-Arrhenius to an Arrhenius dependence upon cooling in a rather narrow temperature window near T_fst, which is accompanied by a breakdown of the Stokes-Einstein relation below T_fst <cit.>.
The specific heat of melts c_p at T>T_fst is substantially above that associated with the rule of Dulong-Petit.
This means that structural rearrangements go beyond (quasi-) harmonic vibrations so that the melt is called (thermodynamically) fragile.
After passing through a maximum near T_fst upon cooling, c_p quickly decreases to values satisfying Dulong-Petit.
This is indicative of vibrations around barely changing reference structures.
The melt is called strong.
The potentially most localized abrupt change at the FST is the jump of the correlation length describing the asymptotic decay of density oscillation <cit.>.
In contrast to the FST, the glass transition refers to a more continuous process in which a melt falls out of equilibrium during cooling, however, with a strong dependence on the cooling rate.
Similar to the FST, the glass transition leads to a reduction of the specific heat toward the rule of Dulong-Petit. 

Ascertaining to what degree FST discontinuities translate into corresponding features in the glass requires equilibrium melts to be quenched instantly from both above and below T_fst to well below T_g in the first step.
This way, the glass represents the frozen-in structure of a fragile and strong melt, respectively, and hence will be called either a fragile glass or a strong glass in the following. 
Whenever T_g and T_fst are not clearly separated, conventional cooling protocols using constant cooling rates plus potentially holding periods below T_g generally produce a mix of fragile and strong glass, since the melt would not be fully equilibrated at T > T_fst, while some low-temperature relaxations would occur at T < T_fst.
The lower the cooling rate q̇, the more the structure leans toward a strong glass, but changes are continuous in q̇.
Thus, previous studies <cit.> revealing that the failure mechanisms evolves smoothly from being homogeneous in a high-q̇ sample to localized shear bands using low q̇ do not contradict our hypothesis that the brittleness of a (bulk metallic) glass is dictated by its state at the FST.

In the remainder of this work, we use molecular dynamics (MD) simulations to contrast the mechanical behavior of strong and fragile glasses during nanoindentation.
To this end we recycle previously produced atomistic equilibrium configurations of a (in-silico model for a) ternary BMG, Zr_0.6Cu_0.3Al_0.1, which had been shown to undergo a FST in thermal equilibrium <cit.>. 
MD is an ideal tool for this study because equilibrium melts can be quenched extremely quickly and homogeneously, preventing temperature-gradient-induced heterogeneities or residual stresses from occurring.



§ METHODS




 §.§ Molecular dynamics

In this work, we simulate the ternary alloy Zr_0.6Cu_0.3Al_0.1 using an embedded-atom potential <cit.> parameterized by Cheng, Ma, and Sheng <cit.>.
This system revealed a transition from a fragile to a strong liquid at T_fst≈830 <cit.>, at which the correlation length of the long-range density fluctuations changed quasi-discontinuously by roughly 10% in a very narrow temperature range similarly as in a related experimental BMG called Vitleroy 106a <cit.>.

Initial configurations were taken from the above-mentioned previous work <cit.>, in which a sample containing ≈ 8800 atoms was equilibrated at various temperatures, including a narrow temperature regime below T_fst.
The full equilibration times used during step-wise cooling were:
30 ns at T = 1,000 K, 64 ns at 900 K, 400 ns at 850 K, 1.25 μs at 812 K, 2.5 μs at 800 K, and 4.0 μs at 787 K. 
In the following, samples obtained by quenching an equilibrium sample from T < T_fst will be referred to as strong glasses, while those obtained from equilibrium simulations at T>T_fst will be called fragile glasses.
All samples were duplicated n_x=3, n_y=3, n_z=3 times in x, y, and z-direction, respectively, equilibrated for 4 at the temperature from which they were obtained, and subsequently quenched to room temperature (T = 300 with cooling rate of 1/, and annealed for another 4. During these simulations, a timestep of 2 was used with periodic boundary conditions applied to all box directions.
Such equilibrated samples were replicated again to obtain a larger sample with 230 x 16.5 x 86 nm^3, used in the nanoindentation simulations. Using these large samples (≈ 18 Mio atoms), a surface was created along the z-axis, and a 1 nm thick layer of atoms at the bottom of the simulation box along the z-direction was fixed. The samples were periodic along the x- and y-direction. Next, the samples were further equilibrated for 200 at room temperature and 0 MPa using a timestep of 2. The temperature was maintained using a Langevin thermostat <cit.> while the pressure was controlled using Nosé-Hoover chain barostat <cit.>. All simulations were conducted using the LAMMPS simulation package <cit.> and the snapshots were prepared using OVITO <cit.>. 


The indentation was performed with a rigid cylindrical indenter exerting a force of magnitude F(r) = -K(r - R)^2 on each atom, where K is the force constant which we chose to be K = 10 eV/Å^3, r is the distance from the center of the indenter to the respective atom, and R is the radius of the indenter which was fixed to R = 100. Lilleodden et al. <cit.> studied the response of single-crystalline gold to nanoindentation while varying the force constant of the indenter and found that the force-displacement curves are unaffected by the change of K.
Displacement-controlled nanoindentation simulations were performed by displacing the indenter along the z-direction towards the sample with a fixed velocity of vi = 5/ until reaching a depth of 20 and using a timestep of 2. 




 §.§ Local strain analysis

The detection of shear bands was performed by using the von Mises shear strain. The von Mises strain was calculated by getting an affine transformation matrix to the relative neighbors through a least-square fitting <cit.>. The undeformed glass was chosen as a reference configuration. More details of the strain calculation can be found in Refs. <cit.>. 
To get the regions with a nonaffine displacement, the current atomic configuration needs to be compared with a reference configuration. The neighbors of each particle are found by defining a sampling radius (i.e., 10 Å in our case). The local strain is then calculated from the displacements of the neighboring particles to the central one and the relative displacements that they would have if they were in a region of uniform strain. The final value of the nonaffine displacement was divided by the number of atoms within that spherical region of the same radius. More details about the calculation of the nonaffine displacements are given in Ref. <cit.>.




 §.§ Image analysis


Snapshots of the local von Mises shear strain taken at different depths during the nanoindentation process were image analyzed.
To this end, pre-trained DINOv2 neural networks <cit.> were used for feature extraction because DINOv2 is known for its robustness in capturing image features through self-supervised learning.
The DINOv2 architecture is based on the so-called vision transformer (ViT) <cit.>, which is currently the de-facto standard for computer vision.
In this work, we mainly use the bare backbones ViT-b/14 with registers <cit.> and ViT-g/14 without registers available at GitHub (<https://github.com/facebookresearch/dinov2>), because they showed the best results for our analysis.
The models had been pre-trained on 142 million images in a contrastive learning setting, which allows the model to capture robust image features, although images supposedly did not show shear bands or flow lines. 

Each image was preprocessed to meet the input requirements of the DINOv2 model.
The preprocessing steps included resizing the image to 224×224 pixels, converting it to a PyTorch <cit.> tensor, and applying the DINOv2-specific preprocessing function.
The latter includes normalizing the images by the mean and standard deviation values of ImageNet <cit.> statistics: mean = [0.485, 0.456, 0.406] and standard deviation = [0.229, 0.224, 0.225].
After putting all images through the model, the cosine similarity of each pair of images was computed.
It ranges from -1 (anti-parallel, minimum similarity) to  1 (parallel, maximum similarity).



§ RESULTS


In this work, we simulate the indentation process of glasses obtained from quenching equilibrium melts from six temperatures T_q to T = 300 K, where all indentation simulations were conducted.
Three of the initial temperatures, that is, T_1 = 787 K, T_2 = 800 K, and T_3 = 812 K were below the fragile-to-strong cross-over temperature T_fst = 830 K, the remaining ones T_4 = 850 K, T_5 = 900 K, and T_6 = 1,000 K were above  T_fst.
To be concise, the sample quenched from T_1 may not have fully reached equilibrium at T_1.
To be specific, the relaxation time of energy fluctuations in the α-relaxation regime can be estimated to be τ = 10 ns and the Kohlrausch exponent to β = 0.73, which yields a 1-exp(-(t_sim/τ)^β) = 0.995 complete α relaxation given an simulation time of t_sim = 100 ns.
For the volume calculation, the relaxation is even more complete.
In this case, τ = 1 ns and β = 0.79 yielding 1-exp(-(t_sim/τ)^β) = 1.

At first glance, all six samples show qualitatively similar force-displacement curves  as revealed in Fig. <ref>:
an initial Hertzian regime up to an indentation of d ≈ 2 nm, followed by a non-elastic regime.
A maximum indentation force of F ≈ 7 ± 0.5 μN is reached at d ≈ 10 ± 1 nm for all  T_q≤ 900 K samples. 
After a short drop, F starts increasing again at d ≈ 12 ± 2 nm. 



Closer inspection of Fig. <ref> reveals qualitative differences between fragile and strong glasses.
All strong samples have similar F_N(d) relations at d < d_max, where d_max is the depth at which the force is a local maximum.
In contrast, F_N(d) of fragile samples reveal some T_q dependence at d < d_max.
The situation reverts when d exceeds d_max by a few nanometers.
This time, F_N(d) converges to a single dependence for all fragile glasses but reveals a weak T_q dependence for the strong glasses.
Moreover, the drop in the indentation force after reaching its maximum is noticeably more pronounced for strong than for fragile glasses. 



Subtle qualitative differences between the fragile and the strong glasses can already be noticed at a relatively small indentation depth.
The strong glass starts to deviate from their linear response by assuming a left-curved F_N(d) relation, while the fragile glasses adopt a minor right-curved F_N(d) dependence. 
In contrast, the contact modulus E^*(T_q) has a smooth dependence on T_q without detectable signs of a discontinuity of the function itself or its derivative, as is revealed in Fig. <ref>
These findings align with the existing literature, which also reports more significant effects of cooling rate on plastic than on elastic properties <cit.>.





Despite subtle differences between fragile and strong plasticity at the early loading stages, effects are more pronounced at large indentation.
To analyze plasticity, we investigated the local von Mises local shear strain, as described in Sec. <ref>.
Exemplary results are shown in Fig. <ref> with representatives corresponding to the three different plasticity regimes of the F(d) curve, i.e., before, near, and after the maximum indentation force.
Fig. <ref> depicts more snapshots of the von Mises strain, i.e., for all six samples at seven depths, including one snapshot taken in the elastic regime. 
We also note that visualization of the shear bands using non-affine displacement squared, defined similarly as by Falk and Langer <cit.> but normalized to the number of neighbors, produced snapshots, which closely resembled those based on the von Mises strain, as is evidenced in Fig. <ref>. 


Shear bands, which are the main carrier of plasticity in metallic glasses <cit.> start to occur once F(d) deviates from the elastic response.
They mature, i.e., the shear bands reach a stage where their growth slows down or stops after F(d) reaches its maximum, at which point they have hit the surface.  
Contrary to previous simulations <cit.> and our fragile glasses, the strong samples develop visible, radially shaped shear bands with an average width of w ≈ 8 nm.
This is on par with w ≈10 nm <cit.>,  which is the intrinsic thickness of mature shear bands in ex-silico metallic glasses.
The shear bands of strong glasses are more localized and more asymmetric than in fragile glasses, to the extent that they could be said to differ qualitatively. 

Going beyond a visual inspection of the shear bands requires a quantitative analysis to be conducted. 
To this end, we study the standard deviation of the atomic von Mises strain divided by the average atomic von Mises strain,  Ψ = ⟨Δη_n^2 ⟩^1/2 / ⟨η_n ⟩, where η_n is the von Mises strain projected to atom n,   ⟨ ...⟩ an average over all atoms, and Δη_n = η_n - ⟨η_n ⟩.
Ψ is similar to a previously defined localization parameter <cit.> but normalized so that it does not change when atoms with zero von Mises strain are added. 
Like the previous parameter, Ψ is sensitive to localization because atoms with large variance tend to cluster in shear bands. 
Results for Ψ are shown in Fig. <ref>.
At very small d, the value for Ψ is essentially identical for all samples.
This can be understood from the fact that elastic deformation is scale-invariant and, thereby, the value for Ψ at d = 0.5 nm is merely 2% 
between the most fragile and the strongest sample compared to a 25% difference in E^*.
 


The values of Ψ(d) bifurcate at a depth of d ≳ 4 nm, where Ψ(d) keeps increasing for the strong samples while they saturate otherwise.
An even stronger qualitative difference of strong and fragile samples becomes apparent at d ≳ 9 nm, where the slope of Ψ(d) suddenly increases near the point of the local F_N(d) minima for the strong samples, while Ψ(d) has no such discontinuity in the fragile samples. 


From the results shown so far, a qualitative difference in the deformation behavior of glasses reflecting the structure of a fragile and a strong melt has become obvious.
To ensure that these differences are not merely a kinetic effect resulting from the ratio of the indentation rate and the relaxation time at T_q surpassing some critical value, we performed another indentation simulation of the T_q = 850 K sample, however, using a ten-fold increased indentation velocity.
This exceeds the ratio of relaxations times between the “hottest strong” and the “coldest fragile” melt, for both energy and volume relaxation. 

The ten-fold increased indentation velocity produced a noticeable increase in the F_N(d) relationship.
However, the higher indentation velocity produces noticeably more symmetric and less localized shear bands or rather flow patterns, as is revealed in Fig. <ref> both visually but also in terms of the strain-localization parameter.
The higher symmetry of the flow images at the higher velocity could be argued to arise from the reduced time for an initial local symmetry breaking to migrate from small scales to large scales.
In our simulations, the symmetry breaking arises due to structural heterogeneities, potentially also, to some degree, by thermal fluctuations.
In case of instability, which is unavoidable in strain-softening materials, numerical round-off errors, even in a perfect symmetric finite-element simulation, might suffice to cause similar asymmetries in the shear bands as in the MD simulations when the loading is slow. 





To explore the degree to which our findings can be tested experimentally if images of flow profiles are available but there is no spatially resolved von Mises strain, we quantified the similarity between the shear-band signature using two different models, as described in the Methods section. 
The result produced with ViT_g/14 is shown in Fig. <ref>.
At each indentation depth, inspection of both colors and numbers included in the figure generally gives the largest difference in similarity between the hottest strong and the coolest fragile samples. 
This claim holds for both models.
Moreover, the ViT_g/14 judges the similarity between the upper left and upper right panel in Fig. <ref> to be more than that between the top right and the bottom right, i.e., 0.94 and 0.88 cosine similarity, respectively. 
This is an interesting finding given that the covered colored area is similarly large for the two samples indented to 15 nm.

Figuratively speaking, the human eye can clearly identify the top left panel as the child and the upper right as its adult, rugose version, while the upper and lower right panels represent two different adults. 



Since neither the color maps nor the numbers included in Fig. <ref> allow us to conclude whether the two models reveal discontinuities between fragile and strong glasses, a normalized similarity S_n(T_q) of a sample quenched from arbitrary T_q and indented to d = 15 nm with that quenched from T_q = 1,000 K is shown in Fig. <ref>.
It is defined as the normalized cosine similarity between the sample quenched from T_q = 1000 K and all other samples at an indentation depth of 15 nm, as defined by

    S_n(T) = S^1000K_cos(T)- min(S^1000K_cos(T_q))/1- min(S^1000K_cos(T_q)),

where S_cos^T_1(T_2) is the cosine similarity of the flow profiles of quench temperature T_1 and T_2.
This measure makes the two studied models operate on a similar domain, because it is normalized such that the similarity between the least similar and the T_q = 1000 K sample is zero.





Although the similarity indices show a clustering for the strong samples and a jump between strong and fragile samples, the judgment appears less clear than that by humans or the von Mises localization parameter. 
It is nevertheless remarkable that both models produce similar results,  although the ViT-b/14 has less than 8% of the parameters compared to ViT-g/14.



A final question to be addressed here is to what extent the shape of the flow profiles and shear bands, particularly those obtained for the strong glasses, is erratic or deterministic. 
All strong samples develop a long shear band to the right with spider-web-like features between the shear band and the surface, plus a shorter but thicker shear band to the left, which is very close to the surface.
The probability for breaking the symmetry when quenching from three independent samples each time in the same direction is 1/4, which is small compared to unity but not very small so that it is unclear whether some hidden memory in the strong melt causes the symmetry to be broken each time in the same direction. 
We remind the reader that all final configurations share the same ancestor, e.g., the sample obtained after equilibrating the melt at T = 812 K was quenched to yield the T_q = 812 K glass at 300 K, but also copied and further equilibrated at 800 K for times much exceeding the energy and volume relaxation time. 

Since equilibration and indentation simulations require several million CPU hours, averaging over independently produced samples is impractical.
To nevertheless ascertain how deterministic features are, we repeated the simulations for a larger cell, in which the quenched samples were not only repeated 14 × 1 × 5 times in x, y, and z-direction respectively but 15 × 1 × 6 times (≈ 21.3 Mio).
Moreover, the original simulation cell kept its orientation one time but was rotated by 90^∘ so that the original yz surfaces became the new to-be-indented xy surface.
Von Mises strains of these three different T_q = 812 K samples are contrasted in Fig. <ref>.
The two new configurations show the same characteristics as the old one, i.e., one prominent long shear band, which breaks the symmetry and has the spider-web feature underneath it. 
Thus, not only the gross features of the shear profile appear to be predetermined. 




§ DISCUSSION

The FST is conjectured to exist in all liquids <cit.>. MGs prepared experimentally are quenched with cooling rates that are fast enough to avoid crystallization and, at the same time, slow enough to allow for this FTS transition <cit.>. However, the glasses prepared in MD simulations are usually cooled with cooling rates several decades faster than those used in experiments <cit.>, preventing them from experiencing the FTS liquid transition <cit.>. 
The results of this work show that this renders a one-to-one comparison between simulation and experiments problematic. 
However, suppose glasses are prepared in a way that makes them undergo FTS liquid transition. In that case, the differences between the simulated and experimentally prepared glasses will reduce, and a one-to-one comparison of the mechanisms will be much trusted. 
This was shown to be the case recently when authors combined MD and Monte Carlo swaps to equilibrate the system and produce stable glasses with properties closer to the experimental ones <cit.>.

Regarding the mechanical behavior of MGs, Packard and Schuh <cit.> showed that for isotropic materials such as metallic glasses, the formation of narrow shear bands in the area with concentrated shear stresses that exceed the yield stress. Nevertheless, from our results presented above, the first shear transformation zones in the bulk sample appeared at an indentation depth of about 2.5 nm and ≈ 15 nm under the indenter, but not all of them are exactly along the axis of indentation, suggesting that the local shear stresses do not uniquely determine the STZ formation. Similar observations were made by Wang et al. <cit.>. This can be explained by the fact that the local variation of the atomic configuration results in BMG having different zones with low or high shear stresses distributed in the sample and with STZ preferring to nucleate at the sites with low shear resistance <cit.>, which dictate the location where STZ form and also the propagation of the shear bands afterward. This is in line with the observations we presented, where asymmetric shear bands are observed in the strong glass, although the cylindrical indenter is symmetric. In addition to the strain field applied by the indenter, the shear band will naturally propagate towards the surface because it provides no constraints on their propagation <cit.>.

The differences between strong and fragile glasses can be explained by the degree of stability of the glasses and the concept of the potential energy landscape (PEL) <cit.>. The strong glass was cooled using a much lower effective cooling rate ≈ 10^7 K/s, giving the atoms more time to relax and move towards deeper energy minima. On the other hand, the fragile glasses were cooled from temperatures above the FTS transition temperature, where the samples did not explore deeper energy minima. Thus, those glasses are less stable than the strong ones. This implies that, when comparing the mechanical properties and the mechanical behavior of the strong and the fragile glasses, the strong glass will have larger mechanical properties and be more brittle, which is consistent with experimental and other simulation results that show that well-relaxed glasses have a rougher PEL, which results in lower atomic mobility, that makes the glasses more brittle and has a higher strength than the less stable glasses <cit.>. 
Although their focus was not related to the FST, experimentally, it has been proven that the processing history of the glass dictates the yielding and the shear band dynamics in the sample <cit.>. Shi and Falk <cit.> made closely similar conclusions using binary Lennard-Jones glasses, where they observed a stronger localization of the local shear strain in the samples cooled slowly. 

The presence of ≈ 8 nm thick shear bands in the strong glass, which are well compared to those found experimentally <cit.>, is reported here for the first time. The radial shape of the shear bands is consistent with the slip-lines that develop during an indentation using a cylindrical indenter and are also observed experimentally <cit.>. Previous MD simulations <cit.> showed the presence of radial shear bands as well, but with a much lower thickness. These interesting differences in the deformation behavior of fragile and strong glasses shed light on the overall deformation mechanisms of MGs, where both localized and homogeneous deformation were observed for the same glass composition.




§ CONCLUSIONS

In this paper, we used large-scale molecular dynamics simulations to explore how the fragility of the melt affects the brittleness of metallic glasses. We showed that melts quenched from just below and just above the fragile-to-strong transition temperature show qualitatively different plasticity. 
The strong glasses exhibit asymmetric and localized shear bands. In contrast, the fragile glasses form broader and less defined shear bands. These differences between the strong and the fragile glasses were quantitatively observed using a modified strain localization parameter as well as by using pre-trained artificial intelligence models for image similarities. The distinctness between the deformation behavior and plastic flow profile of the fragile and strong glasses arises because strong glasses can relax into deeper energy minima, resulting in more stable atomic configurations enhancing their mechanical strength.

These findings highlight a potentially overlooked link between the thermodynamic state of the liquid at the FST and the deformation behavior of the resulting glass. 
This link can be expected to generalize to other classes of glass-forming liquids than BMGs.
In fact, recent work suggested that the propensity for ductility is controlled by the topography of the energy landscape based on the simulation of various glass-forming systems <cit.>.
It changes from being relatively smooth above T_fst (high c_p) to deep quasi-harmonic minima below T_fst (c_p in accordance with Dulong-Petit).
However, it remains to be determined to what extent the brittleness of other glass-forming systems changes abruptly depending on whether they are cooled from above or below T_fst.


The authors gratefully acknowledge the Gauss Centre for Supercomputing e.V. (www.gauss-centre.eu) for funding this project by providing computing time through the John von Neumann Institute for Computing (NIC) on the GCS Supercomputer JUWELS at Jülich Supercomputing Centre (JSC).


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[Glushko et al.(2024)Glushko, Pippan, Şopu, Mitterer, and Eckert]Glushko2024NC
  author author O. Glushko, author R. Pippan,
  author D. Şopu, author C. Mitterer, and author J. Eckert, title
  title How to catch a shear band and explain plasticity of
  metallic glasses with continuum mechanics, journal journal Nature Communications volume 15, https://doi.org/10.1038/s41467-024-49829-2 10.1038/s41467-024-49829-2
  (year 2024)NoStop
[Inoue et al.(2003)Inoue,
  Shen, Koshiba, Kato, and  Yavari]Inoue2003
  author author A. Inoue, author B. Shen,
  author H. Koshiba, author H. Kato, and author
  A. R. Yavari, title
  title Cobalt-based bulk glassy alloy with ultrahigh strength and
  soft magnetic properties, https://doi.org/10.1038/nmat982
  journal journal Nature Materials volume 2, pages 661 (year
  2003)NoStop
[Ritchie(2011)]Ritchie2011
  author author R. O. Ritchie, title title The conflicts between
  strength and toughness, https://doi.org/10.1038/nmat3115
  journal journal Nature Materials volume 10, pages 817 (year
  2011)NoStop
[Şopu(2023)]sopu2023JAC
  author author D. Şopu, title title STZ-vortex model:
  The key to understand STZ percolation and shear banding in metallic
  glasses, https://doi.org/10.1016/j.jallcom.2023.170585 journal journal Journal of Alloys and Compounds volume 960, pages 170585 (year
  2023)NoStop
[Mu et al.(2021)Mu,
  Chellali, Boltynjuk, Gunderov, Valiev, Hahn, Kübel, Ivanisenko, and Velasco]Mu2021ADM
  author author X. Mu, author M. R. Chellali,
  author E. Boltynjuk, author D. Gunderov, author
  R. Z. Valiev, author
  H. Hahn, author C. Kübel, author Y. Ivanisenko, and author L. Velasco, title title
  Unveiling the local atomic arrangements in the shear band regions of
  metallic glass, https://doi.org/10.1002/adma.202007267 journal journal Advanced Materials volume 33, pages 2007267 (year
  2021)NoStop
[Liu et al.(2013a)Liu, Yang, and Liu]Liu2013Acta
  author author Z. Liu, author Y. Yang, and  author C. Liu, title title Yielding and shear banding of metallic glasses,  https://doi.org/10.1016/j.actamat.2013.06.025 journal
  journal Acta Materialia volume 61,  pages 5928 (year 2013a)NoStop
[Xie et al.(2023a)Xie, Chauraud, Atila, Bitzek, Korte-Kerzel, and Guénolé]Xie2023SM
  author author Z. Xie, author D. Chauraud,
  author A. Atila, author E. Bitzek, author
  S. Korte-Kerzel, and author
  J. Guénolé, title title Thermally activated nature of synchro-shockley dislocations in laves
  phases, https://doi.org/https://doi.org/10.1016/j.scriptamat.2023.115588 journal journal Scripta Materialia volume 235, pages 115588 (year
  2023a)NoStop
[Xie et al.(2023b)Xie, Chauraud, Atila, Bitzek, Korte-Kerzel, and Guénolé]Xie2023PRM
  author author Z. Xie, author D. Chauraud,
  author A. Atila, author E. Bitzek, author
  S. Korte-Kerzel, and author
  J. Guénolé, title
  title Unveiling the mechanisms of motion of synchro-shockley
  dislocations in laves phases, https://doi.org/10.1103/PhysRevMaterials.7.053605 journal
  journal Phys. Rev. Mater. volume 7,  pages 053605 (year 2023b)NoStop
[Şopu et al.(2020)Şopu,
  Scudino, Bian, Gammer, and  Eckert]opu2020SM
  author author D. Şopu, author S. Scudino,
  author X. Bian, author
  C. Gammer, and author
  J. Eckert, title title Atomic-scale origin of shear band multiplication in heterogeneous
  metallic glasses, https://doi.org/10.1016/j.scriptamat.2019.11.006
  journal journal Scripta Materialia volume 178, pages 57–61 (year
  2020)NoStop
[Walley(2007)]Walley2007MMTA
  author author S. Walley, title title Shear localization: A
  historical overview, https://doi.org/10.1007/s11661-007-9271-x
  journal journal Metallurgical and Materials
  Transactions A volume 38, pages
  2629–2654 (year 2007)NoStop
[Yang et al.(2022)Yang,
  Zhang, Zhang, Tang, and  Chen]Yang2022JNC
  author author H. Yang, author L. Zhang,
  author J. Zhang, author H. Tang, and author
  S. Chen, title title Effect of sample size and cooling rate on the plastic deformation
  behavior of bulk metallic glasses: A comparative study, https://doi.org/10.1016/j.jnoncrysol.2022.121643 journal
  journal Journal of Non-Crystalline Solids volume 589, pages 121643 (year
  2022)NoStop
[Shi and Falk(2007a)]Shi2007acta
  author author Y. Shi and author M. L. Falk,  title title Stress-induced structural
  transformation and shear banding during simulated nanoindentation of a
  metallic glass, https://doi.org/10.1016/j.actamat.2007.03.029
  journal journal Acta Materialia volume 55, pages 4317 (year
  2007a)NoStop
[Zhang et al.(2022)Zhang,
  Ding, and Ma]Zhang2022PNAS
  author author Z. Zhang, author J. Ding, and  author E. Ma, title title Shear transformations in metallic glasses without
  excessive and predefinable defects, journal journal Proceedings of the National Academy of Sciences volume 119, https://doi.org/10.1073/pnas.2213941119
  10.1073/pnas.2213941119 (year 2022)NoStop
[Angell(1991)]Angell1991NOC
  author author C. Angell, title title Relaxation in liquids,
  polymers and plastic crystals — strong/fragile patterns and problems,  https://doi.org/10.1016/0022-3093(91)90266-9 journal
  journal Journal of Non-Crystalline Solids volume 131–133, pages 13–31 (year
  1991)NoStop
[Angell(1995)]Angell1995
  author author C. A. Angell, title title Formation of glasses from
  liquids and biopolymers, https://doi.org/10.1126/science.267.5206.1924 journal
  journal Science volume 267, pages 1924 (year 1995)NoStop
[Angell et al.(2000)Angell,
  Ngai, McKenna, McMillan,  and Martin]Angell2000LAP
  author author C. A. Angell, author K. L. Ngai,
  author G. B. McKenna, author P. F. McMillan, and author S. W. Martin, title
  title Relaxation in glassforming liquids and amorphous solids,  https://doi.org/10.1063/1.1286035 journal journal Journal of Applied Physics volume 88,  pages 3113–3157 (year 2000)NoStop
[Stolpe et al.(2016)Stolpe,
  Jonas, Wei, Evenson,
  Hembree, Yang, Meyer, and  Busch]Stolpe2016PRB
  author author M. Stolpe, author I. Jonas,
  author S. Wei, author
  Z. Evenson, author W. Hembree, author F. Yang, author A. Meyer, and author R. Busch,  title title Structural changes during a
  liquid-liquid transition in the deeply undercooled
  Zr_58.5Cu_15.6Ni_12.8Al_10.3Nb_2.8
  bulk metallic glass forming melt, https://doi.org/10.1103/PhysRevB.93.014201 journal journal Phys. Rev. B volume 93, pages 014201 (year 2016)NoStop
[Sukhomlinov and Müser(2018)]Sukhomlinov2018PRM
  author author S. V. Sukhomlinov and author M. H. Müser, title title Quasidiscontinuous change
  of the density correlation length at the fragile-to-strong transition in a
  bulk-metallic-glass forming melt, https://doi.org/10.1103/PhysRevMaterials.2.115604 journal
  journal Phys. Rev. Mater. volume 2,  pages 115604 (year 2018)NoStop
[Ghaemi et al.(2022)Ghaemi,
  Jafary-Zadeh, Khoo, and Gao]Ghaemi2022JNCS
  author author M. Ghaemi, author M. Jafary-Zadeh, author K. H. Khoo, and author H. Gao,  title title Chemical affinity can govern notch-tip
  brittle-to-ductile transition in metallic glasses, https://doi.org/10.1016/j.eml.2022.101651 journal journal Extreme Mechanics Letters volume 52,  pages 101651 (year 2022)NoStop
[Jafary-Zadeh et al.(2017)Jafary-Zadeh, Tavakoli, Koh, Aitken, and Zhang]JafaryZadeh2017
  author author M. Jafary-Zadeh, author R. Tavakoli, author J. Koh,
  author Z. Aitken, and author Y.-W. Zhang, title
  title Effect of chemical composition and affinity on the short-
  and medium-range order structures and mechanical properties of Zr-Ni-Al
  metallic glass, https://doi.org/10.1016/j.jnoncrysol.2016.10.042
  journal journal Journal of Non-Crystalline
  Solids volume 456, pages 68 (year 2017)NoStop
[Daw and Baskes(1984)]Daw1984PRB
  author author M. S. Daw and author M. I. Baskes, title title Embedded-atom method:
  Derivation and application to impurities,surfaces,and other defects in
  metals, https://doi.org/10.1103/physrevb.29.6443 journal journal Phys. Rev. B volume
  29, pages 6443 (year 1984)NoStop
[Cheng et al.(2009a)Cheng, Ma, and  Sheng]Cheng2009PRL
  author author Y. Q. Cheng, author E. Ma, and  author H. W. Sheng, title title Atomic level structure in multicomponent bulk
  metallic glass, https://doi.org/10.1103/PhysRevLett.102.245501
  journal journal Phys. Rev. Lett. volume 102, pages 245501 (year
  2009a)NoStop
[Schneider and Stoll(1978)]Schneider1978PRB
  author author T. Schneider and author E. Stoll, title title Molecular-dynamics study of
  a three-dimensional one-component model for distortive phase transitions,  https://doi.org/10.1103/physrevb.17.1302 journal
  journal Physical Review B volume 17,  pages 1302 (year 1978)NoStop
[Shinoda et al.(2004)Shinoda, Shiga, and Mikami]Shinoda2004PRB
  author author W. Shinoda, author M. Shiga,  and author M. Mikami,  title title Rapid estimation of elastic constants
  by molecular dynamics simulation under constant stress, https://doi.org/10.1103/PhysRevB.69.134103 journal journal Phys. Rev. B volume 69, pages 134103 (year 2004)NoStop
[Thompson et al.(2022)Thompson, Aktulga, Berger, Bolintineanu, Brown, Crozier, in 't Veld, Kohlmeyer, Moore,
  Nguyen, Shan, Stevens,
  Tranchida, Trott, and Plimpton]Thompson2022CPC
  author author A. P. Thompson, author H. M. Aktulga, author R. Berger,
  author D. S. Bolintineanu,
  author W. M. Brown, author P. S. Crozier, author
  P. J. in 't Veld, author
  A. Kohlmeyer, author
  S. G. Moore, author
  T. D. Nguyen, author
  R. Shan, author M. J. Stevens, author J. Tranchida, author C. Trott, and author S. J.  Plimpton, title title Lammps - a
  flexible simulation tool for particle-based materials modeling at the atomic,
  meso, and continuum scales, https://doi.org/https://doi.org/10.1016/j.cpc.2021.108171 journal journal Computer Physics Communications volume 271, pages 108171 (year
  2022)NoStop
[Stukowski(2010)]Stukowski2009MSMSE
  author author A. Stukowski, title title Visualization and
  analysis of atomistic simulation data with OVITO-the Open Visualization
  Tool, journal journal Modeling and Simulation
  in Materials Science and Engineering volume 18,  https://doi.org/10.1088/0965-0393/18/1/015012
  10.1088/0965-0393/18/1/015012 (year 2010)NoStop
[Lilleodden et al.(2003)Lilleodden, Zimmerman, Foiles, and  Nix]Lilleodden2003LMPS
  author author E. Lilleodden, author J. Zimmerman, author S. Foiles,  and author W. Nix, title title Atomistic simulations of elastic deformation and
  dislocation nucleation during nanoindentation, https://doi.org/10.1016/s0022-5096(02)00119-9 journal
  journal Journal of the Mechanics and Physics of Solids  volume 51, pages 901 (year
  2003)NoStop
[Shimizu et al.(2007)Shimizu, Ogata, and Li]Shimizu2007MTJ
  author author F. Shimizu, author S. Ogata,  and author J. Li, title title Theory of shear banding in metallic glasses and
  molecular dynamics calculations, https://doi.org/10.2320/matertrans.mj200769 journal journal Materials Transactions volume 48,  pages 2923 (year 2007)NoStop
[Falk and Langer(1998)]Falk1998
  author author M. L. Falk and author J. S. Langer, title title Dynamics of viscoplastic
  deformation in amorphous solids, https://doi.org/10.1103/physreve.57.7192 journal journal Physical Review E volume 57, pages 7192 (year 1998)NoStop
[Oquab et al.(2024)Oquab,
  Darcet, Moutakanni, Vo,
  Szafraniec, Khalidov, Fernandez, HAZIZA, Massa, El-Nouby, Assran, Ballas, Galuba, Howes, Huang, Li,
  Misra, Rabbat, Sharma,
  Synnaeve, Xu, Jegou,
  Mairal, Labatut, Joulin,  and Bojanowski]oquab2023dinov2
  author author M. Oquab, author T. Darcet,
  author T. Moutakanni, author H. V. Vo, author
  M. Szafraniec, author
  V. Khalidov, author
  P. Fernandez, author
  D. HAZIZA, author F. Massa, author A. El-Nouby, author M. Assran, author N. Ballas, author W. Galuba, author R. Howes, author P.-Y.  Huang, author S.-W. Li, author I. Misra, author M. Rabbat, author
  V. Sharma, author G. Synnaeve, author H. Xu, author H. Jegou, author J. Mairal,
  author P. Labatut, author A. Joulin, and author
  P. Bojanowski, title
  title DINOv2: Learning robust visual features without
  supervision, https://openreview.net/forum?id=a68SUt6zFt journal journal Transactions on Machine Learning Research   (year 2024)NoStop
[Dosovitskiy et al.(2021)Dosovitskiy, Beyer, Kolesnikov,
  Weissenborn, Zhai, Unterthiner, Dehghani, Minderer,
  Heigold, Gelly, Uszkoreit,  and Houlsby]dosovitskiy2021imageworth16x16words
  author author A. Dosovitskiy, author L. Beyer,
  author A. Kolesnikov, author D. Weissenborn, author
  X. Zhai, author T. Unterthiner, author M. Dehghani, author M. Minderer, author G. Heigold, author S. Gelly, author J. Uszkoreit, and author N. Houlsby, title title An
  image is worth 16x16 words: Transformers for image recognition at scale,  in https://openreview.net/forum?id=YicbFdNTTy booktitle International Conference on Learning Representations  (year 2021)NoStop
[Darcet et al.(2024)Darcet,
  Oquab, Mairal, and Bojanowski]darcet2024visiontransformersneedregisters
  author author T. Darcet, author M. Oquab,
  author J. Mairal, and author P. Bojanowski, https://arxiv.org/abs/2309.16588 title Vision transformers
  need registers (year 2024), https://arxiv.org/abs/2309.16588 arXiv:2309.16588 [cs.CV] NoStop
[Paszke et al.(2019)Paszke,
  Gross, Massa, Lerer,
  Bradbury, Chanan, Killeen,
  Lin, Gimelshein, Antiga,
  Desmaison, Köpf, Yang,
  DeVito, Raison, Tejani,
  Chilamkurthy, Steiner, Fang,
  Bai, and Chintala]paszke2019pytorchimperativestylehighperformance
  author author A. Paszke, author S. Gross,
  author F. Massa, author A. Lerer, author
  J. Bradbury, author
  G. Chanan, author T. Killeen, author Z. Lin, author N. Gimelshein, author L. Antiga,
  author A. Desmaison, author A. Köpf, author
  E. Yang, author Z. DeVito, author M. Raison, author A. Tejani, author S. Chilamkurthy, author B. Steiner, author L. Fang, author J. Bai, and author S. Chintala, https://arxiv.org/abs/1912.01703 title
  Pytorch: An imperative style, high-performance deep learning library
  (year 2019), https://arxiv.org/abs/1912.01703
  arXiv:1912.01703 [cs.LG] NoStop
[Russakovsky et al.(2015)Russakovsky, Deng, Su, Krause, Satheesh, Ma, Huang, Karpathy, Khosla, Bernstein, Berg, and Fei-Fei]russakovsky2015imagenetlargescalevisual
  author author O. Russakovsky, author J. Deng,
  author H. Su, author
  J. Krause, author S. Satheesh, author S. Ma, author Z. Huang, author A. Karpathy,
  author A. Khosla, author M. Bernstein, author
  A. C. Berg, and author
  L. Fei-Fei, https://arxiv.org/abs/1409.0575 title Imagenet large scale
  visual recognition challenge (year 2015), https://arxiv.org/abs/1409.0575 arXiv:1409.0575 [cs.CV] NoStop
[Hassani et al.(2019)Hassani, Lagogianni, and Varnik]Hassani2019PRL
  author author M. Hassani, author A. E. Lagogianni, and author F. Varnik, title title Probing the degree of
  heterogeneity within a shear band of a model glass, https://doi.org/10.1103/PhysRevLett.123.195502 journal
  journal Phys. Rev. Lett. volume 123,  pages 195502 (year 2019)NoStop
[Greer et al.(2013)Greer,
  Cheng, and Ma]Greer2013MSER
  author author A. Greer, author Y. Cheng, and  author E. Ma, title title Shear bands in metallic glasses, https://doi.org/10.1016/j.mser.2013.04.001 journal journal Materials Science and Engineering: R: Reports volume 74, pages 71 (year 2013)NoStop
[Yang et al.(2016)Yang,
  Luo, Huang, Hu, Vargheese, Shi, and Mauro]Yang2016Acta
  author author Y. Yang, author J. Luo, author L. Huang, author
  G. Hu, author K. D. Vargheese, author Y. Shi, and author J. C.  Mauro, title title Crack
  initiation in metallic glasses under nanoindentation, https://doi.org/10.1016/j.actamat.2016.06.001 journal
  journal Acta Materialia volume 115,  pages 413 (year 2016)NoStop
[Adjaoud and Albe(2021)]Adjaoud2021
  author author O. Adjaoud and author K. Albe,  title title Nanoindentation of nanoglasses tested
  by molecular dynamics simulations: Influence of structural relaxation and
  chemical segregation on the mechanical response, journal
  journal Frontiers in Materials volume
  8, https://doi.org/10.3389/fmats.2021.664220
  10.3389/fmats.2021.664220 (year 2021)NoStop
[Zhao et al.(2021)Zhao,
  Zhu, Wang, Niu, Xu, and Zhao]Zhao2021CMS
  author author D. Zhao, author B. Zhu, author S. Wang, author
  Y. Niu, author L. Xu, and author H. Zhao, title title Effects of
  pre-strain on the nanoindentation behaviors of metallic glass studied by
  molecular dynamics simulations, https://doi.org/10.1016/j.commatsci.2020.110073 journal
  journal Computational Materials Science volume 186, pages 110073 (year
  2021)NoStop
[Sheng et al.(2022)Sheng,
  Şopu, Fellner, Eckert, and  Gammer]Sheng2022PRL
  author author H. Sheng, author D. Şopu,
  author S. Fellner, author J. Eckert, and author
  C. Gammer, title title Mapping shear bands in metallic glasses: From atomic structure to
  bulk dynamics, journal journal Physical Review
  Letters volume 128, https://doi.org/10.1103/physrevlett.128.245501
  10.1103/physrevlett.128.245501 (year 2022)NoStop
[Zhang and Greer(2006)]Zhang2006APL
  author author Y. Zhang and author A. L. Greer, title title Thickness of shear bands
  in metallic glasses, https://doi.org/10.1063/1.2336598 journal journal Applied Physics Letters volume 89, pages 071907 (year
  2006)NoStop
[Cheng et al.(2009b)Cheng, Cao, and  Ma]Cheng2009Acta
  author author Y. Cheng, author A. Cao, and  author E. Ma, title title Correlation between the elastic modulus and the
  intrinsic plastic behavior of metallic glasses: The roles of atomic
  configuration and alloy composition, https://doi.org/10.1016/j.actamat.2009.03.027 journal
  journal Acta Materialia volume 57,  pages 3253 (year 2009b)NoStop
[Ito et al.(1999)Ito,
  Moynihan, and Angell]Ito1999Nature
  author author K. Ito, author C. T. Moynihan,  and author C. A. Angell,  title title Thermodynamic determination of
  fragility in liquids and a fragile-to-strong liquid transition in water,  https://doi.org/10.1038/19042 journal journal Nature volume 398, pages
  492 (year 1999)NoStop
[Lucas(2019)]Lucas2019NOCX
  author author P. Lucas, title title Fragile-to-strong
  transitions in glass forming liquids, https://doi.org/10.1016/j.nocx.2019.100034 journal journal Journal of Non-Crystalline Solids: X volume
  4, pages 100034 (year 2019)NoStop
[Way et al.(2007)Way,
  Wadhwa, and Busch]Way2007Acta
  author author C. Way, author P. Wadhwa, and  author R. Busch, title title The influence of shear rate and temperature on
  the viscosity and fragility of the
  Zr_41.2Ti_13.8Cu_12.5Ni_10.0Be_22.5
  metallic-glass-forming liquid, https://doi.org/10.1016/j.actamat.2006.12.032 journal
  journal Acta Materialia volume 55,  pages 2977 (year 2007)NoStop
[Atila et al.(2020)Atila,
  Kbirou, Ouaskit, and Hasnaoui]Atila2020me
  author author A. Atila, author M. Kbirou,
  author S. Ouaskit, and author A. Hasnaoui, title
  title On the presence of nanoscale heterogeneity in
  Al_70Ni_15Co_15 metallic glass under
  pressure, https://doi.org/10.1016/j.jnoncrysol.2020.120381
  journal journal Journal of Non-Crystalline
  Solids volume 550, pages 120381
  (year 2020)NoStop
[Atila et al.(2019)Atila,
  Ghardi, Ouaskit, and Hasnaoui]Atila2019PRB
  author author A. Atila, author E. M. Ghardi,
  author S. Ouaskit, and author A. Hasnaoui, title
  title Atomistic insights into the impact of charge balancing
  cations on the structure and properties of aluminosilicate glasses,  journal journal Physical Review B volume 100, https://doi.org/10.1103/physrevb.100.144109 10.1103/physrevb.100.144109
  (year 2019)NoStop
[Atila et al.(2022)Atila,
  Ouldhnini, Ouaskit, and Hasnaoui]Atila2022PRB
  author author A. Atila, author Y. Ouldhnini,
  author S. Ouaskit, and author A. Hasnaoui, title
  title Atomistic insights into the mixed-alkali effect in
  phosphosilicate glasses, journal journal
  Physical Review B volume 105, https://doi.org/10.1103/physrevb.105.134101 10.1103/physrevb.105.134101
  (year 2022)NoStop
[Packard and Schuh(2007)]Packard2007acta
  author author C. Packard and author C. Schuh, title title Initiation of shear bands
  near a stress concentration in metallic glass, https://doi.org/10.1016/j.actamat.2007.05.054 journal
  journal Acta Materialia volume 55,  pages 5348 (year 2007)NoStop
[Wang et al.(2010)Wang,
  Hodgson, Zhang, Yan, and  Yang]Wang2010CMS
  author author J. Wang, author P. D. Hodgson,
  author J. Zhang, author W. Yan, and author
  C. Yang, title title Effects of pores on shear bands in metallic glasses: A molecular
  dynamics study, https://doi.org/10.1016/j.commatsci.2010.08.001
  journal journal Computational Materials Science  volume 50, pages 211 (year
  2010)NoStop
[Kosiba et al.(2020)Kosiba,
  Scudino, Bednarcik, Bian,
  Liu, Kühn, and Pauly]Kosiba2020PRB
  author author K. Kosiba, author S. Scudino,
  author J. Bednarcik, author J. Bian, author
  G. Liu, author U. Kühn, and author S. Pauly, title title Guiding
  shear bands in bulk metallic glasses using stress fields: A perspective from
  the activation of flow units, https://doi.org/10.1103/PhysRevB.102.134113 journal journal Phys. Rev. B volume 102, pages 134113 (year 2020)NoStop
[Tang et al.(2021)Tang,
  Liu, Ma, Du, Mousseau, Zhou, and Bauchy]Tang2021MH
  author author L. Tang, author H. Liu, author G. Ma, author
  T. Du, author N. Mousseau, author W. Zhou, and author M. Bauchy, title title The
  energy landscape governs ductility in disordered materials, https://doi.org/10.1039/d0mh00980f journal journal Materials Horizons volume 8, pages 1242 (year 2021)NoStop
[Liu et al.(2013b)Liu, Yang, and Liu]Liu2013ActaMat
  author author Z. Liu, author Y. Yang, and  author C. Liu, title title Yielding and shear banding of metallic glasses,  https://doi.org/10.1016/j.actamat.2013.06.025 journal
  journal Acta Materialia volume 61,  pages 5928 (year 2013b)NoStop
[Shi and Falk(2007b)]Shi2007TSF
  author author Y. Shi and author M. L. Falk,  title title Simulations of nanoindentation in a
  thin amorphous metal film, https://doi.org/10.1016/j.tsf.2006.01.032 journal journal Thin Solid Films volume 515, pages 3179 (year 2007b)NoStop
[Antoniou et al.(2007)Antoniou, Bastawros, and Biner]Antoniou2007JMR
  author author A. Antoniou, author A. Bastawros, and author B. Biner, title title Experimental observations
  of deformation behavior of bulk metallic glasses during wedge-like
  cylindrical indentation, https://doi.org/10.1557/jmr.2007.0061
  journal journal Journal of Materials Research  volume 22, pages 514 (year
  2007)NoStop
[Antoniou et al.(2005)Antoniou, Bastawros, Lo, and  Biner]Antoniou2005MSEA
  author author A. Antoniou, author A. Bastawros,
  author C. Lo, and author S. Biner, title
  title Deformation behavior of a zirconium based metallic glass
  during cylindrical indentation: in situ observations, https://doi.org/10.1016/j.msea.2004.11.031 journal journal Materials Science and Engineering: A volume
  394, pages 96 (year 2005)NoStop
[Su and Anand(2006)]SU2006Acta
  author author C. Su and author L. Anand,  title title Plane strain indentation of a zr-based
  metallic glass: Experiments and numerical simulation, https://doi.org/10.1016/j.actamat.2005.08.040 journal
  journal Acta Materialia volume 54,  pages 179 (year 2006)NoStop








Supplementary material to:
Brittleness of metallic glasses dictated by their state at the fragile-to-strong transition temperature






 









 Paper Title: Energy-lowering symmetry breaking creates a flat-band insulator in paramagnetic Nb3Cl8.





Renewable and Sustainable Energy Institute, University of Colorado, Boulder, Colorado 80309, USA

Renewable and Sustainable Energy Institute, University of Colorado, Boulder, Colorado 80309, USA

alex.zunger@colorado.edu
Renewable and Sustainable Energy Institute, University of Colorado, Boulder, Colorado 80309, USA



§ ABSTRACT 
Ordinary band structure calculations of quantum materials often incorrectly predicted metallic, instead of insulating electronic structure, motivating Mott-Hubbard strong electron correlation as a gapping mechanism. More recently, allowing the formation of local structural symmetry breaking motifs in otherwise ordinary band theory was shown to lower the total energy while predicting insulating gaps when they are observed. An important counter example was recently pointed out whereby the flat band formed in Nb_3Cl_8 by symmetry breaking is by itself partially occupied, thus failing to account for the observed insulating state. It is shown here that a generalized symmetry breaking involving the cooperative structural and magnetic effects produces in mean-field-like density functional theory an energy lowering insulating phase.


Energy-lowering symmetry breaking creates a flat-band insulator in paramagnetic Nb_3Cl_8
    Alex Zunger
    August 5, 2024
========================================================================================



Gap anomaly in Nb_3Cl_8: A range of quantum materials manifest local microscopic degrees of freedom nested within the average crystallographic structure. Such local motifs are experimentally observable by local structural probes [1,2] and calculable quantum mechanically as energy-lowering symmetry breaking motifs [3,4]. Symmetry breaking can split off narrow flat bands [5–8] from a broad conduction band (as in LaTiO_3 and VO_2), or from the broad valence band (as in LaFeO_3 and SrBiO_3), leading in a single step to the observed insulating phases in these quantum oxides. This offers a link between energy gap formation and the local structural motifs  traditional in solid-state chemistry, such as Peierls instabilities, Jahn-Teller distortions, or bond disproportionation, both in d-electron (such as LaMnO_3 [9] and NbO_2 [10]) or in non-d-electron compounds (such as BaBiO_3 [11]). It was recently pointed out [12,13] that α-phase Nb_3Cl_8 is special among other quantum materials in that symmetry breaking (here, Nb-Nb-Nb trimer formation) creates a partially occupied band in the ground state, hence the system remains metallic whereas the observed α-phase [12,13] is insulating. This discrepancy opens the possibility that α-phase Nb_3Cl_8 (see Supplemental Section A on the structures) is the long-sought example where purely electronic strong correlation, encoded by the Mott-Hubbard approach [14] in a fixed crystallographic structure, is not just a textbook illustration, but is required for explaining gaping unobtainable by uncorrelated physics.

The significance of this discovery: This triumph [12] is far from being trivial. It requires avoidance of numerous alternatives explainable via conventional band structure such as density functional theory (DFT). Had the composition of the measured sample been Cl-richer (such as Nb_3Cl_9) or one where one Cl atom would be replaced by Te (such as in Nb_3Cl_7Te [15]), the correct insulating phases would be guaranteed by the trivial closed-shell configuration. Structural and magnetic symmetry breaking must be examined, although they were rejected to establish a compelling alternative scenario for insulation via strong correlation. For example, the sister d-electron compound NbO_2 is predicted by DFT to be a metal without any structural symmetry breaking, but once Peierls-like symmetry breaking of Nb-Nb dimerization is examined [16], it shows total energy stabilization, concomitantly with becoming an insulator, in agreement with experiments without need for strong correlation Hubbard description [Figure 1(a, b)]. Also, other perovskites like SrVO_3 [17] and LaNiO_3 [18] are predicted by DFT to be metals, and structural or magnetic symmetry breaking is too weak to convert them to insulators. However, the predicted metallic states in the former oxides are the correct experimental reality for these two compounds [19–21], unlike the case for the insulating α-phase of Nb_3Cl_8.

The question posed: Can symmetry breaking—either structural or magnetic, or both—lower the total energy, while also opening a band gap? Here we examine the possible role of magnetism in explaining the insulation of Nb_3Cl_8. We use the DFT exchange-correlation functional [22] meta-generalized gradient approximation (meta-GGA) to calculate the electronic structures without Coulomb repulsion U. The DFT method details are provided in Supplemental Section B.





Structural symmetry breaking in nonmagnetic NbO_2 suffices to correctly describe it as an insulator: Before considering Nb_3Cl_8, we first glance again at its simpler sister compound NbO_2. This compound has a metal-insulator transition with two nonmagnetic phases whose crystal structures are both tetragonal but different in formation of Nb-Nb dimers. As shown in Fig. 1(a, b), the high-temperature (high-T) phase has the Fermi level residing in the broad principal conduction band and is metallic because it has no Nb-Nb dimers [Fig. 1(a)]. As shown in Fig. 1(c, d) the low-temperature (low-T) phase has Nb-Nb dimerization [Fig. 1(c)]. This dimerization constitutes structural symmetry breaking which stabilizes the total energy by 152 meV/formula unit relative to the metallic phase and splits off occupied flat bands from the principal conduction band down into the internal energy gap, thereby opening a band gap [Fig. 1(b)]. The states of the split-off flat bands in the low-T insulating phase are mainly localized on the Nb-Nb dimers [Fig. 1(d)], while the states near the Fermi level in the high-T metallic phase surround the Nb atoms [Fig. 1(b)].

The structural symmetry breaking in Nb_3Cl_8 retains the incorrect metallic phase: The compound Nb_3Cl_8 is more complex than its sister compound NbO_2 in terms of the structural symmetry breaking, paramagnetism and thus band gap opening mechanism. Nb_3Cl_8 is a layered material where each layer has Nb trimers forming the distorted Kagome lattice with a breathing mode [12,23]. Each Nb trimer is surrounded by 13 Cl ions, forming an Nb_3Cl_13 cluster [Nb_3Cl_8]^5+[Cl_5]^5-. Here a Nb trimer has 15 electrons but 8 of them are transferred to Cl_8, so 7 electrons are left per cluster. They occupy the molecular orbitals configuration (1a_1)^2(1e)^4(2a_1)^1 so that the highest occupied molecular orbital (HOMO) is singly occupied for the nonmagnetic Nb trimer cluster. By comparing the band structures of monolayer and bilayer structures for the α-phase Nb_3Cl_8, Supplemental Section C shows that the interlayer coupling does not influence the metallic state of the non-magnetic phase. Thus, subsequent calculations are performed by using the monolayer structure. The Nb-Nb-Nb trimerization in α-phase Nb_3Cl_8 is observed experimentally, showing short intracluster Nb-Nb bond length d^short(Nb-Nb) of 2.834 Åand long intercluster Nb-Nb bond length d^long(Nb-Nb) of 3.998 Å[24]. The DFT structural relaxed calculations giving intracluster distance of 2.85 Åand intercluster of 4.02 Åare close to the experimental measurement [24]. We define the “degree of trimerization” (DOT) as:

    DOT=d^long(Nb-Nb)-d^short(Nb-Nb)/d^long(Nb-Nb)+d^short(Nb-Nb)× 100%.

The DOT for the DFT relaxed structure and for the  experimental structure [24] are both estimated to be 17.0%. The calculated total energy gives an enormous stabilization of 3365 meV/f.u. but the system is a band structure metal.

Paramagnetism in α-phase Nb_3Cl_8: The paramagnetic (PM) α-phase of Nb_3Cl_8 was observed experimentally by measuring the magnetic susceptibility as a function of temperature, which was well fitted to the Curie-Weiss function giving rise to an effective PM Bohr magneton of 1.65 μ_B [12]. This paramagnetism is not considered relevant to gapping in the strongly correlated approach. In such Mott model, magnetic exchange J deciding magnetism is modeled perturbatively in terms of Mott-ness strong Coulomb repulsion J1/U, leading to the expected weak or no magnetism when a large Coulomb U is selected. Here, we do not exclude the PM configuration.

Description of paramagnetic symmetry breaking: We compute PM non-perturbatively using magnetic exchange interactions J obtained from spin polarized DFT. The PM configuration is not replaced a priori by the nonmagnetic structure where each site has zero moment. The latter view assumes that the local symmetry equals the average crystal structure of global symmetry. Instead, a PM configuration can be described more generally as a distribution of non-zero local magnetic moment. We use for this purpose a “specially constructed” [25] supercell consisting of n Nb_3Cl_8 formula units (where n is a size convergence parameter and we use n=16 here) with zero global magnetization. Much like “special k-points for Brillouin zone integrating” [26,27], the spin-special quasirandom structure (spin-SQS) method is not a single snapshot but represents a configurational average. Indeed, the spin-SQS searches for a magnetic configuration that best matches the target multisite spin-spin correlation function with the given short-range order. The local magnetic moments are decided by DFT energy minimization and charge self-consistency starting from the optimal spin-SQS magnetic structure. Since the band structure of PM phase in a small Brillouin zone (corresponding to a real space supercell) is inconveniently dense to observe important information such as band dispersion, we unfold the band structure into the original Brillouin zone thereby obtaining effective spectral functions [28] that have, unlike conventional band structure, natural energy-dependent intensities and “fuzziness”.

The important observation regarding magnetic symmetry breaking: Once structural symmetry breaking (Nb trimerization) is accounted for, we find that magnetic symmetry breaking opens a gap in the PM phase, explaining the insulating phase without recourse to strong correlation or to dynamic effects. This is described next by breaking the process into steps that clarify the individual contributions.

Hierachy of symmetry breaking: Figure 2 demonstrates the three successive levels of theoretical description of monolayer Nb_3Cl_8 based on the existence or absence of two symmetry breaking modes—Nb trimerization and magnetic symmetry breaking.



Level 1 (symmetry unbroken) is for the hypothetical structure without Nb trimerization and magnetism. As shown in Fig. 2(a), the Fermi level lies within a broad principal conduction band with its charge density distributed equally on the Nb atoms. This configuration is taken as the total energy reference Δ E=0.

In Level 2 (structural symmetry breaking) with the Nb trimerization, we find that the Nb trimerization stabilizes the total energy by 3365 meV/f.u. relative to Level 1 and induces a single half-occupied isolated flat band at the Fermi level. This isolated flat band with band width of 0.3 eV emerges from the broad principal conduction band of Level 1 and its charge density is dominantly located around the Nb trimers, as shown in Fig. 2(b). For the bilayer structure of nonmagnetic α-phase Nb_3Cl_8, there are two split-off flat bands crossing with the Fermi level that are degenerate at H point (See Supplemental Section B). Evidently, in both monolayer and bilayer structures of α-phase Nb_3Cl_8, the Nb trimerization with the formation of partially occupied split-off flat bands cannot open a band gap.

In Level 3 (structural and magnetic symmetry breaking) that considers both (i) Nb trimers due to structural symmetry breaking and (ii) magnetic symmetry breaking. The latter is depicted as PM spin-SQS with self-consistently determined distribution of local moments, as shown in Fig. 2(c). This shows that the half-filled isolated flat band of Level 2 split forming two isolated flat bands and creating a band gap of 0.3 eV. It increases to 1.2 eV when using the HSE06 exchange correlation functional instead of SCAN. This process from Level 2 to Level 3 reduces the total energy by 105 meV/f.u. The band gap is opened by interaction of (i) structural symmetry breaking (forming trimers) and (ii) magnetic symmetry breaking. We find that the band gap cannot be opened if there is only one of the two symmetry breaking modes. This shows that one can predict the insulating phase of such quantum materials without Hubbard U.

Distribution of local motifs: To further indicate the difference between PM and nonmagnetic phases, Figure 3 shows the distribution of local magnetic moments in PM Nb_3Cl_8 (Level 3). It is  observed that: (i) All of these results present a distribution of non-zero local magnetic moments and an average zero net magnetic moment, which is independent of the exchange correlation functionals and supercell sizes; (ii) The distribution of magnetic moments for SCAN and HSE06 is very different partially due to the strong exchange interaction in HSE06, but the choice of different exchange correlation functionals do not qualitatively change our main point that PM is essential to open the band gap.

Discussion: The PM phase lacks the long-range order of the AFM phase but has short-range order that can still be essential to open the band gap. Indeed long-range antiferromagnetic order is well known [29] to enable insulating gaps. However, the absence of long-range magnetic order as the case in PM phase can, in the presence of short-range order lead either (i) to metallic phase if symmetry breaking is weak, as in SrVO_3 [3,17] and LaNiO_3 [30,31], or (ii) to an insulating PM phase, as in MnO, FeO, CoO, and NiO. Furthermore, symmetry breaking can induce other effects such as effective mass enhancement [17] without strong correlation.

The present results indicate that approximating the PM as nonmagnetic configuration and using primitive unit cell as input to do the calculations is not appropriate, missing energy lowering and gapping. Compared to nonmagnetic NbO_2 that only needs structural symmetry breaking (Nb-Nb dimerization) to open a band gap (Fig. 1), α-phase Nb_3Cl_8 is more complicated because it needs both structural (Nb-Nb-Nb trimerization) and magnetic (PM) symmetry breaking modes (Fig. 2).

The insulating bilayer nonmagnetic β-phase Nb_3Cl_8: Besides the high-T PM α-phase, Nb_3Cl_8 has a low-T nonmagnetic β-phase, which is measured also as an insulator [12]. Because of the proximity of the layers in the β-phase, we allow in this case interlayer interaction via adding the Van der Waals (VdW) dispersive forces to the regular DFT. These dispersive forces have no influence on the α-phase but shorten the interlayer distance and increase the coupling within the bilayer β-phase. We find that in nonmagnetic β-phase Nb_3Cl_8 with two chemical formulae in the primitive cell, there are two split-off flat bands crossing with the Fermi level that are not degenerate at any k-points (see Supplemental Section C). This energy splitting is induced by the interlayer interaction. Therefore, in the β-phase the Nb trimerization and coupling of Nb trimers from adjacent layers have the potential to open a band gap, even without magnetism or strong correlation. Supplemental Section C provides a few tests of the insulating gap under different crystal structural parameters. A detailed study of the insulating gap is β-phase Nb_3Cl_8 is however beyond our scope as the details of the interlayer geometry are unknown.

Many-body physics is certainly to be preferred over single particle theory for understanding multiplet effects in electronically isolated ions in the gas phase [32,33], or for d-electron ions in wide-gap solids [34,35], etc. Many-body effects certainly exist as spectroscopic multiplet effects [36,37] and in multi excitons. Still, multi-electron effects that are often apparent in isolated ions such as carbon [38] often “melt away” when such atoms bond in condensed phases—as is the case when diamond or graphite are formed from carbon atoms. So, the question is: when is a many-electron treatment unescapable? But we see that the outlier phenomenology of gapping, the unusual case of Nb_3Cl_8 is describable by conventional DFT while reproducing also observable structural (Nb-Nb-Nb trimerization) and magnetic (the stabilization of a PM) symmetry breaking. This leads to an interesting duality: How will the band gap change if both the observable symmetry breaking (discussed in the present work) and the strong correlation Mott-Hubbard treatment used successfully in Ref. [12,23] to obtain gapping (without magnetic symmetry breaking) have important influences in producing an insulating phase? This remained question will inspire future research.


    We thank Dr. Shunye Gao for discussions of his paper denoted here as Ref. [12]. This work was supported by the U.S. Department of Energy, Office of Science, Basic Energy Sciences, Materials Sciences and Engineering Division under Grant No. DE-SC0010467. This work used resources from the National Energy Research Scientific Computing Center (NERSC), which was supported by the Office of Science of the U.S. Department of Energy.







§ REFERENCE

[1]	A. N. Beecher, O. E. Semonin, J. M. Skelton, J. M. Frost, M. W. Terban, H. Zhai, A. Alatas, J. S. Owen, A. Walsh, and S. J. L. Billinge, Direct observation of dynamic symmetry breaking above room temperature in methylammonium lead iodide perovskite, ACS Energy Lett. 1, 880 (2016).

[2]	E. S. Božin, C. D. Malliakas, P. Souvatzis, T. Proffen, N. A. Spaldin, M. G. Kanatzidis, and S. J. L. Billinge, Entropically stabilized local dipole formation in lead chalcogenides, Science 330, 1660 (2010).

[3]	O. I. Malyi and A. Zunger, False metals, real insulators, and degenerate gapped metals, Appl. Phys. Rev. 7, 041310 (2020).

[4]	X.-G. Zhao, Z. Wang, O. I. Malyi, and A. Zunger, Effect of static local distortions vs. dynamic motions on the stability and band gaps of cubic oxide and halide perovskites, Materials Today 49, 107 (2021).

[5]	Z. Zhu, H. Peelaers, and C. G. Van De Walle, Electronic and protonic conduction in LaFeO3, J. Mater. Chem. A 5, 15367 (2017).

[6]	M. Eaton, A. Catellani, and A. Calzolari, VO2 as a natural optical metamaterial, Opt. Express 26, 5342 (2018).

[7]	F. El-Mellouhi, E. N. Brothers, M. J. Lucero, I. W. Bulik, and G. E. Scuseria, Structural phase transitions of the metal oxide perovskites SrTiO3, LaAlO3, and LaTiO3 studied with a screened hybrid functional, Phys. Rev. B 87, 035107 (2013).

[8]	J. Varignon, J. Santamaria, and M. Bibes, Electrically switchable and tunable Rashba-type spin splitting in covalent perovskite oxides, Phys. Rev. Lett. 122, 116401 (2019).

[9]	H. Sawada, Y. Morikawa, K. Terakura, and N. Hamada, Jahn-Teller distortion and magnetic structures in LaMnO3, Phys. Rev. B 56, 12154 (1997).

[10]	A. O’Hara and A. A. Demkov, Nature of the metal-insulator transition in NbO2, Phys. Rev. B 91, 094305 (2015).

[11]	S. Sarkar, R. Raghunathan, S. Chowdhury, R. J. Choudhary, and D. M. Phase, The mystery behind dynamic charge disproportionation in BaBiO3, Nano Lett. 21, 8433 (2021).

[12]	S. Gao, S. Zhang, C. Wang, S. Yan, X. Han, X. Ji, W. Tao, J. Liu, T. Wang, S. Yuan, G. Qu, Z. Chen, Y. Zhang, J. Huang, M. Pan, S. Peng, Y. Hu, H. Li, Y. Huang, H. Zhou, S. Meng, L. Yang, Z. Wang, Y. Yao, Z. Chen, M. Shi, H. Ding, H. Yang, K. Jiang, Y. Li, H. Lei, Y. Shi, H. Weng, and T. Qian, Discovery of a single-band Mott insulator in a van der Waals flat-band compound, Phys. Rev. X 13, 041049 (2023).

[13]	Y. Haraguchi, C. Michioka, M. Ishikawa, Y. Nakano, H. Yamochi, H. Ueda, and K. Yoshimura, Magnetic–nonmagnetic phase transition with interlayer charge disproportionation of Nb3 trimers in the cluster compound Nb3Cl8, Inorg. Chem. 56, 3483 (2017).

[14]	J. Hubbard and B. H. Flowers, Electron correlations in narrow energy bands, Proc. R. Soc. A 276, 238 (1963).

[15]	H. Zhang, Z. Shi, Z. Jiang, M. Yang, J. Zhang, Z. Meng, T. Hu, F. Liu, L. Cheng, Y. Xie, J. Zhuang, H. Feng, W. Hao, D. Shen, and Y. Du, Topological flat bands in 2D breathing-Kagome lattice Nb3TeCl7, Adv. Mater. 35, 2301790 (2023).

[16]	M. J. Wahila, G. Paez, C. N. Singh, A. Regoutz, S. Sallis, M. J. Zuba, J. Rana, M. B. Tellekamp, J. E. Boschker, T. Markurt, J. E. N. Swallow, L. A. H. Jones, T. D. Veal, W. Yang, T.-L. Lee, F. Rodolakis, J. T. Sadowski, D. Prendergast, W.-C. Lee, W. A. Doolittle, and L. F. J. Piper, Evidence of a second-order Peierls-driven metal-insulator transition in crystalline NbO2, Phys. Rev. Mater. 3, 074602 (2019).

[17]	Z. Wang, O. I. Malyi, X. Zhao, and A. Zunger, Mass enhancement in 3d and s-p perovskites from symmetry breaking, Phys. Rev. B 103, 165110 (2021).

[18]	G. Gou, I. Grinberg, A. M. Rappe, and J. M. Rondinelli, Lattice normal modes and electronic properties of the correlated metal LaNiO3, Phys. Rev. B 84, 144101 (2011).

[19]	L. Zhang, Y. Zhou, L. Guo, W. Zhao, A. Barnes, H.-T. Zhang, C. Eaton, Y. Zheng, M. Brahlek, H. F. Haneef, N. J. Podraza, M. H. W. Chan, V. Gopalan, K. M. Rabe, and R. E.-Herbert, Correlated metals as transparent conductors, Nat. Mater. 15, 2 (2016).

[20]	Y. Tomioka, T. Ito, E. Maruyama, S. Kimura, and I. Shindo, Magnetic and electronic properties of single crystals of perovskite nickelate oxide LaNiO3 prepared by the laser diode floating zone method, J. Phys. Soc. Jpn. 90, 034704 (2021).

[21]	B. Li, D. Louca, S. Yano, L. G. Marshall, J. Zhou, and J. B. Goodenough, Insulating pockets in metallic LaNiO3, Adv. Electron. Mater. 2, 1500261 (2016).

[22]	J. Sun, A. Ruzsinszky, and J. P. Perdew, Strongly constrained and appropriately normed semilocal density functional, Phys. Rev. Lett. 115, 036402 (2015).

[23]	S. Grytsiuk, M. I. Katsnelson, E. G. C. P. van Loon, and M. Rösner, Nb3Cl8: A prototypical layered Mott-Hubbard insulator, npj Quantum Mater. 9, 1 (2024).

[24]	M. Ströbele, J. Glaser, A. Lachgar, and H.-J. Meyer, Structure and electrochemical study of Nb3Cl8, Z. Anorg. Allg. Chem. 627, 2002 (2001).

[25]	A. Zunger, S.-H. Wei, L. G. Ferreira, and J. E. Bernard, Special quasirandom structures, Phys. Rev. Lett. 65, 353 (1990).

[26]	A. Baldereschi, Mean-value point in the Brillouin zone, Phys. Rev. B 7, 5212 (1973).

[27]	D. J. Chadi and M. L. Cohen, Special points in the Brillouin zone, Phys. Rev. B 8, 5747 (1973).

[28]	V. Popescu and A. Zunger, Extracting E versus k effective band structure from supercell calculations on alloys and impurities, Phys. Rev. B 85, 085201 (2012).

[29]	J. C. Slater, Magnetic effects and the Hartree-Fock equation, Phys. Rev. 82, 538 (1951).

[30]	G. Catalan, Progress in perovskite nickelate research, Phase Transitions 81, 729 (2008).

[31]	J. Varignon, M. N. Grisolia, J. Íñiguez, A. Barthélémy, and M. Bibes, Complete phase diagram of rare-earth nickelates from first-principles, npj Quantum Mater. 2, 1 (2017).

[32]	C. E. Moore, Atomic Energy Levels as Derived from the Analyses of Optical Spectra: The Spectra of Hydrogen, Deuterium, Tritium, Helium, Lithium, Beryllium, Boron, Carbon, Nitrogen, Oxygen, Fluorine, Neon, Sodium, Magnesium, Aluminum, Silicon, Phosphorus, Sulfur, Chlorine, Argon, Potassium, Calcium, Scandium, Titanium, and Vanadium (U.S. Department of Commerce, National Bureau of Standards, 1949).

[33]	C. E. Moore, A Multiplet Table of Astrophysical Interest (U.S. Department of Commerce, Office of Technical Services, 1959).

[34]	M. Imada, A. Fujimori, and Y. Tokura, Metal-insulator transitions, Rev. Mod. Phys. 70, 1039 (1998).

[35]	A. Fazzio and A. Zunger, Many-electron multiplet effects in the optical spectra of NiO, CoO and MnO, Solid State Commun. 52, 265 (1984).

[36]	A. Zunger, Electronic structure of 3d Transition-Atom Impurities in Semiconductors, in Solid State Phys., edit. F. Seitz, D. Tumbull, and H. Ehrenreich (Acad Press NY) 39, 275-464 (1986).

[37]	J. S. Griffith, The Theory of Transition-Metal Ions (Cambridge University Press, 1971).

[38]	G. M. Lawrence and B. D. Savage, Radiative lifetimes of UV multiplets in boron, carbon, and nitrogen, Phys. Rev. 141, 67 (1966).



 Paper Title: Droplet-confined electroplating for nanoscale additive manufacturing: current control of the initial stages of the growth of copper nanowires.


[
 
 Droplet-confined electroplating for nanoscale additive manufacturing: current control of the initial stages of the growth of copper nanowires 


 
 
 
 Mirco Nydegger1
       and Ralph Spolenak1,^*
       


 1Laboratory for Nanometallurgy, Department of Materials, ETH Zürich, Vladimir-Prelog-Weg 1-5/10, Zürich 8093, Switzerland
 
 
  
  ^*To whom correspondence should be addressed; e-mail: ralph.spolenak@mat.ethz.ch

 
 
Droplet-confined electrodeposition enables a precise deposition of three dimensional, nanoscopic and high purity metal structures. It aspires to fabricate intricate microelectronic devices, metamaterials, plasmonic structures and functionalized surfaces. Yet, a major handicap of droplet-confined electrodeposition is the current lack of control over the process, which is owed to its dynamic nature and the nanoscopic size of the involved droplets. The deposition current offers itself as an obvious and real-time window into the deposition. Therefore, the current during droplet-confined deposition is analysed. Nucleation and growth dynamics are evaluated systematically. Our results indicate different deposition regimes and link current to both volume and morphology of deposited copper. This allows for optimized electroplating strategies and to calibrate the slicing algorithms necessary for a controlled deposition of 3D structures. The potential of selecting appropriate solvents further readies this novel technique for the reliable deposition of functional structures with submicron resolution.

 
 Keywords:  microscale, additive manufacturing, metal nanostructures, 3D Nanofabrication, electrohydrodynamic ejection, EHD 

]




§ INTRODUCTION




Intricate, nanostructured metals have exciting applications in integrated circuits and MEMS devices<cit.>, nanorobots<cit.> and plasmonic structures<cit.>. Small-length scale additive manufacturing (AM) aspires to provide access to such three dimensional structures at the nanoscale without the geometrical constraints of two-dimensional lithography. Small-length scale AM techniques based on localized electrodeposition<cit.> are especially well-suited for the deposition of device-grade metals, due to a high purity and a dense microstructure<cit.> of plated metals. Such plated metals often exhibit excellent mechanical and electrical properties<cit.>. Therefore, small-length scale electrochemical AM has drawn significant attention from both fundamental and industrial research. Such small-length scale electrochemical AM techniques rely on a localization of one of the fundamental steps of electroplating: a localized supply of both electrons and cations or a spatial confinement of the electrolyte. 



Localized supply of electrons: The localized supply of electrons for reduction can be achieved by using focused electron beams (such as in electron microscope)<cit.> or by utilizing sharp electrodes (such as sharp scanning tunneling microscope tips)<cit.>. Here, the short mean free path of electrons in solvents will spatially constrict the deposition<cit.>. Despite the potential for layer-by-layer deposition, the fabrication of arbitrarily shaped, three dimensional structures has not yet been demonstrated to our knowledge. 

Localized supply of cations: Cations can be provided locally by immersing hollow AFM tips (refered to as FluidFM) <cit.> or quartz nozzles in a supporting electrolyte, in a setup similar to scanning ion electrochemical microscopy<cit.>. This allows to produce high-fidelity structures from a variety of metals<cit.>. 

Confinement of the electrolyte: A more common approach is to utilize a mask so that only parts of the conductive substrate are exposed to the electrolyte. This will limit the deposition into the exposed area and leave an out-of-plane structure when the mask is removed. This approach is often used to synthesize nanowires<cit.>, but also more intricate structures <cit.>. Electroplating confined by a meniscus between the substrate and a nozzle offers a flexible mask and therefore a higher flexibility for intricate designs. This technique further allows for the fabrication of high resolution structures (<40nm)<cit.>. 

A fundamental limitation shared by all ECAM techniques is the low deposition speed compared to other small-length scale AM techniques due to the necessary migration of cations. The migration of cations can be sped up by a forced mass-transfer by ejection of ion-loaded solvent droplets in an electric field. The forced mass-transfer is then followed by a spatially confined electroplating in droplets<cit.>. This technique, known as electrohydrodynamic redox 3D printing (EHD-RP), allows for a fabrication of complex structures with speed of around 100 voxels/s, the voxel size being around 60 nm<cit.>. The voxel size directly depends on the size of the solvent droplet, which is generated from a quartz capillary (Fig. <ref> a-e). Therefore, EHD-RP combines high deposition speed typical for material-transfer AM techniques with the dense microstructure of electroplated metals. Combining this technique with an electron microscope allows to electroplate metals directly onto insulating substrates<cit.>.

It is important to note that droplet confinement is conceptually not merely an extension of standard bulk electrochemical processes, but works in a different regime of electroplating with different governing mechanisms. In particular, the transient nature of the droplet as well as the absence of a liquid connection between the electrodes are unique features of EHD-RP. This allows, for example, direct deposition of alloys with a controlled chemical composition <cit.> without the need for additives. Furthermore, the deposited metal can be switched in a voxel-by-voxel fashion on-the-fly<cit.> to create chemically architected materials. Yet, several fundamental questions still need to be addressed to fully leverage this novel technique routinely in nanofabrication. Specifically, to ready this technique for the fabrication of complex structures, a direct feedback mechanism for the printed volume has to be established. This is fundamental to optimize the discretization of a planned design into individual voxels to be printed (slicing) and is therefore crucial for the reliable fabrication of intricate nanostructures. Further, the nucleation and current-microstructure relationships in this novel regime of electroplating have only partially been described<cit.>, despite their importance for the fabrication of metals with dense microstructures. Further, an accurate current control promises to optimize the resolution of droplet-confined electroplating.

In this work, we first investigate the deposition current by mapping the relationship between applied potential and measured current in EHD-RP for different solvent systems and different distances between nozzle and substrate. We then investigate the nucleation in droplet-confined electroplating at different currents. Lastly, we establish a relationship between deposition current and the deposited voxels. We limit the number of voxels that are investigated to 4 to limit the influence of other effects such as field focusing<cit.>.



§ RESULTS

 


 




 §.§ Voltammogram of electrohydrodynamic redox printing

Deposition in EHD-RP is assumed to be confined into transient solvent-droplets without a liquid connection between working- and counterelectrode. This unique working principle is reflected by a distinctive shape of the voltammograms (Plots of applied potential versus the measured current) of EHD-RP (Fig. <ref>f and Fig. <ref>). The voltammograms are measured by applying a constant potential between the immersed electrode (which acts as the anode in the setup) and the substrate (the cathode), while measuring the current for 1 s.The applied potential is then increased by Δ V (Fig. <ref>b). The applied potential is cycled between 0 and 160 V with a Δ V of 0.5 V (resulting in a scan speed of 500 mV/s, unless stated otherwise). For these experiments, the printed geometry consisted of parallel lines while maintaining a substrate-nozzle distance of 10 µm (or between 5–15 µm when investigating the influence of the distance). To verify that the observed current-potential relationship is not an artefact of the used amperemeter or setup, the anode was replaced with a 100 GOhm resistor (selected to match the current at 100 V) and connected to the substrate. With the wire replaced by a resistor, the current did not show such change in the slope (Supporting Information (SI) Fig. <ref>a). Additionally, no current was detected when no solvent was added into the nozzle. We therefore conclude that the observed I-V curves are not due to artefacts in the setup.

The I-V curve in Figure <ref>f shows three distinct regimes characterized by different slopes of the curve. When a potential below the minimal ejection potential (V < Vejection) is applied, no current above noise level (± 2*10^-11 A) is measured. Here, the applied potential is not strong enough to start the hydrodynamic ejection of ion-loaded droplets. Note that in contrast to standard EHD printing no pressure is applied to the liquid in the nozzle <cit.>. The absence of a detectable current above noise-level includes the region around 0 (0  ±  4 V) where a current from the dissolution of the Cu electrode could be expected, but is not observed (SI Fig. <ref>b). The only current that is detectable for V < Vejection is an induced current from changing the potential, since the experimental setup acts fundamentally as a capacitance. This induced current decays quickly if the Δ V is small (Δ V ≤ 1 V). When the potential is raised to above the minimal ejection potential Vejection a sudden increase in current is observed. Usually, a potential of 75–90 V is necessary to start ejection. However, this minimal potential depends on further parameters, such the distance between nozzle and substrate, ion concentration and and the diameter of the nozzle (Fig. <ref>). For 5 µm  distance and acetonitrile (ACN) as the solvent, a potential of Vejection = 82 V is necessary, for 10 µm Vejection = 92 V, and for 15 µm Vejection = 100 V. Upon start of the ejection, the current raises instantaneous to ca. 0.5 nA for ACN. For water, the current fluctuates in the beginning between 0 and 1 nA (Fig. <ref>c), indicating an unstable ejection mode, but then stabilises at around 0.2 nA. A further increase of the potential leads to a linear increase of the current in both solvents (ACN: 3.9*10-11 S, 1 mM CuSO4: 5*10-11 S). Above a current of around 0.8–1 nA the conductivity decreases for both solvents (ACN: 1.8*10-11 S, 1 mM CuSO4: 1.4*10-11 S). A change in the conductivity is observed at the same current when the potential is reduced (Δ V < 0, Fig. <ref>f). The main difference with Δ V < 0 is that the ejection is continuously until a current of effectively 0 is reached. Therefore, deposition at low currents (I < 0.3 nA for water and I < 0.5 - 0.8 nA in ACN) can be achieved by first setting the potential so that a current of < 0.5 nA is measured and then lowering the potential again. Droplets can also be ejected with a negative potentials as shown in SI Fig. <ref>, but the ejection is not stable as indicated by the current curve. So far, no structures have been printed using negative applied potentials.

 

 

We observed that the current depends on a variety of parameters. During the second cycle between 0 and 160 V a higher current compared to the first cycle is measured for both acetonitrile and water as the solvent (Figure <ref> a,c). A third and forth cycle each lead to a further increase (SI <ref>c). An increase in the nozzle-to-substrate distance, however leads to a decrease of the current and necessitates a higher Vejection to start the ejection. Water without an added electrolyte can be ejected with the same three different regimes of the conductivity (except for the highest nozzle-substrate distance, where the current stays too low), but requires high potentials to start the ejection. Consequently, an increased concentration of metal ions in the solvent lowers the potential needed to initiate ejection. The orifice size of the nozzle also has a pronounced influence on the ejection behaviour. While with a small aperture the above described behaviour is observed, a large nozzle orifice necessitates a large potential to start the ejection and the current increases steeply. Large orifices render the control over the current difficult due to this steep increase of the current.

Important for future depositions is the observation that the same applied potential can result in different currents depending on the elapsed time of the deposition and on parameters that are difficult to control precisely. For example, laser-based nozzle pulling systems produce orifices with a small variation in size and the distance between nozzle and substrate varies when out-of-plane structures are deposited. Therefore, the second part of this article investigates nucleation and growth dynamics under constant currents. To stabilize the current, the applied potential V is varied in such a way that the deposition current stays within ± 5 % of the set value. 



 §.§ Nucleation and growth of Cu on Au

Figs. <ref> and <ref> show the nucleation of copper on gold substrates printed using ACN and 1 mM CuSO4 dissolved in H2O, respectively, as solvent and electrolyte. The nucleation is shown for five different currents (0.25, 0.5, 0.75, 1 and 1.25 nA) and arrested at different time steps. These currents were selected to represent both sections of the CV graph that exhibit different conductivities. From previous studies it is also known that droplet-confined electroplating achieves fast deposition rates of around 3 µm s-1<cit.>. Therefore, the necessary time steps to illustrate nucleation are in the order of few milliseconds. We found that the most reliable approach to such short time steps is to use a lateral translation of the substrate with different velocities (note that we keep the nozzle constant but move the substrate). The nominal residence time is then approximated by dividing the orifice diameter (ACN: 200–230 nm; H2O: 60–90 nm) by the velocity of the substrate (4–200 µm s-1). This approach was chosen as the reaction time of the control software and power source as well as parasitic capacities induced when switching on/off the system did not allow to reliably access such short time steps.

The nucleation behaviour of Copper in ACN (Fig. <ref>) shows distinct differences between low currents (0.2–0.5 nA) and high currents (≥ 0.75 nA). The current and the necessary applied potential during deposition are shown in <ref>. At 0.25 nA only patches of Cu are deposited. Number and size of the deposited patches increases with increasing residence time. At 0.5 nA very small grains are observed that  successively start to merge with longer residence times and form a rough, but continuous structure, as seen from SEM images and later crosssections. At currents of 0.75 nA and higher, progressively large grains can be observed when the residence time is larger than 20 ms. At 1 nA, few but very large grains can be observed, while at 1.25 nA dense lines without apparent grain boundaries can be observed. The area over which nucleation is observed (indicated by the dashed white lines when nuclei could be identified) depends strongly on the set current. At low currents, the width of the band stays constant, independent of the residence time. At high currents the nuclei are spread over a larger area (larger than the SE micrograph shows) and the area decreases with longer residence time. 

For copper that is deposited using water (Fig. <ref>) generally smaller grains are observed than in ACN. Further, the width of the band, in which nuclei are observed stays constant over different currents. For high currents (I ≥ 1 nA) after a seeding stage, a few grains grow preferably before merging and forming a layer. This is shown with higher temporal resolution in Fig. <ref>b. We identify three subsequent stages of the nucleation of Cu in water with 1.25 nA: first, very small nuclei are seeded homogeneously. In a second stage, the largest nuclei grow preferrably, until in a third stage, these grains start to coalesce and form a continuous line. Yet, a porous interface between Au substrate and the the Cu line remains. 

SI figure <ref> shows the nucleation after the same number of time steps achieved by single or by multiple overpasses of a fraction of the residence time each, using 1 mM CuSO4 as the electrolyte. The structures look similar after 10 ms residence time. The results indicate that for 1 nA using multiple, but shorter overpasses leads to an increase in the number of nuclei. Nevertheless, we conclude that using multiple short overpasses does not fundamentally change the growth behaviour and the mechanisms during the deposition.



 §.§ Illustrating the voxel during initial stages of the deposition

The nucleation of Cu in different solvent and at various currents indicated strong differences in the resulting microstructure. These differences are shown in detail in Figs. <ref>a and <ref>a, that present the cross-sections of Cu lines printed with residence times of 50, 100, 150 and 200 ms at the indicated currents for both solvents. 50 ms residence time refers to a translational speed of 4 and 2 µm/s with a nozzle with 200 nm and 100 nm orifice size respectively. At slower translational speeds, the lines are not adhering to the substrate anymore and out-of-plane growth is observed<cit.>, because the growth-rate exceeds the velocity of the substrate. Therefore, multiple overpasses of 50 ms each are used to further increase the residence time without introducing out-of-plane growth. Further, this approach allows to illustrate the material deposited in 50 ms increments.

A current of 0.25 nA and ACN as the solvent deposits, as already shown in Fig. <ref>, only a small amount of material. This carbon-rich structure has a low contrast in SEM. However, it is still visible that this deposit shows a high porosity, resembling a network structure. At 0.50 nA a dense microstructure is observed. At 0.75 nA the deposition is confined to a narrower structure (after the first layer), meaning that material is predominantly added to the top section. Further, larger grains than in lower currents are observed. However, porosity starts also to emerge between the grains. A further increase of the current leads to wider (horizontal width) structures, larger grains and also more porosity in between the grains. In previous reports, a constriction of the base of printed pillars was reported<cit.>. This is visible here as well for currents of 1 nA: It appears that the upper part of the structure is not connected to the substrate. This is probably due to the presence of porosity at the interface with a limited number of connecting grains between substrate and printed structure. 

In water, for currents of 0.75 nA and lower (and a nozzle-to-substrate distance of 7.5 µm), a flat, unconfined deposition is observed (refered to as a spraying mode). Spraying occurs when the droplets reach the coulomb limit of the droplets leading to a spray of the droplet and hence a deposition in a larger area. Note that currents of 0.75 nA refer to the second regime in the I-V curves, before the slope changes. For currents of 1 nA and higher a very fine grainstructure is observed. This is in agreement with previous reports of Cu pillars printed with this technique <cit.> using 1 mM CuSo4. At 1.25 nA the structure exhibits high porosity, especially close to the interface with the substrate. This porosity was expected from Fig. <ref> and similar porosity has been reported previously<cit.>. 

The shape of the voxels, however, are a function of not only the current, but also the nozzle-to-substrate distance (Fig. <ref>c). An decrease in the distance leads to more porous structure, while at high distance spraying instead of confined deposition is observed. Therefore, optimal deposition parameter need to balance current and distance to achieve optimized material quality.



§ DISCUSSION

A possible explanation of the observed current-potential characteristic as well as the differences in nucleation and microstructure of deposited Cu is the transition from a transient to a stationary electrolyte droplet on the substrate. A transient solvent droplet would evaporate before the next electrolyte droplet arrives. Such a transition has been hypothesized already in previous work based on an observed relationship between applied potential and microstructure<cit.> for Cu and ACN, but not studied as a function of the current. 

At V < Vejection no droplets are ejected and therefore no current is measured. Strikingly, the necessary potential to start any deposition is up to two order of magnitudes higher than in standard electroplating. This is due to the high electric fields necessary to drive the elecdrohydrodynamic ejection of the droplets. Further, the Vejection depends on the orifice size of the nozzle. For optimized deposition performance, an ideal orifice size leads to a linear I-V curve with a flat slope, which allows to finely control the current and does not lead to a deposition in a spray mode. This ideal orifice size was observed to differentiate between the utilized solvents. The ideal orifice size is probably unique for each solvent system as electrohydrodynamic ejection depends on a variety of parameters of the solvent<cit.> (such as the viscosity, surface tension, density, permittivity and conductivity). 

At V > Vejection & I < 0.75 - 1 nA, droplets are ejected but evaporate quickly, so that the solvent droplet evaporates completely before the next droplet reaches the substrate (or the droplets already evaporate before they impact the substrate), while at (V > Vejection &  I >0.8-1 nA:), there is a stationary solvent droplet. The solvent droplet is partially screening the electric field between the substrate and the nozzle, requiring a higher applied potential for the ejection. This solvent droplet therefore acts as an additional resistance in the system, leading to a lower conductivity. Further, this hypothesis explains the observed changes in the nucleation between I ≤ 0.5 nA  and I ≥ 0.75 nA. Electroplating in a stationary droplet is less kinetically dominated and therefore leads to thermodynamically more favourable microstructures (i.e. larger grains), as observed for high currents in ACN. In water, however, the trend towards larger grains is less visible. This could be due to higher levels of contamination (leading to frequent nucleation) or the formation of hydrogen on the substrate.

An alternative explanation for the observed different conductivities that we rejected is a change in the ejection mode<cit.>. There are different EHD modes described in the literature, which are, in order with ascending necessary electric field: dripping mode, pulsating cone, cone-jet, tilted-jet and twin- or multi-jet. A dripping mode is unlikely for our setup as this mode is generally associated with gravitational pull and the applied pressure to the solvent in the nozzle pushing, instead of an electrostatic pulling of the droplets. A tilted-jet is apparent in droplet-confined deposition by a deposition with varying offset from the nozzle position, while multi-jet deposition is apparent from multiple deposition sites simultaneously. Such an offset or multiple deposition locations were not observed for the discussed current-potential range. Hence, the two possible modes are pulsating-cone and cone-jet mode. Yet, a change in the mode would not explain the observed change in nucleation and growth. However, different ejection modes could explain the observed behaviour for too-large orifices (Fig. <ref>f): for a small nozzle, individual solvent droplets are ejected, while for a large nozzle only a jet can be sustained that connects nozzle and substrate (leading to the fast increase in current). A jet is most probably too small to be observed in the utilized camera.

The difference in the observed I-V characteristic between the first and the second cycle probably originates from the generation of a surplus of charge carriers during the first cycle (i.e. more charge carrier generated than ejected, possibly due to the involved high potentials). This is further supported by the observation that the necessary Vejection is higher in pure water that in a solution of 1 mM CuSO4. Therefore and unsurprisingly a higher number of charge carriers increases the current at a given potential.



§ CONCLUSIONS AND OUTLOOK

Droplet-confined electroplating must be both predictable and controllable to be a versatile fabrication technique. Therefore, we investigated the current-potential relationship and explored how the current can serve as an in situ feedback mechanism for the deposition. An analysis of the current unraveled a characteristic I-V curve. This curve allows to investigate and quantify the influence of a variety of parameters, such as the elapsed time, the distance between nozzle and substrate and the orifice size on the deposition. The deposition at constant currents allowed to catalogue the nucleation and growth of copper in both water and acetonitrile as an electrolyte.

For acetonitrile, a current of 0.75 nA indicates best shape of the voxel and a dense microstructure, although higher currents allow for lines without visible grain boundaries. An analysis of the current indicates that this is a transient regime for both studied electrolytes, where the droplets potentially still evaporate before the next droplet impacts. At current of 1 nA, the growth inside the solvent droplet leads to large (and sometimes very long grains). For water, the volumetric deposition speed is much lower at the same current, indicating a lower efficiency of the process, possibly due to other charge carriers, such as protons getting reduced on the substrate and forming hydrogen. 

The results presented here could also serve as a template protocol for testing novel candidate solvents for EHD-RP: first, a CV curve is measured to optimize the orifice size for the given solvent, similar to Fig. <ref>f. Then, lines are printed and cross-sections analyzed if material has been deposited. No deposition, but a measured ejection current can indicate no dissolution of the sacrificial anode <cit.> or a decomposition of the electrolyte. If deposition in a spraying mode is observed as in Fig. <ref> for low currents, the distance should be reduced, while dendritic growth indicates too close a distance, too high a current (Fig. <ref>) or an insufficient vapor pressure of the solvent that prevents timely evaporation of the solvent. An iterative approach will allow to quickly optimize the deposition parameters (orifice size, set current and distance between nozzle and substrate) for a chosen solvent and metal combination. Such a study will provide insights into the interplay and importance of dielectric constant, surface tension and vapor pressure of the candidate solvent for successful droplet-confined electroplating.

In conclusion, the current offers a self-evident tool to predict the properties of the deposited copper. Since any slicing algorithm requires knowledge about the geometrical shape and volume of an individual voxel, utilizing the current as an indicator for the volumetric deposition rate is crucial for the controlled fabrication of intricate 3 dimensional shapes<cit.>. This readies the technique significantly to deposit functional structures.



§ EXPERIMENTAL



 §.§ Materials

Nozzles for deposition were fabricated on a P-2000 micropipette puller system (Sutter Instruments) from filamented Quartz capillaries (Sutter Instruments, Item QF100-70-15). Nozzle diameters were determined on a Quanta 200F (Thermo Fisher Scientific), equipped with a Schottky type field emission gun (FEG) in low vacuum mode (30 Pa) to avoid charging. Nozzles with diameters of 60–90 nm were used for salt solutions and 200-230 nm for acetonitrile. The nozzles were filled with acetonitrile (Optima, Fisher Chemical)  or copper sulfate (1 mM CuSO4 (Sigma Aldrich, 99.999% metal basis) in water (LC/MS-Grade, Fisher Chemical)) by using gas-tight glass syringes with custom made PEEK tips. The nozzles were cleaned on the inside by rinsing with the solvent. Substrates were 0.4 cm × 2.0 cm pieces of Au coated (80 nm) Si wafers. The substrates were cleaned in technical acetone, analytical isopropanole and subsequently blow-dried with compressed air before use. Cu wires (Alfa Aesar, 0.25 mm diameter, 99.999% metal basis) were etched in concentrated nitric acid (65% HNO3, Sigma Aldrich) for 10 s, then dipped in water and stored in analytical ethanol until use (maximal 10 minutes), while Au wires (Metaux Precieux SA, 99.999%) were immersed for 30 s, dipped in water and stored in air.



 §.§ Setup
 
Fundamentally, the setup consists of piezo stages that move the substrate in X, Y, and Z direction (QNPXY-500, QNP50Z-250, Ensemble QL controller, Aerotech). Additionally, stage translations in X and Y direction larger than 500 µm were enabled by additional long-range stages (M112-1VG, PI for Y direction, manual micrometre screw, Mitutoyo for X direction). Piezo stages and power source were controlled through a custom Matlab script. The nozzle is mounted on a motorized nozzle holder (Z825B, controlled with Kinesis, both Thorlabs). The deposition is observed trough an optical microscope composed of a ×50 objective lens (LMPLFLN, Olympus) and a CMOS camera (DCC1545M, Thorlabs). The lens is mounted at an inclination of 60° to the substrate normal and the substrate is illuminated from the behind using a green LED (LEDMT1E, Thorlabs). A power source (B2962, Keysight) with triaxial cable connectors was used for polarizing the anodes. The metal wires were connected using a mechanical clamp. The complete printing setup is mounted inside a gas tight box to provide an oxygen-free atmosphere (Controlled with an oxygen sensor, Module ISM-3, Dansensor) and is placed on a damped SmartTable (Newport) to provide a vibration-free environment.



 §.§ Deposition

Deposition was performed at room temperature in an argon atmosphere (<100 ppm O2). Typical potentials applied to the anode during printing were 80–160 V. For printing with a certain current the potential was adjusted to match this ejection current by averaging the current over 40 data points (spaced 0.2 s) and adjusted in 0.2 V increments if the average deviated more than 5%. Nozzle substrate distance was controlled to be 5–15 µm by analysing the light microscope images and adjusting the position of the z-axis piezo stage accordingly.
 


 §.§ Analysis
 
High Resolution Scanning electron microscopy (SEM) was performed with a Magellan 400 SEM (Thermo Fisher Scientific, former FEI) equipped with an Octane Super EDXsystem (EDAX, software: TEAM). Tilt angles were 55°. HR-SEM images were taken in immersion mode with an acceleration voltage of 5 kV. A dual-beam Helios 5UX (Thermo Fisher Scientific) with a focused Ga+ liquid metal ion source was used for focused ion beam (FIB) milling of  cross-sections. Prior to FIB-milling the pillar was coated by a protective carbon layer, which is visible in the SEM image as a black background. 



§ AUTHOR CONTRIBUTIONS

M.N. and R.S. devised the concept. M.N. performed deposition experiments and provided SEM and FIB analysis. M.N. visualised the data and wrote the original paper draft. Both authors discussed the results and reviewed the manuscript.



§ CONFLICTS OF INTEREST

The Authors declare no conflicts of interest.



§ ACKNOWLEDGEMENT

This work was funded by grant no. SNF 200021-188491. The authors thank Noa Bernasconi for initial deposition experiments. Electron-microscopy analysis was performed at ScopeM, the microscopy platform of ETH Zürich.

 
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Supplementary Information





Nanoscale droplet-confined electroplating: nucleation, growth and coulombic efficiancy 



Mirco Nydegger and Ralph Spolenak






 

 

 




