Paper Title: Fast and scalable finite-element based approach for density functional theory calculations using projector-augmented wave method.












Department of Computational and Data Sciences, Indian Institute of Science Bengaluru, India




Department of Mechanical Engineering, University of Michigan, Ann Arbor, Michigan 48109, USA


phanim@iisc.ac.in

Department of Computational and Data Sciences, Indian Institute of Science Bengaluru, India




             

§ ABSTRACT 









Gaining ab initio insights into complex  material systems at nanometer length-scale demand enormous computational resources due to the strict accuracy requirements of density functional theory (DFT) calculations compounded with the cubic scaling complexity of DFT with number of atoms. 
In this work, we present a computationally efficient methodology that utilizes a local real-space formulation of the  projector augmented wave (PAW) method  discretized with finite-element (FE) basis to enable accurate and large-scale electronic structure calculations. This is achieved through the synergy of the PAW method's ability to transform the underlying all-electron DFT problem into a form that involves smoother electronic fields and the potential of systematically convergent higher-order finite-element basis to achieve significant computational gains. In particular, we employ higher-order quadrature rules to evaluate integrals involving PAW-atomic data accurately, allowing the use of coarser FE meshes for electronic fields. 
Further, we have developed efficient computational strategies for solving the underlying FE discretized PAW generalized eigenproblem by employing the Chebyshev filtered subspace iteration approach to compute the desired eigenspectrum in each self consistent field iteration. These strategies leverage the low-rank perturbation of FE basis-overlap to invert FE discretized PAW overlap matrix in conjunction with exploiting the FE-cell level structure of both the local and nonlocal parts of the discretized PAW Hamiltonian and overlap matrices. Using the proposed approach we benchmark the accuracy and performance on various representative examples involving periodic and non-periodic systems with state-of-the-art plane wave based PAW implementations.  Furthermore, we also demonstrate a significant computational advantage over state-of-the-art plane-wave basis for large-scale systems comprising of around 35,000 electrons. Our approach () significantly reduces the degrees of freedom required to achieve desired accuracy when compared to norm-conserving pseudopotentials, thereby enabling large-scale DFT simulations at an order of magnitude lower computational cost.





                              
Fast and scalable finite-element based approach for density functional theory calculations using projector-augmented wave method
    Phani Motamarri
    August 5, 2024
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§ INTRODUCTION 


Electronic structure calculations based on Kohn-Sham density functional theory (DFT) have significantly advanced our understanding of material properties. The Kohn-Sham approach to DFT <cit.> reformulates the many-body problem of interacting electrons into an equivalent computationally tractable problem of non-interacting electrons within an effective mean field governed by electron density.  This reformulation has made it possible to predict a wide range of material properties, including mechanical, transport, electronic, magnetic and optical properties, over the past six decades. Pseudopotential approximations in DFT <cit.> have been crucial in making these calculations computationally efficient, as they focus only on valence electrons that participate in chemical bonding. To this end, these approximations transform an all-electron DFT calculation into an equivalent problem involving smooth pseudo-wave-functions for valence electrons, using an effective external potential (pseudopotential) that lumps the effect of core-electrons and the singular Coulomb nuclear potential. In a numerical setting, pseudopotential DFT calculations are computationally efficient since they only compute valence electronic wavefunctions, allowing for the use of coarser grids or lower plane-wave energy cutoffs to resolve these smooth pseudo electronic fields. Over the years, pseudopotentials have evolved from norm-conserving <cit.> to ultrasoft potentials <cit.>, and now to the state-of-the-art projector augmented wave (PAW) <cit.> method, enabling accurate predictions of material properties across diverse material systems without the need for expensive all-electron DFT calculations. However, the stringent accuracy required for DFT calculations to compute meaningful properties, combined with the cubic-scaling computational complexity with the number of electrons, demands significant computational resources even for pseudopotential DFT calculations, typically limiting their application to material systems with at most few thousands of electrons. Overcoming the current limits on ab-initio calculations has been a decades-long pursuit by researchers around the world, and efforts over the past three decades have focussed on developing accurate and computationally efficient methods for solving the Kohn-Sham DFT problem. These include efficient discretization methods for solving DFT equations, reduced-scaling algorithms for lowering the computational complexity of the DFT problem as well as efforts to improve the parallel scalability of the widely used codes. Among the various discretization methods, plane wave (PW) basis is the most popular method of choice  for metallic and heterogenous material systems due to the computational efficiency afforded by these basis sets owing to its spectral convergence. Despite the popularity of PW basis for DFT calculations, the limitations of these Fourier methods stemming from the poor scalability on parallel computing architectures, the inefficiency in treating non-periodic systems, and the inability to handle generic boundary conditions led to the development of systematically improvable, efficient, and scalable real-space discretization techniques based on finite-difference <cit.>, finite-elements <cit.>, wavelets <cit.> and other reduced-order basis techniques <cit.>. 

The finite-element (FE) basis, a compactly supported piece-wise polynomial basis, is a relatively new approach in real-space methods for DFT, offering several advantages over other commonly used basis sets.  FE basis sets are systematically convergent for any material system, can accommodate generic boundary conditions including fully periodic, semi-periodic, non-periodic. Their locality allows one to exploit fine-grained parallelism on modern heterogenous architectures while providing excellent parallel scalability on distributed systems<cit.>. As demonstrated in previous works <cit.>, finite-element-based methods have significantly outperformed plane-wave basis in the case of norm-conserving pseudopotential DFT calculations, as the system size increases for a given accuracy of ground-state energy and forces. Furthermore, adaptive resolution offered by FE basis sets enables all-electron calculations <cit.> but mixed all-electron and pseudopotential calculations <cit.>. The open-source code DFT-FE <cit.>, the workhorse behind the ACM Gordon Bell Prize 2023, inherits these features while incorporating scalable and efficient solvers for the solution of Kohn-Sham equations. The DFT-FE code has shown excellent scalability massively parallel many-core CPU and hybrid CPU-GPU architectures (tested up to  200,000 cores on many-core CPUs and  40,000 GPUs on hybrid CPU-GPU architectures), while simulating material systems as large as 600,000 electrons <cit.>. Furthermore, DFT-FE has been recently employed in a range of scientific studies involving large-scale DFT calculations on material systems with tens of thousands of electrons – computing the electronic structure of large DNA molecules <cit.>, predicting energetics of extended defects in crystalline materials <cit.>, investigating phase transformations in doped nanofilms <cit.>, evaluating spin Hamiltonian parameters of systems with defects <cit.> and solving inverse DFT problem <cit.> 



We note that the aforementioned finite-element-based DFT calculations are all limited to the use of norm-conserving pseudopotentials. These pseudopotentials can become very hard for material systems with elements involving d or f electrons and these DFT calculations can be computationally demanding. Further, these often require semi-core states to be treated as valence states for better transferability, resulting in harder pseudopotentials with more valence electrons, particularly for alkali, alkaline earth metals, or early transition metals. Previous works <cit.> have shown that these hard norm-conserving pseudopotentials require a large number of degrees of freedom (30,000 to 50,000 finite-element basis functions per atom) to achieve chemical accuracy, leading to high computational costs for solving the Kohn-Sham DFT eigenvalue problem. While ultrasoft pseudopotentials relax the norm conservation condition <cit.>, resulting in softer potentials, their construction is complex and requires extensive testing to ensure accuracy and transferability. To address these issues, Blöchl <cit.> developed the projector augmented wave (PAW) method, formally an all-electron approach that generalizes the LAPW method and the pseudopotential approach. The PAW method recovers all-electron wavefunctions by introducing a linear transformation operator (𝒯) acting on smoothly varying pseudo wavefunctions. Within the frozen core approximation, it treats core states as frozen and retrieves the oscillatory behavior of all-electron valence wavefunctions near atomic centers through the action of 𝒯 on the smooth pseudo-valence wavefunctions. This allows one to reformulate the all-electron Kohn-Sham DFT eigenproblem into an equivalent PAW eigenvalue problem involving only smooth pseudo-valence wavefunctions, striking a balance between accuracy and computational efficiency. As a result, the PAW approach within the frozen core approximation has become widely adopted in popular plane-wave based DFT codes <cit.>.




The real-space discretization techniques have also been employed in the context of PAW implementations and are primarily based on finite-difference approximations <cit.>, wavelet based methods <cit.> and psinc basis <cit.>. However, most of these PAW based DFT calculations deal with material systems involving few thousands of electrons excepting the linear scaling DFT approach <cit.> using the PAW method applied to material systems with a band-gap. The computational efficiency and scalability of these real-space implementations compared to state-of-the-art plane-wave-based PAW implementations remains unknown. This work introduces the local real-space formulation of the PAW method amenable for finite-element (FE) discretization, proposing efficient strategies to solve the underlying FE-discretized PAW generalized eigenvalue problem using a self-consistent field iteration approach. Our methodology significantly reduces FE degrees of freedom required to achieve chemical accuracy using the PAW method, outperforming the optimized norm-conserving (ONCV) pseudopotential based DFT calculations employed in previous FE based approaches for a given system size. Additionally, we demonstrate that our method can handle fully periodic, non-periodic, and semi-periodic boundary conditions for generic material systems as large as 50,000 electrons. Moreover, our method offers substantial computational efficiencies over state-of-the-art plane-wave-based approaches for systems with over 5,000 electrons.  Notably, to the best of our knowledge this represents the first development of a fast and scalable approach for the PAW method within the framework of finite-element discretization.

We adopt the original Blöchl's <cit.> PAW method in our current work and do not employ the Kresse's modifications <cit.> that include the compensation charge contribution for computing the exchange-correlation term, which increases computational costs and introduces numerical instabilities <cit.>. The key aspects of the proposed finite-element (FE) approach for the PAW method include: (i) local reformulation of the extended PAW electrostatics involving smooth pseudo-electron density and the compensation charge making it amenable for FE discretization; (ii) deducing the discretised governing equations by evaluating the variational derivative of the locally reformulated all-electron DFT energy functional in the PAW method; (iii) employing higher-order quadrature rules to efficiently and accurately evaluate integrals involving atom-centered functions, allowing the use of coarser FE grids for electronic fields and yet capture the relevant PAW atomic data; (iv) implementation of a self-consistent field (SCF) iteration approach based on Anderson mixing of total charge density using an efficient real-space strategy to minimize the weighted total charge density residual norm; (v) an efficient scheme to evaluate the inverse of FE discretized PAW overlap matrix  using Woodbury's formula in conjunction with Gauss-Lobatto-Legendre quadrature rules and a block-wise matrix inversion for the overlap of atomic projectors in real-space; and (vi) leveraging this efficient computation of ^-1 to design a Chebyshev polynomial based filter <cit.> in each SCF iteration to compute the space rich in occupied eigenstates, and solving the PAW generalized eigenvalue problem by projecting onto this filtered space. The proposed formulation has been implemented in a distributed setting on CPUs using the message passing interface (MPI) for communication across multiple nodes to enable large-scale PAW based DFT calculations. We benchmark the accuracy and performance of our method against plane-wave-based PAW implementations on various representative non-periodic, semi-periodic, and fully periodic systems. Compared to reference data from plane-wave calculations, our results show excellent agreement in ground-state energies, energy variation with bond length and lattice parameters, formation energies, surface energies, and band structures. Relative to widely used plane-wave based PAW implementations, our finite-element based approach demonstrates significant computational advantage both in terms of CPU node-hrs and minimum wall time with increasing systems sizes reaching up to 5× and 10×, respectively, for non-periodic systems (∼18,000 electrons) and  3× and 6×, respectively, for periodic systems (∼34,000 electrons). Our implementation also exhibits excellent parallel scaling efficiency (∼70%) at 1200 degrees of freedom per MPI rank for material systems, achieving nearly a 10-fold speed-up over the state-of-the-art DFT-FE code using norm-conserving ONCV pseudopotentials for system sizes close to 50,000 electrons, with a significant reduction in computational resource requirements.


 
 The remaining sections of the manuscript is organised as follows. Section <ref> briefly introduces the Blöchl PAW method followed by a discussion on the local real-space formulation of the PAW method and the governing equations to be solved. Subsequently, Sec. <ref> introduces the finite-element discretization for the PAW governing differential equations and Sec. <ref> details the proposed computational methodology in achieving an efficient and scalable solution procedure for solving the FE discretized PAW generalised eigenvalue problem using a self-consistent field iteration approach. Section <ref> presents  comprehensively accuracy and performance benchmarks of our implementation comparing with state of the art PAW based plane-wave(PW) codes on representative benchmark systems. Additionally, this section presents a comparative study of the current work and the norm-conserving pseudopotential (ONCV) in the DFT-FE code in terms of parallel scalability and time to solution. Finally, we conclude with a short discussion summarizing the key findings and outline future prospects arising from this work in Sec. <ref>




§ REAL-SPACE FORMULATION OF PROJECTOR AUGMENTED WAVE METHOD









 
The ground-state properties of a material system comprising of N_a nuclei and N_e electrons is governed by the following all-electron Kohn-Sham DFT energy functional <cit.>



    E[{_i }, {^a}] = T_s[{_i }] + E_xc[n()] + J[ρ(),{^a} ]

where {_i } and {^a} denote the set of single-electron orthonormal wavefunctions and the set of nuclear position vectors respectively. We restrict our discussion here to spin-unpolarized case, while the extension to spin-polarized systems is straightforward and not elaborated here. The term T_s[{_i }] in Eqn. (<ref>) denotes the kinetic energy of the non-interacting electrons where

    T_s[{_i }] = 2∑_i^Nf_i∫_i^*()(-1/2∇^2)_i()d,

with f_i denoting the orbital occupancy function and the integer N ≥ N_e/2 denotes the number of single-electron wavefunctions. Further, E_xc[n()] in Eqn. (<ref>) is the exchange-correlation energy that accounts for the quantum-mechanical many-body effects. Adopting the generalized gradient approximation (GGA) <cit.> for the exchange-correlation functional description in this work, we have

    E_xc[n()] = ∫ϵ_xc[n(),∇ n()]d.

where n()= 2∑_i^Nf_i|_i()|^2 represents the all-electron density. The total electrostatic energy in Eqn. (<ref>) is denoted by J[ρ(),{^a}] = E_el[ρ(),{^a}] - E_self[ {^a}]. Here, E_el is the electrostatics energy associated with the total charge ρ()=n()+ ∑_ab^a(), where b^a()=-𝒵^aδ( - ^a) denotes the nuclear charge density with 𝒵^a being the atomic number of  a^th atom.  This electrostatics energy E_el involves a convolution in real-space given by:

    E_el[ρ,{^a} ] = 1/2∫∫ρ()ρ()/|-|dd

where the domain of integration for  in the above Eqn. (<ref>) is ℝ^3, while the domain of integration for  in the case of an infinite-periodic system is over the unit-cell of interest, whereas in the case of a finite system, it is over ℝ^3. Further, for a periodic system the index a refers to atoms at all lattice sites with the compact support of b^a() lying in the unit cell of interest, while for a finite system, it refers to all the atoms in the system. Finally, the self-interaction energy corresponding to the nuclear charge density b^a() is denoted as E^a_self[^a] and hence the total self-interaction energy is given by E_self[ {^a}] = ∑_aE^a_self[^a].



Within the frozen-core approximation in PAW method, the valence all-electron Kohn-Sham (KS) wavefunctions({_i}) that are orthogonal to the core wavefunctions are recovered from pseudo-smooth(PS) wavefunctions ({_i}) using a linear transformation operator(𝒯)  <cit.> given by:

    |_i⟩ = 𝒯|_i⟩ = (ℐ + ∑_a 𝒯^a) |_i⟩

The index i = {1,2...,N} in Eqn. (<ref>) runs over the valence states with  N ≥ N_v/2 where N_v being the number of valence electrons. Furthermore, the linear transformation operator (𝒯) in Eqn. (<ref>) is defined such that it differs from the Identity operator (ℐ) by a set of atom-specific transformations 𝒯^a that has an effect only inside a certain atom-centered ball Ω_a with radius | - _a| < r_c^a referred to as augmentation sphere. Consequently, the all-electron valence Kohn-Sham wavefunctions can be expressed as

    _i() = _i()  + 
       ∑_a∑_α(^a_α()-^a_α())⟨^a_α|_i|⟩

where {^a_α} are the all-electron partial waves, {^a_α} are the pseudo-smooth partial waves that are used to expand {_i()} and {_i()} inside the augmentation sphere. Further, {^a_α} in Eqn. (<ref>) are the projector functions that are orthogonal to the pseudo smooth partial waves  i.e., ⟨^a_α|^a_β|=⟩δ_αβ. In the above Eqn. (<ref>) and subsequently in the manuscript, the atom-centered fields are denoted with superscript `a', implicitly denoting their functional dependence on ( - ^a). Furthermore, the atom-centered functions {^a_α}, {^a_α}, {^a_α} in Eqn. (<ref>) are of the separable form ϑ_α(r)S_lm  (θ), with ϑ_α(r)  being a radial function and S_lm(θ) is the real-spherical harmonic function that denotes the angular part of these atom-centered functions.
We note that r=|-^a| and  θ∈ℝ^2 represents the azimuthal and polar components of (-^a). 
Moreover, we note that the compact support of the functions ^a_α() and (^a_α()-^a_α()) lies in the augmentation sphere Ω_a, and consequently the decomposition in Eqn. (<ref>) transforms the orthonormality requirement of all-electron Kohn-Sham wavefunctions to 𝒮-orthonormality (𝒮=𝒯^†𝒯) condition on the pseudo-smooth(PS) wavefunctions {_i} i.e. ⟨_i|𝒮|_j⟩=δ_ij. 
 Eqn. (<ref>), further allows the decomposition of the all-electron density n() into a pseudo-smooth (PS) electron density () and an atom-centered correction which replaces the PS electron-density inside the augmentation sphere with the true all-electron density and is given by

    n()    = () + ∑_a(n^a() - ^a())
    with,  ()    = 2∑_i^Nf_i|_i()|^2 + _c() 
    while,  n^a()    = n^a_c()+ ∑_αβ^a_α()^a_β()D^a_αβ
    and  ^a()    =^a_c()+ ∑_αβ^a_α()^a_β()D^a_αβ

In the above equations, n^a() represents the atom-centered all-electron density and ^a(), the atom-centered PS electron density that matches with n^a() outside the augmentation sphere. Further, the matrix-components of the atom-dependent Hermitian matrix ^a (often referred to as channel occupancy matrix <cit.> or spherical density matrix <cit.>) are given by

    D^a_αβ = 2∑_if_i ⟨_i|^a_α|⟨%s|%s⟩⟩^a_β|_i

Using the decompositions in Eqns. (<ref>) and (<ref>), the all-electron kinetic energy T_s[{_i}] and the exchange correlation energy E_xc[n()] can be split into a smooth contribution  and an atom-centered correction term as follows:

    T_s[{_i }]    = T_s[{_i}] + ∑_aT^a_core + ∑_a∑_αβΔ T^a_αβD^a_αβ
     
    E_xc[n()]    = E_xc[()] +
     ∑_aΔ E^a_xc

where T^a_core is the kinetic energy contribution from the core states while the terms Δ T^a_αβ, Δ E^a_xc are atomic corrections computed as

    Δ T^a_αβ =  -1/2∫_Ω_a( ^a_α()∇^2^a_β() - ^a_α()∇^2^a_β()) d
       Δ E^a_xc = ∫_Ω_a(ϵ_xc[n^a,∇ n^a]-ϵ_xc[^a,∇^a])d


Finally, the all-electron electrostatics energy is decomposed into a smooth contribution and an atom-centered correction term by introducing a compensation charge ^a() compactly supported in Ω_a so that the multipole moments of (n^a() - ^a() + b^a() -^a()) vanish  <cit.> inside Ω_a. To this end, the compensation charge ^a() in terms of atom-centered spherical shape functions ^a_lm() is given by

    ^a() = κ^a^a_0() + ∑_lm,αβΔ^a_lmαβD^a_αβ^a_lm()

The atom-centered shape functions ^a_lm() in the above Eqn. (<ref>) are compactly supported in Ω_a and are usually constructed in the form of Gaussians, sinc-squared or Bessel functions <cit.> to satisfy the condition ∫_Ω_a r^lS_lm(θ)^a_l'm'()d = δ_ll'δ_mm'. Furthermore κ^a and Δ ^a_lmαβ in Eqn. (<ref>) are given by

    κ^a    = ∫_Ω^a(n^a_c(r)-^a_c(r))r^2dr-𝒵^a/√(4π)
    Δ ^a_lmαβ   = ∫_Ω_ar^l(^a_α()^a_β() - ^a_α()^a_β() )S_lm(θ̂)d


Now, the all-electron electrostatic energy can be computed as

    J[ρ(),{^a}] =  E_el[(),{^a} ] + ∑_aΔ E^a_el

where the smooth total density is given by () = () + (), with ()=∑_a^a(). The electrostatic energy involving () is given by

    E_el[(),{^a} ] = 1/2∫∫()()/|-|dd

and the atom-centered correction term is given by

    Δ E^a_el = 
    E^a_el[ρ^a()] - E^a_el[^a()]  - E^a_self[^a]   where
       E^a_el[ρ^a()] = 1/2∫_Ω_a∫_Ω_aρ^a()ρ^a()/|-|dd
       E^a_el[^a()]   = 1/2∫_Ω_a∫_Ω_a^a()^a()/|-|dd

with ρ^a() = n^a()+b^a() and ^a() = ^a()+^a(). The choice of the compensation charge in Eqn. (<ref>) allows one to evaluate  Δ E^a_el only within the augmentation sphere Ω_a.

Combining the various atom-centered correction terms from different energy contributions into Δ E^a, the all-electron energy in Eqn. (<ref>) in PAW method is reformulated as follows:

    E[{_i }, {^a}] = E[{_i}, {^a}] +∑_aΔ E^a[{D^a_αβ}]

 where we define E[{_i}, {^a}] and Δ E^a[{D^a_αβ}] as,

    E[{_i}, {^a}]    =  T_s[{_i } ] + E_xc[()] + E_el[(),{^a} ]  
    Δ E^a[{D^a_αβ}]    = Δ E^a_xc + Δ E^a_el + ∑_αβΔ T^a_αβD^a_αβ + T^a_core

and the details of evaluating Δ E^a[{D^a_αβ}] are mentioned in Appendix-<ref>.
At this juncture, we note from Eqn.(<ref>) that the all-electron Kohn-Sham energy functional of {_i} has been decomposed into an energy functional of smooth wavefunctions {_i} and a term that includes atomic corrections. The validity of this decomposition is reliant on two key assumptions. First, it is assumed that the augmentation spheres are non-overlapping and to achieve this, specialised PAW data sets are generated based on the application of interest <cit.>. Second assumption is that the set of partial waves ({_α} and {_α}) forms a complete basis within the augmentation sphere. However, in practice, only a finite number of partial waves are used. The error arising from this incompleteness is mitigated by introducing an atom-specific zero potential v̅^a()  with a compact support only inside augmentation sphere <cit.>. 

The electrostatic interaction energy involving the smooth total density E_el[(),{^a}] in Eqn. (<ref>) is extended in real-space and can be reformulated as a local variational problem in the spirit of <cit.>. Subsequently, the variational problem involved in computing this electrostatic energy contribution is given by:

    E_el[,{^a}] = -max_φ∈γ{1/8π∫|∇φ()|^2 d - ∫()φ()d}

where φ(,{^a}) denotes the trial function corresponding to electrostatic potential due to smooth electron-density (()) and the sum of atom-centered compensation charges (()). Finally, the problem of determining the all-electron ground state energy and the all-electron density for a given position of nuclei is expressed as the following variational problem involving the smooth electronic fields:



    E_GS = min_{_i}∈𝒳max_φ∈γ{ T_s[{_i}] - 1/8π∫|∇φ()|^2 d + ∫()φ()d + E_xc[()] + ∑_aΔ E^a[{D^a_αβ}] }


where the function space 𝒳 = {{_i} | <_p,_q >_𝕏 = δ_pq} with <_p,_q> denoting the 𝒮-inner product ⟨_p|𝒮|_q|$⟩ defined on𝕏. The space𝕏denotes a suitable function space that guarantees the existence of the minimisers. The domains used for numerical computations that are non-periodic corresponds to a large enough domain such that the wavefunctions{_i}decays to 0 and in periodic calculations, these correspond to supercells with periodic boundary conditions. Denoting such an appropriate domain byΩ, the appropriate function spaces for𝕏andγare𝕏=γ=H^1_0(Ω)in the case of non-periodic problems and𝕏 = γ= H^1_per(Ω)in the case of periodic problems.



 §.§ Governing Equations

The stationarity condition with respect to smooth fields{_i}corresponding to the PAW variational problem in Eqn. (<ref>) gives rise to nonlinear generalized Hermitian eigenvalue problem (GHEP) of the following form:

    ℋ_i = ε_i𝒮_i

whereℋis a Hermitian operator with

    ℋ = -1/2∇^2 + V_eff(,{^a}) + ℋ_nloc
       ℋ_nloc_i :=  ∑_a∑_αβ^a_α()Δ h^a_αβ∫^a_β() _i()d

and𝒮is a positive-definite Hermitian operator with

    𝒮 = ℐ + 𝒮_nloc
       𝒮_nloc_i :=  ∑_a∑_αβ^a_α()Δ s^a_αβ∫^a_β() _i()d

The set({ε_i},{ _i})are the eigenvalue and eigenfunction pairs of the generalised eigenvalue problem (<ref>) and theNsmallest eigenvalue and eigenvector pairs are used to compute the PS density()and the matrix^aemploying the Eqns. (<ref>) and Eqn. (<ref>) respectively. Further, the effective potential (V_eff) comprising of the total electrostatic potential, exchange-correlation potential and zero potential is computed as

    V_eff  = V_el + V_xc + V̅ =  δ E_el/δ + δ E_xc/δ + ∑_av̅^a

 where the electrostatic potentialδE_el/δ=φ(,{^a})is computed as the solution of the Poisson equation associated with the smooth total density()as given by eq.(<ref>).
Furthermore, the atom centered contributionsΔh^a_αβandΔs^a_αβinℋ_nlocand𝒮_nlocare expressed as   
    
    Δ h^a_αβ = δΔ E^a/δ D^a_αβ + ∑_lmΔ^a_lmαβ∫^a_lm()φ()d
        - ∫_Ω_av̅^a()^a_α()^a_β()d

    Δ s^a_αβ  = ∫_Ω_a(^a_α()^a_β()-^a_α()^a_β())d = √(4π)Δ^a_00αβ

Finally, the set of governing differential equations to be solved in the PAW method are:

    (-1/2∇^2 + V_eff(,{^a} )+ℋ_nloc)_i =ε_i(ℐ+𝒮_nloc)_i
      
        2∑_if(ε_i,μ) = N_v,   f(ε_i,μ) = 1/1+ exp(ϵ_i-μ/σ)
       () = 2∑_if(ε_i,μ)|_i()|^2, () = ∑_a^a()
       ^a() = κ^a^a_0() + ∑_lm,αβΔ^a_lmαβD^a_αβ^a_lm()
       Û^a_α i = ∫_α^a()_i()d, D^a_αβ = 2∑_if(ϵ_i,μ)Û^a^*_α iÛ^a_iβ
       () = () + (),  -1/4π∇^2φ(,{^a}) = ()

whereÛ^a^*_αiis the complex conjugate ofÛ^a_αi,μis the Fermi-energy andf_i:=f(ϵ_i,μ)is the orbital occupancy function lying in the interval[0,1]which is computed -via- the Fermi-Dirac distribution withσ= k_BTwhereTis the smearing temperature andk_Bis the Boltzmann constant.

When dealing with periodic crystals, 2D periodic slabs, or surfaces, we invoke the Bloch theorem <cit.> along the periodic axes. Instead of solving the DFT problem on a large periodic supercell, we solve the reduced problem on smaller unit cells with periodic boundary conditions. By invoking Bloch theorem, the eigenfunctions of the PAW generalised eigenvalue problem can be expressed as

    _i,() = e^i._i,()

wherei = √(-1)and_i,()is a unit-cell periodic complex valued function satisfying_i,(+_r) = _i,()for all lattice vectors_rwithdenoting a point in the first Brillouin zone(BZ) of the reciprocal lattice. The set of governing equations for the PAW formalism with Bloch wavefunctions_i,()can be recast as:

    (-1/2∇^2 - i·∇ + 1/2||^2 + V_eff(,{^a} )+ℋ^_nloc)_i,    = ε_i,(ℐ+𝒮^_nloc)_i,  on  Ω_p,  ∀∈ BZ 
    ℋ^_nloc_i, :=  ∑_a∑_αβ∑_re^-i· (-_r) ^a_α(-(^a+_r))Δ h^a_αβU^a_β i, 
    𝒮^_nloc_i, :=  ∑_a∑_αβ∑_re^-i· (-_r) ^a_α(-(^a+_r))Δ s^a_αβU^a_β i
    
    2∑_i_BZ2f(ε_i,,μ) d = N_v,   f(ε_i,,μ)  = 1/1+ exp(ϵ_i,-μ/σ), () = ∑_i_BZ2f(ϵ_i,,μ)|_i,()|^2 d, () = ∑_a^a()
    U^a,_α i := ∫_Ω_p∑_r'e^i· (-_r')p_α^a( - (^a+L_r'))_i,()d,  D^a_αβ = 2∑_i_BZf(ϵ_i^,μ)U^a,*_α iU^a,_i βd
    ^a() = κ^a^a_0() + ∑_lm,αβΔ^a_lmαβD^a_αβ^a_lm(), () = ()+(), -1/4π∇^2φ(,{^a}) = ()

where_BZdenotes the volume average of the integral over the first Brillouin zone (BZ) corresponding to the periodic unit cellΩ_p. We further note that the equations corresponding to non-periodic case in Eqn.(<ref>) can be easily recovered from the above Eqn. (<ref>) by considering (= 0,_r= 0) and replacing the periodic unit-cell domainΩ_pwith a non-periodic simulation domainΩ. We note that semi-periodic boundary conditions can be handled in a similar way depending on the number of periodic directions. Furthermore, we note that the set of equations in Eqn.(<ref>) or Eqn.(<ref>) involves a nonlinear generalized Hermitian eigenvalue problem (GHEP) that needs to be solved self-consistently in conjunction with the Poisson equation. In particular, we seek to solve the fixed point iteration[(),{ ^a}] = F([(),{ ^a}])using a self-consistent field (SCF) iteration procedure where the evaluation of the mapF([(),{ ^a}])involves the solution of the GHEP and the Poisson equation in Eqn.(<ref>) or Eqn.(<ref>) at every SCF iteration.









§ PAW-FE: FINITE ELEMENT DISCRETIZATION FOR PAW METHOD









Finite-element method involves decomposing the spatial domain of interest into non-overlapping subdomains called finite-elements (or cells) using a finite-element (FE) mesh. The underlying FE basis functions spanning a finite-element subspace are piecewise polynomials that exhibit systematically convergent behaviour <cit.>. Furthermore, these basis functions are strictly local, having a compact support only in the finite-element cells sharing a finite-element node, making them well-suited for massive parallelization. In contrast to conventional finite-element basis typically constructed from a tensor product of Lagrange polynomials interpolated through equidistant nodal points in a given finite-element, in this work, we specifically employ spectral finite-element basis that areC^0continuous Lagrange polynomials generated using Gauss-Lobatto-Legendre(GLL) nodal points that are the roots of derivatives of Legendre polynomials. We refer to previous works <cit.> about the advantages provided by higher-order adaptive spectral finite-element-based methods to solve the Kohn-Sham DFT problem in the context of norm-conserving pseudopotentials and all-electron DFT calculations. A crucial aspect of the computational methodology described below involves the use of these higher-order spectral finite-elements in conjunction with the reduced order Gauss-Lobatto Legendre (GLL) quadrature rules combined with the non-overlapping nature of augmentation spheres that enables the efficient inversion of the PAW overlap matrix. We discuss here the finite-element discretization of the governing equations deducing the algebraic generalized eigenvalue problem to be solved in the PAW method.


As seen from Eqn.(<ref>), the real-space formulation of the PAW method presented in this work results in a min-max problem involving the smooth pseudo-wavefunctions and the electrostatic potential. Employing a single FE discretization for both the wavefunctions and the electrostatic potential  can result in non-variational behaviour of the electronic ground-state energy. Therefore, we seek a solution for the electrostatics problem -via- the Poisson equation with a more stringent accuracy than the eigenvalue problem solution employing different orders of FE interpolating polynomial basis for the electrostatic potential and the wavefunctions. To this end, we represent the electronic fields in Eqn.(<ref>) in the FE basis and are given by 

    ^h_i,() = ∑_J=1^MN^h,p_J()^J_i,, φ^h,p_el() = ∑_J=1^M_elN^h,p_el_J()φ^J

where^J_i,,φ^Jdenote the FE discretized fields, whileN^h,p_J:1 ≤J ≤MandN^h,p_el_J:1 ≤J ≤M_eldenote the strictly local Lagrange polynomial basis functions of degreepandp_el, respectively spanning the finite-element subspace, generated using the nodes of the FE triangulation𝒯^hwith the characteristic mesh size denoted byhandp_el > pas discussed. To this end, the finite-element discretization of the Kohn-Sham generalized eigenvalue problem in Eqn.(<ref>) results in an algebraic generalized Hermitian eigenvalue problem^_i, = ε^h_i,^_i,, where^and^denote the discretized Hamiltonian matrix and the PAW overlap matrices of sizeM ×Mwithε^h_i,denoting theith eigenvalue corresponding to the discrete eigenvector_i,( see Appendix-<ref> for derivation of the FE discretized algebraic EVP). As discussed in  Appendix-<ref>, the finite-element discretized Hamiltonian and PAW overlap matrices has both local and non-local contributions and hence can be decomposed as^=^,loc+ ^,nlocand^=+ ^,nlocwhere the matrix entries of the local contributions^,locandare given by

    H_IJ^,loc = _Ω_p{1/2∇ N^h,p_I()·∇ N^h,p_J()   
       + V^h_eff() N_I^h,p()N_J^h,p()
       + 1/2||^2N_I^h,p()N_J^h,p() - i· N^h,p_I()∇ N^h,p_J()
       +^h_GGA() ·(N_I^h,p()∇ N^h,p_J()+N^h,p_J()∇ N^h,p_I() )} d
       M_IJ = ∫_Ω_pN_I^h,p()N_J^h,p()d

with

    V^h_eff =  (φ^h,p_el()+ V̅() +∂ϵ_xc(, ∇)/∂|_ = ^h) 
       ^h_GGA() = ∂ϵ_xc(, ∇)/∂∇|_ = ^h

and the discrete electrostatic potentialφ^h,p_el()inV^h_effabove is computed by solving the finite-element discretized Poisson equation= wheredenotes the Laplace operator discretized using the finite-element basisN^h,p_el_J:1 ≤J ≤M_elwhiledenotes the electrostatic potential vector whose entries are the expansion coefficientsφ^Jin Eqn.(<ref>) anddenotes the forcing vector corresponding to, the right-hand side of the Poisson equation in Eqn.(<ref>). To this end, the entries ofandare given by

    L_IJ   = 1/4π∫_Ω_p∇ N^h,p_el_I()·∇ N^h,p_el_J()d,
    
    c_I   = ∫_Ω_p^h()N^h,p_el_I()d

Finally, the matrix entries corresponding to the non-local contributions in^and^are given by

    ^,nloc = ∑_a^a,Δ^a_^a,†, ^,nloc = ∑_a^a,Δ^a_^a,†
       C^a,_J α = ∫_Ω_p∑_re^-i·(-_r)^a_α(-(^a+_r))N^h,p_J()d

where the atom-dependent coupling matricesΔ^a_andΔ^a_are of sizen^a_pjn^a_pjwithn^a_pjdenoting the number of projectors for atom indexaand the corresponding matrix entries denoted asΔh^a,h_αβandΔs^a_αβrespectively are given by


    Δ h^a,h_αβ = δΔ E^a,h/δD^a_αβ - ∫_Ω_av̅^a()^a_α()^a_β()d
        + ∑_lmΔ^a_lmαβ∫_Ω_p∑_r^a_lm(-(^a+_r))φ^h_el()d
       Δ s^a_αβ  = √(4π)Δ^a_00αβ

    
Since the FE basis functions are localised in real-space, the matrices^,locand^,locare sparse. Since the projectors^a(-^a)have a compact support within the augmentation sphereΩ_a, we note that theM ×n^a_pjmatrix^a,is also a sparse matrix for every atoma. Consequently, we can now rewrite the discretized Hamiltonian matrix and the overlap matrix within the PAW formalism as 
 
    ^ = ^_loc + ^Δ_^^†, ^ =  + ^Δ_^^†

where theM ×n_projmatrix^ = [^1,... ^a,...^N_a, ], withn_proj = ∑_a n^a_pjdenoting the total number of atomic projectors in the system. In the above Eqn.(<ref>),Δ_andΔ_are the block diagonal matrices of sizen_proj ×n_projwith each block corresponding to the atom-dependent coupling matricesΔ^a_andΔ^a_respectively as introduced in Eqn.(<ref>).



    
     
       
    
    
 Finally, the discretized governing equations corresponding to Eqn. (<ref>) that need to be solved in PAW-FE are given by:

    ^_i, = ε^h_i,^_i,
    
     2∑_n^N_BZf(ε^h_i,,μ)d = N_v,  f(ε^h_i,,μ) = 1/1+ exp(ε^h_i,-μ/σ)
    ^h() = 2∑_i^N_BZf(ε^h_i,,μ)|^h_i,()|^2d, 𝖴^a,_i = ^a,†_i,
    
    D^a,h_αβ = 2∑_i^N_BZf(ε^h_i,,μ)𝖴^a,^*_α i𝖴^a,_β i d
    ^h() = ∑_a^a,h(), ^h() = ^h()+^h(), =


We note that the discretized generalized Hermitian eigenvalue problem (GHEP) in the above Eqn.(<ref>) is a nonlinear eigenproblem and is solved forNsmallest eigenvalue-eigenvector pairs, which in turn are used to evaluate the smooth pseudo density^h()and the atom-centred density matrix{ D^a,h_α,β}. This nonlinear GHEP is self-consistently solved
in conjunction with the discretized Poisson equation subject to the applied boundary conditions in order to obtain the ground-state electronic fields and the energy. As noted before, replacing the periodic unit-cell domainΩ_pwith a non-periodic simulation domainΩand considering=0,_r=0, we recover the finite-element discretized equations corresponding to non-periodic boundary conditions from Eqn.(<ref>). Subsequently, the total all-electron energy in the PAW method,E^hin terms of the finite-element discretized ground-state solution fields(ε^h_n,,^h(), D^a,h_αβ, φ^h,p_el )is given by
    
    E^h =    2∑_i=1^N_BZf(ε^h_i,,μ)ε^h_i, d - ∑_a∑_αβΔ h^a_αβD^a,h_αβ
       - ∫_Ω_p[ ^h()V^h_eff() + ^h_eff,GGA() ·∇^h() ] d
       + 1/2∫_Ω_p^h()φ^h,p_el()d + E_xc[^h()] 
       + ∑_aΔ E^a[{D^a,h_αβ}]

The various computational aspects involved in efficiently solving the nonlinear generalized eigenvalue problem (Eqn. (<ref>)) using a self-consistent field (SCF) iteration procedure are discussed in the subsequent section. In the following section, we choose to drop theindex in the FE discretized electronic fields, vectors and matrices introduced in Eqn. (<ref>) for brevity, and we refer to them asũ_i^h(),_i,,andwithout theindex.  







§ SELF CONSISTENT FIELD ITERATION

In this section, we begin by describing the self-consistent field iteration (SCF) procedure adopted in our work to solve the nonlinear PAW generalized eigenvalue problem, highlighting the real-space mixing strategy employed in our work suited towards solving the Kohn-Sham fixed point problem inherent in the PAW method. Subsequently, we describe the Chebyshev filtering subspace iteration (ChFSI) approach and the related modifications to adapt ChFSI for efficiently solving the underlying generalized eigenvalue problem within a given SCF iteration. Additionally, we elaborate on the key computational strategies that leverage the underlying sparse finite-element structure and the compact support of the PAW projectors in real-space to facilitate fast and scalable calculations of computationally intensive steps in the ChFSI approach.




The finite-element discretized governing equations (<ref>) in the PAW method involves the solution of a nonlinear generalized eigenvalue problem that needs to be solved self-consistently as alluded to before. Since the total charge density is the sum of the smooth pseudo density () and the compensation charge () which is a function of the spherical density matrix^a, we note that the Kohn-Sham fixed point iteration map in the PAW method is written in terms of total charge density i.e_out^h() = F(_in^h()). To this end, we start with an input guess for the total charge density_in = _in^h + ^h(^a_in)and the mapF(_in^h())involves evaluating the effective potential (V^h_eff(),^h_GGA()), the non-local Hamiltonian coupling matrix (Δ^a_) using^h_into compute the eigenpairs (ε^h_i,_i) of the PAW generalized eigenvalue problem which are in turn used to  output total charge density_out = _out^h + ^h(^a_out). The above procedure is summarised in steps 2-5 of the Algorithm(<ref>) that describes the SCF iteration and these steps are continued until||^h_out() - ^h_in()||is lesser than the required tolerance in some  appropriate  norm. 
The computationally intensive step involved in the SCF iteration is solving the discretized generalised eigenvalue problem (step-4 of algo.(<ref>)). This step asymptotically scales cubically with the system size, making it crucial to employ efficient numerical strategies to delay the onset of cubic scaling and also reduce the number of SCF iterations. 

The following subsection describes the mixing strategy employed in our work to accelerate the convergence of the SCF procedure.


 §.§ Total charge density mixing

In contrast to DFT calculations involving norm-conserving pseudopotentials and all-electron methods, where pseudo-valence electron density or the all-electron density is mixed during the SCF iteration, in the PAW method, the total charge density (() = () + ()) needs to be updated self consistently. This is because the compensation charge()depends on{^a}which in turn depends on pseudo-smooth wavefunctions (see Eqn.(<ref>)) and hence needs to be mixed at every SCF iteration.

We employ an-stage Anderson <cit.> mixing scheme to accelerate the SCF iteration in the current work, with appropriate modifications relevant to the real-space PAW framework as discussed in this section. Denoting input total charge densities and output total charge densities at then^thSCF iteration as^(n)_in()and^(n)_out()respectively, the input total charge density for(n+1)^thSCF iteration^(n+1)_in()is computed as

    ^(n+1)_in = γ_mix_out + (1-γ_mix) _in

where_in(out)is expressed as a linear combination of input and output total charge densities fromnprevious SCF iterations. To this end, we have

    _in(out) = ^n_in(out) + ∑_k=1^n-1 c_k (^(n-k)_in(out) - ^(n)_in(out))

where we determine the linear combination coefficientsc_kby minimizing the norm square of the residualR = ||_res||^2_W = ||_out() - _in()||^2_Wwith the weighted norm defined as 

    ||_res||^2_W := ⟨_res|ℐ+κ_c|_res|⟩eq:weightnorm

 whereκ_cinbasis is the Coulomb kernel1/|- '|andℐis the Identity operator.κ_cin this weighted norm definition penalizes the term⟨_res|_res|$⟩ during the Anderson minimization step at longer wavelength changes of total charge density (i.e. at smaller magnitude of wavevectors, || → 0 in Fourier representation) in order to suppress charge sloshing in material systems with vanishing bandgaps as  explored in prior works <cit.>. Since the objective function  (<ref>) in Anderson mixing is non-local in real-space, we now provide a prescription to recast this equation into a local-form using an auxiliary potential corresponding to residual charge density _res() -via- the solution of the Poisson problem. To this end, we first note that  

    ⟨_res|κ_c|_res|   ⟩=  ∫∫_res()_res(')/|-'|dd'
       = ∫_res() φ_res() d

where φ_res() = ∫_res(')/|-'|d' denotes the auxiliary potential and is computed by solving the Poisson problem -1/4π∇^2 φ_res() = _res().  

 
 

Finally, from this Poisson equation and Eqn. (<ref>), we have the following

    ⟨_res|κ_c|_res|=⟩∫() φ_res() d =  -1/4π∫∇^2 φ_res() φ_res() d, 
       = -1/4π∫(∇.(φ_res∇φ_res) - |∇φ_res()|^2] ) d, 
       = 1/4π∫|∇φ_res()|^2 d,

where we used the divergence theorem for zeroing out the boundary term to get the last equality in the above equation.

Employing the finite-element discretization for the underlying electronic fields, we solve the following minimization problem in the Anderson mixing scheme.

    ||^h_res()||^2_W  :=  1/4π∫|∇φ^h_res()|^2d + ∫|_res^h()|^2d

where, φ^h_res() is the solution of the FE discretized Poisson equation _res=_res with  denoting the discretized Laplacian operator as given in Eqn. (<ref>) and
_res is the forcing vector given by ∫^h_resN_I^h,p_el d while the entries of the solution vector _res denotes the linear combination coefficients of the expansion of φ^h_res() in FE basis. Finally, the Anderson mixing coefficients c_k in Eqn. (<ref>) obtained by minimizing the weighted norm defined in Eqn.(<ref>) are employed to determine the input total charge density _in^h,(n+1) using Eqn. (<ref>). In other words, the linear combination coefficients c_k are used to determine ^h_in() and {^a,h_in} for the subsequent (n+1)^th SCF iteration. 

Using ^h(), the Hamiltonian() is computed and subsequently we solve the FE discretized generalized eigenvalue problem _i=ε_i_i. The following subsections describes the details of the efficient solution strategies employed in the current work.






 











 §.§ Subspace iteration procedure for PAW generalised eigenvalue problem

We employ the Chebyshev filtered subspace iteration approach(ChFSI) <cit.> to solve the FE discretized generalized non-linear eigenvalue problem in Eqn.(<ref>) for the smallest N eigenvector and eigenvalue pairs. ChFSI belongs to a class of iterative orthogonal projection methods and has been successfully employed in conducting large-scale real space DFT calculations over the past few years <cit.>. This approach relies on a Chebyshev filtering procedure that constructs a subspace rich in the wanted eigenvectors exploiting the fast growth property of Chebyshev polynomials outside the interval [-1,1], followed by a Rayleigh-Ritz step. The ChFSI approach can be viewed as a nonlinear subspace iteration procedure in the context of self-consistent field (SCF) iteration employed in solving the DFT nonlinear eigenvalue problem, allowing the reuse of the filtered subspace V spanned by the eigenvectors of a given SCF iteration to a subsequent iteration progressively improving it. To this end, the eigenspace of interest is refined at every SCF iteration, leading to a close approximation of the desired eigenspace of the non-linear eigenvalue problem as the SCF iterations proceeds towards convergence. Most importantly, the ChFSI method is naturally amenable for efficient use of high-performance computing architectures since the construction of the filtered subspace does not involve any interaction amongst the vectors, allowing for matrix-multivector multiplications to be done blockwise or in parallel over a few blocks, reducing the peak memory requirement <cit.>. 






Unlike most of the prior works <cit.> that employed ChFSI for standard eigenvalue problems, we need to solve here a large sparse generalized Hermitian eigenvalue problem (GHEP) as given in Eqn. (<ref>).  In the following, we describe the ChFSI approach adopted in the current work to solve this GHEP arising out of the finite-element (FE) discretization of the PAW method. 



  
Construction of Chebyshev filtered subspace: To solve the GHEP in Eqn. (<ref>), we consider the matrix 
^-1 that has the same eigenpairs as =ϵ.  To amplify the eigenspace of the interest corresponding to the N smallest eigenvectors (wanted spectrum) and dampen the unwanted spectrum, we seek to construct a Chebyshev polynomial filter involving  ^-1. This requires us to efficiently evaluate the inverse of , a large FE discretized sparse matrix. We now recall from Eqn.(<ref>) that  is of the form =  + Δ_^†, a rank-n_proj perturbation of .  We now use the Woodbury formula <cit.> to compute the inverse of , that can be expressed as

    ^-1   = M^-1 - ^-1Δ __inv^†^-1
    where  Δ__inv   = (Δ_^-1 + ^†^-1)^-1

We note that the above Equation (<ref>) requires the computation of inverse of the large M × M sparse FE basis overlap-matrix (), and the inverse of a smaller matrix (Δ_^-1 + ^†^-1) of size n_proj n_proj that scales with number of atoms. To evaluate ^-1, Gauss-Lobatto-Legendre (GLL) quadrature rules coincident with the finite-element nodes of spectral finite-elements are employed in this work. These GLL points are reduced-order quadrature rules and are used only to construct  in Eqn. (<ref>), rendering the matrix  diagonal denoted by _D. This leads to a trivial evaluation  <cit.> of ^-1 by inverting the diagonal entries of _D to be denoted as _D^-1 in the subsequent text. The other term in ^-1 which needs careful treatment, is the inverse of  (Δ_^-1 + ^†^-1). Recall from Eqns. (<ref>) and (<ref>), 
Δ_ is block diagonal with each block corresponding to each atom `a' and hence Δ_^-1 can be evaluated as the inverse of individual blocks Δ_^a. Furthermore, using the fact that the PAW augmentation spheres are localized in real-space and seldom overlap, the matrix ^†_D^-1 can be  approximated  as a block diagonal matrix with each block corresponding to an atom in the given material system. Hence, the block diagonal form of the Δ^a __inv matrix in real-space allows us to compute its inverse atom-by-atom. To this end, the approximate ^-1 can be recast as follows:

    ^-1   = _D^-1 - ∑_a_D^-1^aΔ^a __inv^a†_D^-1
    where  Δ^a __inv   = (Δ_^a^-1 + ^a^†_D^-1^a)^-1

In this work, we seek to construct a subspace rich in the desired eigenvectors of the PAW generalized eigenvalue problem =ϵ in Eqn. (<ref>)  by building a Chebyshev polynomial filter of =^-1 a close approximation to ^-1. To this end, the filtering procedure first relies on mapping the unwanted spectrum of  to [-1,1] and the desired spectrum to (-∞,-1). This is accomplished by computing a shifted and scaled matrix  through the affine transformation given by = 2/b-a -b+a/b-aI where a and b denote the upper bounds of wanted and unwanted  spectrum respectively. The upper bound b of the unwanted spectrum is obtained inexpensively by using a few steps of generalized Lanczos iteration <cit.> as discussed in the Algorithm (<ref>) which employs = _D + ∑_a^aΔ^a_^a^† and its inverse ^-1 as discussed in Eqn. (<ref>). A good approximation to the upper bound a of the wanted spectrum is obtained as the highest generalized Rayleigh quotient of  in the Chebyshev filtered subspace of the previous SCF iteration.

We now construct the filtered subspace by computing the action of a degree-m Chebyshev polynomial filter T_m() on an input subspace _in where the subspace _in is usually the eigenspace from the previous SCF iteration and is a matrix of size M × N. This action is accomplished -via- the three-term recurrence relation of Chebyshev polynomials  as given below:

    _f = T_m()_in where  T_m+1() = 2 T_m() - T_m-1()

The rapid growth property of Chebyshev polynomials outside [-1,1] amplifies the components in _in along the direction of the desired eigenvectors (occupied states) while dampening the components along the unwanted eigenvectors (unoccupied states), leading to the filtered subspace _f to be a close approximation of the eigenspace of interest of the PAW eigenproblem =ϵ in a given SCF iteration.
 



  
Rayleigh-Ritz step: Upon computing the filtered subspace _f in a given SCF iteration, we project the Hamiltonian() and PAW-overlap() onto the filtered subspace _f to construct projected matrices ^p and ^p of size N N as shown in step 5 of algo.(<ref>). The Rayleigh-Ritz step now involves solving a subspace eigenvalue problem for N eigenpairs and is of the form

    ^p = ^pΛ

where  and Λ  denote the eigenvector matrix and the diagonal eigenvalue matrix of the smaller subspace eigenvalue problem Eqn.(<ref>) respectively. Finally, a subspace rotation step (see step 7 of Algo.(<ref>)) is carried out to obtain the -orthonormal eigenvectors  in the finite-element subspace which is in turn used to compute the output total charge density ρ^h() = n^h + b().

A good approximation to the eigenspace of interest in the ChFSI approach employed in this work relies on the choice of the degree m of the Chebyshev polynomial filter. This choice depends on the separation of eigenvalues within the desired region of eigenspectrum and also, the magnitude of the largest eigenvalue (λ_max) of the matrix ^-1 that determines the ratio of spectral widths of the wanted to the unwanted regions of the eigenspectrum. We note that this λ_max is dictated by the fineness of the mesh size of the underlying finite-element discretization, i.e., it increases with decreasing mesh size. In the PAW method within the frozen core approximation, the electronic fields {ψ̃_i} (or equivalently {ũ^_i} in the periodic case) and φ vary smoothly compared to the electronic fields in the case of ONCV pseudopotentials for most atomic species. This allows for the use of coarser finite-element meshes in the case of PAW formalism, leading to the choice of Chebyshev polynomial degree between 10 and 30 for accurately approximating the desired eigenspace. The reduced degrees of freedom and the lower Chebyshev polynomial degree required to reach the desired chemical accuracies in the proposed finite-element framework for the PAW approach gives four to five fold computational advantage compared to current state-of-the-art finite-element code for DFT using ONCV pseudopotentials (DFT-FE) as will be demonstrated in the subsequent sections.

The ChFSI approach described here involves computational intensive operations notably in two key steps: first, constructing a subspace by computing , where  represents any one of the matrices  , , ^-1, and second, performing a Rayleigh-Ritz step by computing matrix-matrix products of the form ^†. We will now elucidate efficient strategies that exploit the compact support nature of the finite-element basis to evaluate these operations in a computationally efficient manner.











 §.§ Efficient computational methodologies

In the proposed ChFSI approach in Algorithm (<ref>), the efficient computation of matrix-matrix products of the form  and ^† is critical for enabling fast, accurate and scalable PAW-FE calculations on parallel computing architectures. We first consider the case of evaluating the matrix multivector products of the form  arising in the construction of Chebyshev filtered subspace, where the matrix  can be one of discretized Hamiltonian matrix(), PAW overlap() matrix or its inverse. We note here that the matrix  is a large sparse matrix of dimension M × M with M ≈10^3 - 10^8 while  is a large dense matrix of dimension M × N  with N ≈10 - 10^4.  A naive strategy often adopted in the finite-element literature to compute  would be to construct the FE discretized global sparse matrix  using sparse-matrix storage data-structures and then compute the sparse matrix-dense matrix product . This strategy would be highly inefficient on modern parallel computing systems involving multi-threaded and multi-node architectures owing to the expensive data movement costs involved in the underlying sparse-matrix algorithms. To this end, we employ a cell-matrix approach <cit.> that avoids the construction of global sparse-dense matrix-products and instead employs dense-dense matrix-products at FE cell level to reduce memory requirements, enhance cache locality, and improve arithmetic intensity of computation. We describe below the key mathematical aspects used in this implementation strategy to efficiently evaluate   during the ChFSI procedure in solving the PAW generalized eigenvalue problem.




Denoting the simulation domain by Ω_p, we begin by partitioning Ω_p into subdomains Ω^(t) ∀ t = 1,2,...,n_t, where n_t is the number of subdomains with each subdomain Ω^(t) assigned to an MPI task t. Further, let E_t be the number of finite-element (FE) cells and m_t be the number of basis functions (number of finite-element nodes) in each subdomain Ω^(t) such that Ω^(t) = ∪^E_t_e=1Ω^(e,t), where the notation (e,t) refers to the index of the FE cell 'e' in an MPI task 't'. This allows us to decompose the integrals over the domain Ω_p in Eqns. (<ref>), (<ref>) to a series of integrals over smaller domains Ω^(e,t) as

    ∫_Ω_pf()d = ∑_t∑_e∫_Ω^(e,t)f()d

Finally, the integrals over each domain Ω^(e,t) are computed using a suitable quadrature rule as:

    ∫_Ω^(e,t)f()d = ∑^N_q_qf(_q)w_q

where N_q is the total number of quadrature points in the domain Ω^(e,t) while w_q, _q are the quadrature weights and sampling points respectively. 3D Gauss Legendre quadrature rules that are tensor products of 1D quadrature rules <cit.> are employed in this work, and the order of these rules employed are determined such that errors due to the quadrature rule are of higher order than the discretization error. The flexibility in the choice of quadrature rules for evaluating various integrals involving PAW atomic data in  Eqns. (<ref>),(<ref>) allows the use of coarser finite-element grids to represent electronic fields and yet capture the relevant atom-centered data, a notable advantage compared to finite-difference approach, a popular real-space discretization DFT method, where special schemes need to be devised to retain the use of coarser grid resolution for smoother electronic fields without egg-box effects <cit.>. 

We note that the operation  can be decomposed as = _loc + _nloc (where := , , ^-1 and _loc:= _loc, , _D^-1 respectively). To this end, the mathematical expressions for  relevant to our implementation strategy within the finite-element framework can be written as 
 

    = [∑_t^n_t^(t)^T ^(t)^T∑_e^E_t^(e,t)^T(^(e,t)_loc + ∑_a^(a,e,t)Δ^a _(∑_t^a∑_e^a^(a,e^a,t^a)^†δ_e^aeδ_t^at) )^(e,t)^(t)^(t)]


    = [∑_t^n_t^(t)^T ^(t)^T∑_e^E_t^(e,t)^T(^(e,t) + ∑_a^(a,e,t)Δ^a _(∑_t^a∑_e^a^(a,e^a,t^a)^†δ_e^aeδ_t^at) )^(e,t)^(t)^(t)]


    ^-1 = ^-1_D - [∑_t^n_t^(t)^T ^(t)^T∑_e^E_t^(e,t)^T  ^-1_D^(e,t)∑_a^(a,e,t)Δ^a __inv(∑_t^a∑_e^a^(a,e^a,t^a)^†δ_e^aeδ_t^at) ^(e,t)^(t)^(t)]^-1_D
      

In the Eqns.(<ref>)-(<ref>), the Boolean sparse matrix ^(t) represents the partitioner matrix that acts on  (or ^-1_D) to provide the subdomain level matrix ^(t) while preserving the continuity of the nodal field across the subdomain boundaries. The m_t  m_t sparse matrix ^(t) represents the constraint matrix that constrains the values of the m_t × N matrix ^(t) at the specific FE nodes. These constraints either satisfy the necessary boundary conditions (periodic, non-periodic, or semi-periodic) imposed on the discretized electronic wavefunctions or address constraints from non-conforming meshes <cit.>. Furthermore, the n_p^3× m_t Boolean matrix ^(e,t) denotes the sub-domain to FE cell map associated with the sub-domain Ω^t whose action on ^(t)^(t) results in the extraction of the finite-element cell-level entries from ^(t)^(t) while ensuring the continuity condition of the discretized electronic wavefunction across the finite-element cells within the partitioned sub-domain Ω^(e,t). Efficient computation of  illustrated in Eqns.(<ref>)-(<ref>) can be summarized as a sequence of the following steps:

    
  * Precompute the entries of the cell-level matrices ^(e,t)_loc, ^(e,t)_loc ^(a,e) and Δ^a_, the atom dependent coupling matrices (Δ^a_:= Δ^a_, Δ^a_, Δ^a__inv) using the quadrature rules as discussed in Eqns.(<ref>) and (<ref>))
    
  * Extract the finite-element cell-level matrix:
 ^(e,t)= ^(e,t)^(t)^(t),∀ (e,t)
    
  * Evaluate the partial non-local operator action at the FE cell level: 
    
Δ^a_  = ∑_t^a∑_e^a^(a,e^a,t^a)^†δ_e^aeδ_t^at^(e,t) where δ_e^a e indicates that this computation is done in only those FE cells e^a lying in the augmentation sphere (Ω_a) centered around an atom `a'. Furthermore, δ_t^a t indicates that this computation is done only on those MPI tasks t^a which are associated with Ω_a.
    
  * Evaluate the local operator action along with the impending non-local operator action:

    ^(e,t) = ^(e)_loc^(e,t) + ∑_a^(a,e,t)Δ^a _Δ^a_
    
  * Assembly of the global vector =:

    = ∑_t^n_t^(t)^T ^t^T∑_e^E_t^(e,t)^T^(e,t)

We remark that the finite-element cell-level matrices ^e,t_loc and the coupling matrix Δ^a_ are computed at every SCF iteration while ^e,t_loc is computed for a given finite-element mesh. Further Δ^a _ is computed only once while Δ^a_^-1 is evaluated
only once per ground-state calculation.

Following the evaluation of =, the key computationally intensive operation in the construction of Chebyshev filtered subspace, the subsequent step is the Rayleigh-Ritz step that requires the evaluation of matrix-matrix products of the form ^† (see Algorithm 3). To this end, the matrices , are partitioned row wise into _p,_p across various MPI tasks. To this, the local _p^†_p is evaluated to obtain the smaller matrix N N in each MPI tasks and finally, the local contribution is summed up across all MPI tasks to obtain ^† = ∑_p_p^†_p. 

So far, in this section we discussed the various computational methodologies employed in the proposed PAW-FE approach that can enable accurate large-scale DFT calculations on material systems up to tens of thousands of electrons in a computationally efficient manner. The following section discusses the accuracy and performance results of the proposed computational approach while benchmarking with state-of-the-art plane Wave(PW) codes Abinit<cit.> and Quantum Espresso<cit.>.











§ RESULTS AND DISCUSSION






In this section, we demonstrate the accuracy, performance and parallel scalability of the proposed method  on representative benchmark problems involving periodic, semi-periodic and non-periodic boundary conditions. We first discuss the accuracy of our implementation on non-periodic systems involving molecular systems and small metallic nano-clusters by comparing energetics with plane-wave-based PAW implementations. We then demonstrate the performance of  in terms of both computational node-hours and minimum wall time with respect to plane-wave calculations for increasingly large sizes of metallic Cu nanoclusters containing up to 17537 electrons (923 atoms). Subsequently, we demonstrate the accuracy of our method compared to plane-wave calculations on semi-periodic and fully periodic systems involving unit-cells, large super-cells, surfaces. We further discuss the performance of our method in terms of computational node-hrs and minimum wall time for increasingly large sizes of Cu_3Pt periodic supercells containing up to 34,304 electrons (2048 atoms) compared to plane-wave DFT calculations using PAW. Finally, we showcase the computational efficiency and parallel scalability of  compared to norm-conserving ONCV pseudopotential DFT calculations using  on large-scale material systems involving ∼50,000 electrons.

All the DFT calculations reported in this work employ GGA <cit.> exchange-correlation functional of the PBE <cit.> form. Furthermore, we employ the PAW data-sets from the pseudo-dojo database <cit.>. In all our simulations, we use Fermi-Dirac smearing with T=500 K.  Additionally, for the mixing of the total charge density involving (()) and {^a}, we employ the n-stage Anderson mixing <cit.> as discussed in the Section <ref>.

For all accuracy validation studies in this work, we employ refined finite element (FE) meshes with FE interpolating polynomial (p) of degree 7. These meshes are constructed such that the differences in the all-electron ground-state energies (Eqn. (<ref>)) between successively sub-divided meshes reach 10^-6 Ha/atom. In addition, the choice of quadrature integration rules and FE interpolating polynomial for electrostatics p_el is chosen such that the energy variation with respect to these parameters is an order of magnitude lower than this discretization error. In the case of plane-wave calculations we use [ version  is used for the accuracy validation] code and the cut-off energy for wavefunctions E^wfc_cut is chosen so that the ground-state energy is converged up to 10^-6 Ha/atom while simultaneously ensuring that energy change with respect to the cut-off energy for total charge density E^rho_cut is an order of magnitude lower.

The comparative study involving the performance of our  implementation with that of plane-wave calculations has been conducted by solving a given DFT problem to the same level of accuracy. To this end, we choose the discretization parameters such that the discretization error with respect to a very refined calculation in each case to be around ∼2 × 10^-4 Ha/atom for our comparisons. We employ () code[ version  with ELPA is used for the performance benchmarking study. The authors observed  was computationally efficient than  on few benchmark examples.] for conducting our performance studies using plane-wave DFT calculations. Furthermore, we use two metrics to compare the performance of PAW-FE and the plane-wave calculations: (i) computational cost (η_c) in node-hrs, and (ii) minimum wall time (τ^min_c) in secs. The computational cost η_c is obtained by multiplying the minimum number of compute nodes required to fit a given problem with the average wall-time per SCF iteration[Average wall-time per SCF is obtained by dividing the wall time taken for the self-consistent field iteration procedure with the number of SCF iterations taken to reach a default convergence threshold for self-consistency]. Further, the minimum wall time (τ^min_c) is obtained by computing average wall time per SCF by increasing the number of compute nodes till a parallel scaling efficiency of around 35% is reached. All the simulations involved in the accuracy and performance benchmarking studies were performed on high-memory CPU nodes of PARAM Pravega[PARAM Pravega is one of India's fastest supercomputers stationed at Indian Institute of Science comprising of 156 High Memory Intel Xeon Cascade-Lake based CPU nodes (7,488 Cores) where each node consists of 48 Intel Xeon 6248 processor, 768 GB memory and Mellanox HDR Infiniband interconnect between all the nodes for fast MPI communication.]. 



 §.§ Non-periodic systems

In this subsection, we consider the case of non-periodic systems for our accuracy and performance benchmarking. In particular, for accuracy validation studies, we consider isolated molecules such as CO, CH4, H2O and also metallic nano-clusters such as Cu13, Al13. We benchmark the accuracy of   with  by comparing the energy vs bond length curves and the ground state energies computed at the minimum energy configuration. Following the accuracy validation study, we assess the performance of our implementation by comparing the computational cost (η_c) and minimum wall time(τ^min_c) of   with Quantum Espresso(), a widely used code employing plane-wave basis. In this performance benchmarking study, we consider three-dimensional icosahedron<cit.> shaped Cu nanoparticles of various sizes ranging from 55 atoms (1045 electrons) to 923 atoms (17537 electrons). Homogeneous boundary conditions have been imposed on the electrostatic potential (φ^h()) for these calculations using   while in plane-wave codes, periodic boundary conditions (PBCs) are only admissible and hence PBCs are employed using a suitable vacuum to minimise image-image interactions. 



  §.§.§ Accuracy benchmarking

 In all the accuracy benchmarking studies reported in this subsection, we choose refined finite-element (FE) meshes with FE interpolating polynomial degree p to be 7 and the mesh size h to be around 1.0 Bohr while in the case of [. We use the =0 in  to enforce Blöchl formulation of exchange-correlation(XC) contribution with =0 to avoid any approximation in the XC correction term computation.], we use a plane-wave cut-off energy E^wfc_cut=60 Ha. Figure (<ref>) reports the energy as a function of inter-atomic distance obtained from  and   for the molecular systems O2,  H2O, NH3 and CH4. Since the scale of ground-state energies is different in the two implementations compared here, the energies in the plots of Fig. (<ref>) are measured relative to the minimum ground-state energy obtained in this study using  and  for each of the above molecular systems. We now describe the details of the simulations conducted for our comparative study: (i) Oxygen: The energy as a function of O-O bond length is plotted in Fig. (<ref>)(a) and the bond length for the minimum energy configuration is found to be 2.30 Bohr in   which is in excellent agreement with , (ii) Ammonia: The trigonal pyramidal geometry of ammonia is considered where the N-H bond lengths are varied such that the H-N-H bond angle is maintained to be 106.2^∘. In the Fig. (<ref>)(b), we plot the energy variation as a function of this N-H bond length and we observe the N-H bond length at the minimum energy configuration to be around 1.93 Bohr using   which matches very closely to the value obtained with , (iii) Water: The O-H bond lengths in the water molecule are varied while ensuring that the planar H-O-H bond angle of 104.46^∘ is maintained. Energy of the molecule is plotted as a function of this O-H bond length in Fig. (<ref>)(c) and we observe from our   calculations that the O-H bond length at the minimum energy configuration is 1.83 Bohr which is in close agreement with , (iv) Methane: The tetrahedral geometry of methane is considered where the C-H bond lengths are varied such that H-C-H angle is maintained at 109.5^∘. Fig. (<ref>)(d) shows the comparison between  and  calculations. These results indicate that the C-H bond length at the minimum energy configuration is 2.07 Bohr in both cases.
 
 To summarize the observations from Fig. (<ref>), we find an excellent agreement in all the energy vs bond-length curves between  and  for the representative molecules considered in this work. Furthermore, the bond length at the minimum energy configuration obtained from our implementation shows a very close match with that of plane-wave calculations as summarized in Table <ref>. We now compare the ground-state energies of a few representative systems obtained using   with that of  in Table<ref>. To this end, we consider the above four molecular systems at the minimum energy configuration obtained from Fig. <ref>. Additionally, we consider the 13-atom icosahedron nanoclusters Cu_13 and Al_13 with 4.8 Bohr and 5.3 Bohr as the nearest neighbour distance, respectively <cit.>. To have a one-on-one comparison of ground-state energies with  for these non-periodic systems considered in this work, we compute pseudo valence energy in  using the expression in Eqn.(<ref>) as opposed to all-electron energy in Eqn.(<ref>). We note that Table (<ref>) shows an excellent match between the ground-state energies obtained from   and , and the differences in the energies are O(10^-5)Ha/atom. Having established the agreement of  with a plane-wave code in terms of energetics, we now focus on performance benchmarking of  against the plane-wave implementations.
 
  
 






  §.§.§ Performance benchmarking

We benchmark the performance of  with Quantum espresso <cit.>, a widely used open-source plane-wave (PW) code, by comparing the average CPU time per SCF in terms of computational node-hrs(η_c) and minimum wall time(τ^min_c) with increasing sizes of Cu nanoclusters<cit.> solved to similar level of accuracy between the two approaches. These icosahedron-shaped nanoclusters considered in this study are constructed with 4.8 Bohr nearest neighbour distance<cit.>, and the nanocluster size is increased by varying the number of shells. To this end, non-periodic DFT calculations using the PAW method are conducted on Cu3-shell (2793 electrons), Cu4-shell (5871 electrons), Cu5-shell (10,659 electrons), Cu6-shell (17,537 electrons). The two metrics (η_c,τ^min_c) used to benchmark the performance results are discussed at the beginning of this section. In , the degree of the FE interpolating polynomial p is chosen to be 5, and a mesh size h=1.2 Bohr is employed with a Chebyshev polynomial degree of 20 in the ChFSI step. While in the case of  , the plane-wave cut-off energy E^wfc_cut=25 Ha is used with Γ-point for k-point sampling and default settings are used for the eigensolver. These discretization parameters are chosen so that the discretization error in the ground-state energies obtained in both   and   is ∼ 2 × 10^-4Ha/atom. Furthermore, we used a vacuum of around 20 Bohr and 15 Bohr from the farthest atom of the nanocluster in  and  respectively, ensuring that ground-state energies are converged up to O(10^-5)Ha/atom with vacuum size.

Table (<ref>) reports the average CPU time per SCF iteration in terms of computational node-hrs (η_c) and the number of basis functions in  and  for various sizes of Cu nanoclusters studied here. From this table, we find that for system sizes greater than ∼ 1000 electrons,  becomes more efficient than . We observe the efficiency gains for  increases with system size over , achieving gains of around 3.1×, 4.3× and 5.1× for Cu_4-shell, Cu_5-shell and Cu_6-shell respectively, the benchmark systems considered for CPU-time comparison.  
 This increase in computational gains is attributed to the need for using more processors to satisfy the peak memory requirement, where the efficient parallel scalability of  provides the necessary advantage. Further, in this case of non-periodic systems, the spatial adaptivity of finite-element basis allows the FE mesh to be coarse-grained into the vacuum in contrast to a uniform spatial resolution of the plane-wave basis in , which is also a key factor for the superior gains observed in . Additionally, we estimate the computational complexity of  from Table(<ref>) and is found to be of N_e^1.8 while for  we observe it to be of N_e^2.6. Hence, we expect to see further gains of  over  with increasing system size.
 
 Finally, the Table <ref> compares the average time per SCF in terms of minimum wall time(τ^min_c) obtained in both  and . We note that the minimum wall time of  is measured at ∼ 30%-40% scaling efficiency while a scaling efficiency of ∼ 50%- 55% is reached for [  could be scaled further on more nodes, however, the maximum available 140 high-memory nodes on PARAM-Pravega limited the further reduction of minimum wall time]. Even for a small system involving Cu3-shell comprising of 2793 electrons, we observe a gain of around 3.2 in the minimum wall time for  compared to , and we see almost a staggering speed up of 10 for Cu6-shell comprising of 17,537 electrons, the largest system considered in this study. The increased speedup with system sizes is attributed to the superior scaling of  which requires only nearest neighbour communication and the various efficient computational methodologies employed in solving the PAW eigenvalue problem using the Chebyshev filtered subspace iteration approach (ChFSI) in  (see Section <ref>).












  


 §.§ Periodic/Semi-periodic systems

In this subsection, we consider the case of periodic and semi-periodic systems for our accuracy and performance benchmarking. In particular, we benchmark the accuracy of  with  by comparing energy versus lattice parameters of representative bulk systems involving Li2O, Cu, Cu3Pt and Li unit cells. Further, we compute and benchmark with  the formation energy of Li2O and the surface energies of the semi-periodic systems – Zr(001) and La(011) terminated Lithium Lanthanum Zirconium Oxide (LLZO), an electrolyte material. Finally, we compare the band structure of C diamond primitive cell obtained from  and .  Following the accuracy benchmarking study, we assess the performance of  with   on two metrics (η_c, τ^min_c) on increasing sizes of Cu3Pt supercells.






  §.§.§ Accuracy benchmarking

In all the accuracy benchmarking studies of periodic/semi-periodic systems reported here, we compute the all-electron PAW ground state energy using Eqn. (<ref>) in . Further, we choose refined finite-element (FE) meshes with FE interpolating polynomial degree to be 7 and the mesh size h to be around 1.0 Bohr while in the case of  , we use a plane-wave cut-off energy E^wfc_cut=60 Ha.



  
Energy vs lattice parameter: Figure (<ref>) shows the variation of ground-state energy as a function of lattice parameter for the case Li2O(cubic), Cu(fcc), Cu3Pt(fcc) and Li(bcc) unitcells. We use orthogonal unit-cells in each case and employ shifted 888 Monkhorst-Pack k-point grid for sampling the Brillouin zone <cit.>. In Fig.(<ref>), we plot the energies relative to the minimum ground-state energy obtained in this study using  and  for each of the above material systems. This is done to account for different energy scales in the two implementations considered here. From Fig.(<ref>), we observe an excellent correspondence between  and . Furthermore, we also observe from Table. <ref> that the lattice parameters at the minimum energy configurations show an excellent match for all the representative periodic systems considered here.


 


  
Formation energy: Using the ground-state energies at the minimum energy configuration obtained from Fig. <ref>, we compute the formation energy Δ E^Li2O_f of Li2O as 
Hartree-per-formula unit(Ha/f.u.) using the following expression:

    Δ E^Li2O_f = 1/4E_Li2O - E_Li - 1/2E_O2

To this end, Δ E^Li2O_f is found to be -0.223095 Ha/f.u. in , while in the case of , we obtain a value of -0.223126 Ha/f.u. This results in a difference of ∼ 3.210^-5 Ha/f.u., establishing a good agreement in the formation energy computed between the two codes




  
Surface energies: Next, we compute and validate the surface energies of LLZO (Lithium Lanthanum Zirconium Oxide), a material of interest for solid electrolytes, focusing on La(011) and Zr(001) terminations. We obtain the coordinates from a study of LLZO particle morphology<cit.>. The slab is placed at the center of the simulation domain with 25 Bohr vacuum on either side. In  , semi-periodic boundary conditions are employed for the electronic fields, with homogeneous Dirichlet boundary conditions for φ^h() on the boundary parallel to the surface. While in , periodic boundary conditions are employed. The surface energy(γ) is computed using the following expression in the units of J/m^2 

    γ = 1/2A_surface( E_surface - E_bulk)

where A_surface is the surface area, E_surface is the ground-state (GS) energy of the material systems with the surface of interest and E_bulk is the GS energy of the corresponding bulk system and the values obtained from  and  are shown in Table <ref>. From this table, we observe a close match in surface energies computed from both codes. The difference in surface energies are ∼ 210^-3 J/m^2. Additionally, it is observed that the Zr(001) surface has higher surface energy in both codes, consistent with observations from previous studies<cit.>.





  
Band structure comparisons: We now compute the band-structure of the diamond primitive cell of Carbon with lattice constant 3.373 Bohr. The Brioullin Zone(BZ) and the k-point path are shown in Fig.(<ref>) and were obtained using seeK-path<cit.>. A shifted 888 Monkhorst-Pack k-point grid was used for the initial SCF calculation for sampling the Brillouin zone, following which a non-self-consistent field (NSCF) calculation was performed using the k-points specified along the high symmetry points of Fig.(<ref>) to extract the band structure. Figure (<ref>) shows the overlay of the band structure computed using  and .  We observe an excellent match in the eigenvalues and, hence, the band structure between the two codes.  
    

The comprehensive benchmarking studies discussed here establish the accuracy of our  computational methodology and implementation for periodic/semi-periodic systems. We now proceed to discuss the performance aspects of  for periodic systems with increasing system sizes.






















  §.§.§ Performance benchmarking

We benchmark here the performance of  with  similar to the non-periodic systems discussed previously and compare the computational efficiency and minimum wall time on increasingly large supercells of Cu3Pt crystal solved to similar levels of accuracy between the two approaches. The supercells are constructed by stacking of Cu3Pt unit cells along the three lattice dimensions and to this end, we consider supercells of sizes 4× 4 × 4 (4288 electrons), 5 × 5 × 5 (8375 electrons), 6 × 6 × 6 (14472 electrons), 7 × 7 × 7 (22981 electrons), 8 × 8 × 8 (34304 electrons). We perform a Γ-point calculation with periodic boundary conditions imposed on all electronic fields in both  and . Further, the discretization parameters are chosen such the discretization error in the ground-state energies obtained in both cases is 210^-4Ha/atom. In particular, in  , the degree of the FE interpolating polynomial p is chosen to be 6, and a mesh size of h=1.8 Bohr is employed with Chebyshev polynomial degree of 20 in the ChFSI step. While in the case of  , the plane-wave cutoff energy E^wfc_cut = 25 Ha is used with default settings for the eigensolver.

Table(<ref>) and (<ref>) presents the computational cost[The computing system used for conducting the performance benchmarking studies are high memory nodes ( 768 GB per node) which favors   as the minimum number of nodes required to fit peak memory requirement is lesser.] and the minimum wall time comparison between  and . We recall that the computational cost (η_c) is computed as the product of minimum number of nodes required to fit the problem and the average wall time per SCF iteration. From the Table.(<ref>), we observe that the number of basis functions required by  is a factor of 6 higher than  and we begin to see computational gains in  for systems greater than 8000 electrons. The efficiency gains we achieve with an increase in system sizes are around 1.7×, 2.3× and 3.2× for 6 × 6 × 6, 7 × 7 × 7, 8 × 8 × 8 Cu3Pt supercells respectively, the benchmark systems considered for CPU-time comparison in this work. We note that the computational gains obtained in the periodic case are lesser than the case of non-periodic systems because of the degrees of freedom advantage that plane-wave (PW) codes provide for bulk systems (∼ 500 PW basis functions per atom) compared to non-periodic systems (∼ 5000 PW basis functions per atom) to reach the desired accuracy.  However, on estimating the computational complexity of  from Table (<ref>) we find that  scales as N_e^2.2 while  scales as N_e^2.8 indicating that we will obtain an increase in computational gains for  compared to  as the system size.

Finally, the Table (<ref>) compares the average time per SCF in terms of minimum wall time (τ^min_c) between  and . To this end, we observe a gain of around 2.3× for Cu3Pt 5 × 5 × 5 supercell with 8375 electrons and the computational gains increase to almost 7.7× for system size comprising of 34,304 electrons. Even for fully periodic calculations, these increased gains with system size underscore the importance of fast and scalable computational methods developed in this work.


 
 





 




 §.§ Scalability and parallel efficiency

In this subsection, we compare the computational cost and parallel scaling efficiency of the proposed computational methodology  relative to the norm-conserving pseudopotential (ONCV) calculations <cit.> using the  code solved to a similar level of accuracy. We remark here that the transferability of ONCV pseudopotential DFT calculations have been shown to be on par with PAW approaches <cit.> for most material systems in contrast to the early norm-conserving Troullier-Martins (TM) pseudopotentials <cit.> and have become one of the popular method of choices in most DFT implementations including . However, as alluded to before, these ONCV pseudopotentials can become very hard, requiring a large number of degrees of freedom to reach the desired accuracy for most material systems involving elements with d or f electrons. Towards this study, we consider 999 Cu3Pt supercell and an icosahedral-shaped Pt8-shell nanocluster <cit.>, with the nearest neighbour distance being 5.27 Bohr, as representative systems for periodic and non-periodic calculations. We obtain the PAW data set and ONCV pseudopotential data from pseudodojo <cit.> repository for our comparative study. The finite-element mesh parameters in both cases are determined such that the discretization error is around 210^-4 Ha/atom. Table (<ref>) shows the details of number of electrons, finite-element (FE) mesh parameters, number of FE basis functions between the ONCV based  and  calculations and the Chebyshev polynomial degree employed during the ChFSI step. The simulations for this study were conducted on the OLCF supercomputer Frontier [Frontier is the world's fastest exascale computer. Each compute node of Frontier has  64-core AMD Epyc 7713 processor with 512 GB memory and multiple slingshot NICs for communication between nodes.] CPUs. 

To assess the strong scaling behaviour between  using ONCV pseudopotentials and the  calculations, we plot the average wall time per SCF iteration τ_c against the number of compute nodes in Fig. (<ref>).
We observe that  is almost an order of magnitude (∼ 10× to 12×) faster than the  calculations using the ONCV pseudopotentials in terms of computational node-hrs for the large-scale representative systems considered here. Furthermore, we observe similar or better parallel scaling efficiency, while requiring 4× to 7× less number of compute nodes to fit the given problem in the case of . The efficiency gains obtained in  can be attributed to (i) coarser finite-element mesh size leading to much lesser degrees of freedom compared to the ONCV pseudopotential-based  calculations to reach the desired chemical accuracy, (ii) reduced Chebyshev polynomial degree during the ChFSI step due to the coarser mesh size in  calculations, (iii) lesser number of valence electrons within the frozen-core approximation of the PAW method in the case of Pt leading to better parallel scaling efficiency of ChFSI due to reduced communication costs in the Rayleigh-Ritz(RR) step. We further note that the computational gains of  compared to   using ONCV pseudopotential calculations may vary depending on the pseudopotential repository used, and the current comparisons use the data sets obtained from pseudo-dojo<cit.> repository, as mentioned before.








    










§ SUMMARY

In this work, we present a computationally efficient real-space methodology for large-scale density functional theory (DFT) calculations using finite-element (FE) discretization of the projector-augmented wave (PAW) method. To the best of our knowledge, this represents the first finite-element-based approach for DFT calculations, combining the efficiency of PAW formalism involving smooth electronic fields with the ability of systematically convergent higher-order finite-element basis to achieve significant computational efficiencies on modern computing systems. The methodology exploits the compactly supported nature of FE basis functions and further takes advantage of the adaptive resolution offered by the FE basis to handle generic boundary conditions efficiently. Furthermore, this work demonstrates a substantial computational advantage over the current state-of-the-art plane-wave-based PAW implementations for medium to large-scale material systems, thereby pushing the boundaries of length-scales that can be realised from ab initio calculations than possible today. 

The development of the proposed approach is based on the amalgamation of the following key ideas. First, we deduced the finite-element (FE) discretized governing equations corresponding to Blöchl's PAW method by making use of a locally reformulated electrostatics functional involving the smooth electron density and compensation charge. Further, higher-order quadrature rules were utilized to accurately evaluate the underlying integrals involving atom-centered functions in these governing equations, thereby allowing the use of coarser FE grids for the various electronic fields. Secondly, we employed a self-consistent field (SCF) iteration approach based on Anderson mixing of smooth electron density and atom-dependent spherical density matrix. This is accomplished by minimizing the weighted norm of the total charge density residual, which is locally reformulated using the gradients of the electrostatic potential corresponding to this residual. Next, we developed an efficient strategy to compute the inverse of the FE-discretized PAW overlap matrix  which is essential for solving the underlying generalized eigenvalue problem. Leveraging this strategy for computing ^-1, we designed a Chebyshev polynomial-based filter within each SCF iteration to compute the desired eigenspace. The FE-cell-level structure of both the discretized Hamiltonian () and the inverse PAW overlap matrix (^-1) is exploited to devise computationally efficient strategies that  employ cache-friendly matrix-matrix products to evaluate the action of the Chebyshev filter on the trial subspace during the course of Chebyshev filtered subspace iteration procedure.

The accuracy of the proposed method was validated against a plane-wave implementation of PAW across various non-periodic, semi-periodic, and fully periodic representative benchmark systems. Compared to the plane-wave calculations, our proposed methodology() showed excellent agreement in ground-state energies, energy variation with bond length and lattice parameters, formation energies, surface energies, and band structures. Furthermore, compared to the state-of-the-art plane-wave implementation, the efficient computational strategies employed in  enabled a 5 reduction in computational cost in non-periodic systems(∼18000 electrons) and a 3 in periodic systems (∼34000 electrons) with more significant gains observed as system size increases. Additionally, in terms of minimum wall time we observe substantial speed-ups for both periodic and non-periodic systems (∼ 7×-10×) using . Additionally, the proposed computational strategies exhibit excellent parallel scaling efficiency (∼70%) even at 1200 degrees of freedom per MPI task and achieves a substantial ten-fold reduction in computational cost over current state-of-the-art  code using norm-conserving pseudopotentials for system sizes close to 50,000 electrons.

The proposed work, , offers a systematically convergent basis capable of handling generic boundary conditions while substantially reducing the number of basis functions required to achieve desired chemical accuracy compared to the current state-of-the-art finite element-based DFT methods using norm-conserving pseudopotentials. Furthermore, the computational methodologies developed are well-suited for modern GPU architectures, which can significantly accelerate  calculations and reduce computational time—a focus of future investigation. This work constitutes a significant step towards enabling ab initio calculations of complex systems at larger length scales, thereby aiding in tackling a diverse set of key scientific and technological problems. Additionally, it paves the way for longer time-scale molecular dynamics simulations with ab initio accuracy by providing a more comprehensive energy landscape for robust statistical and machine learning-based computational models.



§ ACKNOWLEDGEMENTS

The authors would like to thank Prof. Vikram Gavini for many helpful discussions and his valuable suggestions. The authors gratefully acknowledge the seed grant from Indian Institute of Science and SERB Startup Research Grant from the Department of Science and Technology India (Grant Number: SRG/2020/002194). The research used the resources of PARAM Pravega at Indian Institute of Science, supported by National Supercomputing Mission (NSM) R&D for exa-scale grant (DST/NSM/R&D Exascale/2021/14.02). This research also used resources of the Oak Ridge Leadership Computing Facility at the Oak Ridge National Laboratory, which is supported by the Office of Science of the U.S. Department of Energy under Contract No. DE-AC05-00OR22725. K.R. would like to acknowledge Prime Minister Research Fellowship(PMRF) from the Ministry of Education India for financial support. S.D acknowledges the support of Toyota Research Institute. P.M. acknowledges Google India Research Award 2023 and Indo-Korean Science and Technology center Bengaluru for financial support during the course of this work.  




§ ENERGY EVALUATION

The reformulated all-electron energy functional Eqn.(<ref>) in the PAW method is expressed in terms of E[{_i}, {^a}] and Δ E^a[{D^a_αβ}].  To this end, the atom-centered correction term is expressed as

    Δ E^a[{D^a_αβ}] = Δ E^a_xc + Δ E^a_el + ∑_αβΔ T^a_αβD^a_αβ + T^a_core

  In this section, we discuss the evaluation of various terms involved in computing Δ E^a[{D^a_αβ}] for completeness which is inline with prior works<cit.>. 


  
Kinetic Energy correction: 


    


    Δ T^a_αβ =    (1/2∑_l_α l_β∑_m_α m_β(∫dϕ_α(r)/drdϕ_β(r)/drr^2drδ_l_α l_βδ_m_α m_β + ∫_Ω_aϕ_α(r)ϕ_β(r)dr(l_α+1)(l_α)δ_l_α l_βδ_m_α m_β). 
       -. 1/2∑_l_α l_β∑_m_α m_β(∫_Ω_ad_α(r)/drd_β(r)/drr^2drδ_l_α l_βδ_m_α m_β + ∫_Ω_a_α(r)_β(r)dr(l_α+1)(l_α)δ_l_α l_βδ_m_α m_β))



  
Electrostatics energy correction


    


    Δ E^a_el = 
    E^a_el[ρ^a()] - E^a_el[^a()]  - E^a_self[^a] = Δ𝒞^a + ∑_αβΔ𝒞^a_αβD^a_αβ + ∑_αβα'β'D^a*_αβΔ𝒞^a_αβα'β'D^a_α'β'


    where,  Δ𝒞^a =    1/2( ∫_Ω_a∫_Ω_an_c^a()n_c^a()/|-|dd - ∫_Ω_a∫_Ω_añ_c^a()ñ_c^a()/|-|dd - (κ^a)^2∫_Ω_a∫_Ω_a_00^a()_00^a()/|-|dd)  
        - κ^a ∫_Ω_a∫_Ω_añ_c^a()_00^a()/|-|dd -√(4π)𝒵^a∫_Ω_an^a_c(r)/rdr


    Δ𝒞^a_αβ =   ∫_Ω_a∫_Ω_aϕ^a_i()ϕ^a_j()n^a_c()/|-|dd - ∫_Ω_a∫_Ω_a^a_i()^a_j()ñ^a_c()/|-|dd  -𝒵^a∫_Ω_aϕ^a_i()ϕ^a_j()/||d
        -κ^a ∫_Ω_a∫_Ω_a^a_i()^a_j()g_00^a()/|-|dd - Δ^a_00αβ(κ^a ∫_Ω_a∫_Ω_a_00^a()_00^a()/|-|dd + ∫_Ω_a∫_Ω_añ^a_c()_00^a()/|-|dd)


    and  Δ𝒞^a_αβα'β' =   1/2[ ∫_Ω_a∫_Ω_aϕ^a_α()ϕ^a_β()ϕ^a_α'()ϕ^a_β'()/|-|dd - ∫_Ω_a∫_Ω_a^a_α()^a_β()^a_α'()^a_β'()/|-|dd]  
        -∑_L[ 1/2Δ^a_lmα'β'∫_Ω_a∫_Ω_a^a_α()^a_β()_lm^a()/|-|dd + 1/2Δ^a_lmαβ∫_Ω_a∫_Ω_a^a_α'()^a_β'()_lm^a()/|-|dd]
       -∑_lm1/2Δ^a_lmαβ∫_Ω_a∫_Ω_a_lm^a()_lm^a()/|-|ddΔ^a_lmα'β'

where, r = |-^a| and κ^a,Δ^a_lmαβ are defined in Eqn.(<ref>). We note that 1/|-| in the above equations can be expanded in terms of real spherical harmonics as

    1/|-| =
    ∑_ℓ∑_m=-ℓ^ℓ4π/2ℓ+1min(r,r')^ℓ/max(r,r')^ℓ+1
    S_ℓ,m(θ)S_ℓ,m(θ')

where r = |-^a| and θ∈ℝ^2 represents the azhimuthal and polar coordinates of  in spherical coordinates, with r' = |-^a| and θ' represents the azhimuthal and polar coordinates of  in spherical coordinates. Using the above Eqn.(<ref>), the integrals involved in Eqns. (<ref>), (<ref>), (<ref>) can be
evaluated in spherical coordinate system as discussed in detail in the Appendix of <cit.>.




  
Exchange Correlation Energy correction:


    

The evaluation of atom centred exchange-correlation is evaluated on a radial grid using the Lebedev quadrature rule<cit.> for the angular integration and Simpson's rule for the radial integration.

    Δ E^a_xc = ∫_Ω_a(ϵ_xc[n^a,∇ n^a]-ϵ_xc[^a,∇^a])d

where

    n^a() = n^a_c(r)S_00(θ) + ∑_αβϕ^a_α(r)ϕ^a_β(r)S_l_α m_α(θ)S_l_β m_β(θ)D^a_αβ


    ∇ n^a() = [          ∂ n^a_c(r)/∂ rS_00(θ) + ∑_αβ(∂ϕ^a_α(r)/∂ rϕ^a_β(r) + ∂ϕ^a_β(r)/∂ rϕ^a_α(r) )S_l_α m_α(θ)S_l_β m_β(θ)D^a_αβ;        ∑_αβ1/r(ϕ^a_α(r)∂ S_l_α m_α(θ)/∂ϑϕ^a_β(r)S_l_β m_β(θ) + ϕ^a_β(r)∂ S_l_β m_β(θ)/∂ϑϕ^a_α(r)S_l_α m_β(θ))D^a_αβ; ∑_αβ1/r sin(ϑ)(ϕ^a_α(r)∂ S_l_α m_α(θ)/∂ςϕ^a_β(r)S_l_β m_β(θ) + ϕ^a_β(r)∂ S_l_β m_β(θ)/∂ςϕ^a_α(r)S_l_α m_β(θ))D^a_αβ ]

where θ = [ϑ,ς] ∈ℝ^2 represents the polar and azhimuthal coordinates in spherical coordinate system. Further Eqn.(<ref>) can be simplified  using the below expressions

    ∂ S_lm(θ)/∂ϑ = - √(2l+1)/√(2l+3)√((l+1)^2 - m^2)S_l+1,m(θ)/sin(ϑ)


    ∂ S_lm(θ)/∂ς = -|m|S_lm̅(θ),  where,  m̅ = -m

The expression for ^a() can be derived following the above steps in a similar fashion. Finally, the exchange-correlation energy correction term is evaluated as

    Δ E^a_xc = ∑_qw_q∫(ϵ_xc[n^a(r,θ_q),∇ n^a(r,θ_q)]-ϵ_xc[^a(r,θ_q),∇^a(r,θ_q)]) r^2dr

Further, the atom-centered exchange correlation potential correction (Δ V^a, αβ_xc) is expressed as

    Δ V^a,αβ_xc = ∑_qw_q(∫δ E^a_xc[n^a,∇ n^a]/δ D^a_αβϕ^a_α(r)ϕ^a_β(r) r^2dr - ∫δ E^a_xc[^a,∇^a]/δ D^a_αβ^a_α(r)^a_β(r) r^2dr)S_l_α m_α(θ_q)S_l_β m_β(θ_q)

where, q refers to the Lebedev quadrature points used to sample θ and w_q are the corresponding quadrature weights while the above radial integrals are evaluated until augmentation radius (r^a_c) using the adaptive Simpsons integration rule. We note that the Eqns. (<ref>), (<ref>)  are efficiently evaluated by recasting the integrands into a dense matrix-matrix multiplication in our  implementation.




  
PAW Valence Energy for non-periodic systems:
Recall from Eqn.(<ref>), the FE discretized all-electron PAW energy for a non-periodic system corresponding to a simulation domain Ω can be computed using the following expression

    E^h =    2∑_i=1^Nf(ε^h_i,μ)ε^h_i - ∑_a∑_αβΔ h^a_αβD^a,h_αβ 
            - ∫_Ω[ ^h()V^h_eff() + ^h_eff,GGA() ·∇^h() ] d
       + 1/2∫_Ω^h()φ^h,p_el()d + E_xc[^h()]+ ∑_aΔ E^a[{^a,h}]  where  ^h() = ^h() + ^h()

    We note that the above expression for E^h includes the energy contribution of the frozen core, and in order to exclude the frozen core contribution in E^h and compare the ground-state energies one-on-one with  <cit.> in our benchmarking studies, we introduce the PAW valence energy E^h_v for non-periodic systems as follows
    
    E^h_v = E^h - ∑_aE_xc[n^a_c() ] -∑_aC^a_corr,  where,  E_xc[n^a_c() ] = ∫_Ω_aϵ_xc[n^a_c,∇ n^a_c]d
       and 𝒞^a_corr = ∫_Ω∫_Ω^a_c()^a_c()/|-|dd - (E^a_el[n^a_c()+b^a()] - E^a_el[^a_c()]), ^a_c() = ^a_c() + κ^a_0^a()



§ VARIATIONAL FORMULATION OF THE PAW ENERGY FUNCTIONAL AND FINITE-ELEMENT DISCRETIZATION

In this section, we present the variational formulation of the PAW energy functional and subsequently derive the finite-element discretized equations to be solved in the PAW method. For brevity, the expressions derived below assume the case of a non-periodic isolated system or a periodic system involving a large supercell with Γ-point sampling. To this end, we make the assumption that the wavefunctions are real; however, the derivation is trivially extendable for unit-cell calculations with multiple -point sampling of the Brillouin zone involving complex wavefunctions. We begin with the all-electron Kohn-Sham energy functional in Eqn.(<ref>) to be minimized subject to 𝒮-orthonormality constraints. Consequently, we introduce the Lagrange multiplier matrix (Λ) for enforcing these constraints and the functional to be minimized to determine the ground-state energy for given positions of nuclei {^a} in the PAW method can be written as follows:

    F[{_i },φ,Λ] = E[{_i },φ] - ∑_ijλ_ij(∫_i()𝒮_j()d-δ_ij)

Now, taking the variational derivative of the above functional with respect to _n, we have

    2 f_n∫1/2∇_n()·∇δ_n()d + 2 f_n∫(φ()+V̅)_nδ_n d + ∑_a∑_αβΔ^a_lmαβ∫δ D^a_αβ/δ_n()δ_n()d∫^a_lm()φ()d
        + ∫δ E_xc/δ()(∫δ()/δ_n()δ_n()d)d + ∫δ E_xc/δ∇()·[∫(δ∇()/δ_n()δ_n() + δ∇()/δ∇_n()δ∇_n() )d] d
        +∑_a∑_αβδΔ E^a/δ D^a_αβ∫δ D^a_αβ/δ_n()δ_n()d = 2∑_j(λ_nj+ λ_jn)/2∫δ_n()𝒮_j()d

where,

    δΔ E^a[{D^a_αβ}]/δ D^a_αβ = (Δ T^a_αβ + Δ E^a_el,αβ + 2∑_α'β'Δ E^a_el,αβα'β'D^a_α'β' + Δ V^a, αβ_xc - ∫_Ω_av̅^̅a̅(r)S_00(θ)^a_α()^a_β()d) =  Δ h^a_αβ

with

    ∫δ D^a_αβ/δ_n()δ_n()d = 2 f_n∫_α^a() δ_n()d∫_β^a() _n()d

Finally, we have

    f_n∫1/2(∇δ_n()·∇_n() +(δ E_xc/δ() + φ())δ_n()_n())d + f_n∫δ E_xc/δ∇() ·( _n()δ_n() + δ_n()_n() ) d
       + f_n∑_a∑_αβ∫^a_α()δ_n()d(Δ h^a_αβ+∑_lmΔ^a_lmαβ∫_lm^a()φ()d)∫^a_β()_n()d= ∑_jλ^S_nj∫δ_n()𝒮_j()d

where the entries λ^S_nj = λ_nj+λ_jn/2 are the entries of a symmetric matrix Λ^S = 1/2(Λ + Λ^T).
Similarly, taking the variational derivative of Eqn. (<ref>) with respect to φ(), we get

    1/4π∫∇φ()·∇δφ()d = ∫()δφ()d
 
where the simulation domain for Eqns.(<ref>) and (<ref>) is represented by Ω. The variatonal quantities are such that _n() ∈𝕏 and φ()∈γ with 𝕏=γ = H^1_0(Ω) in the case of non-periodic problems and 𝕏=γ=H^1_per(Ω) in the case of periodic problems.
 Now, we let 𝕍^h ∈𝕏, 𝕍^h_el∈γ denote the finite-dimensional subspaces of dimensions M and M_el respectively, and in the discrete setting, we seek the approximate solution _n^h()∈ 𝕍^h, φ^h()∈𝕍^h_el  such that 

    f_n∫_Ω1/2(∇δ_n()·∇^h_n() +(δ E_xc/δ() + φ^h())δ_n()^h_n())d + f_n∫δ E_xc/δ∇() ·( ^h_n()δ_n() + δ_n()^h_n() ) d
       + f_n∑_a∑_αβ∫^a_α()δ_n()d(Δ h^a_αβ+∑_lmΔ^a_lmαβ∫^a_lm()φ^h()d)∫^a_β()^h_n()d= ∑_jλ^S_nj∫δ_n()𝒮^h_j()d
       ∀ δ_n() ∈𝕍^h

and 

    1/4π∫∇φ^h()·∇δφ()d = ∫()δφ()d   ∀ δφ∈𝕍^h_el

Further, let the finite dimensional subspace 𝕍^h be spanned by the finite-element basis { N^h,p_I() }  for   1 ≤ I ≤  M and the subspace 𝕍^h_el be spanned by the FE basis {N^h,p_el_I()}, for  1 ≤ I ≤   M_el. This allows us to express the approximate solution ^h_n() uniquely as ^h_n() = ∑_JN^h,p_J()u_n^J and, subsequently φ^h() as φ^h() = ∑_JN_J^h,p_el()φ^J.
 Moreover, since Eqn.(<ref>) is valid for all δ_n() ∈𝕍^h, choosing δ_n() = N^h,p_I() successively for every I, we now obtain the FE discretized equations to be solved in the form of the following matrix equations 

    f_n∑_J∫1/2∇ N^h,p_I()·∇ N^h,p_J()u_n^J d + f_n∑_J∫V^h_eff()N_I^h,p()N_J^h,p()u_n^J d
     +     f_n∑_J∫V^h_GGA·(N^h,p_I()∇ N_J^h,p() + ∇ N_I^h,p()N^h,p_J())u_n^J d
       + f_n∑_J∑_a,αβ∫N^h,p_I()^a_β()d(Δ h^a_αβ+∑_lmΔ^a_lmαβ∫^a_lm()φ^h()d)∫_α^a()N_J^h,p()u_n^Jd
       = ∑_jλ^S_nj∑_J(∫N^h,p_I()N^h,p_J()u_j^Jd + ∑_a,αβ∫N^h,p_I()^a_β()d√(4π)Δ ^a_00αβ∫_α^a()N_J^h,p()u_j^Jd)

Rewriting the above expression in the matrix-form we have

    𝖴 = 𝖴Λ^S

where, the matrix 𝖴 = [𝗎_1... 𝗎_n ...𝗎_N] with the entries of the vector 𝗎_n given by u^J_n for 1 ≤ J ≤  M which are the linear combination coefficients of ^h_n() when expressed in FE basis. Now, we note that the matrix Λ^S in the Eqn. (<ref>) is a symmetric matrix which is orthogonally diagonalizable to be of the form Λ^S = ^T
 and let ϵ^h_nf_n be the entries of the diagonal matrix . Furthermore, introducing the unitary transformation of the discretized wavefunction matrix 𝖴 using the orthogonal matrix  diagonalizing Λ^S, we have = 𝖴 which can then be used to transform the matrix equation Eqn. (<ref>) to obtain the following generalised eigenvalue problem in the PAW method

    _n = ϵ^h_n_n

where the entries of the FE-discretized Hamiltonian () and the PAW overlap matrix are given by Eqns.(<ref>) to (<ref>) in Section (<ref>). Additionally, since Eqn.(<ref>) is satisfied ∀ δφ∈𝕍^h_el, we choose δφ() = N_I^h,p_el() successively for all I to finally obtain the following FE discretized Poisson's equation

    1/4π∑_J∫∇ N_I^h,p_el()· N_J^h,p_el()φ^Jd = ∫^h()N_I^h,p_el()d
 




§ CHEBYSHEV FILTERED SUBSPACE ITERATION

As discussed in Section (<ref>), we use a self-consistent field iteration (SCF) procedure to solve the nonlinear PAW generalised eigenvalue problem. This involves the solution of a linearized generalised eigenvalue problem at each SCF iteration for which we employ the Chebyshev filtered subspace iteration(ChFSI) approach. As outlined in Algo.(<ref>), ChFSI requires upper and lower bounds of the eigenspectrum corresponding to the FE discretized eigenvalue problem = ϵ^h_n for which we use the generalized Lanczos procedure discussed in Algo.(<ref>).
     

   









 Paper Title: Weak $\mathbb{Z}_2$ Supertopology.










k.parshukov@fkf.mpg.de
Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany



RIKEN Center for Emergent Matter Science, Wako, Saitama 351-0198, Japan


Max Planck Institute for Solid State Research, Heisenbergstrasse 1, D-70569 Stuttgart, Germany

§ ABSTRACT 
Crystal symmetries can enforce all bands of a material to be topological, a property that is commonly referred to as “supertopology".
Here, we determine the symmetry-enforced ℤ_2 supertopologies of non-magnetic centrosymmetric materials with weak
and strong spin-orbit coupling (SOC). For weak (i.e., negligible) SOC, crystal symmetries can enforce Dirac nodal lines
protected by a π-Berry phase, 
while for strong SOC, crystal symmetries can give rise to nontrival weak ℤ_2 topologies in 2D subplanes
of the 3D Brillouin zone. We catalogue all centrosymmetric space groups whose symmetries enforce these  ℤ_2 supertopologies.
Suitable material realizations are identified and experimental signatures of the supertopologies, such as surfaces states and quantum Hall responses,
are being discussed. 


Weak ℤ_2 Supertopology
    Andreas P. Schnyder
    August 5, 2024
=======================





§ INTRODUCTION


In the last decade and a half, the study of materials with non-trivial band topology has become an important field of physics <cit.>. 
The exotic properties of these materials, such as chiral band crossings, anomalous surface states, and topological transport, are described by topological invariants, e.g., Chern numbers or
ℤ_2
indices <cit.>.
In order to understand the origin of these topological band features, it is necessary to investigate the connection between  symmetry and topology: 
The crystal symmetries do not only lead to symmetry-related copies of the topological band features, but
 they enforce by themselves certain nontrivial band topologies. For example, screw rotations
 enforce Weyl points on rotation axes <cit.>, while glide mirrors 
 enforce nodal lines <cit.>.
This property is referred to as “symmetry-enforced topology" <cit.> and guarantees that bands must be topological irrespective of the chemical composition and other details of the material. 
The principle of symmetry-enforced topology can be used to design or search for new materials with the desired topological characteristics <cit.>.
Another intriguing class of materials is called “supertopological materials" <cit.>, for which all connected bands isolated in energy have a stable topological invariant.
To obtain topological
responses in supertopological materials, a precise tuning of the electronic filling is not necessary,
since all bands are topologically nontrivial. 

So far, symmetry-enforced band topologies due to screw and glide symmetries have been enumerated <cit.> and 
their origin has been explained in terms of three fundamental theorems <cit.>. Systematic catalogues of topological materials have revealed a large number of supertopological compounds, where every band has a nontrivial ℤ_2 Kane-Mele index <cit.>. However, these surveys do not explain the underlying mechanism for the supertopology and, moreover, weak ℤ_2 invariants are not  considered for partial fillings of connected sets of bands.


In this paper, we derive the symmetry-enforced  ℤ_2 supertopologies in nonmagnetic centrosymmetric materials with weak (i.e., negligible) and strong spin-orbit coupling (SOC).
We find that for strong SOC, crystal symmetries can enforce weak ℤ_2 topologies in 2D subplanes of the 3D Brillouin zone (BZ), while for weak SOC crystal symmetries
can guarantee the existence of Dirac nodal lines. 
We enumerate all centrosymmetric space groups (SGs), whose symmetries enforce these weak ℤ_2 topologies and Dirac nodal lines, see Tables <ref> and <ref>. 

Our approach makes use of the Fu-Kane ℤ_2 symmetry indicator for centrosymmetric systems <cit.>, which expresses the Kane-Mele ℤ_2 index <cit.> in terms of inversion eigenvalues at the time-reversal invariant momenta (TRIMs).
This index can be defined for both strong and weak SOC <cit.>, indicating ℤ_2 topology of a (partial) band gap or the existence
of Dirac nodal lines, respectively.  
Centrosymmetric non-magnetic materials exhibit a space-time inversion symmetry P𝒯, which for  strong (weak) SOC squares to -1 (+1). As a consequence,
the electronic bands are doubly degenerate, either due to Kramers theorem (when (P𝒯)^2 = -1)  or due to spin degeneracy (when (P𝒯)^2 = +1). 
Hence, at the TRIMs the ℤ_2 indicator is only well-defined for even fillings. In addition, depending on the considered space group, extra crystal symmetries
can lead to fourfold degeneracies at the TRIMs, such that the ℤ_2 indicator is only defined for fillings that are multiples of four.
In our analysis we need to treat these two cases separately, see Secs. <ref> and <ref>, respectively. 

The remainder of this paper is organized as follows.
In Sec. <ref> we review the definition and significance of the ℤ_2 indicator both for strong and weak SOC.
The space groups whose crystal symmetries enforce nontrivial ℤ_2 topologies
are determined in Secs. <ref> and <ref>, see Tables <ref> and <ref>.
In both of these sections, we focus first on a particular example, i.e., space groups 15 (C2/c) and 135 (P4_2/mbc), respectively, and then generalize the deliberations to other space groups.
Material examples are discussed throughout all of these three sections and summarized in Table <ref>. 
We conclude in Sec. <ref> with some final remarks and directions for future research.
Some technical details are given in the appendices. 
 







§ ℤ_2 SYMMETRY INDICATOR





For inversion-symmetric materials with time-reversal symmetry (𝒯^2=± 1) one can define a ℤ_2 indicator in terms of inversion eigenvalues at the TRIMs <cit.>.
This works for any filling,
for which there is a full band gap
at all considered TRIMs.
For a given 2D plane containing the four TRIMs Γ_i (i ∈{a,b,c,d}) the ℤ_2 indicator
is defined as

    (-1)^ν = ∏_i ∈{a,b,c,d}δ_i,   δ_i = ∏_m=1^N/2ξ_2m(Γ_i),

where δ_i is a product of inversion eigenvalues ξ_2m at the TRIM Γ_i of the doubly degenerate bands 2m, and N is the number of occupied bands. In the following, we assume N to be even to ensure an insulating gap at the TRIMs.

In the case of negligible SOC with (P𝒯)^2=+1,  
the indictor ν, Eq. (<ref>), describes the value of  the quantized Berry phase <cit.>, evaluated along any time-reversal symmetric loop C passing through the four TRIMs <cit.>

    C = c_ab - c_ab - (c_cd - c_cd),

where c_ab and c_ab are time-reversed paths between the two TRIMs Γ_a and Γ_b. Specifically, the Berry phase γ∈{ 0, π} is given by (see Ref. PhysRevLett.115.036806 and Appendix <ref>)

    e^iγ = (-1)^ν.

The non-trivial value protects a nodal line piercing a surface enclosed by the contour. In other words, the contour and line degeneracy are linked. The degeneracy is protected by the (P𝒯)^2=+1 symmetry.

SOC splits spin degeneracies in general. For time-reversal symmetric systems, it implies 𝒯^2=-1. Therefore, the symmetry (P𝒯)^2=+1 changes to (P𝒯)^2=-1, and
the line degeneracy discussed above is gapped out. At the same time, all the electronic bands stay doubly-degenerate due to the Kramers theorem. 
If the indicator [Eq. (<ref>)] is defined for four TRIMs in a plane and the SOC opens a gap, the value determines the Kane-Mele ℤ_2 invariant.
The topological phase implies the emergence of helical states at the material surface.




§ TWOFOLD DEGENERACIES AT TRIMS





Crystal symmetries can enforce degeneracies at high-symmetry points in the Brillouin zone <cit.>. To begin with we consider the space groups with four two-fold degenerate TRIMs due to the P𝒯 symmetry. Inversion eigenvalues for degenerate states are the same and cannot be determined by the symmetries. However, the ℤ_2 indicator can be enforced if the eigenvalues are related between different TRIMs <cit.>.



 §.§ Space Group 15 (C2/c)


We consider space group 15 as an example of a space group enforcing a ℤ_2 invariant to be non-trivial for 4ℕ+2 occupation. 

The monoclinic space group 15 (C2/c) is generated by translations along x-, y-, z-axes t(1, 0, 0), t(0, 1, 0), t(0, 0, 1) respectively, glide mirror symmetry M_010(0, 0, 12) along the y-axis, inversion P, and the translation t(12, 12, 0), which is characteristic for the base-centered lattice <cit.>. 
The action of symmetries can be represented as follows

    M_010(0, 0, 12): (x, y, z)    → (x, -y, z+12), 
    
        P: (x, y, z)    → (-x, -y, -z),

where we consider the spinless representations.

A sketch of the BZ is shown in Fig. <ref> (a) following Bilbao crystallographic server notation <cit.>. For more information about the basis vectors, see Appendix <ref>. We are interested in a plane in the BZ with four TRIMs L_1(-π, -π, π), L_2(-π, π, π), V_1(-π, -π, 0) and V_2(-π, π, 0).



The little groups for all of L and V points contain only inversion symmetry. Generally, it means that the little group is abelian with 1D irreducible representations. 
It ensures the absence of any symmetry-protected degeneracies.


The mirror symmetry M_010(0, 0, 12) acts on a vector 𝐤=(k_x, k_y, k_z) in the BZ relating it with R_M_010𝐤=(k_x, -k_y, k_z) = 𝐤̃, where R_M_010 is a transformation matrix in k-space. Therefore, the mirror reflection relates V_1 and V_2 points, L_1 and L_2 points.

Now we consider a Bloch state |ψ(𝐤)⟩ at momentum 𝐤. It is related by mirror reflection symmetry with the state D_M_010|ψ(𝐤)⟩ = |ψ̃(R_M_010𝐤)⟩, where D_M_010 is symmetry representation in Hilbert space. These two states have the same energy.

At TRIMs parities ξ of Bloch states |ψ(Γ_i)⟩ are well-defined, because Γ_i = -Γ_i + G, where G is a reciprocal vector. Importantly, the parities at |ψ(𝐤)⟩ and D_M_010|ψ(𝐤)⟩ are connected. The connection can be deduced from the commutation relation D_P D_M_010 = D_M_010 D_P D_t(0, 0, 1), where D_P, D_t(0, 0, 1) are representations of inversion and translation operators, respectively. For proof of the statement, see Appendix <ref>. When applied to a Bloch state at 𝐤 the expression is written as

    D_P D_M_010 |ψ(𝐤)⟩ = e^i k_z D_M_010 D_P |ψ(𝐤)⟩.

Now if the parity of the state |ψ(𝐤)⟩ is defined at 𝐤 and equal to ξ, then parity of D_M_010|ψ(𝐤)⟩ is e^i k_zξ.

Applying the argument to L and V points we conclude that 

    ξ(L_1) = -ξ(L_2),   ξ(V_1)    = ξ(V_2), 
    ξ(L_1)ξ(L_2)ξ(V_1)ξ(V_2)    = -1.

The Berry phase Eq. (<ref>) for the contour Eq. (<ref>) passing the four TRIMs is enforced to be π for every band. The quantized non-trivial Berry phase implies the presence of the nodal line piercing the surface enclosed by the contour.

A SOC-free material with symmetries of the SG 15 has a nodal line between the conduction and valence bands if an odd number of Kramers partners is occupied N = 4ℕ + 2. 
The presence of the degeneracy is due to the band inversion at one of the TRIMs. The SOC lifts the line degeneracy opening a gap inside the plane with L and V points. The two bands are necessarily topological, and the gap between them is non-trivial. Formally, the relation between inversion eigenvalues Eq. (<ref>) is true in both regimes. For systems with strong SOC, the actions of time-reversal and mirror acquire a spin-part, transforming the spin degree of freedom. However, the inversion is not affected, and the relation Eq. (<ref>) is satisfied. 
The Dirac nodal line can be considered as a parent state for the topological insulating phase.
Every disconnected band in the plane with L, V points has the non-trivial ℤ_2 invariant. Therefore, every second gap in the plane (4ℕ + 2 occupation) has a gapless surface state.


We constructed a tight-binding (TB) model with symmetries of the space group 15 with and without SOC Fig. <ref> (b)-(c) [see Appendix <ref>]. There is an enforced crossing inside the plane, corresponding to the piercing nodal line. The degeneracy is split under SOC opening a gap inside the L-V plane. 
For the constructed TB models, we introduced a termination direction along the y-axis to observe surface states due to non-trivial topology, see Fig. <ref> (b)-(c). 

As TRIMs are defined by Γ_i = -Γ_i + G, there are several options to choose a plane (contour) with four discussed points and non-trivial ℤ_2 invariant. For SG 15 we can consider L_1(π, π, -π) and V_1(π, π, 0) instead, then the contour with π Berry phase encloses a nodal line pinned to high-symmetry line AM.



 §.§ Generalizations to other space groups

To generalize the approach, we list criteria that we used to find all centrosymmetric space groups, whose symmetry enforces a non-trivial invariant in a 2D submanifold of the 3D BZ.

    
  * The δ_i product should be well-defined at four TRIMs [see Eq. (<ref>)]. For the 4ℕ+2 occupation it is possible only if there is no enforced degeneracy at the TRIMs, except the spin-degeneracy. It means there are only one dimensional irreps of the TRIMs single-valued little groups in case of the negligible SOC, and all double-valued group irreps must be two-dimensional for strong SOC. 
    
  * The space group must contain a symmetry element, that relates the TRIMs. Moreover, any TRIM among the four should be related to at least one another. This allows us to relate inversion eigenvalues of electron states at the TRIMs. Importantly, these states also have the same energy due to the symmetry. It is common in the literature to give the same label to such high-symmetry points.
    
  * The space group must be non-symmorphic. Only in this case, it is possible to pair states with opposite parities at the symmetry-related TRIMs [see Eq. (<ref>)]. This condition is necessary to obtain a non-trivial value of the invariant Eq. (<ref>).
    
  * In the case of strong SOC the ℤ_2 invariant is defined for a gapped phase, therefore the space group must not enforce any additional degeneracies inside the plane with non-trivial invariant.

We found eight space groups with a non-trivial invariant in a 2D submanifold without SOC, see Table <ref>. Three of them SG 70, 88, and 141 have four non-equivalent TRIMs related by a four-fold symmetry. Five SGs 15, 66, 74, 84, and 131 have two pairs of TRIMs related by two-fold symmetries.
Moreover, two space groups enforce additional degeneracy at the TRIMs, which is not compatible with the first criterion. However, when  SOC is strong, the degeneracy is lifted.



§ FOURFOLD DEGENERACIES AT TRIMS



If crystal symmetries enforce four-fold degeneracies at TRIMs and relate the momenta, the
δ values [Eq. (<ref>)] must be the same at two related points. Therefore, to enforce the invariant to be non-trivial, symmetries of a TRIM's little group must relate inversion eigenvalues of degenerate states. In Refs. <cit.> the authors developed this approach for two possible cases of finite weak and strong SOC. For the strong SOC all considered TRIMs must have a four-fold degeneracy enforced by crystal symmetries. If the SOC is weak the degeneracies can be two-fold, however, in the absence of SOC the bands must be four-fold degenerate. The latter case is unstable, as strong SOC might exchange bands. 
In this work, we complete the space group catalog. 





 §.§ Space Group 135 (P4_2/mbc)


We discuss the approach considering the space group 135. The spinless representations of glide mirror symmetries are given by

    M_100(12,12,0)   : (-x +12, y+12, z),
    
        M_010(12,12,0)   :  (x +12, -y+12, z) ,
    
        M_110(12,12,12)   :  (-y +12, -x+12, z+12).

With this we can write relations between representations acting on Bloch states

    D_PD_M_100   =e^i(-k_x+k_y)D_M_100D_P, 
    
        D_PD_M_010   =e^i(k_x-k_y)D_M_010D_P, 
    
        D_PD_M_110   =e^i(-k_x-k_y+k_z)D_M_110D_P, 
    
        D_M_100D_M_010   =(-1)^fe^i(k_x-k_y)D_M_010D_M_100,
    
        D_M_010^2   =(-1)^fe^i k_x,
    
        D_𝒯^2   =(-1)^f,

where f = 1 for spin-12 particles and f = 0 for the spinless case.
The TRIMs R_1(π,0,π), R_2(0,π,π), X_1(π,0,0) and X_2(0,π,0) are invariant under P,   M_100 and M_010 symmetries. For these points the relation {D_P,D_M_100}=0 enforces the degeneracy with (+-) inversion eigenvalues. In both cases of strong and negligible SOC the degeneracy is fourfold, and δ = -1 in Eq. (<ref>).

At the A(π, π,π) and M(π,π,0) points the same symmetries give the relations [D_P,D_M_010]=0, D_𝒯M_010^2 = -1. For the spinless case one already can conclude the presence of the degeneracy due to the Kramers theorem with the same inversion eigenvalues. In the spinful case we additionally can write [D_P,D_M_100]={D_M_100,D_M_010}=0. Together the relations enforce four orthogonal states with the same inversion eigenvalue at the same energy ψ,   P𝒯ψ,  M_010ψ,  𝒯M_010ψ. For the M point it implies δ = 1. 

At the A(π, π,π) and Z(0,0,π) points the symmetry M_110 enforces degeneracy with opposite inversion eigenvalues due to the relation {D_P,D_M_110}=0. δ_Z = -1, however, at A point there is already fourfold degeneracy with the same inversion eigenvalues, therefore, the total degeneracy at the A point is eight.

Bands in the plane R-Z-X-M host the non-trivial ℤ_2 invariant. In case of the negligible SOC the plane is pierced by a nodal line, pinned to the k_z=π plane. For a TB model (see Appendix <ref>) the location of the line in BZ is shown in Fig. <ref> (a). In a model with termination in y direction, the bulk bands are projected onto k_x-k_z plane. The projected nodal line crosses MZ at Z. The SOC opens the gap at the point and the helical surface states emerge [Fig. <ref> (b-c)].
The discussed results are summarized in Table <ref>.








 §.§ Generalizations to other space groups

Four-fold degeneracies at TRIMs can be enforced by the non-symmorphic symmetries. The symmetries relate inversion eigenvalues between degenerate states. We look for the centrosymmetric space groups that have TRIMs with the non-symmorphic little groups. We say that a 2D surface with four 4-fold degenerate TRIMs without SOC is topological, if the ℤ_2 invariant and the corresponding Berry phase are non-trivial for the 8ℕ+4 occupation.
If the surface is the plane we also consider the effect of the SOC.
There are two possibilities, as the degeneracies can persist or be lifted. In the former case the ℤ_2 Kane-Mele invariant is enforced only by the symmetries and does not depend on a specific material. An example will be the SG 135 with four TRIMs in the plane R-Z-X-M. In the latter case, the invariant is enforced for a small, but finite SOC. The coupling can lift the degeneracies, but should not invert bands. The argument becomes weaker here. For instance, in the SG 60, the four-fold degeneracy at the S point is lifted under the SOC. 

Moreover, symmetries can enforce degeneracies larger than four. If it happens without SOC, the system with the interaction can still have four TRIMs in the plane with four-fold degeneracies. In SG 61, the eight-fold degeneracy at the R point is lifted to four two-fold degenerate bands if the SOC is present.

In all examples of space groups with the enforced ℤ_2 invariant that we found, see Table <ref>, there is one TRIM, where inversion commutes with a symmetry that enforces degeneracy. At the other three TRIMs, the inversion anticommutes with the symmetry operator, enforcing degeneracy with opposite inversion eigenvalues.






§ MATERIAL EXAMPLES

We implemented a material search to look for compounds with symmetry-enforced ℤ_2 quantum spin Hall states (see Table <ref>). We considered examples of compounds tabulated in the Inorganic Crystal Structure Database (ICSD) <cit.> with a number of different elements less than four and a heavy element with atomic number Z>54. In this case, the SOC is large enough to open a gap. We did not consider compounds with f-electrons close to the Fermi level to avoid the effects of electron correlations. As mentioned before, every second gap must have the non-trivial ℤ_2 invariant, so we restricted our search to materials with the non-trivial gap close to the Fermi level (with only one exception of PbWO_4 – a trivial insulator). Also, the number of bands near the Fermi level should not be large to resolve the non-trivial gap.

The compound TaP in SG 109 (I4_1md) is a Weyl semimetal <cit.>. However, not much is known about its other phase with the symmetries of SG 141 (I4_1/amd) and the structure type β-NbP <cit.> (see ICSD 108656 <cit.>).
Its BZ has four N points N_1(π,0,0), N_2(0,π,0), N_3(0,π,π) and N_4(π,0,π) in the primitive reciprocal basis. 
In a slab geometry with termination in the z direction N_1 point projects onto N_4, and N_2 onto N_3. We computed the band structure (Fig. <ref>) with the Quantum ESPRESSO software <cit.> for a supercell with 10 unit cells stacked in the z direction. In every second bulk gap there are helical surface states corresponding to the symmetry-enforced ℤ_2 invariant. 



Another representative of materials in SG 141 (I4_1/amd) is BW <cit.>. It becomes superconducting at T_c = 4.3 K <cit.> and can be an example of a symmetry-enforced topological superconductor. 



We found an intriguing material example PbWO_4 used in electromagnetic detectors for high-energy physics research <cit.>. It grows in a single crystal form with symmetries of SG 88 (I4_1/a) <cit.>. Though it is a trivial insulator, the next gap below the Fermi level is enforced to be topological.






§ CONCLUSIONS



In this work, we investigated the origin of supertopology. We found 17 SGs enforcing topological invariants for every band independently of material characteristics. In several space groups, the symmetry-enforced Dirac nodal line (and its surface state) is the parent state for the quantum spin Hall state if SOC is added. 

In the description of the enforced Kane-Mele invariant, we assumed the presence of the bulk band gap in a plane. This assumption does not demand the bulk gap everywhere in the full BZ. This implies the possible coexistence of the bulk Fermi surface with boundary resonances <cit.>.
If the origin of the Fermi surface is the symmetry-enforced connectivity between the bands <cit.>, the bulk gap can be opened with the breaking of the crystalline symmetries. Unless there is a band inversion in the non-trivial plane, the invariant must be preserved and non-trivial. This can be an approach to create symmetry-enforced weak topological insulators.

Approaches used in the paper can be applied to study representation-enforced topology <cit.>. In this case, one can consider a particular Wyckoff position and assume that bands induced from each of the site symmetry groups do not mix. With these conditions the classification of the space groups can be extended to the classification of the Wyckoff positions that induce bands carrying weak and strong ℤ_2, and higher-order ℤ_4 topologies <cit.>. 

The discussed symmetry indicators can be applied to magnetic systems with P and anti-unitary symmetries. The anti-unitary symmetry (time-reversal with a crystalline operation) must quantize ℤ_2 topological invariant <cit.>, and the inversion eigenvalues indicate a non-trivial phase. Thus, the material search can be extended to magnetic phases.

With the addition of chiral symmetry, 2D electron systems can host higher-order topology <cit.>. Therefore, symmetry-enforced higher-order topological phases are possible and require further investigation. 

We study the 2D symmetry indicators as the 3D counterparts can not be enforced due to material dependence of the inversion eigenvalues at Γ point. However, in higher dimensional systems, the approach can be applied to enforce 3D symmetry indicators. 



§ ACKNOWLEDGMENTS

We wish to thank Hans Peter Büchler, Kirill Alpin, and Andreas Leonhardt for useful discussions.
K.P. and A.P.S. are funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – TRR 360 – 492547816.
M.M.H. is funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – project number 518238332.

apsrev4-1














This appendix contains details on symmetries of SG 15, tight-binding models for SG 15 and 135. In the last section we discuss the connection between Berry phase and inversion eigenvalues, and give two examples of paths with symmetry-enforced π Berry phase.




§ SG 15 – CONVENTIONAL AND PRIMITIVE BASES


To transform reciprocal vectors from a primitive basis g_1, g_2, g_3 to a conventional k_x, k_y, k_z one has to use the matrix 

    C = [  1 -1  0;  1  1  0;  0  0  1 ].

In the main text we consider TRIMs L_1^g(π, 0, π), L_2^g(0, π, -π), V_1^g(π, 0, 0), and V_2^g(0, π, 0). The superscript g corresponds to the primitive basis g. The coordinates of TRIMs in primitive are related to the conventional basis by 𝐤_Γ_i = C𝐤_Γ_i^g. 



§ PARITIES UNDER TWO-FOLD ROTATION


Here we prove the commutation relation Eq. (<ref>). Inversion and glide mirror representations are given by

    (x, y, z)     (-x, -y, -z) (-x, y, -z+12),
    
        (x, y, z)    (x, -y, z+12)
        (-x, y, -z-12),

where M_010 corresponds to M_010(0,0,12) glide mirror reflection. 
With these two expressions we conclude 

    D_P D_M_010   = D_t(0, 0, -1) D_M_010 D_P 
        =  D_M_010 D_t(0, 0, -1) D_P 
       = D_M_010 D_P D_t(0, 0, 1).

The translation operator D_t(0, 0, 1) acts on a Bloch state as a phase multiplication e^i k_z, and D_P D_M_010 = e^i k_z D_M_010 D_P.



§ BERRY PHASE AND INVERSION EIGENVALUES

In this section, we follow Ref. PhysRevLett.115.036806 to show the relation between inversion eigenvalues of occupied states and the Berry phase. 
We consider two Bloch states related by inversion symmetry P|u_n(-𝐤)⟩ = e^i β_n(𝐤)|u_n(𝐤)⟩. The Berry connection can be defined as A(𝐤) = -i ∑_n ⟨ u_n(𝐤)|∇_𝐤u_n(𝐤)⟩, where we sum over occupied bands n.
Connection difference at 𝐤 and -𝐤 is A(𝐤)-A(-𝐤) = -∇_𝐤∑_n β_n(𝐤).
The Berry phase along a path c_ab - c_ab connecting two TRIMs Γ_a and Γ_b is an integral of the Berry connection

    γ_ab = -∑_n [β_n(Γ_b) - β_n(Γ_a)]  2π.

The path c_ab represents the time-reversal partner of c_ab.

At the same time the product of inversion eigenvalues at a TRIM Γ_i

    ∏_nξ_n(Γ_i) =∏_n⟨ u_n(Γ_i)|P| u_n(Γ_i)⟩ =
        e^-i ∑_n β_n(Γ_i),

where i∈{a,b}.
Therefore, the inversion eigenvalues of occupied states determine the Berry phase along the contour c_ab - c_ab

    e^iγ_ab = ∏_nξ_n(Γ_a)ξ_n(Γ_b).




§ BERRY PHASE CONTOURS

We give several examples of contours with symmetry-enforced π Berry phase for systems with negligible SOC. 
The contours contain two paths related by time-reversal symmetry. In Fig. <ref>(a) the Berry phase along contour c_UZ - c_UZ-(c_RT - c_RT) is equal to π. In the figure we show path c_UZ - c_UZ with zero Berry phase in green. Berry phase along gray lines is compensated, and along red path -(c_RT - c_RT) is π. The shape of paths c_UZ,  c_RT can be arbitrary, but they must not encounter additional crossings between TRIMs.

In SG 56 and SG 138, the non-trivial invariant is enforced only along contours connecting TRIMs that can not be embedded in a 2D plane. One of the possible closed paths is shown in Fig. <ref>(b). 

We want to note that in SG 60 there are no closed contours with symmetry-enforced ℤ_2 invariant (see Table <ref>). In this group, T, U, and R points are 8-fold degenerate, and the contour must not pass these points. Z points and X, Y, and S points have different k_z values (see BZ in Fig. <ref>). Therefore, a non-trivial contour includes paths connecting the Z point with one of the X, Y, and S TRIMs, and the total number of these paths quanta (that are going through only one BZ) is even. It is possible only if the total number of these quanta (with k_z change) in k_x and k_y directions are even, too. We will refer to it as the “k_z condition".
π Berry phase can be acquired along a path that passes S (like YSY) or starts at S (SYS). In the second case, to connect S points along a line with a trivial phase we have to add paths at the boundary of BZ (gray paths). These lines end at X or Y points. As a result, in both cases, we have to connect two X points (or Y points) separated by one BZ along a path that passes S an even number of times. This condition together with the “k_z condition" seems impossible. However, the SG 60 has non-trivial planes in the presence of a small SOC. If the coupling is enough to lift 8-fold degeneracies at T, U, and R, but is not sufficient to mix the split bands at the S point, then there are 2D planes with non-trivial ℤ_2 invariant (see unstable ℤ_2 invariant in Ref. PhysRevMaterials.5.124202).





§ TIGHT-BINDING MODEL



 §.§ SG 15

We consider the Wyckoff position 4a and electron orbitals at (0,0,0) and (0,0,12) sites.
In k space the Hamiltonian with the spin degrees of freedom will take the form

    H(k) = [  H_↑(k)   Δ_SOC; Δ_SOC^†  H_↓(k) ],

where H_↑(k)=H_↓(k)=d·τ. Pauli matrices τ correspond to the orbital degree of freedom, 

    d_x    = 8 cosk_z2(2 cosk_y2cosk_x+k_z2+cosk_z2), 
    
    d_y    = 8 sink_z2(2 cosk_y2cosk_x+k_z2+cosk_z2), 
    
    d_z    = -4 sink_x2sink_y2,
    
    d_0    = 4 cosk_x2cosk_y2.

Δ_SOC is the SOC and takes the form

    Δ_SOC =     (
    [   0 Δ_1; Δ_2   0;     ]),

where k-dependent functions Δ_1,Δ_2 are given by

    Δ_1    =√(2)(e^-π i2sink_z2+e^π i2cosk_z2) f_k_x k_y k_z, 
    Δ_2    = √(2)(e^-π i2sink_z2-e^π i2cosk_z2) f_k_x k_y k_z, 
    
    f_k_x k_y k_z   = cosk_x+k_y+k_z2 +i cosk_x-k_y+k_z2.



 §.§ SG 135

For SG 135 we choose the Wyckoff position 4a with coordinates (0, 0, 0), (0, 0, 12), (12, 12, 0), and (12, 12,12). The tight-binding model takes the form

    H(k) = 
            [  H_↑(k)   Δ_SOC; Δ_SOC^†  H_↓(k) ],

where blocks correspond to spin ↑ and ↓, and are given by the matrix

    H_↑(k)   =(
    [   h_↑11   h_↑12   h_↑13   h_↑14; h_↑12^*   h_↑22   h_↑23   h_↑13; h_↑13^* h_↑23^*   h_↑11   h_↑12; h_↑14^* h_↑13^* h_↑12^*   h_↑22;         ]).

The matrix elements are k-dependent functions

    h_↑11   = h_↓11 = 2cos k_x t_7+2cos k_y t_6+2cos k_z t_8+t_1,
    
    h_↑22   =  h_↓22 = 2cos k_x t_6+2cos k_y t_7+2cos k_z t_8+t_1,
    
    h_↑12   = h_↓12 = (1+e^-i k_z) t_2,
    
    h_↑13   =(1+e^i k_x) (1+e^i k_y) t_3^* e^-i (k_x+k_y),
    
    h_↑14   =(1+e^i k_z) e^-i (k_x+k_y+k_z)
       [t_4^* (1+e^i (k_x+k_y))+t_5^* (e^i k_x+e^i k_y)],
    
    h_↑23   =(1+e^i k_z) e^-i (k_x+k_y)
       [t_4^* (e^i k_x+e^i k_y)+t_5^* (1+e^i (k_x+k_y))],
    
    h_↓13   =(1+e^i k_x) (1+e^i k_y) t_3 e^-i (k_x+k_y),
    
    h_↓14   =(1+e^i k_z) e^-i (k_x+k_y+k_z)
       [t_4(1+e^i (k_x+k_y))+t_5(e^i k_x+e^i k_y)],
    
    h_↓23   =(1+e^i k_z) e^-i (k_x+k_y)
       [t_4(e^i k_x+e^i k_y)+t_5(1+e^i (k_x+k_y))],

with parameters given by

    t_1    = 0, t_2 = -0.024, t_3 = 0.594 - 0.366 i, 
    
    t_4    = 0.372 - 0.368 i,
    t_5 = -0.395 + 0.895 i, 
    
    t_6    = -1.414, 
    t_7 = 1.962, 
    t_8 =1.966.

The SOC is described by the matrix

    Δ_SOC = (
    [   0   0   0 Δ_1;   0   0 Δ_2   0;   0 Δ_3   0   0; Δ_4   0   0   0;     ])

with k-dependent elements

    Δ_1    = (-1+e^i k_z) e^-i (k_x+k_y+k_z)
       [t_9(-1+e^i (k_x+k_y))+t_10(e^i k_x-e^i k_y)],
    Δ_2    =  i(-1+e^i k_z) e^-i (k_x+k_y)
       [t_9(e^i k_x-e^i k_y)-t_10(-1+e^i (k_x+k_y)))],
    Δ_3    = e^-i k_z(-1+e^i k_z) 
       [t_9^* (e^i k_x-e^i k_y)+t_10^* (-1+e^i (k_x+k_y))],
    Δ_4    = -i (-1+e^i k_z) 
       [t_9^* (-1+e^i (k_x+k_y))-t_10^* (e^i k_x-e^i k_y)].

The parameters for our model are given by

    t_9 = -0.688 - 0.688 i,
    t_10 = -0.098 + 0.098 i.






 Paper Title: Comparative analysis of structural, elastic, electronic, phonon, thermal and optical properties of two $\text{Na}_6\text{Ge}_2\text{Se}_6$ phases from first principles calculations.






Department of Physics, Missouri University of Science and Technology, Missouri, Rolla 65401, USA

Department of Chemistry, Missouri University of Science and Technology, Missouri, Rolla 65401, USA

Department of Physics, Missouri University of Science and Technology, Missouri, Rolla 65401, USA

§ ABSTRACT 

The demand for new alkali metal chalcogenide materials is continuously increasing due to their potential applications across various technological fields. Recently, a new compound, Na6Ge2Se6, was computationally predicted, representing a new phase distinct from the experimentally observed Na6Ge2Se6 reported in 1985. Notably, this newly predicted phase displays a lower total energy compared to the previously known experimental phase, as determined by first-principles calculations. In this study, we undertake a thorough comparative analysis of the structural, elastic, electronic, phonon, thermal, and optical properties of these two Na6Ge2Se6 phases. Our results show that both phases meet mechanical and dynamical stability criteria. The electronic band structure analysis confirms the semiconducting nature of both materials, with a 2.97 eV indirect band gap for the predicted phase and a 2.93 eV direct band gap for the observed phase. Optically, both phases exhibit strong absorption in the ultraviolet region. Thermal properties analysis reveals that the predicted phase is more thermodynamically stable below 907 K, while the observed phase shows greater thermodynamic stability above this temperature.




Comparative analysis of structural, elastic, electronic, phonon, thermal and optical properties of two Na6Ge2Se6 phases from first principles calculations
    Aleksandr Chernatynskiy
    August 5, 2024
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§ INTRODUCTION


Ternary alkali metal-based chalcogenides are characterized by the general formula ABZ where A signifies an alkali metal (Li, Na, K), B includes main group elements (e.g., Ge, Bi) or transition metals (e.g., Fe, Cu), while Z represents chalcogen elements (e.g., S, Se). These compounds have gained considerable interest recently for their potential applications in energy conversion/storage, optoelectronics, and thermoelectrics. These areas are vital for supporting the advancement of current technologies and industries, especially in addressing the global energy challenge as society aims to transition from fossil fuels to renewable, efficient, and eco-friendly energy solutions. In response to these demands, numerous ABZ compounds have been synthesized and explored.

In the quest for improved energy storage solutions, the focus on ion-battery anodes is crucial due to their significant impact on battery performance. ABZ materials like LiLnSe2 and NaFeS2 have shown high storage capacities and efficient energy cycling in recent studies <cit.>, with NaFeS2 showcasing excellent cycling stability as a Li-ion battery anode, suggesting potential for enhanced battery lifespan and reliability. Optoelectronically, ABZ compounds such as NaSbS2 offer strong absorption in the visible spectrum, and their affordability, abundance, and non-toxicity make them attractive for solar energy applications<cit.>. Furthermore, in thermoelectric applications, ABZ materials like CsAg5Te3 exhibit promising mid-temperature performance with high ZT values<cit.>, thanks to their intrinsically low thermal conductivity<cit.>. Ternary chalcogenides have also played a crucial role in enabling the synthesis of more complex multinary chalcogenides in a rational manner<cit.>. Many of these compounds have demonstrated significant technological potential. For instance, outstanding electric thermal properties have been observed in materials like Na_0.95Pb_19SbTe_22, achieving a ZT value greater than 1 across one of the broadest temperature ranges (475 to 650 K) reported for any single material<cit.>. Additionally, a study on Na_1-xK_xAsQ_2 (Q=S,Se) highlighted its potential applications in signal processing and data transmission, attributed to its nonlinear optical properties<cit.>.

Although many ABZ compounds have been synthesised, the existing collection of these ternary ABZ compounds still lacks many possible combinations. The process of discovering new materials through purely experimental means is often prolonged, as the growth and characterization of their properties can be time-consuming. With the increase of  computational power, simulations are increasingly employed in material discovery, for both screening of the properties of already known materials as well as prediction of the new compounds. Recently, a new structure of the chalcogenide, Na6Ge2Se6 was computationally predicted using simulated annealing and first-principles methods<cit.>. This compound represents a new phase distinct from the experimentally observed compound reported in 1985 with the same composition (Na6Ge2Se6)<cit.>. The latter has been utilized recently in synthesizing a potential nonlinear optical material, Na8Mn2(Ge2Se6)2<cit.>. Notably, the newly predicted Na6Ge2Se6 phase shows a lower total energy compared to the observed phase from first-principles calculations, suggesting it as a potential meta-stable state of Na6Ge2Se6 that has not been reported before. Furthermore, the observed Na6Ge2Se6 phase has received little attention, with no comprehensive analysis conducted to the best of our knowledge. This gap in research has motivated us to perform a comparative analysis of the structural, elastic, electronic, phonon, thermal, and optical properties of these two compounds, which could serve as a valuable reference for future studies.




















§ COMPUTATIONAL METHODOLOGY


In this study, we examined the ground state properties of both predicted and observed Na6Ge2Se6 phases using density functional theory (DFT) as implemented in the Vienna Ab initio Simulation Package (VASP)<cit.>. This involved solving the Kohn-Sham equations<cit.> to determine the ground state energies of a crystalline system. Given the close proximity of the ground state energies between the predicted and observed phases, calculations were performed using two distinct approaches for the exchange-correlation potential: the Generalized Gradient Approximation (GGA) within the Perdew-Burke-Ernzerhof framework (GGA-PBE)<cit.> and the hybrid functionals from Heyd-Scuseria-Ernzerhof (HSE06)<cit.>. In both Na6Ge2Se6 structures,  Integration over the Brillouin zone was executed via the tetrahedron method with Gaussian smearing, employing a 5×5×5 Monkhorst–Pack k-point mesh<cit.>. The energy cutoff for the plane wave basis set was set to 600 eV. The Ge 3d^104s^34p^2, Se 4s^2p^4, Na 2p^63s^1 electrons are treated as valence electrons. convergence thresholds were 1e-8 eV for the electronic self-consistent loop and 1e-7 eV for structure optimization.

Further, we explored the elastic and dynamical properties of optimized structures using first-principle calculations with the GGA-PBE functional within VASP. This analysis allowed for the calculation of elastic constants C_ij and several elastic properties, including the bulk modulus B, Young's modulus E and shear modulus G, along with optical properties such as the dielectric function ϵ(ω), absorption coefficient α(ω) and conductivity σ(ω) using the VASPKIT<cit.> tool. The Phonopy package<cit.> facilitated the calculation of phonon dispersion relations, phonon density of states, and related thermal properties. Considering the tendency of GGA-PBE to underestimate band gaps in semiconductors and insulators, the HSE06 functional<cit.> was applied to investigate electronic and optical properties. The electronic band structures and density of states for both phases were visualized using the open-source Python library Pymatgen<cit.>.





§ CRYSTAL STRUCTURE

We analyzed the Na6Ge2Se6 structures using two different exchange-correlation energy functionals: GGA-PBE and the HSE06 hybrid functional. The calculated energies per atom are presented in Table <ref>. Both PBE and HSE06 functionals indicated lower energies for the predicted Na6Ge2Se6 phase by 16 meV/atom. To confirm the convergence of our DFT calculations, we conducted two additional structure relaxations with the GGA-PBE functional: one with a 400 eV energy cutoff and a 3×3×3 k-point mesh, and another with a 500 eV energy cutoff and a 4×4×4 k-point mesh. For the predicted phase, both of these settings produced energy per atom within 0.006% of the most accurate result listed in the Table <ref>. For the observed phase, the difference is even smaller, within 0.002%. The energy difference between the predicted and observed phases listed above is an order of magnitude larger than 0.06%, thus numerical accuracy is sufficient to make conclusions about the energy order of two structures.

This leads us to conclude that the predicted Na6Ge2Se6 phase is energetically more favorable than the observed one at 0 K.

The predicted Na6Ge2Se6 crystallizes in the space group R3 (No.148), whereas the observed phase is found in the space group P2_1/c (No.14). Both structures exhibit identical ethane-like selenide [Ge2Se6] units, featuring a Ge-Ge bond length of 2.49 Å for the predicted phase and 2.47 Å for the observed phase, and an average Ge-Se bond length of 2.39 Å  as detailed in Table <ref>. These bond lengths are slightly greater than the originally observed Na6Ge2Se6 data<cit.>, which recorded a Ge-Ge bond length of 2.43 Å  and a Ge-Se bond length of 2.33 Å. The [Ge2Se6] ethane-like dimer configuration bears similarity to the [P2Se6] in Mg2P2Se6<cit.>, and [Si2Se6] in Na4MgSi2Se6<cit.>. For both Na6Ge2Se6 phases, the formed [Ge2Se6] dimers are isolated from each other, similar to the cases in K6Ge2Se6<cit.> and Cs6Ge2Se6<cit.>. The predicted Na6Ge2Se6 phase exhibits cell parameters of a=11.349 Å, b=11.349 Å, c=10.900 Å, α=90, β=90 and γ=120, while the observed Na6Ge2Se6 has a=8.445 Å, b=12.038 Å, c=8.316 Å, α=90, β=118.877 and γ=90 with a smaller unit cell volume. The predicted Na6Ge2Se6 features only one crystallographically unique Na atom, one Ge atom, and one Se atom, contrasting with the observed Na6Ge2Se6, which has three unique Na atoms, one Ge atom, and three Se atoms. In the predicted Na6Ge2Se6, each Na atom coordinates with five Se atoms to form a [NaSe5] pyramid. This pyramid shares corners with its neighboring [NaSe5] pyramids, whereas in the observed phase, each Na atom bonds with six Se atoms to form a [NaSe6] slightly distorted octahedron. These octahedra are connected by face-sharing with their neighboring [NaSe6] octahedron, as illustrated in Fig. <ref>(c) and Fig. <ref>(d). The electron localization function (ELF) provides a visual representation of valence electron distribution<cit.>, highlighting features that correspond to chemical bonding characteristics. Fig. <ref> shows that areas near the center of Se atoms and between Ge/Ge atoms exhibit ELF values close to 1.







§ ELASTIC PROPERTIES




Analyzing the elastic properties offers insights into the mechanical behavior of compounds, including their stability, ductility, and brittleness. Such information is especially valuable in industrial and device manufacturing sectors, enhancing our understanding of the forces acting in solids. To calculate elastic constants, we employed the energy-strain method<cit.>, as facilitated by VASPKIT<cit.> based on DFT calculations with GGA-PBE functional.

For the predicted Na6Ge2Se6 phase, which crystallizes in the trigonal crystal system, There are five independent
elastic constants: C_11, C_33, C_44, C_12, C_13 with the added relation

    C_66 = (C_11 - C_12)/2

This contrasts with its observed counterpart, which crystallizes in monoclinic crystal system with lower symmetry that has thirteen independent elastic constants: C_11, C_12, C_13, C_15, C_22, C_23, C_25, C_33, C_35 , C_44, C_46 ,C_55 and C_66. The configurations of these constants constitute the elastic matrices for each respective phase as follows:

    C_predicted =
    [ 32.06 10.86 13.31     ·     ·     ·;     · 32.06 13.31     ·     ·     ·;     ·     · 21.06     ·     ·     ·;     ·     ·     · 12.09     ·     ·;     ·     ·     ·     · 12.09     ·;     ·     ·     ·     ·     · 10.60 ]


    C_observed =
    [ 28.28 10.82 15.82     ·  2.45     ·;     · 32.08 15.21     ·  1.55     ·;     ·     · 40.55     ·  4.76     ·;     ·     ·     · 10.43     ·  2.81;     ·     ·     ·     · 10.67     ·;     ·     ·     ·     ·     · 10.75 ]






These elastic constants are pivotal for assessing other elastic properties. A crystal must satisfy the Born-Huang criteria<cit.> to be deemed mechanically stable. For the predicted structure, the necessary and sufficient Born-Huang criteria include:

    C_11 > |C_12|, C_44 > 0, C_66 > 0


    2C_13^2 < C_33(C_11 + C_12)
























Due to the intricacy of these criteria for the monoclinic system, their detailed equations are not listed here. Crucially, the calculated elastic constants for both Na6Ge2Se6 phases fulfill their respective stability criteria, indicating mechanical stability for both the predicted and observed phases. 

Constants C_11, C_22 and C_33 quantify the response to uniaxial strain along the three principal axes. The differences between these constants measure the extent of anisotropy in linear compressibility. From matrix C_predicted and C_observed, we can see that for predicted phase, C_11=C_22>C_33 which means linear compressibility is isotropic in the x- and y-directions and greater than in the z-direction. 
For the observed phase, C_33>C_22>C_11, indicating that the observed Na6Ge2Se6 phase is characterized by anisotropic linear compressibility. While both phases have similar resistance to deformation along the x- and y-directions, the observed phase is significantly stiffer in the z-direction.

On the other hand, C_44, C_55 and C_66 assess the material’s resistance to shear deformation about these axes. The elastic constant matrix for the predicted Na6Ge2Se6 phase exhibits C_44=C_55 and slightly smaller C_66, indicating its shear modulus is weakly anisotropic. For this phase, the resistance to shear deformation is isotropic about the x- and y-directions and greater than that in the z-direction. Comparing both phases, the observed phase shows slightly higher resistance to shear deformation about the principal axes.

The off-diagonal terms indicate coupling between different deformation modes. For instance, C_12 correlates the stress applied in the x-direction to the resultant strain in the y-direction. Analysis of the matrices suggests that both phases have a similar response to deformation in the y-direction when stress is applied in the x-direction. However, the predicted phase shows greater shear strain in the z-direction induced by stress in both the x- and y-directions.





Utilizing the independent elastic constants, we can derive key mechanical properties such as the bulk modulus (B), shear modulus (G), Young's modulus (E), and Poisson's ratio (ν) through the following relations, with the calculated values presented in Table <ref>:

    B=B_V+B_R/2


    G=G_V+G_R/2


    E=9BG/3B+G


    ν=3B-2G/2(3B+G)

Here, B_V, B_R and G_V, G_R represent the Voigt<cit.> and Reuss<cit.> values of the bulk modulus and shear modulus respectively. Furthermore, we evaluated Kleinman's parameter (ζ) using VASPKIT<cit.>. 

The bulk and shear moduli offer insights into a crystal's mechanical behavior, indicating its resistance to volumetric and shape deformations, respectively. The data in the Table <ref> reveal that the observed Na6Ge2Se6 phase possesses a greater resistance to deformation. This is expected as the unit-cell volume of the predicted phase is greater than that of the observed phase. Pugh's ratio (B/G) serves as a gauge for a material's ductility or brittleness, with a threshold value of 1.75 distinguishing between the two behaviors where the material is claimed to be brittle, if B/G<1.75 and classified to be ductile if B/G>1.75<cit.>. Our findings suggest that both phases exhibit ductility. The stiffness and thermal shock resistance of these materials are inferred from Young's modulus (E)<cit.>, indicating superior stiffness in the observed phase. Conversely, a lower Young's modulus hints at enhanced thermal shock resistance<cit.> for the predicted phase. Poisson's ratio (ν) sheds light on the nature of bonding forces within a crystal and aids in evaluating its mechanical properties, including stability against shear<cit.>. Values for the predicted and observed phases are 0.268 and 0.283, respectively, falling within the central force solids' expected range of 0.25 to 0.50<cit.>. This metric also predicts the material's ductility or brittleness, classifying those with ν>0.26 as ductile and those below 0.26 as brittle<cit.>. Thus, our analysis corroborates both phases' ductility, aligning with conclusions drawn from Pugh's ratio. Lastly, the Kleinman parameter (ζ), varying between 0 and 1, indicates the dominance of bond stretching or bending in resisting external stress<cit.> where the lower value of the Kleinman parameter ζ indicates a minimal role of bond bending in resisting external stress, whereas a higher value suggests a negligible contribution of bond stretching or contracting to resist externally applied stress. For both Na6Ge2Se6 phases, a ζ>0.5 indicates a major role for bond bending in its mechanical strength.

Fig. <ref> shows the 3D and 2D representations of Young’s modulus and linear compressibility for both Na6Ge2Se6 phases. For the predicted phase, the Young’s modulus and linear compressibility in the xy-plane form a circle, as shown in Figures <ref>(e) and (f), suggesting both properties are isotropic within the xy-plane. However, both properties show significant deviation from a circular form within the yz- and xz-planes, indicating that Young’s modulus and linear compressibility are anisotropic overall for the predicted Na6Ge2Se6. For the observed phase, Figures <ref>(g) and (h) show that both properties are anisotropic, with the 2D representation of Young’s modulus in the xy- and yz-planes and linear compressibility in the xy-plane more closely resembling a circle. This suggests relative isotropy for both properties within the corresponding planes.












§ ELECTRONIC PROPERTIES

We present the electronic band structure along with the partial and total density of states (DOS) in Fig. <ref>. These calculations were performed using optimized lattice parameters along high-symmetry directions in the first Brillouin zone. The partial density of states (PDOS) on orbitals and elements are depicted in Fig. <ref>. The band structures and DOS for both Na6Ge2Se6 phases were determined using the HSE06 hybrid exchange-correlation functional<cit.> to ensure enhanced accuracy<cit.>. The energy reference point is set at the top of the valence band. The band structure analysis reveals that the predicted Na6Ge2Se6 phase possesses an indirect band gap, with the conduction band minimum (CBM) located at the X-point and the valence band maximum (VBM) at the Z-point, resulting in a band gap of 2.972 eV. In contrast, the observed Na6Ge2Se6 phase exhibits a direct band gap of 2.932 eV, with both the CBM and VBM situated at the Y_2-point. 

From Fig. <ref>, we observe that both phases share similar orbital and element contributions to the total DOS. Regions close to the Fermi level are predominantly influenced by contributions from Se-p and Ge-s orbitals, while Na’s involvement near the Fermi level is minimal. This observation suggests that the band gaps in both Na6Ge2Se6 phases are largely attributable to the [Ge2Se6] dimers. Within the valence band, ranging from -4 to 0 eV, the primary contributions come from Se-p and Ge-p orbitals in both phases. This trend extends to the lower regions of the conduction band, from 4 to 8 eV, where Na-p displays a slight contribution to the total DOS in this region.







§ PHONON AND THERMAL PROPERTIES


Investigating phonon properties is fundamental for understanding crystalline materials, as it provides insights into structural stability, phase transitions, and how vibrations influence their thermal behavior. In our study of the phonon dispersion and phonon band structures for both Na6Ge2Se6 phases, we employed the Phonopy package<cit.> to create a series of 2×2×2 supercell structures with various displacements. We then conducted force calculations using VASP, employing the GGA-PBE exchange-correlation functional. The phonon frequencies and eigenvectors were determined from the dynamical matrices, which were calculated based on the force constants derived using Phonopy.

The phonon dispersion relations along high-symmetry paths for both phases are depicted in Fig. <ref>. The unit cell of the predicted phase comprises 14 atoms, resulting in 42 phonon branches, while the observed phase has 28 atoms, resulting in 82 branches, which include three acoustic and the rest optical branches. The frequency spectrum of these modes spans from 0 to 10 THz, without any discernible gap between the acoustic and optical modes for either phase.
In the lower frequency domain, below 7 THz, Na and Se atoms predominantly contribute to the optical branches. In the higher frequency range, the contributions mainly stem from Ge and Se. This distribution underscores the strong bonding present within the [Ge2Se6] dimers, reflecting their structural integrity across both phases of Na6Ge2Se6. The lack of negative frequency branches within the dispersion plots affirms the dynamic stability of these phases at zero pressure.





The Debye temperature (θ_D) represents the temperature corresponding to the highest energy vibrational mode in a solid, derived from the equation<cit.>:

    θ_D=h/k_B[3n/4πV_0]^1/3v_m

Here, h is the Planck’s constant, k_B is the Boltzmann’s constant, n is the number of atoms in the unit cell, V_0 is the unit cell's equilibrium volume, and v_m is the material's average speed of sound. The average speed of sound v_m can be determined from the material's mass density (ρ), along with its bulk (B) and shear modulus (G), through the equations for longitudinal (v_l) and transverse (v_t) sound speeds:

    v_m=[1/3(3/v_t^3 +1/v_l^3)]^1/3


    v_t=√(G/ρ)


    v_l=√(3B+4G/3ρ)

The calculated values for v_t, v_l, v_m and θ_D are provided in Table <ref>. These values indicate that the predicted Na6Ge2Se6 phase exhibits a slower longitudinal speed of sound but a higher transverse one than observed phase, although the values differ only slightly. The Debye temperature of the predicted phase is approximately 0.89% lower than that of its counterpart.

Phonon calculations offer insights into additional thermal properties such as entropy (S), constant-volume heat capacity (C_v), and Helmholtz free energy (F), which are depicted in Fig. <ref>. The heat capacity curves, as one can see from Fig. <ref>(b), are almost indistinguishable as one expects from similar speeds of sound and Debye temperatures. They show standard Debye theory behavior with the increasing temperature: quantum T^3 at low temperature, saturating to the constant classical Dulong-Petit limit<cit.> above Debye temperature.

The calculation of the Helmholtz free energy (F) within the harmonic approximation<cit.> facilitates an evaluation of the thermal contributions to the relative stability of two phases, as described by the following equation:

    F=   E_total + 1/2∑_qνħω(qν) 
       +k_BT∑_qνln[1-exp(-ħω(qν)/k_BT]

Here, E_total denotes the total energy of the crystal, available in Table <ref>. The summed terms represent the Helmholtz free energy attributable to phonons<cit.>, with the initial sum reflecting the zero-point energy (ZPE) that is independent of temperature. The predicted phase's ZPE is approximately 1 meV/atom higher than that of the observed phase. The subsequent sum accounts for the temperature-dependent term referring to the thermally induced occupation of phonon modes. The Helmholtz free energies, plotted in Fig. <ref>(d) for temperatures ranging from 0 to 1000 K, with a closer look at the range of 905 to 912 K in Fig. <ref>(c), illustrate that the predicted phase possesses lower free energy from 0 to approximately 907 K. Since its rate of decrease with temperature is more gradual than that of the observed phase, the free energy of the observed phase becomes lower when the temperature exceeds 907 K. This suggests that the predicted phase is more thermodynamically stable than the observed phase in the temperature range from 0 to 907 K, indicating that this predicted phase is very likely to be experimentally observable. 













§ OPTICAL PROPERTIES

A material's optical behavior is characterized by several energy/frequency-dependent parameters, such as the dielectric function ϵ(ω), absorption coefficient α(ω), conductivity σ(ω), energy-loss function L(ω), reflectivity R(ω) and refractive index n(ω). The outcomes of these calculated properties are depicted in Fig. <ref>, based on formulas integrated within VASPKIT<cit.>.

    ϵ(ω)=ϵ_1(ω)+iϵ_2(ω)


    α(ω)=√(2)ω/c[√(ϵ_1^2+ϵ_2^2)-ϵ_1]^1/2


    L(ω)=ϵ_2/ϵ_1^2+ϵ_2^2


    n(ω)=[√(ϵ_1^2+ϵ_2^2)+ϵ_1/2]^1/2


    R(ω)=(n-1)^2+k^2/(n+1)^2+k^2



To examine the anisotropic characteristics, we calculated the optical parameters along the three principal axes and presented the optical spectra for incident photon energies up to 25 eV. For the predicted phase, these parameters are isotropic in the x- and y-directions. Therefore, Fig. <ref> displays the optical parameters and the corresponding derived optical properties plotted only in the y- and z-directions for the predicted phase.

The complex dielectric function is essential for describing a material's optical properties, as it serves as the foundation from which other energy-dependent optical constants are derived. We illustrate the dielectric functions for both predicted and observed Na6Ge2Se6 phases in Fig. <ref>(c) and (i), respectively. The real part of the dielectric function (ϵ_1) initially increases, reaches a peak, then sharply decreases, eventually dropping below zero. The imaginary part (ϵ_2), indicative of dielectric loss, closely aligns with the optical absorption coefficient. From the figure we can see ϵ_1 and ϵ_2 peak in different directions. For the predicted phase, ϵ_1 peaks in z-direction, and ϵ_2 peaks in y-direction. This contrasts with the observed phase, where both ϵ_1 and ϵ_2 peak in the same y-direction. The material's electronic characteristics can be inferred from the absorption coefficient (α(ω)), depicted in Fig. <ref>(a) and (g). Both phases begin to absorb at photon energies of approximately 3.1 eV and 2.9 eV respectively, indicating their optical band gaps. The absorption coefficients in three different directions follow a similar trend, with peaks at 7.4 eV for the predicted phase and 9.5 eV for the observed phase, both within the ultraviolet spectrum. The material's reflection and absorption properties, related to its loss function, are illustrated in Fig. <ref>(e)(k) and (d)(j). The edge trailing of the material's reflection and absorption spectra matches the peak in the loss function, showing a sharp increase around 12 eV for both phases. This spike indicates enhanced absorption and reflection of electromagnetic waves in the ultraviolet region, with the predicted phase exhibiting much notable anisotropy along y- and z-directions with the maximum appearing in y-direction for both phase. Optical conductivity which represents electrical conductivity over specific photon energy ranges is shown in Fig. <ref>(b) and (h). Conductivity sharply increases from zero at photon energies of 3.1 eV and 2.9 eV for the predicted and observed Na6Ge2Se6 phases respectively, mirroring the optical band gap observed in the absorption curves in Fig. <ref>(a)(g). In summary, the predicted and observed phases display similar optical properties, with the predicted phase exhibiting a higher degree of optical anisotropy in the y- and z-directions, while being isotropic in the x- and y-directions.







§ CONCLUSION

A comprehensive comparison was conducted between two Na6Ge2Se6 phases: a computationally predicted phase and an experimentally observed phase. This comparison spans their structural, elastic, electronic, phonon, thermal, and optical properties, employing first-principles methods.

The predicted Na6Ge2Se6 phase is the energetically favored structure at zero pressure and temperature based on first-principles calculations. Both phases feature the identical [Ge2Se6] dimer within their crystal structures, with significant differences observed in the bonding environment between Se and Na atoms (five Na atoms bond with one Se atom in the predicted phase, whereas six Na atoms bond with one Se atom in the observed phase).

Regarding elastic properties, both phases meet the criteria for mechanical stability and share similar characteristics. These include their resistance to volumetric and shape deformations, ductility, nature of bonding forces (central force), and bond bending as a major role in resisting external stress. The differences arise from their anisotropic characteristics. While the predicted phase is isotropic in the x- and y-directions, it exhibits greater anisotropy in the y- and z-directions compared to the observed phase.

Electronic structure analyses reveal a closely matched energy band gap between the two phases, primarily contributed by the [Ge2Se6] dimers, with only a 1.3% difference. However, the electronic band structure indicates that the predicted phase exhibits an indirect band gap, while the observed phase has a direct band gap.

The optical behavior between the phases is similarly aligned, with both exhibiting a comparable optical band gap (3.1 eV for the predicted phase and 2.9 eV for the observed phase) and enhanced absorption and reflection of electromagnetic waves in the ultraviolet region. Although the predicted phase is isotropic in the x- and y-directions, it displays a greater degree of optical anisotropy in the y- and z-directions, aligning with the anisotropic nature identified in the elastic properties.

Phonon dispersion analyses affirm the dynamic stability of both phases. Further calculations of thermal properties indicate that the predicted phase is more thermodynamically stable in the temperature range from 0 to 907 K, suggesting a high likelihood of experimental observation.



§ ACKNOWLEDGEMENTS

The authors thank NSF, USA (grant No. DMR1809128) for the funding of this project. The authors also acknowledge the usage of the HPC cluster "Foundry" at Missouri S&T funded by NSF award OAC-1919789.



§ DATA AVAIABILITY STATEMENT

The data that support the findings of this study are available upon request.









































 Paper Title: Anomalous strain rate sensitivity of a Cu/Al$_2$O$_3$ multi-layered thin film.















myaddress1,myaddress3]Szilvia Kalácskamycorrespondingauthor2[mycorrespondingauthor2]Corresponding author szilvia.kalacska@cnrs.fr
myaddress3]László Pethő
myaddress1]Guillaume Kermouche
myaddress3]Johann Michler
myaddress2]Péter Dusán Ispánovity



[myaddress1]Mines Saint-Etienne, Univ Lyon, CNRS, UMR 5307 LGF, Centre SMS, 158 cours Fauriel 42023 Saint-Étienne, France
[myaddress3]Empa, Swiss Federal Laboratories for Materials Science and Technology, Laboratory of Mechanics of Materials and Nanostructures, CH-3602 Thun, Feuerwerkerstrasse 39. Switzerland
[myaddress2]ELTE Eötvös Loránd University, Department of Materials Physics, Pázmány Péter sétány 1/a, 1117 Budapest, Hungary







§ ABSTRACT 
To study the size and strain rate dependency of copper polycrystalline microstructures, a multi-layered copper/Al_2O_3  thin film was deposited on a Si substrate using a hybrid deposition system (combining physical vapour and atomic layer deposition). High temperature treatment was applied on the “As Deposited" material with ultrafine-grained structure to increase the average grain size, resulting in a “Heat Treated" state with microcrystalline structure. Focused ion beam milling was employed to create square shaped micropillars with two different sizes, that were subjected to compressive loading at various (0.001/s – 1000/s) strain rates. The detected two distinct anomalies in the strain rate sensitivity behavior appearing at high strain rates could be related to the pillar diameter and the grain size of the deformed samples. The Al_2O_3 interlayer studied by transmission electron microscopy showed excellent thermal stability and grain boundary pinning by precipitation, also resulting in the homogeneous deformation of the pillars and preventing shear localization. Geometrically necessary dislocation densities estimated by high (angular) resolution electron backscatter diffraction presented inhomogeneous dislocation distribution within the deformed pillar volumes, that is attributed to the proximity of the sample edges. Finally, the Al_2O_3 interlayers successfully suppressed any possible recrystallization processes, contributing to the excellent film stability, that makes the proposed coating ideal to be operating under extreme conditions.









thin films, micromechanical testing, microstructure, high strain rate, plastic deformation, size effect















§ INTRODUCTION


Thin films are employed by a large variety of applications. Protective coatings can improve corrosion or scratch resistance, they can hold decorative purposes (to change the color/roughness of the surface), or they can have other functional objectives. Creating multi-layered thin films with alternating dissimilar sublayers promises unusual (electrical <cit.>, thermal <cit.>, optical <cit.>, etc.) properties to be experimentally investigated. Such systems, where grain size and texture can be controlled by the deposition / annealing process, represent an outstanding opportunity to focus on a few key aspects of the deformation processes driven by the collective behavior of dislocations.


Copper coating is important in several applications due to its excellent conductivity, corrosion resistance <cit.>, and antimicrobial properties. Copper is the most widespread choice as a conducting material for connections in integrated circuits with submicron features <cit.>. The microstructure of the coating not only influences the thermal and electrical conductivity <cit.>, but also its mechanical properties, too. Therefore, it is important to study the deposited coating from the structural engineering point of view too.

Hybrid thin film coatings emerged due to their enhanced mechanical properties, such as in Al/Al_2O_3 depositions <cit.>. Here, the combination of physical vapor deposition (PVD) is combined with atomic layer deposition (ALD). PVD is employed to grow high purity thick layers in a low oxygen environment, while ALD is used to obtain precise thickness control over thinner (typically in the nm range) layers.

Investigating the mechanical stability under various extreme conditions is crucial for defining the margins of the coatings' applicability. Recent innovations in experimental micromechanical testing now enables probing at a wide range of deformation speeds <cit.> at the relevant (application) scale. The two essential parameters for understanding deformation kinetics are the strain rate sensitivity (SRS, m) of the flow stress and the activation volume (Ω) <cit.>. Analyzing these two parameters can provide a clearer understanding of the rate-controlling deformation mechanisms. However, there is limited experimental information on the high strain rate (HSR) regime mechanical behavior as a function of size <cit.> and microstructure in the literature <cit.>.








The aim of this study is, therefore, to design a hybrid (Cu/Al_2O_3) thin film and tailor its microstructure by heat treatment to become a model material in order to study the SRS behavior as a function of sample size and microstructure. The purpose of the Al_2O_3 interlayer was to limit grain growth between the subsequent Cu depositions, allowing to investigate the “As Deposited" (ultrafine-grained) coating parallel to its “Heat Treated" counterpart containing
microcrystalline grains. 







§ MATERIALS & METHODS








 §.§ Sample preparation

For the creation of the desired multilayers, a hybrid thin film deposition system combining both ALD and PVD methods was applied (Swiss Cluster). The sequential deposition consisted of alternating (5 times) ∼ 1 m thick pure Cu separated by 10 nm thin Al_2O_3 interlayers deposited on a Si (100) substrate (Fig. <ref>a) at room temperature. Cross-sectional electron backscatter diffraction (EBSD) confirmed the average grain size of this “As Deposited - AD" film to be d^AD∼ 120 nm (area weighted mean, see in Fig. <ref>c,e), qualifying this coating as ultrafine-grained (UFG).

Heat treatment was performed in a furnace at 800^∘C for 4 hours in Ar gas to evade oxidation, that resulted in the “Heat Treated - HT" sample. The roughness of the film surface along with the average grain size (d^HT∼ 1.1 m) has increased considerably, as it can be seen in Fig. <ref>d,f. The increased grain size defines this coating having microcrystalline (MC) structure.

In order to investigate the cross-section of the coating a sharp edge was prepared by broad beam Ar ion polishing (Jeol IB-1953CP) with beams of 6 kV (60 mins) and 2 kV (60 mins). For the micromechanical experiments square based taper-free micro-samples were created close to the sharp edge by focused ion beam (FIB) using a FEI Helios 600i and a Tescan Lyra3 Ga^+ scanning electron microscope (SEM) system (Fig. <ref>b). Sequential preparatory beams of 30 kV 21 nA (rough milling for removing large amount of material around the region of interest) down to 2.5-0.4 nA polishing were utilized with a 45^∘ incident angle direction with respect to the prepared surfaces. Two batch of different sized cuboid pillars (∼ 4.2 m and ∼ 2.9 m side length) were fabricated to study possible size related effects in the SRS.






 §.§ Micromechanical testing


Micro-samples were deformed using an Alemnis ASA nanodeformation system equipped with three different load cells (strain gage based standard and mini load cells for quasi-static strain rates (SR), piezo-based SmarTip for high SRs). For pillar compressions, conductive diamond flat punches (Synton AG) with diameters of 5 m and 10 m were utilized. The recorded load-displacement curves were baseline (drift) and compliance corrected using the AMMDA evaluation software. During HSR testing, due to the high acceleration following the tip actuation (achieving up to 6 mm/s tip velocity), an oscillating noise appeared, superimposed to the signal (“ringing effect"). In order to determine the yield stresses σ_y in such difficult cases, the elastic and plastic parts of the stress-strain curves were fitted using a linear equation, and their intersection was used to obtain the apparent yield stress (see Fig. <ref>a,b and Suppl. Fig. S<ref> for the full process). This method enabled to avoid further data filtering or averaging in the σ_y determination. For the consistency of the yield stress values at low and high strain rates, all mechanical data were treated the same way.








 §.§ Microstructure characterization

EBSD was employed to get statistical information on the grains of the samples. A Zeiss Supra 55 VP FEG-SEM equipped with an Oxford Instruments Symmetry 2 detector was used to record the EBSD diffraction patterns (2x2 binning, frame averaging: 4, step: 80-100 nm). High (angular) resolution (HR-) EBSD was utilized for the geometrically necessary dislocation (GND) density estimation prior to and following the deformation. This technique is based on image cross-correlation, linking each point on the map to a reference stress state <cit.>. The stored patterns were evaluated using the BLG Vantage CrossCourt Rapide v4.6 HR-EBSD software. Post mortem FIB-assisted 3D tomography <cit.> was performed on two pillars (HT big pillars, deformed at 0.01/s and 1000/s) to study the effect of SR on the microstructure close to the surface and inside the deformed volume. 

Transmission electron microscopy (TEM) was used to gain information on the stability of the hybrid layers after high temperature processing. As a result of the heat treatment, the grains were expected to grow, that can disturb the continuity of the deposited thin (10 nm) oxide layer. For this, a Jeol-ARM200F Cold FEG NeoARM Cs-corrected TEM was utilized at 200 kV. This instrument is equipped with an energy dispersive spectroscopy (EDS) detector (SDD CENTURIO-X) too. Multiple TEM-lamellae were prepared by FIB milling, following the standard preparation recipe, using sequential settings from 30 kV, 21 nA–40 pA (rough milling and liftout) down to 5 kV–1 kV, 20 pA (thinning and final cleaning).















§ RESULTS




The engineering stress (σ_eng) - engineering strain (ε_eng) curves resulting from several pillar compressions at various strain rates (0.001/s≤ε̇≤ 1000/s) are shown in Fig. <ref>. It is expected that the AD pillars exhibit higher yielding than the HT specimen, according to grain boundary strengthening. This phenomenon is confirmed in Fig. <ref>a,b for the two types of coatings (AD and HT). Work hardening seems not to be affected by the SR change. It is important to note that in the HT samples, due to the increased coating roughness, the uneven surface can introduce an artefact to the mechanical data. As it can be observed in Fig. <ref>b (P02, ε̇=0.01/s), if the surface of the pillar contains a part exceeding the average height, upon contact with the flat punch, this grain will start to deform, resulting in a less steep elastic loading (due to the fact that the σ_eng - ε_eng curves are calculated with the full pillar dimensions). However, once the surface is levelled by the indenter tip, the elastic loading part is recovered while the full pillar cross-section is being compressed beyond this point. The primarily deformed grain might locally modify the deformation state of the neighboring grains, however, due to the multilayered nature of the thin film, we expect to see negligible influence of this effect in the global mechanical behavior. To avoid this, either i) additional FIB polishing could be applied that levels the pillar surface, or ii) a Pt deposition cap may be added. Both solution could potentially modify the recorded mechanical data <cit.>, so in this work, we avoid using any of the aforementioned corrections, and restrict ourselves to the analysis of the unmodified HT coating.

Smaller pillars exhibit lower yielding, that can be attributed to the i) surface damage caused by FIB milling <cit.> or ii) it can be a grain size effect enhancing strain localization in the smaller HT specimen. As it was mentioned previously, to evade tapering, a higher incident angle of the bombarding ions were utilized, that may result in a higher concentration of implanted Ga close to the pillar surfaces. In smaller pillars, the effect of Ga presence is enhanced due to the increased surface-to-volume ratio. This explanation is the most probable cause of the slight yield stress decrease in smaller pillars rather that other  microstructure related effects, since it is apparent in both the AD (UFG) and HT (MC) pillars in a similar manner at both low and high strain rates (see in Fig. <ref>).





In order to determine the SRS-factor (m), the following equation is used:


    m=∂ (lnσ_y)/∂ (lnε̇).


In Fig. <ref>a, the determined average σ_y values are plotted on a linear scale as a function of ε̇ on a logarithmic scale. The error bars show the spreading of the calculated values from repeated pillar compression tests (typically three), that presents good repeatability of the experiments at almost all applied strain rates. As a result of the employed linear fitting method, σ_y can be determined fairly consistently.  The shaded area represents the noise during loading, indicating that when a load cell is switched from strain gage based (ε̇≤ 1/s) to piezoelectric based detection (ε̇≥ 10/s), the noise increases significantly. The drawback of using a noisier (but much stiffer) load cell <cit.> is compensated by the ability to go beyond the previously achievable deformation speeds and sampling rates. The noise becomes extremely large then the load signal starts to have a overwhelming oscillation due to the ringing effect (ε̇ = 1000/s). To reduce this outcome originating from the immense acceleration during the operation (6 mm/s), certain pillars were compressed in a way that the tip acceleration was initiated far (∼ 14 m) away from the surface, anticipating that at this distance the tip resonance decreases to minimal upon impact, when the flat punch is already travelling at its maximum speed <cit.>. To further improve the mechanical data, some pillars were compressed to very high strains to allow the tip to drive through these pillars at a constant speed without stopping, thereby completely destroying the samples and preventing them from being examined by post mortem EBSD analysis (hence the notation “pancake" in Suppl. Fig. S<ref>-S<ref>).

Strain rate sensitivity is plotted in Fig. <ref>b for all the tested pillars. The fitted dotted lines are used to determine the m values, calculated using Eq. <ref>. At lower strain rates, the HT samples are expected to behave similarly as the AD samples based on the work of Ramachandramoorthy and Kalacska et al. (for more details see the Discussion section.) <cit.>, therefore, only one additional ε̇ state was tested in this regime. Increased repeatability error in the HT samples are attributed to the surface roughness increase, as stated previously.


In order to understand the effect of heat treatment on the Cu/Al_2O_3 multi-system, TEM-lamellae were prepared from the HT sample via FIB milling. The results shown in Fig. <ref> confirm the enlarged grains due to high temperature ageing. Precipitates along grain boundaries were observed with sizes varying between 2-5 nm (Fig. <ref>c). EDS line profile collected through a Cu/Al_2O_3/Cu interface confirmed the integrity of the Al_2O_3 deposition after heat treatment, and the estimated layer thickness was calculated to be ∼ 13.5 nm (Fig. <ref>d). It is mentioned that this result may depend on the thickness of the lamella.



Finally, the effect of SR on the evolved dislocation structure was investigated by cross-sectional HR-EBSD. The pillars' surfaces were mapped before and after deformation, and the resulting GND density maps are plotted in Fig. <ref>. Here, the small grains (containing less than 20 pixels) were excluded from the analysis (green areas). Also, patterns with lower band contrast (BC) value than a given threshold (BC <100) were also removed from the evaluation (i.e. where patterns overlap close to grain boundaries). Afterwards, each identified grain was assigned to a reference pattern, chosen from the same grain with the lowest kernel average misorientation value (black dots). GND densities (ρ_GND) were then estimated based on 20 regions of interest for three pillars (P02 – P04). 










§ DISCUSSION


Initially, the FIB-milled pillars were chosen to be square shaped for two main reasons: i) to allow the monitoring of the ρ_GND evolution by HR-EBSD mapping, and ii) to create perfect uniaxial loading conditions and identical cross-sections along the height of the pillars for more accurate mechanical data analysis. Looking at the already deformed samples (Suppl. Fig. S<ref>) it can be concluded that the pillars deform in a “barrelling" manner, achieving the largest shape change closer to the top of the specimens at the low strain rate (LSR) regime (softer top). When a pillar is compressed at HSR (≤ 100/s), the shape of the pillars become slightly different, as here the middle of the coating extrudes the most. 





Slip plane localization is successfully suppressed by the consecutive Al_2O_3 layers in both the AD and HT samples, hence the resulting smooth σ_eng - ε_eng curves in Fig. <ref>, that show the absence of large strain bursts (or stress drops). The surface analysis of the deformed pillars support these findings in Suppl. Figs. S<ref> and S<ref>. Flow stresses reach quite high average values (σ_y^AD=758 MPa and σ_y^HT=460 MPa) during quasi-static deformation, much higher than pure Cu in literature, approaching the regime of nanocrystalline copper <cit.>. This increased yielding could be partly attributed to the grain boundary (Hall-Petch) strengthening (using the already established d values for the AD and HT samples with the equation σ_GB = σ_0 +kd^-0.5 with σ_0^Cu = 25 MPa material constant and k=0.12 MPa m^1/2 strengthening coefficient <cit.>, resulting in σ_GB^AD=371 MPa and σ_GB^HT=139 MPa theoretical values). Further strengthening mechanisms also play a key role in raising the σ_y values, such as dislocation and potentially (based on Fig. <ref>c) precipitation strengthening. As suggested by Fig. <ref>a, nanotwinning also occurs in this thin film system <cit.> that could be significant in the HSR behavior in case of possible dynamic recrystallization <cit.>. Additionally, high angle grain boundaries in the AD sample tend to run along the deformation axis of the pillars (somewhat columnar grain growth resulting in ellipsoid grain shapes), that will also increase the material's resistance towards compressive/tensile stresses.

When we compare the yield stress values to the 3D printed Cu UFG/MC system published by Ramachand­ra­moorthy and Kalacska et al. (<cit.>), we found these results to be in good agreement with both the AD/HT yielding, meaning that the multi-layered thin film proposed here is a fitting model material to further elaborate on the anomalous behavior of the polycrystalline Cu strain rate sensitivity. 

In Fig. <ref>, two anomalous trends in the SRS can be observed: the first one being the saturation of σ_y values of the AD samples at HSR in case of the smaller pillars. This behavior was already reported in Ref. <cit.>, where circular Cu polycrystalline pillars with 2 m gauge diameters were tested. There it was found that only the pillars with UFG structure exhibited such anomaly, modifying the SRS factor from m_UFG^LSR =0.06 to m_UFG^HSR =0.003. Here, we detected similar behavior, but remarkably, only in the smaller (2.9 m side length) pillars, leading to the conclusion that the pillar microstructure is not the only driving factor in the anomalous SRS behavior, but also the size of the sample has an additional influence too. In our AD thin films, the SRS factors were found to be close to the already reported values (m_AD^LSR =0.03, m_AD^HSR =0.01). When pillars were fabricated with bigger (4 m) side length, this anomalous behavior disappeared, and a single m value could describe the SRS within the whole investigated ε̇ regime.

The apparent activation volume at the yield point is calculated by the following equation <cit.>:

    Ω=√(3)k_B T∂(lnε̇)/∂ (lnσ_y),

where k_B and T are the Boltzmann constant and the temperature, respectively. By using b = 2.56 Å as the Burgers vector of a {111}<110> dislocation in copper, the calculated activation volumes are Ω_AD^big∼ 7.5b^3, Ω_HT^LSR∼ 5.5b^3 and Ω_AD^small-HSR∼ 13b^3. This suggests that in the LSR regime, the deformation is controlled by the movement of dislocations (1b^3-10b^3) nucleated at high angle grain boundaries and at surface sources. In the small AD pillars, the increase in Ω suggests a collective dislocation dynamic process (10b^3-100b^3) <cit.>, meaning that dislocations are created at multiple sources in large numbers upon impact, but their movement is restrained.

The second anomaly in Fig. <ref> was discovered upon testing the pillars above 100/s, where the σ_y values unexpectedly increase in the HT samples, regardless of their size. At LSRs, HT pillars exhibited similar SRS (m_HT^LSR =0.02) as the AD samples. This value matches the m factor reported in Ref. <cit.> for the MC structured pillars, where a monotonous SRS was also observed in this ε̇ regime. However, upon testing the HT pillars at ε̇=1000/s, the detected m value has increased significantly (m_HT^big-HSR =0.19, m_HT^small-HSR =0.11), showing that at HSR, a transition in the dislocation dynamics happen, resulting in a strain rate hardening regime. This crossover has already been proposed by simulations <cit.> and demonstrated by large scale impact testing <cit.>, but, to our knowledge, it is the first time to be presented at such small scale, during micromechanical testing. According to Ref. <cit.>, at LSRs, dislocations are thermally activated, while when  ε̇ exceeds 10^2/s, dislocation velocities become so high that the phonon-drag (thermal elastic vibrations) on their motion can no longer be  neglected <cit.>, making dislocations much more difficult to move, due to which we observe such an outstanding increase in the σ_y values. Just like in case of the first anomaly, the microstructure plays a crucial part in the SRS, as this deviation will most likely present itself in the AD samples too, only at somewhat higher SRs than what is currently reachable by small scale experiments <cit.>.














 §.§ Microstructure evolution



As it can be seen in Fig. <ref>,and Suppl. Figs. S<ref> and S<ref>, the heat treatment has increased the initial grain size, that has caused the previously flat Al_2O_3 layers to become rippled, following the curvature of the grains growing in their vicinity. This process can also affect the thickness of the Al_2O_3 layer, too, as the increased grain surface can thin down the interlayer, just like the material of a balloon upon its inflation. One of the striking things to notice was that the grain growth has seemingly reached a final state and further grain growth was blocked even at longer ageing (72 hours was the longest achieved). Even after repeated attempts (initially at 600^∘C, then 700^∘C and finally 800^∘C treatment using different testpieces), the grain growth process was halted. In this system, the Cu grain boundaries could not disappear due to an unknown phenomena. As a possible explanation, the diffusion of the Al (or Al_2O_3) has been proposed, by either i) precipitation along grain boundaries, or ii) atomic diffusion, as in an intra- or inter-granular manner. High resolution TEM imaging has confirmed the first hypothesis, showing that Al_2O_3 precipitates are located between large grains (marked by white arrows in Fig. <ref>c), where the curvature of the oxide layer has significantly increased. In order to confirm the second hypothesis based on atomic level diffusion, further experiments will have to be performed, that is out of the scope of the current work. In summary, the aluminum-oxide layer has successfully served its purpose as a thickness (and grain size) controlling barrier, but if larger grains are needed to be designed in the future, the composition or the thickness of the interlayer will need to be modified in the currently investigated system.


Furthermore, the decrease of the initial total dislocation density (ρ_Total) is also anticipated as a result of high temperature processing. Consequently, hardening rates during deformation could be different for the AD and HT samples. However, it is difficult to separate the effect of ρ_Total difference from the influence of the grain boundaries (that also act as dislocation sources) on the yielding and hardening behaviors, therefore this aspect is not discussed further. On the other hand, the resulting dislocation densities from low and high strain rate compressive testing can be investigated in case of the HT samples, where grains are large enough to perform meaningful GND density analysis by HR-EBSD. As the average GND densities shown in Fig. <ref> and in Table <ref>, the initial ρ_GND=1.3 × 10^14 m^-2 substantially increases due to external loading. When we compare similar maximum strains at two different SRs (ε∼ 18%, ε̇∈ 0.01/s, 1000/s), the resulting ρ_GND values are approximately 10 times higher at the surface. By performing post mortem FIB cross sectioning, we confirmed that in the middle of these pillars (∼ 1.6 m below the surface), the average ρ_GND values are somewhat lower, that could be attributed to the “surface effect", allowing facilitated dislocation nucleation (mainly due to the proximity of free surface and FIB damage) close to the facets of the pillars. This is in good agreement with previous GND density measurements on Cu micropillars <cit.>, but it should be noted that during pillar compression,  only a fraction of the generated dislocations is geometrically necessary. Additionally, material extrusion happens at the edges without further material constraints, therefore more lattice rotation and more GND generation can occur.






 §.§ Strain rate related microstructure transformation


In Ref. <cit.>, when UFG pillars were compressed at HSR, some pillars exhibited grain coarsening in the gauge section, that was attributed to static recrystallization due to the extremely high dislocation densities, and not resulting from dynamical processes during the compression itself <cit.>. In order to verify the appearance of recrystallization in this thin film (that could potentially modify the deposited coating's properties after impact), the deformed pillars were mapped by EBSD immediately (∼ 2 hours) after micromechanical testing (Suppl. Figs. S<ref>- S<ref>), and after ageing the samples under ambient conditions for 1.5 years (Fig. S<ref>). We found no evidence of abnormal grain growth in these pillars, regardless of the applied SR. The absence of recrystallization possibly originates from the applied Al_2O_3 interlayers, that conclusively not only exhibit excellent high temperature stability, but could potentially deactivate unwanted microstructure transformation, if the coating is to be operating under extreme conditions. 






§ CONCLUSIONS


In summary, the created Cu/Al_2O_3 multi-layered thin film was successfully tested in its “As Deposited" and “Heat Treated" form, that allowed us to create a model material system to study strain rate sensitivity as a function of the micropillar size and microstructure. The 10 nm thin Al_2O_3 interlayers showed excellent thermal stability, even when long heat treatment was applied. Due to the high temperature processing, Cu grain growth has reached a stabilized state with a microcrystalline structure, grown from the deposited ultrafine-grained formation. After thermal processing, only limited and localized dewetting was observable (while no delamination has occured) that had, therefore, no influence on the micromechanical characterization, as the coating remained in good mechanical contact with the Si substrate.

During the mechanical data analysis, two anomalous phenomena was identified in the strain rate sensitivity behavior, namely i) yield stress saturation in the AD samples when the pillar diameters were below a given threshold, and ii) increased strain rate hardening in the HT samples at very high (ε̇ = 1000/s) strain rates. This behavior clearly demonstrates that dislocation nucleation and motion is strongly deformation rate dependent in Cu at high strain rates.

Analysing the post mortem surfaces of the deformed pillars it was concluded that the Al_2O_3 interlayers have successfully prevented shear localization (even in the HT pillars), resulting in smooth stress-strain curves and homogeneous deformation throughout the height of the pillars, that produced shape changes in a barrelling manner.

All these conclusions were made on FIB-milled pillars, bearing in mind that the Ga^+ implantation has likely impacted the resulting yield stresses. However, the proposition of such a coating to be considered as a model material to investigate the mechanical behavior of other complex (i.e. 3D printed) materials can finally be accepted by performing a large number of tests in identical conditions, and finding a good correlation between the previously reported yielding behavior and the current work.

GND densities were also studied in detail before and after compression, leading to the observation of slightly higher dislocation densities in the vicinity of the pillars' edges, resulting in an inhomogeneous distribution within the samples. This most likely plays a key role in the mechanical behavior and results in a transition of the strain rate sensitivity from monotonous to anomalous, when the sample sizes are reduced below a characteristic (microstructure dependent) length scale. 

Finally, no recrystallization occured in the deformed samples, regardless of their size, microstructure or ageing time, that leads to the deduction that the employed Al_2O_3 interlayers successfully prohibit such transformation even when the samples were subjected to high strain rates. This means that the proposed Cu/Al_2O_3 multi-layered system can be applied as a coating to withstand extreme conditions. 





§ ACKNOWLEDGEMENT

Matthieu Lenci (Ecole des Mines, St. Etienne) is acknowledged for the TEM imaging.




§ CREDIT STATEMENTS

S. Kalácska: Conceptualization, Methodology, Validation, Formal analysis, Investigation, Resources, Data Curation, Supervision, Project administration, Writing – Original Draft, Writing – Review & Editing, L. Pethő: Investigation, Resources, Writing – Review & Editing, G. Kermouche: Resources, Writing – Original Draft, Writing – Review & Editing, J. Michler: Resources, Writing – Review & Editing, P. D. Ispánovity: Conceptualization, Writing – Original Draft, Writing – Review & Editing



§ FUNDING SOURCES

SK and GK were funded by the French National Research Agency (ANR) under the project No. ANR-22-CE08-0012-01 (INSTINCT) and No. ANR-20-CE08-0022 (RATES). PDI acknoledges the support by the National Research, Development and Innovation Fund of Hungary under project No. NKFIH-FK-138975.



§ DATA AVAILABILITY

Experimental data generated in this study have been deposited in the Zenodo database at <www.doi.org/10.5281/zenodo.12782449>.








Supplementary Materials for 

Anomalous strain rate sensitivity of a Cu/Al_2O_3 multi-layered thin film



Szilvia Kalácska^a,b,*, László Pethő^b, Guillaume Kermouche^a, Johann Michler^b, Péter D. Ispánovity^c 




^a Mines Saint-Etienne, Univ Lyon, CNRS, UMR 5307 LGF, Centre SMS, 158 cours Fauriel 42023 Saint-Étienne, France

^b Empa, Swiss Federal Laboratories for Materials Science and Technology, Laboratory of Mechanics of Materials and Nanostructures, CH-3602 Thun, Feuerwerkerstrasse 39. Switzerland

^c Eötvös Loránd University, Pázmány P. stny 1/a, 1117 Budapest, Hungary









 

 




Data repository available at: <www.doi.org/10.5281/zenodo.12782449>















































 Paper Title: Atomic Structure of Self-Buffered BaZr(S,Se)$_3$ Epitaxial Thin Film Interfaces.




§ ABSTRACT 


Understanding and controlling the growth of chalcogenide perovskite thin films through interface design is important for tailoring film properties. Here, the film and interface structure of BaZr(S,Se)3 thin films grown on LaAlO3 by molecular beam epitaxy and post-growth anion exchange is resolved using aberration-corrected scanning transmission electron microscopy. Epitaxial films are achieved from self-assembly of an interface “buffer” layer, which accommodates the large film/substrate lattice mismatch of nearly 40% for the alloy film studied here. The self-assembled buffer layer, occurring for both the as-grown sulfide and post-selenization alloy films, is shown to have rock-salt-like atomic stacking akin to a Ruddlesden-Popper phase. Above this buffer, the film quickly transitions to the perovskite structure. Overall, these results provide insights into oxide-chalcogenide heteroepitaxial film growth, illustrating a process that yields relaxed, crystalline, epitaxial chalcogenide perovskite films that support ongoing studies of optoelectronic and device properties. 




CEERS Key Paper. IX. Identifying Galaxy Mergers in CEERS NIRCam Images Using Random Forests and Convolutional Neural Networks
    [
    August 5, 2024
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§ MAIN TEXT


Growth of high-quality semiconductor thin films, needed for applications in microelectronics and optoelectonics, requires the availability of substrates that are well-matched structurally and chemically. This presents problems for many emerging materials for which suitable substrates are not readily available. For instance, many  chalcogenide semiconductors have chemistry, crystal structure, and lattice constants that are substantially different than available oxide and compound semiconductor substrates. Further, many proposed device concepts - including two-dimensional materials for transistors and chalcogenide perovskites for solar cells - may depend on better control over oxide-chalcogenide hetero-interfaces. Better control hinges on a better understanding of interfaces with many elements (here, La, Al, O, Ba, Zr, S, and Se) in atomic structures that may deviate substantially from nominal bulk phases and that are difficult to predict a priori. Advancing semiconductor heteroepitaxial growth of new materials, therefore, depends on overcoming substantial challenges in advanced characterization. Here, we take on this challenge in the context of the growth of chalcogenide perovskite thin films on complex oxide substrates by molecular beam epitaxy (MBE). 

Chalcogenide perovskites of type ABX3 (A=Ca, Sr, Ba, B=Ti, Zr, Hf, X=S,Se), with a direct band gap that is tunable within visible to near-infrared (VIS-NIR), strong optical absorption, low toxicity, and excellent environmental stability, are of interest for future solar cell technologies.<cit.> The most-studied chalcogenide perovskite, BaZrS3, features strong dielectric response, strong optical absorption, exceptional thermal stability, and has been made as powders, crystals, nanocrystals, and thin films.<cit.> Epitaxial thin films in the perovskite structure have been grown by the authors across the full BaZrS_3-ySe_y, y=0-3 alloy range, with direct band gap spanning the range 1.5 - 1.9 eV.<cit.> Notably, despite a very large lattice constant mismatch of BaZrS3 with the LaAlO3 substrates, the formation of relaxed, epitaxial BaZrS3 films was observed to be accommodated by a self-assembled buffer layer at the interface.<cit.>

By comparing the bulk pseudocubic lattice parameters for the BaZrS3 film (497.5 pm) and LaAlO3 substrate (381.1 pm), a coincident site lattice (CSL) was proposed to explain the observed epitaxial relationship.<cit.> The appearance of structural perturbations at the interface and the potential for surface passivation and reconstruction in the presence of H2S at high temperature, however, indicate that the interface is too complex to be described only by a CSL.<cit.> More recently, the authors realized thin film sulfide-selenide perovskite alloys by both direct growth and post-growth selenization methods.<cit.> Beyond modifying the bandgap, the higher atomic mass of selenium compared to sulfur also enables a more detailed atomic-scale study of the interface. 

Here, the epitaxial interface between BaZrS3 and BaZr(S,Se)3 films with LaAlO3 oxide substrates is characterized using aberration-corrected scanning transmission electron microscopy (STEM). Post-growth selenization of BaZrS3 films through annealing in H2Se gas, used to make the BaZr(S,Se)3 films, exchanges a fraction of the chalcogen ions without changing the interface structure or film microstructure. The self-assembled buffer layer is shown to have a rock-salt-like structure, similar to stacking faults often found in perovskites with imperfect cation stoichiometry and in Ruddlesden-Popper (RP) layered structures. This buffer layer accommodates the film-substrate mismatch and explains the structural deviations at and near the interface, which quickly diminish away from the interface.

A high-angle annular dark-field (HAADF) STEM image of a 40-nm-thick selenized film is shown in Figure <ref>a. As previously reported for the pure sulfide, a self-assembled buffer layer and a “cube-on-cube” epitaxial relationship are seen at the interface.<cit.> Elemental quantification with energy dispersive X-ray spectroscopy (EDS) indicates a uniform distribution of Ba, Zr, S, and Se throughout the film cross-section, with an overall composition of BaZrS_3-ySe_y, y=2.21.<cit.> Similar to the pure sulfide BaZrS3 films, the selenized film is composed of 90 rotation variants, which originate due to the nearly-equivalent in-plane pseudocubic lattice parameters along the orthorhombic [010] and [101] projections.<cit.> These two domain types or rotation variants are shown in STEM HAADF images in Figure <ref>b,c and compared to STEM image simulations Figure <ref>d,e  assuming the orthorhombic structure with randomly-distributed anion occupancy according to the y=2.21 average composition. Experiment and simulation are in agreement, where tilting of the S/Se octahedra and alternating Ba-Ba displacements are visible in the [010] and [101] projected variants, respectively (Figure <ref>f,g). 







The atomic structure of the film/substrate interface is highlighted in the simultaneously acquired annular dark-field (ADF) and integrated differential phase contrast (iDPC) images (Figure <ref>a,b). Approaching the interface, the anion octahedral tilt angle (Figure <ref>c) and Ba-Ba displacement angle (Figure <ref>d) are reduced compared to their values measured from the film bulk (6.4 and 3.0, respectively). These changes near the interface are reminiscent of previous reports of perovskite oxide heteroepitaxy, where oxygen octahedra tilting is either dampened or propagated in the thin film through boundary constraints with the substrate.<cit.> 






The first two atomic planes of the selenized film - immediately at the interface with LaAlO3 - exhibit structure contrasting with the rest of the film (Figure <ref>a,e). This observation is similar to that previously reported for the pure sulfide BaZrS3, suggesting that selenization changes the chalcogen composition without affecting the interface structure. ADF, iDPC, and inverse Fourier filtered images (Figure <ref>e-g) illustrate an approximate coincidence of every five LaAlO3 unit cells with every four BaZr(S,Se)3 unit cells, in agreement with the overall periodicity observed from nano-beam electron diffraction (Supplementary Materials). At the coincident sites (connected by white lines in Figure <ref>e,f), the Ba and La atom columns are displaced closer to one another, suggesting stronger bonding at these locations. Based on this interface model, the top and cross-sectional views for the two 90 rotation variants are shown in Figure <ref>h-i, with in-plane lattice parameters determined from experiment (Supplementary Materials). The corners of the projected 5×5 square La sub-lattice are close to the 4×4 near-square Ba sub-lattice, exhibiting a mismatch of 0.5% along [10-1] and 0.4% along [010].

Despite this, the asymmetry of Ba positions in the orthorhombic structure yields imperfect coincidence for both of the 90 rotation variants. This is visible by the misalignment of Ba and La sub-lattices and incompatibility of both structures with the direct bonding coincident sites seen in experiment Figure <ref>h-k. From the structural model, the displacements of Ba are symmetric and opposite with respect to the La sub-lattice of the substrate, resulting in an additional mismatch of ≈±0.3Å, yet in experiment, these displacements in addition to anion octahedral tilting are absent near the interface. In addition, from comparing images from experiment with the structural models (Figure <ref>e,f versus Figure <ref>j,k), additional atom columns are found within the interface gap. These atom column positions exhibit significant displacements compared to the bulk structures of both the BaZr(S,Se)3 film and LaAlO3 substrate. From atomic number-sensitive HAADF images of the interface (Figure <ref>a), the contrast of the atom columns found within the interface gap suggest occupation by S (Z=16) or Se (Z=34). This is evidenced by significantly brighter contrast than that of Al (Z=13) columns in the LaAlO3 substrate (both marked by the white arrows) and lower contrast than those of Ba (Z=56), La (Z=57), or Zr (Z=40). Further, the substrate surface is terminated by this plane of atom columns, suggesting that these atom columns and the rest of the buffer layer self-assemble during H2S exposure of the substrate immediately before and during film growth.














The self-assembled buffer layer is further characterized using EDS elemental mapping at the interface (Figure <ref>a,b and Supplementary Materials). The profile across the interface, shown in Figure <ref>b, is integrated along the plane of the interface to improve the signal-to-noise ratio with a comparatively reduced dose. A well-defined film is observed, with negligible diffusion beyond 1-2 nm across the film/substrate interface. Further, the HAADF image intensity is decreased within a 0.5 nm wide region at the interface (shaded region in Figure <ref>b) compared to the substrate and the rest of the film. Likewise, reduced HAADF intensity is observed at the terminating plane of the substrate immediately below the film, which can be explained by vacancies and/or static displacements of La along the atom columns. Immediately adjacent to this plane of La atoms is a notable peak in the sulfur signal (triangle marker Figure <ref>a,b) whose location coincides with the unidentified atom columns extending into the interface gap (indicated with arrows in Figure <ref>a) These observations suggest that H2S exposure at high temperature produces a subtle reconstruction of the LaAlO3 surface, including La-S bond formation.  



The anion octahedral distortions and cation displacements in the self-assembled buffer layer are similar to the structure of the Ba3Zr2S7 RP phase (space group P4_2mnm), which is shown in Figure <ref>a,b overlaid on HAADF images acquired on selenized and pure-sulfide films.<cit.> To highlight similarities in structure, interplanar distances, calculated from the in-plane averaged atom column positions, are measured from the acquired images of the grown films and simulated images of the RP phase and shown in Figure <ref>c. These spacings are numbered beginning at the film/substrate interface (#0). In the first plane of the films, the anion atom columns are displaced towards the interface gap (Figure <ref>a,b). This displacement is characteristic of the rock-salt layers in the RP phase, in which anion planes terminate each perovskite slab and extend into the gap, and connects the structure of the perovskite film to that of the passivated substrate.

Further quantifying the interplanar distances within the buffer layer(Figure <ref>c) shows agreement with the RP structure. The average separation of 380 pm in the substrate is consistent with the pseudocubic lattice parameter of LaAlO3. Across the interface, or spacing #0, the distances between La and Ba planes in the selenized and pure sulfide films are 331 pm and 325 pm, respectively, close to the ideal rock-salt layer separation of 350 pm in Ba3Zr2S7 (Figure <ref>c). The next interplanar distances (spacing #1) are 250 pm in the selenized and 205 pm in the pure sulfide films. This contraction is consistent with the RP structure, in which the planar spacing immediately adjacent to the rock-salt later is contracted (to 200 pm) due to the polar displacements of Ba and the ZrS6 octahedra.<cit.> A subsequent increase in interplanar distance at the Ba-ZrS6 octahedra (spacing #2) is again similar to the RP structure. Beyond spacing #2 we observe a reduction in the oscillation of planar separations, converging to an average out-of-plane interplanar distance of 251 pm for BaZrS3 and 266 pm for the selenized film, matching the expected perovskite structure (Pnma).<cit.> Patterns of anion octahedral tilts and Ba-Ba angle variations (Figure <ref>e), absent in the RP structure, emerge and become correlated within 1-3 unit cells of the buffer layer, similar to observations at perovskite oxide interfaces.<cit.> Finally, even with anion exchange of S with Se, the structure of the RP buffer is preserved, with aligned Ba and La coincident planes at the interface. 




In conclusion, the formation of epitaxial and relaxed BaZrS3 and BaZr(S,Se)3 chalcogenide perovskite films on LaAlO3 substrates with large lattice mismatch is made possible by a self-assembled buffer layer. This layer likely forms during the initial stages of BaZrS3 film growth, including annealing and H2S gas exposure of the substrate. The buffer layer resembles the rock-salt-like layers found in RP phases, and the interface structure is quantitatively comparable to Ba3Zr2S7. Notably, the selenization process, used to convert the pure sulfide to an alloy film, appears to substitute Se for S throughout the film, including in the buffer layer, without affecting the buffer layer structure or its strain relief functionality. The epitaxial growth and facile selenization process associated with this buffer layer enables study of the optoelectronic properties of chalcogenide perovskite alloys with a wide range of composition, band gap, and lattice constant, for which matching substrates may not be available. Finally, the gas-source film growth method used here may be more broadly applicable to heteroepitaxial growth of chalcogenide semiconductors on non-matched, non-chalcogenide substrates, if the self-assembled buffer layer is found to generalize beyond the LaAlO3/BaZr(S,Se)3 system.








§ METHODS



 §.§ Film Growth

Thin films were grown and annealed using a chalcogenide molecular beam epitaxy (MBE) system (Mantis Deposition M500). Further details on the deposition system and protocols are described in Ref Ye2024-va. Thin films were grown on 10 x 10 x 0.5 mm^3 (001)_PC-oriented LaAlO3 substrates (MTI). PC stands for pseudo-cubic; (001)_PC corresponds to the (012) family of reflections in a rhombohedral system. Prior to growth, the substrates were outgassed in the MBE chamber at 1000 C under H2S gas. Epitaxial BaZrS3 films were grown at 900-1000 C with elemental Zr and Ba sources, under 0.8 sccm H2S flow. BaZrS3 epitaxial thin films were alloyed with selenium by annealing them under a mixture of 0.2 sccm H2S and 0.2 sccm H2Se at 800 C for 60 minutes in the MBE chamber. 



 §.§ STEM Characterization

Cross-sectional samples for electron microscopy were prepared by nonaqueous mechanical wedge polishing followed by single sector Ar+ ion milling (Fischione 1051 TEM Mill).<cit.> A probe aberration-corrected Thermo Fisher Scientific Themis Z S/TEM operated at 200 kV was used for STEM imaging and spectroscopy. Imaging was performed using a probe semi-convergence angle of 18 mrad and probe current of 30 pA. High-angle annular dark-field (HAADF), annular dark-field (ADF), and integrated differential phase contrast (iDPC) images were acquired using collection angles of 65-200 mrad, 25-153 mrad and 6-24 mrad, respectively, and a dwell time of 1 µs. The revolving STEM (revSTEM) method was used to correct for image distortions from sample drift during imaging for the 12-frame 1024×1024 image series.<cit.> STEM image simulations were performed using a custom implementation of the multislice method.<cit.> STEM energy dispersive X-ray spectroscopy (EDS) maps were acquired using SuperX detectors and quantified using the Thermo Fisher Scientific Velox software. A 5 pixel averaging pre-filter and 3-sigma Gaussian post-filter were used for noise reduction in the EDS maps. 



§ DATA AVAILABILITY STATEMENT

The data supporting this study's findings are available from the corresponding author upon reasonable request.


M.X. and J.M.L acknowledge support for this work from the Air Force Office of Scientific Research (FA9550-20-0066) and the MIT Research Support Committee. We acknowledge support from the National Science Foundation (DMR-1751736). K.Y. acknowledges support from the National Science Foundation Graduate Research Fellowship, grant no. 1745302. This research was partly supported by a grant from the United States-Israel Binational Science Foundation (BSF), under grant no. 2020270. This research was supported in part by the Sagol Weizmann-MIT Bridge Program. This research was supported in part by the Skolkovo Institute of Science and Technology as part of the MIT-Skoltech Next Generation Program. This work made use of the MIT.nano Characterization Facilities. 






See supplementary materials for additional film structure characterization, including in-plane/out-of-plane distances, nano-beam electron diffraction, and EDS maps. 









 Paper Title: Quantum Order by Disorder: A Key to Understanding the Magnetic Phases of BaCo$_2$(AsO$_4$)$_2$.








Present Address: Department of Physics and National High Magnetic Field Laboratory, University of Florida, Florida 32611, USA.
National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.


National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.


Los Alamos National Laboratory, Los Alamos, NM 87545, USA.


Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA.


Oak Ridge National Laboratory, Oak Ridge, TN 37830, USA.


National High Magnetic Field Laboratory, Florida State University, Tallahassee, Florida 32310-3706,  USA.


[Corresponding authors:]vzapf@lanl.gov
National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.


[Corresponding authors:]winters@wfu.edu
Department of Physics and Center for Functional Materials, Wake Forest University, Winston-Salem, NC 27109, USA.


[Corresponding authors:]ml10k@lanl.gov
National High Magnetic Field Laboratory, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA.




§ ABSTRACT 
 (BCAO), a honeycomb cobaltate, is considered a promising candidate for materials displaying the Kitaev quantum spin liquid state. This assumption is based on the distinctive characteristics of  Co^2+ ions (3d^7) within an octahedral crystal environment, resulting in spin-orbit-coupled J_ eff = 1/2 doublet states. However, recent experimental observations and theoretical analyses have raised questions regarding this hypothesis. Despite these uncertainties, reports of continuum excitations reminiscent of spinon excitations have prompted further investigations. In this study, we explore the magnetic phases of BCAO under both in-plane and out-of-plane magnetic fields, employing dc and ac magnetic susceptibilities, capacitance, and torque magnetometry measurement. Our results affirm the existence of multiple field-induced magnetic phases, with strong anisotropy of the phase boundaries between in-plane and out-of-plane fields. To elucidate the nature of these phases, we develop a minimal anisotropic exchange model. This model, supported by combined first principles calculations and theoretical modeling, quantitatively reproduces our experimental data. In BCAO, the combination of strong bond-independent XXZ anisotropy and geometric frustration leads to significant quantum order by disorder effects that stabilize colinear phases under both zero and finite magnetic fields. 


Quantum Order by Disorder: A Key to Understanding the Magnetic Phases of 
    Minseong Lee
    
=========================================================================









§ INTRODUCTION
 



The interplay between crystal field symmetry, spin-orbit coupling, and the electron configuration of cobalt ions (Co^2+, 3d^7) in a nearly ideal octahedral oxygen environment stabilizes the J_ eff = 1/2 state, a fundamental building block of numerous intriguing physical phenomena in various configurations. One of the intriguing theoretical proposals is that cobaltates with a J_ eff = 1/2 state can potentially exhibit bond-dependent exchange interactions, including what is lately called the Kitaev exchange interaction <cit.>. It is well known that when the Kitaev exchange interaction dominates the spin Hamiltonian, the ground state can become a Kitaev quantum spin liquid (KQSL) due to strong magnetic frustration <cit.>. The KQSL is highly sought after among frustrated magnets because it has the potential to host exotic emergent phenomena such as Majorana Fermions and topological order, with significant applications in quantum computing <cit.>. Therefore, several honeycomb cobalt-based magnets, including Na_2Co_2TeO_6 <cit.>, and Na_3Co_2SbO_6  <cit.> have been investigated intensively. 

Recently, another cobalt-based honeycomb-lattice material  (BCAO) has been proposed as an ideal platform for realizing a KQSL <cit.> (Figure  <ref>a). BCAO shows a series of magnetic phase transitions under applied magnetic field (H). The zero-field order is identified as either an incommensurate spiral or double-stripe (DS), which is suppressed in favor of a collinear up-up-down (UUD) phase under field <cit.>. These spin structures are illustrated in Figure <ref>b. For magnetic fields in the crystallographic ab-plane, the latter phase is subsequently suppressed by relatively weak fields (0.5 T), above which a KQSL phase has been suggested <cit.>. For field in the out-of-plane direction, which is theoretically more conducive to a field-induced KQSL phase <cit.>, a broad magnetic continuum was observed <cit.>. However, conflicting reports exist regarding whether BCAO is best described by an extended Kitaev model with strongly bond-dependent magnetic couplings or as an XXZ-J_1-J_3 model with weakly bond-dependent couplings; recent inelastic neutron scattering measurements, as well as ab initio calculations <cit.>, align more with the XXZ-J_1-J_3 model <cit.>. It has also been pointed out that strong distortions of oxygen octahedra, whose the trigonal term in crystal-field splitting matrix is larger than one fifth of spin-orbit coupling, lead to Ising- and XY-like spin behavior <cit.>. Additionally, the weak ligand-assisted exchange interactions weaken bond-dependent exchange interactions <cit.>. Hence, further investigation is still required to characterize the phases of BCAO comprehensively. Investigating the angular dependence of phase boundaries is an effective method to understand the origin of magnetic anisotropy <cit.>. Recently, Safari et al. explored the phase diagram of BCAO using magnetotropic susceptibility measurements and theoretical models <cit.>. Their findings indicate that while classical models qualitatively describe the system's behavior, quantum corrections are essential for precise critical field predictions <cit.>.






In this study, we explored the details of magnetic field-induced ground states in BCAO using ac and dc magnetic susceptibility, capacitance measurement, and torque magnetometry measurements at various temperatures (T). With the aim of establishing the origin of the magnetic anisotropy, we study the angular dependent T-H phase boundaries by tuning the magnetic field orientation. With the application of a magnetic field, multiple-phase transitions were identified for H along the in-plane direction, consistent with previous studies <cit.>. In comparison, for H close to the c-axis, the critical field for suppression of ordering is drastically increased, and intermediate phase transitions disappear or are washed out at the lowest temperatures.

To understand the full details of the phase diagram, we employ first-principles-based calculations to provide a starting point for analyzing the magnetic couplings. The resulting model is refined by fitting the experimental phase diagram employing second-order energy corrections <cit.> to account for quantum fluctuation effects on the stability of various magnetic phases. This combined approach allows for the precise reproduction of the experimental observations. Consistent with recent assertions <cit.>, we conclude that quantum order-by-disorder (QOBD) effects dominate in BCAO, influencing both the critical field values and specific progression of field-induced phases.







§ RESULTS
 


 §.§ Magnetic properties and phase diagrams for in-plane fields




Figure <ref>a illustrates the magnetization measured at various temperatures as a function of the field applied along the a-axis. Hysteresis loops emerge between 0.1 T (H_1) and 0.2 T (H_2), which become smaller with increasing temperature and are almost invisible above 4.0 K. Beyond 0.5 T, there is a steep increase in magnetization before reaching saturation around 0.6 T (H_3). The saturation magnetization is about 2.3 μ_B/Co^2+, from which we extracted g_ab∼ 4.6. The magnetization between the saturation field and ∼0.15 T constitutes a one-third magnetization plateau. These findings are consistent with prior studies <cit.>. When the temperature exceeds T_N∼ 5.5 K, all features disappear, and a monotonic increase of the magnetization with field is observed, consistent with a thermal paramagnetic state. Figure <ref>b shows the ac magnetic susceptibility as a function field. The temperature dependent ac magnetic susceptibility is in Supplementary Figure 1. Three peaks are observed: The first two peaks at ∼H_1 and ∼H_2 are consistent with the jump in magnetization in the hysteresis loop and the end of the hysteresis loop. The third peak above ∼H_3 corresponds to the saturation field. ac susceptibility at 6 K does not show any anomalies. The magnetic phase diagram with field along the a-axis deduced from these measurements is presented in Figure <ref>c. We observed three thermodynamic phases (I, III, and IV) and one intermediate state (II) due to the hysteresis loop <cit.>. Neutron scattering experiments identified phase I as the DS phase or closely related spiral phase with in-plane wavevector close to Q = [1/4,0] <cit.>. Phase III is the UUD phase with 1/3 saturation magnetization and Q = [1/3,0] <cit.>. Given the differing wavevectors and the discontinuous change in wavevector through this phase transition, it is not surprising that these phases are separated by a first order phase transition. Phase II corresponds to a coexistence region of Phase I and III. Lastly, phase IV is the asymptotically polarized paramagnetic phase that is smoothly connected to high temperature paramagnetic phase.

Figures <ref>d-<ref>f show analogous results for field along b^*-axis. For this field direction, we observe the same progression of phase transitions. The hysteresis loops are wider but less pronounced, and H_3 is slightly lower in comparison with H||a. Therefore, the width of 1/3 plateau (phase III) is slightly narrower with magnetic field along b^*-axis. 



 §.§ Magnetic properties and phase diagrams for out-of-plane field


Figure <ref>a shows the temperature dependent dc magnetic susceptibility (M_c/H (T)) measured at various constant external magnetic fields. At 1 T, a clear phase transition from the paramagnetic to DS state is observed and marked with an empty circle around 5.3 K. Raising the field to 14 T only suppresses T_N to 4 K. The phase transition is therefore very robust against out-of-plane magnetic field, which is in sharp contrast with the behavior for in-plane fields. 

Figure <ref>b shows the magnetization versus out-of-plane field up to 14 T. At 2 K, the magnetization linearly increases without showing any anomaly, demonstrating a gradual spin canting towards the field direction without phase transitions. At 4 K, we observe a weak slope change around 14 T, denoted as a black arrow, which is consistent with the thermal phase transition between the DS and thermal paramagnet observed in Figure <ref>a. With increasing temperature to 4.5 K, the slope change shifts to slightly lower field of about 11.5 T. Suppression of the phase boundary between AFM and paramagnetic phase with increasing temperature is a typical behavior of antiferromagnets. 



Since 14 T is not enough to observe the magnetic saturation, we measured the magnetization with field along c-axis up to 50 T at Pulsed Field Facility in Los Alamos National Laboratory. 
Results from pulsed field magnetization experiments at various temperature are shown in Figure <ref>c and  <ref>d. The data are admittedly noisy, but the important features are clearly discernible. In the paramagnetic state at 10 K, the magnetization curve resembles the Brillouin function. 
Below T_N, the magnetization increases linearly and shows a slope change in the vicinity of 25 T, which we assign as the out-of-plane saturation field. Therefore dM_c/dH is a constant below ∼20 T and above ∼35 T. Due to the small crystal field excitation gap of Co^2+, it is common to observe a linearly increasing magnetization above the saturation field, due to Van Vleck paramagnetism <cit.>. By extrapolating the Van Vleck contribution to the zero field, we found a saturation moment of about 0.71 μ_B/Co^2+, from which we extract g_c∼ 1.5. We note that the saturation field with out-of-plane configuration in this study is higher than the field claimed by Zhang et al. <cit.> and Safari et al. <cit.>. As we demonstrated in Supplementary Figure 4 and 5, a tiny misalignment of the crystal can lead to a drastically suppressed saturation field. For example, only 6^∘ degrees off from the c-axis suppresses the saturation field 27 T to 4.5 T. 

The 25 T phase transition is smoothly connected to the zero field AFM to paramagnetic phase transition, additionally confirming that it is the saturation field. Thus, the out-of-plane magnetization measurements demonstrate that BCAO is highly anisotropic between in-plane and out-of-plane fields, with a saturation field ratio H_sat^c/H_sat^a of about 50. 




 §.§ Angular dependence of magnetic phase transitions


To investigate how the phase boundaries evolve with the field direction, we measured the magnetization, capacitance, and magnetic torque below T_N while varying the magnetic field orientation across the c-b^* plane. In Figure <ref>a, we first show the evolution of the magnetization. For an out-of-plane field (θ = 0^∘), the magnetization does not show any features up to 14 T. However, with a small rotation of the field direction towards the b^* direction, the transitions to the intermediate UUD and polarized phases appear. At θ = 6.3^∘, the phase transitions to the UUD and polarized phases occur at 1 T and 4 T, respectively. These move towards ∼H_1 and ∼H_2 as the field direction approaches b^*-axis, consistent with Figure <ref>d. The magnetic moment also monotonically increases as the field is rotated towards b^* direction, consistent with the anisotropy of the g-values.



Capacitance and magnetostriction measurements can be sensitive probes of magnetic phase transitions as the change in spin structure accompanies the change in electric susceptibility and lattice parameter of magnetic insulators. We found this to be the case for . Figure <ref>b shows the change in capacitance as a function of magnetic field at various angles. We applied the electric field at 16 kHz along b^*-axis. At 27^∘, the capacitance change shows clear two phase transitions corresponding to the phase transition from DS to UUD, and UUD to polarized phase. They evolve to higher fields as the field becomes close to c-axis. The anomaly from the phase transition from DS to UUD phase becomes weaker while that from UUD to polarized phase is more or less similar at all angles. The capacitance change and magnetostriction data with in-plane magnetic field can be found in Supplementary Figure 2 and 3.

Figure <ref>c and <ref>d show magnetic torque (τ) and its differentiation with magnetic field (dτ/dH) up to 30 T as a function of magnetic field at different fixed angles. Both τ and dτ/dH display a clear anomaly at phase transitions. When the field is only at 0.36^∘ from c-axis, a sharp increase in τ and a peak dτ/dH appear at about 23 T. This is reasonably well consistent with the saturation field observed in pulsed field magnetization shown in Figure <ref>c. The anomaly shifts quickly to lower field as the angle becomes larger than 1^∘ and ends up being only 2 T at 19.7^∘. 

In Figure <ref>a, we show the magnetic torque measured for continuous field-angle sweeps between the c and b^* axes at T = 0.5 K at various fixed magnitudes of H. All curves exhibit 180^∘ periodicity and a saw-tooth shape due to the strong magnetic anisotropy. The sharp change of sign of the torque near 0^∘ indicates the c-axis is the hard axis. Additional step-like anomalies are observed near θ = 0^∘ marking the first order transitions H_1 and H_3. These two features are only clearly observable up to 10 T and they merge into one above 10 T. 

In general, the magnetic torque may be expanded as <cit.>:

    τ/H = ∑_n = 1^∞ c_2nsin(2nθ) = c_2sin(2θ) + c_4sin(4θ) + ⋯

The experimental field-dependence of |c_2| and |c_4| is plotted in Figure <ref>c. Both coefficients increase steeply up to 4 T and then slowly decrease above 4 T up to the maximum field. The ratio of c_2 to c_4 remains more or less constant at ∼0.4 across the entire field range due to the dominant saw-tooth curve shapes.  In the limit of large field, such that the majority of the field angles correspond to the polarized phase, the c_2n coefficients have two contributions. The first contribution is field-independent, and depends exclusively on the g-anisotropy. For the implied range of g-values (g_ab∼ 4.6, g_c = 1.5 - 2.5), the limiting ratio for this contribution leads to |c_4/c_2| = 0.17 - 0.27. The next order contribution to the c_2n scales as 1/H, and is related to both the XXZ exchange anisotropy and  <cit.>.  The findings that c_2 and c_4 decay with increasing field, and that their ratio exceeds the g-only value both demonstrate significant exchange anisotropy is necessary to explain the observed torque. Below, we develop an exchange model for BCAO, and use these observations to verify the model.

From the phase transitions observed in the angular dependence of magnetic properties in Figure <ref> and <ref>a, an angular dependence of the magnetic phase diagram is constructed, as shown in Figure <ref>e. The red empty symbols represent the phase transition from the DS to the UUD phase (H_1), and the black empty symbols indicate the saturation field (H_3). As the field direction moves from the b^*-axis to the c-axis, both phase boundaries sharply increase below 10^∘. The saturation field reaches above 23 T, whereas the phase boundary between the DS and the UUD merges with the saturation field within 5^∘ above about 10 T.



 §.§ Microscopic magnetic model

To first define a realistic range of magnetic couplings, we employ the des Cloizeaux effective Hamiltonian (dCEH) approach <cit.>, based on exact diagonalization of the electronic d-orbital Hamiltonian for clusters include one and two Co sites [see Methods]. 

We first estimate the g-tensor by diagonalizing the electronic Hamiltonian on one Co site, and projecting the Zeeman operator into the low-energy space of j_1/2 states. This yields g_ab = 5.1 and g_c = 2.4, which further confirms accurate reproduction of the trigonal crystal field splitting in BCAO. To estimate the magnetic couplings, we exactly diagonalize the electronic Hamiltonian on pairs of sites and project the resulting low-energy Hamiltonian onto j_1/2 states. The relatively low symmetry crystalline environment allows for a large number of independent exchange constants. In particular, in the R3̅ space group, each first and third neighbor bond is constrained only by inversion symmetry, allowing six independent exchange parameters per bond:


    ℋ =    ∑_ij J_ij^xy (S_i^x S_j^x + S_i^y S_j^y) + J_ij^z S_i^zS_j^z  
        + 2J_ij^±±(S_i^xS_j^x - S_i^y S_j^y) + Γ_ij^yz (S_i^y S_j^z + S_i^z S_j^y) 
        + Γ_ij^xz(S_i^x S_j^z + S_i^z S_j^x) + Γ_ij^xy(S_i^y S_j^x + S_i^x S_j^y)

where the bond-dependent local x-axis is oriented parallel to the bond, and the z-axis is parallel to the crystallographic c-axis. Second neighbor couplings tend to be weaker and are ignored in first approximation. By varying the strengths of the Coulomb interactions within a realistic range [see Methods], we establish reasonable ranges for the first and third neighbor couplings: 
J_1^xy∼ -8 to -12 meV, 
J_1^z/J_1^xy∼ 0.5 to 0.6, 
J_1^±±∼ -0.6 to -0.8 meV, 
Γ_1^yz∼ 0.35 to 0.5 meV,
J_3^xy/J_1^xy∼ -0.25 to -0.4, and 
J_3^z/J_3^xy∼ 0.2. The remaining couplings are all found to have magnitude smaller than 0.1 meV.


There are several aspects of the computed couplings that bear mentioning. In BCAO, the local trigonal distortion of the CoO_6 octahedra has a significant impact on the magnetic Hamiltonian, inducing both the large g-anisotropy, and bond-independent XXZ anisotropy of the intersite exchange. Microscopically, the third neighbor couplings are dominated by a single exchange path involving the e_g electrons, which leads to antiferromagnetic interactions that are rendered anisotropic only by the crystal field effects. As such, the off-diagonal anisotropies should be nearly vanishing and the third neighbor interactions should be parameterized by a pure XXZ anisotropy, with J_3^xy being the same sign as J_3^z. For the first neighbor couplings, the dominant anisotropy is also of the bond-independent XXZ type. While Co^2+ materials can, in principle, host strongly bond-dependent couplings <cit.>, in BCAO we find that these are suppressed by a combination of factors. First, the large trigonal distortion from a pure octahedral environment significantly modifies the composition of the J_ eff = 1/2 moments, and imposes significant XXZ anisotropy on all couplings instead of bond-dependent exchange interactions. Second, the largest contribution to the first neighbor exchange comes from ferromagnetic “charge-transfer” contributions arising from hybridization of the Co e_g orbitals with the oxygen p-orbitals. Unlike the couplings between t_2g electrons, these interactions do not display large bond-dependent anisotropies. As such, the bond-dependent anisotropies (J_1^±± and Γ_1^yz) are subleading. Although some of these features are found in previously proposed models for BCAO <cit.>, not all are reproduced. This raises the question of whether the phase diagram can be understood by employing a model that follows more closely these microscopic considerations. 






To address this question, we made a comprehensive search of the parameter space around the predicted range, with a focus on fitting the experimental phase diagram. BCAO necessarily lies in a parameter region with multiple competing phases, making it necessary to consider quantum corrections to the classical state energies. We employ the method outlined in Ref. jackeli2015quantum [see Methods], which considers states defined by classical spin vectors, but accounts for local fluctuation effects on the state energies at second order in perturbation theory. This naturally includes the leading quantum order-by-disorder (QOBD) effects. From this search, we propose a simplified, but extremely finely tuned, six-parameter model:

J_1^xy = -11.75, J_1^z = -5.89, J_1^±± = -0.33, Γ_1^yz = -0.15, J_3^xy = +3.37, and J_3^z = +0.79 meV. The parameters are more consistent with microscopic considerations but otherwise broadly compatible with previously proposed models for BCAO with dominant XXZ anisotropy <cit.>.

In Figure <ref>, we show the computed evolution of the state energies and magnetization per site of the above model for in-plane fields. We evaluate m = |∑_i g·𝐒_i| from the classical spin vectors for which the second order energies correspond. The zero-field ground state is found to be a colinear DS phase, with spins oriented in the plane, as shown in Figure <ref>b. Under applied field, there are two first order transitions. The first is to the magnetization plateau UUD state, and the second to the polarized state. We estimate for H  ||  a: H_1 = 0.16 T, H_3= 0.55 T, and for H  ||  b^*: H_1 = 0.18 T, H_3 = 0.52 T. Thus, the model reproduces the slight anisotropy in the in-field critical fields observed in experiment. The predicted magnetization as a function of field is shown in Figure <ref>c and <ref>d, and reproduces well the experimental magnetization curves in Figure <ref>.


In Figure <ref>e and  <ref>f, we show a comparison of the experimental and theoretical phase diagram for fields oriented in the b^*c-plane. For strictly out-of-plane fields H||c, the model reproduces the direct transition from the DS state to the polarized phase. The critical field depends very strongly on the field angle; for θ = 0^∘, we compute H_3 = 36 T, while for θ = 1^∘, we compute H_3 = 18 T. This range is comparable to the broadened saturation observed in the pulsed-field measurements in Figure <ref>.
The sharp increase of the critical fields in the vicinity of the c-axis is mainly due to the large bond-independent XXZ anisotropy, which leads to an easy ab-plane. Overall, the model reproduces the phase boundaries well, confirming that the specific progression of field-induced phases can be understood within a microscopically compatible model. 

Finally, to further check that the model reproduces the characteristic magnetic anisotropy of BCAO, we simulated the magnetic torque data from the second-order corrected energies, where τ = ∂ E/ ∂θ. Results are shown in Figure <ref>b alongside the experimental data. The simulations reproduce very well both the positions and magnitudes of the first order jumps in τ corresponding to the various phase boundaries, as well as the rounding of τ(θ) at high field. A precise comparison can be made on the basis of the coefficients c_2n. In Figure <ref>d, we show the theoretical evolution of these coefficients with field. The finding that c_2 and c_4 are well reproduced by the model implies the relative magnitudes of both the g-anisotropy and XXZ exchange anisotropy are well captured. 

Overall, we conclude that the microscopically motivated model provides a very good reproduction of the experimental phase diagram of BCAO and therefore may serve as a starting point for future refinement. The present results confirm the validity of recently proposed models with primarily bond-independent XXZ anisotropy <cit.>, but show that the complex phase diagram of BCAO illuminated by various recent studies is compatible with the microscopic expectations for this material. 



§ DISCUSSION

As a consequence of the exchange anisotropy and frustration between first and third neighbor couplings, the phase diagram of BCAO is uniquely dominated by QOBD effects. Over the region of couplings we considered for BCAO, we observe several trends when comparing classical state energies with the second order corrected energies. First, primarily due to the strong XXZ anisotropy, both antiferromagnetic and polarized (ferromagnetic) states have finite energy corrections. Similar to the DMRG studies in Ref. <cit.>, for the physically relevant magnitudes of the XXZ anisotropy, we find that fluctuations tend to stabilize the polarized phase over the other phases. 
Overall, we find that quantum fluctuations tend to reduce the critical fields by favoring the polarized state, consistent with the conclusions of <cit.>. Second, we emphasize that the collinear DS and UUD phases only appear as ground states due to QOBD effects. We did not find any realistic parameters for which these are classical ground states for any magnetic field. Conversely, the spiral phase is never found to be stable once fluctuation effects are included. These findings are consistent with <cit.>. Finally, we emphasize that the present model for BCAO is extremely finely tuned. As can be anticipated from the (necessarily) small differences of the state energies in Figure <ref>a and  <ref>b deviations of any coupling on the order of ± 0.01 meV may be sufficient to ruin the agreement with experiment. This sensitivity likely also leads to variations in the results of different numerical approaches. In this work, we opted to use the inexpensive approximation of second-order energy corrections to facilitate a fine parameter search. Future refinement of the model could employ more sophisticated numerical methods, provided those approaches adequately treat the QOBD effects. 


In conclusion, this work is a comprehensive study of the magnetic field-induced ground states of BCAO, employing various techniques such as dc magnetization, capacitance, ac magnetic susceptibility at low temperatures. We mapped out the full T-H phase diagram for different magnetic field orientations, including H along a, b^*, c, and c'. To understand the specific progression of field-induced phases, we employed first-principles-based calculations to derive a model that reproduces all essential features of the low-temperature phase diagram. The stabilization of two colinear phases (I and III) in both zero and finite fields is attributed to the significant influence of QOBD effects. BCAO presents an intriguing material where quantum fluctuation effects dominate the phase diagram at both zero and finite field. This investigation highlights new avenues for understanding the complex behavior of BCAO under external magnetic fields.




§ METHODS
 


 §.§ Experimental details


High-quality single crystals of BaCo_2(AsO_4)_2 synthesized through a flux method at Oak Ridge National Laboratory in Oak Ridge, for which the detailed methods are described elsewhere  <cit.>. At low temperature, dc magnetization (M) is measured up to 60 T through millisecond timescales in pulsed magnet using custom-built ^3He pumping (400 mK–3 K), and ac magnetic susceptibility (χ_ac) and capacitance (C) measurements are shown here performed in PPMS for 2 K–30 K under the magnetic fields up to 14 T at Pulsed Field Facility in Los Alamos. For dc magnetization and ac magnetic susceptibility measurements, the sample with 1 mm^2 was aligned along a, b^*, c, and c' and mounted on the sample holder inside the compensation and pick-up coil, where c'-axis is 6^∘ from c-axis on the b^*-c plane.
The electrical contacts were made by painting silver epoxy on the two parallel opposite surfaces for the capacitance measurement and Andeen-Hagerling AH-2700A capacitance bridge was employed to measure the capacitance.

Magnetic torque was measured with the 31 T resistive magnet at the National High Magnetic Field Laboratory in Tallahassee, Florida using a home-made torque magnetometer. The torque magnetometer was formed from two parallel BeCu plates. A tiny aligned crystal with the mass of 0.13 mg was attached using epoxy on the cantilever plates. The capacitance change was monitored while the field or angle was swept using an AH-2700A capacitance bridge. More details can be found in https://nationalmaglab.org/user-facilities/dc-field/measurement-techniques/torque-magnetometry-dc/. 

We then converted the measured capacitance to the deflected angle (α) using the following equation

    C(α) = ϵ_0ϵ_rw/tanαlnltanα-h_0/h_0,

where α is the defected angle that is proportional to the magnetic torque, ϵ_0 and ϵ_r are vacuum and relative permittivity, respectively, and w, l, and h_0 are width, length of the BeCu plate and the gap distance between two BeCu plates at zero magnetic field. We extracted the coefficients c_2n using the following equation

    c_2n = 2/π∫_θ_1^θ_i+πτ/Hsin(2nθ) dθ,

where θ_i is the initial angle, τ is the measured magnetic torque signal.




 §.§ Theoretical phase diagrams


In order to account for quantum fluctuation effects, we employ the method of second-order energy corrections outlined in Ref. jackeli2015quantum. At each site i, the local quantization axis (denoted ẑ_i^') is along the local spin vector, such that ⟨ S_i^z'⟩ = +1/2 for all sites. Fluctuations are then driven by terms acting like J_ij^–S_i^-'S_j^-'. The second-order energy correction due to such fluctuations is:


    Δ E^(2) = ∑_ bonds   ijJ_ij^++ J_ij^–|⟨ m| S_i^-'S_j^-'|gs⟩|^2/E_gs, classical - E_m,  classical

where E_gs, classical is the classical energy of the spin configuration, and E_m, classical is the classical energy of the state with the two spins along the bond ij flipped. For the example of a Heisenberg honeycomb antiferromagnet, the classical ground state energy is E_gs, classical = -0.375  J per site, whereas the present approach provides E_gs, classical + Δ E^(2) = -0.563  J per site. The latter estimate (which includes local singlet fluctuations) is in much better agreement with E_gs = -0.544  J per site from series expansion <cit.> and quantum Monte Carlo <cit.>. 

In order to capture the energetically competitive magnetic orders depicted in Figure <ref>b, we employ a combination of 4-, 6-, and 8-site clusters of spins with appropriate boundary conditions, and numerically minimize E_ classical + Δ E^(2) with respect to the spin orientations. 
We consider the commensurate spiral state with Q = [1/4, 0] as a proxy for the proposed incommensurate spiral with Q = [0.27,0] <cit.>. We find the Q = [1/4, 0] spiral to be the classical ground state for a range of parameters, but it is completely replaced by the colinear ferromagnetic and Q = [1/4, 0] DS phases once Δ E^(2) is included. 



 §.§ First principles calculations




To estimate the magnetic couplings, we consider an electronic model for the Co 3d-orbitals, which is a sum of, respectively, one- and two-particle terms: ℋ_ el = ℋ_1p+ℋ_2p. The one-particle terms include intersite hopping, intrasite crystal field, and spin-orbit coupling, ℋ_1p = ℋ_hop+ℋ_ CF + ℋ_ SO:


    ℋ_hop =    ∑_ijαβσt_ij^αβ c_i,α,σ^† c_j,β,σ
    ℋ_ CF =     ∑_iαβσd_i^αβc_i,α,σ^† c_i,β,σ
    ℋ_ SO =     ∑_iαβσσ^'λ_ Mn⟨ϕ_i^α(σ)|𝐋·𝐒|ϕ_i^β(σ^')⟩ c_i,α,σ^† c_i,β,σ^'

where c_i,α,σ^† creates an electron at Co site i, in d-orbital α, with spin σ. To parameterize the one-particle terms, we performed fully relativistic density functional theory (DFT) calculations at the GGA (PBE <cit.>) level using FPLO <cit.>, a 12×12×12 k-grid, and the structure from Ref. djordevic2008baco2. The resulting Kohn-Sham Bloch functions were projected onto Co d-orbitals to yield single-particle terms from the Kohn-Sham Fock matrix written in the Wannier basis <cit.>. For BCAO, this approach yields accurate crystal-field excitation energies <cit.>, confirming the validity of the computed single-particle parameters. 

For the two-particle terms, we consider a combination of on-site and nearest neighbor intersite contributions: ℋ_2p = ℋ_U + ℋ_J^ nn, respectively. The on-site contributions are given by:


    ℋ_U = ∑_iαβδγ∑_σσ^'U_αβγδ c_i,α,σ^† c_i,β,σ^'^† c_i,γ,σ^' c_i,δ,σ


In the spherically symmetric approximation <cit.>, U_αβγδ are parameterized by the Slater parameters F_0^dd, F_2^dd, F_4^dd. For the (screened) on-site metal Coulomb interactions, we use F_2^dd = 11.55 eV, and F_4^dd = 7.68 eV, following the fitted parameters for LiCoO_2 from Ref. van1991electronic. We consider U_t2g= F_0^dd + (4/49)(F_2^dd+F_4^dd) within the realistic range of 5.0 to 5.5 eV.

The nearest neighbor intersite Hund's coupling is given by:


    ℋ_J^ nn = ∑_ijαβσσ^' J_H,ij^αβ c_i,α,σ^† c_j,β,σ^'^† c_i,α,σ^'c_j,β,σ


A significant contribution to the intersite magnetic couplings comes from ligand “charge transfer” exchange terms <cit.>, which account for perturbative processes where two holes meet on a single oxygen ligand. When downfolded into the d-orbital Wannier basis, these appear as intersite Hund's coupling terms. They give an overall ferromagnetic contribution to the intersite magnetic couplings between the low-energy j_1/2 doublets, which may be rendered anisotropic (bond-independent XXZ anisotropy) by the crystal-field splitting of the t_2g orbitals. In a previous study of BCAO <cit.>, some of the authors accounted for this term via an estimated correction added to the computed couplings. Here, we employ the improved approach utilized in <cit.>  that captures these effects at the level of the electronic Hamiltonian. The d-orbital Wannier functions can be written:


    |d̃_i,α,σ⟩ = ϕ_i,α^0|d_i,α,σ⟩ + ∑_n,βϕ_i,α^n,a|p_n,a,σ⟩


where i labels a given Co metal site, α is a d-orbital, n labels an oxygen ligand site, and a∈{x,y,z} is a p-orbital. The tilde on the left side of the equation indicates a Wannier function, whereas the absence of a tilde indicates a bare atomic orbital. Rotating the p-orbital Coulomb terms into the d̃ Wannier basis results in residual intersite Hund's coupling given by:


    J_H,ij^αβ =  ∑_n ∑_αβ    
    (U^p-J_H^p) ϕ_i,α^n,aϕ_i,α^n,bϕ_j,β^n,aϕ_j,β^n,b
        
    + J_H^p  ϕ_i,α^n,aϕ_i,α^n,aϕ_j,β^n,bϕ_j,β^n,b


where U^p and J_H^p are the atomic ligand Hubbard and Hund's couplings, respectively. The computed magnetic couplings depend most strongly on the overall magnitude of J_ij^αβ, rather than the specific orbital dependence, which provides for some inconsequential flexibility in the choice of U^p and J_H^p.  
We take U^p = 2J_H^p, and 
consider J_H^p within the range of 0.4 to 0.6 eV. This choice corresponds to roughly 60% to 80% of the atomic value for oxygen <cit.>. The sum over oxygen sites in Eq.  (<ref>) is then evaluated from the ab-initio Wannier function coefficients for the bridging oxygen atoms for each bond. 




§ ACKNOWLEDGEMENTS
 
The authors would like to thank A. L. Chernyshev, P. A. Maksimov, Y. B. Kim, F. Desrochers, E. Z. Zhang, K. A. Modic, and S. A. Crooker for insightful discussions. 
The experimental work, analysis and manuscript writing  (S.L., S.Z., S.M.T., L.P., C.A.B., V.S.Z., and M.L.) was funded by the US Department of Energy, Office of Science, National Quantum Information Science Research Centers. The theoretical work, analysis and manuscript writing of S. M. W. was funded by the Oak Ridge Associated Universities
(ORAU) through the Ralph E. Powe Junior Faculty Enhancement Award. Computations were performed using the Wake Forest University (WFU) High-Performance Computing
Facility <cit.>, a centrally managed computational resource available to WFU researchers including
faculty, staff, students, and collaborators. A portion of this work was performed at the National High Magnetic Field Laboratory, which is supported by National Science Foundation Cooperative Agreement No. DMR-2128556, the State of Florida, and the U.S. Department of Energy.



§ AUTHOR CONTRIBUTIONS
 
V.S.Z. and M.L. conceived and designed the project. S.L., S.Z., S.M.T., E.S.C. and M.L. performed measurements. S.M.W. performed the theoretical calculations.
L.P. and C.A.B. prepared high quality samples. S.L., V.S.Z., S.M.W., and M.L. wrote the manuscript, with contributions from all authors.



§ COMPETING INTERESTS
 
The authors declare no competing interests.




Correspondence and requests for materials should be addressed to
Vivien S. Zapf, Stephen M. Winter or Minseong Lee.







§ SUPPLEMENTARY MATERIAL ­– 
 “QUANTUM ORDER BY DISORDER: A KEY TO UNDERSTANDING THE MAGNETIC PHASES OF ”






















 Paper Title: Ge-based Clinopyroxene series: first principles and experimental local probe study.







IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and Photonics, Departamento
de Física e Astronomia da Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n,
4169-007 Porto, Portugal.


IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and Photonics, Departamento
de Física e Astronomia da Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n,
4169-007 Porto, Portugal.
High Performance Computing Chair, University of Évora, Largo dos Colegiais 2, 7004-516 Évora, Portugal


IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and Photonics, Departamento
de Física e Astronomia da Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n,
4169-007 Porto, Portugal.


IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and Photonics, Departamento
de Física e Astronomia da Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n,
4169-007 Porto, Portugal.


CNR-SPIN c/o Università degli Studi dell’Aquila, Via Vetoio 10, 67010 Coppito, L’Aquila, Italy.



Université Grenoble Alpes, CNRS, Institut Néel, 38000, Grenoble, France


Université Grenoble Alpes, CNRS, Institut Néel, 38000, Grenoble, France


C2TN, DECN, Instituto Superior Técnico, Universidade de Lisboa, Bobadela, Portugal.



Instituto de Física, Universidade de São Paulo, CP 66318, 05315-970, São Paulo-SP, Brazil


Instituto de Física, Universidade de São Paulo, CP 66318, 05315-970, São Paulo-SP, Brazil


IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and Photonics, Departamento
de Física e Astronomia da Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n,
4169-007 Porto, Portugal.


jearaujo@fc.up.pt
IFIMUP, Institute of Physics for Advanced Materials, Nanotechnology and Photonics, Departamento
de Física e Astronomia da Faculdade de Ciências da Universidade do Porto, Rua do Campo Alegre s/n,
4169-007 Porto, Portugal.




§ ABSTRACT 
The structural and electronic properties of the CaMnGe2O6 and SrMnGe2O6 clinopyroxene systems have been investigated by means of perturbed angular correlation (PAC) measurements, performed at ISOLDE, 
combined with  ab initio electronic structure calculations within the density functional theory (DFT) framework. The partial density of states (PDOS) 
of the CaMnGe2O6 and SrMnGe2O6 stable compounds 
has been 
determined, and it has been observed that the requirement of including an on-site Hubbard-U potential was necessary  
in order to describe the highly correlated 
Mn 3d-states. By considering U_eff=4 eV, we obtained  
a band gap width  
of 1.82 eV and 1.70 eV, for the CaMnGe2O6 and SrMnGe2O6, respectively. 
Combining electric field gradient (EFG) first principles calculations, using a supercell scheme,  with experimental PAC results,  we were able to infer that the Cd probe can replace either the A (Ca, Sr) or the Mn sites in the crystalline structures. We also showed that Cd substitution is expected to lead to a reduction in the width of the band gap in these systems, evidencing opportunities for potential band-gap engineering.







Ge-based Clinopyroxene series: first principles and  experimental local probe study
    João P. Araújo
    August 5, 2024
===================================================================================







§ INTRODUCTION




Pyroxenes are a class of materials with general formula AMX2O6,  where A is a monovalent or a divalent cation, M is respectively a trivalent or a divalent cation, and X is typically either Si or Ge. They are well known in mineralogy and geology as they are one of the main rock-forming minerals of the Earth's crust <cit.>. Within this rich family of compounds, monoclinic pyroxenes (clinopyroxenes), with a 3d transition metal at the M-site, have recently been the subject of interest in the field of condensed matter physics due to the diversity of their magnetic properties, 
namely: multiferroicity in SrMnGe2O6 <cit.> and NaFe(Si/Ge)2O6 <cit.>; magnetoelectric effect in CaMnGe2O6 <cit.> and Li(Cr/Fe)Si2O6 <cit.>; and also ferrotoroidal ordering in LiFeSi2O6. 

In spite of their natural prevalence, pyroxenes have not been the subject of much computational or theoretical investigations. It is noteworthy of mentioning, that recent density functional theory (DFT) studies have shown that substitution of Al3+ by Tl^3+ in NaAlSi2O6  results in a reduction in the band gap width from 5.32 eV to 2.05 eV. A further reduction is observed when replacing Na by the organic cation CH3SH2, showing the high tunability of the band gap in these materials<cit.>. Other works have shown their potential usefulness in batteries, demonstrating that pyroxenes CaFe/Mn/Si2O6 could display high theoretical energy densities in Ca-based batteries<cit.>, whereas LiFeSi2O6 has been shown to undergo a reversible electrochemical reaction against Li, thus exhibiting potential for use as an electrode material in Li-ion batteries<cit.>. Lastly, the magnetic properties of CaMnGe2O6 were studied by Temnikov et al. <cit.> who computed the exchange parameters 
and showed that the magnetic frustration in the system is weak, which could explain the frequent occurrence of commensurate collinear antiferromagnetic structures in Ca2+-bearing pyroxenes.


The ground state structure for the clinopyroxenes considered in this study is a monoclinic Bravais lattice with space group C2/c (#15), where the M-cations are in octahedral sites (Wyckoff 4e atomic positions), the A-cations are in 8-coordinated sites (Wyckoff 4e atomic positions) and the X-cations are in tetrahedral sites (Wyckoff 8f atomic positions). Their structure is characterised by quasi-one-dimensional chains with edge-sharing MO6 octahedra running along the c-axis, and connected by corner-sharing XO4 tetrahedra, along the c-axis, as schematically represented in Fig.<ref>.



When the M-site is occupied by a magnetic cation, such an arrangement leads to low-dimensional magnetic properties and to magnetic frustration, due to the competition between interchain and intrachain interactions <cit.>. It is the existence and the possible interplay between the low dimensionality and frustration that is thought to give rise to the aforementioned diversity of magnetic properties in these materials <cit.>. Moreover, it has been suggested that the magnetic frustration can lead to spiral spin structures, which may be favourable towards magnetically driven ferroelectricity, making these compounds good candidates for multiferroic behaviour <cit.>. Since these properties are thought to arise from local features that are often not accurately described by long-range order based techniques, such as X-ray diffraction and Raman spectroscopy, it is important to utilise techniques that are more sensitive to local order to aid in the characterisation of these intriguing systems. Several works have, in the past, studied pyroxene systems by means of nuclear techniques, such as nuclear magnetic resonance spectroscopy <cit.> and Mössbauer spectroscopy <cit.>; however, and to the best of the authors' knowledge, this is the first work being reported where the pyroxene systems are studied via perturbed angular correlation (PAC) spectroscopy. In particular, none of the  
nuclear techniques above mentioned have been used to characterise the systems here under study.

In the present work, the AMnGe2O6 germanate clinopyroxene series were explored,  where the A-site is occupied by periodic table group-II elements, i.e.,  Be, Mg, Ca, and Sr. The former two compounds have so far not been experimentally realised, and they were thus studied solely through ab initio calculations. (Ca, Sr)1-xCdxMnGe2O6 and (Ca, Sr)Mn1-xCdxGe2O6 were explored by means of ab initio computational simulations, from which we were able to estimate the band gaps, as well as the effective masses of the electrons and holes of the systems; and by local Cd probe experiments using PAC spectroscopy, allowing one to closely follow the temperature evolution of the electric field gradient (EFG) tensor.



§ METHODS





 §.§ Experimental Methods


For the purpose of performing the PAC measurements, pelletised samples of CaMnGe2O6 and SrMnGe2O6 materials were implanted at ISOLDE-CERN by a 30 keV beam of 111mCd probes [111mCd→111Cd, t_1/2 = 48.6 minutes], which have a nuclear spin I=5/2 and an electric quadrupole moment of Q = 0.83(13) b  in the intermediate state of decay <cit.>. Small doses of the order of 1011 atoms/cm2 were implanted and subsequently an annealing was performed at a temperature of ∼1123 K for 20 minutes in air to recover from implantation damage. Confirmation for the recovery of point defects and for the incorporation of Cd into the lattice was given by the PAC measurements themselves, which yielded well-defined experimental R(t) anisotropy functions. The measurements were performed on 6-BaF2 detectors PAC spectrometers<cit.>, equipped with either a closed-cycle refrigerator or a special high-temperature furnace for temperature control. Due to the relatively short lifetime of the parent probe, each temperature point required the implantation of a new sample for satisfactory activity to be maintained during measurement.
The experimental R(t) function was fitted with exact numerical methods that build the expected observable by solving the hyperfine interaction Hamiltonian's characteristic equations<cit.>. 
The EFG, that characterizes the surrounding charge density that interacts with a
probe nucleus, can be represented as a rank-2 symmetric and traceless tensor. 
As such it is always possible to diagonalise it by rotating the axis system into the EFG eigenbasis,  that is referred to as the principal reference frame. The experimentally observable EFG is conventionally described by two parameters, the principal component V_zz and the axial asymmetry parameter:

    η=V_xx-V_yy/V_zz,

with V_zz, V_yy and V_xx being the eigenvalues of the EFG tensor, defined such that |V_zz|≥|V_yy|≥|V_xx|<cit.>.


The signature for the interaction between the probe atoms and their local environment is the perturbation function G_kk(t). In the case of static electric quadrupole interactions, the interaction can be written as<cit.>:

    G_kk(t)=∑_n s_k_ncos(ω_nt)e^-δω_nt/2,

where s_k_n and ω_n are functions of η and the fundamental frequency ω_Q. The exponential term arises from randomly distributed vacancies, defects, and lattice strains that result in an attenuation of the experimental R(t) function. In the case of the 111mCd probes, the intermediate level in the decay has a nuclear spin of I=5/2, which leads to a split of the intermediate level into three sub-levels due to the quadrupole interaction and three transition frequencies, ω_1, ω_2, and ω_3=ω_1+ω_2. The fundamental frequency is usually defined as <cit.>:

    ω_Q=eQV_zz/4I(2I-1)ħ,

 where Q is the nuclear quadrupole moment, e is the electron charge, and ħ is the reduced Planck constant.



 §.§ Computational Methodology


DFT calculations were performed by using the Quantum Espresso (QE) <cit.> suite with Projector-Augmented Wave (PAW) datasets<cit.>. The atomic valence configurations that were considered for the calculations were: Be[2s2], Ca[3s23p64s2], Cd[4d9.55s25p0.5], Ge[3d104s24p2], Mg[2s22p63s2], Mn[3s23p63d54s2], O[2s22p4], and Sr­[4s24p65s2]. 
The kinetic energy cutoffs were converged to the values of 70 and 540 Ry for the wavefunctions and the charge density, respectively. However, in order to converge the EFGs to the experimental data, and for the calculations that included the Cd substitutional atom, these values were raised to 113 and 900 Ry, for the wavefunctions and the charge density, respectively. Regarding the exchange-correlation (XC) functional, the generalised gradient approximation (GGA) with Perdew-Burke-Ernzerhof<cit.> (PBE) parametrization was employed. In order to treat the strongly correlated 3d-states of Mn, an effective Hubbard-U potential was considered by defining U_eff = 4 eV <cit.>, by employing the simplified version of Cococcioni and de Gironcoli <cit.>, with PBE (PBE+U). 


In order to obtain the ground-state properties, total energy minimization with respect to the changes in the atomic positions and cell parameters were taken into account, until the total force did not exceed 0.05 eV/Å, and constraining the system to zero internal pressure, with the variable cell shape relaxation, considering the damped Beeman ionic dynamics and the Wentzcovitch extended Lagrangian for the cell dynamics <cit.>.
For the four compounds, the enthalpies of formation were determined by comparing respective total energies to that of the respective DFT total energy of the AGeO3 and MnGeO3 stable constituent oxides (Eq. <ref> in Appendix  <ref>).
With respect to the CaMnGe2O6 and SrMnGe2O6 compounds, the EFGs were evaluated for the pristine unit-cell, as well as for the unit-cell and the 1 × 1 × 2 supercell by considering Cd substitution of either A or Mn cation sites  (Fig. <ref>), in order to compare with previously obtained PAC measurements. The computation of the EFGs was performed by using the Gauge Including Projector Augmented Waves (GIPAW)<cit.> package routine implemented in QE. 

For the supercell calculations, only the atomic positions were relaxed, by using the lattice parameters obtained from the pristine unit-cell structural minimization(by employing the variable cell shape relaxations). The Brillouin-zones (BZ) 
were sampled with a (9 × 9 × 18) Monkhorst-Pack k-point grids for unit-cell calculations, and a (9 × 9 × 9) for calculations on the 1 × 1 × 2 supercells; whereas for the calculation of EFGs and PDOS, k-point grids of (11 × 11 × 22) and (11 × 11 × 11) were used, for the unit-cells and supercells, respectively.



§ RESULTS AND DISCUSSION




 §.§ Perturbed Angular Correlation

The PAC measurements on the SrMnGe2O6 (SMGO) system were performed at selected temperatures in the 13-902 K range. 
Figure <ref> shows a few representative experimental γ - γ R(t) perturbation functions obtained for the SrMnGe2O6 compound along with their corresponding Fourier transforms. Several models were tested in order to numerically obtain the best fit to the experimental R(t) spectra before arriving at the presented model, wherein two distinct fractions (two distinct local environments) are evident across the whole temperature range, indicating the probes interacted with two distinct EFG distributions, henceforth referred to as EFGSMGO1 and EFGSMGO2, and represented by the green and blue lines in figure <ref>, respectively. 







The temperature behaviour of the experimental R(t) fit parameters of SMGO  is displayed in figure <ref>. One can observe, in figure <ref> a), green circles, that the V_zz value for the EFGSMGO1 fraction has a noticeable increase at low temperatures, with increasing temperature, reaching a maximum; whereupon it starts to decrease. On the other hand, the axial asymmetry η(1) parameter increases as the temperature increases, as displayed in figure <ref> b) by the green circles.  The EFG aforementioned  
behaviour is typical for a phonon contribution to V_zz and, accordingly, a temperature dependence least-squares fit by the Bayer-Kushida<cit.> model, with a single phonon frequency, was done, according to

    V_zz(T)=V_zz(0){ 1-3/8π^2hA/ν(1/2+1/e^hν/kT-1)}  ,

 where h is the Planck constant, ν is the phonon frequency, k is the Boltzmann constant and A is a quantity that in general can only be said to have the dimensions and order of magnitude of the inverse of the moment of inertia associated to the molecular vibration, except in specific cases where it is indeed the inverse of the moment of inertia associated with a normal mode of molecular vibration<cit.>. 
 From this fitting, we have obtained ν = 8.01(7) × 10^12 Hz, which is within the range of frequencies that are typical for intra-molecular vibrations and is comparable to the value of 6.60(85) × 10^12 Hz reported for Ca2MnO4<cit.>, suggesting that the 
 V_zz temperature behaviour for the EFGSMGO1 local environment 
 is indeed due to a phonon contribution. Regarding the axial asymmetry parameter η(1)  at the  EFGSMGO1 site, it displays an intermediate asymmetry, which increases as temperature raises. 
 
For the EFGSMGO2 fraction, depicted by the blue triagles 
 in figure <ref> a),  the V_zz shows an atypical behaviour of increasing as temperature also increases, though such behaviour has nevertheless been reported in other compounds. Specifically, analogous behaviour was experimentally observed   in the A21am phase of Ca3Mn2O7 for which DFT simulations indicated that such behaviour is actually due to structural effects, that prevail over the phonon contributions <cit.>. 
  The asymmetry parameter η(2) for the EFGSMGO2 fraction, as displayed in figure <ref> b) by the blue triangles, 
 on the other hand, remains approximately constant, up to 474 K, whereupon an anomalous increase occurs. It is unclear if such behaviour could be related to structural phenomena, as there are no measurements reported for temperatures above 350 K. It is worth noting that the probes distribution between both fraction remained fairly constant across the whole temperature range, suggesting that they are
related to the probes occupying two distinct lattice sites.
 



With respect to the CaMnGe2O6 (CMGO) system, the PAC measurements were likewise performed at selected temperatures, in the 11-898 K range. Representative fits to the experimental R(t) perturbation functions for CMGO are presented in figure <ref>, along with the corresponding Fourier transforms. Similar to how the data from the previous compound was treated,  several models were also tested in order to numerically obtain the best fit to the experimental R(t) spectra. Distinctly from SrMnGe2O6, wherein two noticeable fractions are present across the whole temperature range, for the CaMnGe2O6 compound there is only one fraction present from low temperatures, up to 351 K, hereby referred to as EFGCMGO1 and represented by the green lines in figure <ref>. It can be noticed in this figure that a second fraction, labeled EFGCMGO2, appears for temperatures above 351 K, represented by the blue lines. However, it is likely that this second fraction is likewise present at lower temperatures, but  owing to the very small observable frequency associated with it, that is not well defined in the time window available in the PAC experiments, it is not possible to fit this fraction at lower temperatures. Even at high temperatures the EFGCMGO2 fitting parameters have nonetheless an associated  high uncertainty.  



The temperature behaviour of the fit parameters relative to the CaMnGe2O6 system are shown in figure <ref>. One can observe, in figure <ref> a), green circles,  that the EFGCMGO1 distribution has a V_zz value of similar magnitude to EFGSMGO1 (figure <ref> a)), with an analogous decrease as temperature raises. 


Also η(1), displayed in figure <ref> b), green circles,  similarly  increases with increasing temperature, when compared to SMGO η(1) variation (figure <ref> b)). 
As was done in the case of the SrMnGe2O6 compound, a least-squares fits was likewise performed to V_zz, according to equation <ref>. From the fit, a frequency of ν = 10.0(2) × 10^12 Hz was obtained, once more indicating that this behaviour is likely attributable to a phonon contribution to V_zz. Similarly to what was observed for SrMnGe2O6, we note that the magnetic hyperfine interactions were not apparent in the fits to the experimental R(t) functions.






 §.§ Density Functional Theory Calculations




  §.§.§ Structural Parameters


We begin this section by summarizing in Table <ref> the calculated lattice parameters for the SrMnGe2O6 and CaMnGe2O6 systems, computed by applying both the PBE and PBE+U methodologies, as well as the reported experimental values,<cit.> for comparison purposes. 




Regarding the SrMnGe2O6 system, we can observe that 
the lattice parameters computed with the PBE functional are overestimated, with respect to the experimental values, corresponding to a relative increase of ∼1.7%, ∼0.8%, and ∼1.6%, respectively,  for a, b, and c parameters, which is in line with the well-known tendency of the PBE functional.<cit.> The inclusion of the on-site effective Hubbard-U potential results in a further relative increase of these parameters with ∼2.1%, ∼1.7%, and ∼2.1%, when compared to the experimental values. Though the β monoclinic angle is overestimated for both PBE and PBE+U approximations, when comparing to experimental data, the difference is in fact smaller when considering the PBE+U calculations, resulting in a value closer to the experimental one.

With respect to CaMnGe2O6, the a and b theoretical lattice parameters obtained from PBE calculations likewise are overestimated, relative to experimental data, by ∼1.2% and ∼0.5%, respectively. On the other hand we observed that the c parameter is underestimated  by ∼0.9%. Analogously to what was observed for SrMnGe2O6, the a and b lattice parameters that resulted from the PBE+U approach display an increase with respect to the PBE values, presenting an overestimation of ∼1.7% and ∼1.6%, respectively,  with respect to experimental results; whereas c parameter is still underestimated, though smaller than the one obtained by the PBE functional, of  ∼0.4%. Also, and distinctly from what was observed for SrMnGe2O6, the theoretical monoclinic angle is underestimated  relative to the experimental angle, though again, results from the PBE+U approximation is closer to the experimental value.




  §.§.§ Enthalpies of Formation


The enthalpy of formation of the constituent oxides of the AMnGe2O6 series has also been examined through DFT calculations and we have determined that only the CaMnGe2O6 and SrMnGe2O6 clinopyroxene compounds are thermodynamically stable. This is evidenced by their negative enthalpies of formation of  -0.19 eV/f.u. and -0.11 eV/f.u., respectively (calculated by considering equation <ref> of Appendix <ref>). On the other hand, the BeMnGe2O6 and MgMnGe2O6 systems both evidenced positive enthalpies of formation of 0.12 eV/f.u., supporting that these are not stable against dissociation into the respective stable constituent oxides BeGeO3 or MgGeO3 and MnGeO3. This observation is consistent with the fact that only the former two compounds have been experimentally synthesised, whereas the synthesis of the latter two has not yet been reported. Such observation implies that different reaction pathways towards synthesis will have to be explored if these systems were to be experimentally realised. Details on how the enthalpies of formation were obtained are presented in the Appendix <ref>.




  §.§.§ Partial Density of States


The PDOS for the CaMnGe2O6 and SrMnGe2O6 compounds, obtained considering the PBE calculation, without the Hubbard-U correction (Ueff = 0 eV), were computed and are displayed in figure <ref>, in which the valence band maximum (VBM) is aligned at the Fermi energy and is set to zero.
Two narrow bands form the VBM, and these are mainly composed by the Mn occupied 3d-states, hybridized with a smaller contribution from the O p-states. Energetically below these narrow bands (< -2.5 eV), a broader band is observed and it is mostly dominated by the O p-states, with a small mixture of 3d-states from Mn. The conduction band minimum (CBM), on the other hand, presents low density of states and hybridization between the p-states of O and Ge. For increasing energies, the DOS significantly increases forming a narrow resonant band, between 2.5 and 3.0 eV, that is mainly composed by the Mn 3d-states unoccupied states. 

For both studied systems the PDOS are qualitatively very similar, although at the conduction band the observed resonant peaks of Mn for SrMnGe2O6 are much broader when compared to those of CaMnGe2O6.

 






The electronic band gaps are also noted to be fairly narrow, being 0.46 eV and 0.51 eV, respectively, for CaMnGe2O6 and SrMnGe2O6, which is narrower than what has been reported for similar Si-based compounds<cit.>.

Since the band gap widths are likely underestimated in the absence of the Hubbard-U correction, we have also considered the PDOS by including the Ueff = 4 eV, applied to the Mn 3d-states. These are shown in figure <ref> along with the respective electronic band structures in the high-symmetry directions of the primitive cell first Brillouin zone
 (figure <ref>). 

 





When including the Ueff = 4 eV, and comparing to the PDOS results when Ueff = 0 eV, differences are mainly evidenced at the VBM, with the Mn 3d-states. The two narrow bands observed in figure <ref>, now overlap and form one narrow band for CaMnGe2O6 (figure <ref> a)). As for the SrMnGe2O6 system, the narrow bands are energetically very close, almost forming also one narrow band (figure <ref> b)). Moreover, these narrow bands are now dominated by a stronger hybridization between Mn d-states with the O p-states.




Since the inclusion of the U parameter localises and positions the unoccupied narrow 3d-states of Mn higher in energy (between 5.0 and 5.5 eV), the CBM shows slightly increased p hybridized states of O and Ge, than when U is not considered. The PBE+U theoretical band gaps are now wider, when compared to the previous case, with 1.82 eV and 1.70 eV for the CaMnGe2O6 and SrMnGe2O6 compounds, respectively. Despite this, the gap widths are still below the range of other similar Si-based clinopyroxenes<cit.>.




  §.§.§ Electronic Bandstructures





The electronic bandstructures for the pristine CaMnGe2O6 and SrMnGe2O6 systems, as well as the respective Cd-containing supercells were also computed through DFT calculations, in a path along the high-symmetry points depicted in figure <ref>, generated with the SeeK-path tool.<cit.>



The bandstructures for the pristine systems are presented in figure <ref>, whilst those for the supercells can be found in Appendix <ref>.

Differences between both pristine systems is evidenced by the character of the band gap: while SrMnGe2O6 exhibits a direct band gap, CaMnGe2O6 has an indirect gap. We must stress that subtleties persist with respect to the analysis of the VBM, since the valence bands are quite flat, and thus the electronic bands along the different high symmetry segments/points are energetically very close.   


From figure <ref>, one can observe that the top of the valence band is much flatter than the bottom of the conduction band, which shows narrow parabolas around the zone-centre, indicating that the effective mass of the holes, m^*_h,  is much larger than the effective mass of electrons, m^*_e. Such an observation is an indication of a higher mobility of the electrons comparatively to the holes of the system. Since the effective mass of the charge carriers is an essential parameter for the performance of photoelectric and photo-catalytic activities<cit.>, and also for semiconductor radiation detectors<cit.>; we have considered further analysing respective properties (table <ref>). Therefore and to quantify the effective masses of holes and electrons, we employ the well-known definition of the effective mass as

    m^*=ħ^2/∂^2 E/∂ k^2,


with the second derivative of the electronic dispersion relation being obtained from a quadratic fit around the energy range where the band edges are located.



In the case of CaMnGe2O6, we obtain a m^*_e,  around the Γ-point, very close to that of the free electrons, with m^*_e=1.028 m_0 (where m_0 is the mass of the free electron with 9.11×10^-31 kg). While for the hole carriers, and as expected by considering the flatness-shape of the VBM, the mass of holes proved to be substantially larger (in absolute terms) with m^*_h=26.125 m_0, and calculated along the high-symmetry segment of AΓ→ A (table <ref>). Regarding the Cd-containing supercells of CaMnGe2O6, when Cd substitution was considered at the Ca site, we found that the effective masses of the electrons only yielded mild variation when compared to the pristine system. For the electrons, the m^*_e showed very little to no increase
with  m^*_e=0.996 m_0 and m^*_e=1.028 m_0, for the Ca3/4Cd1/4MnGe2O6 and Ca7/8Cd1/8MnGe2O6 systems, respectively. In the case of the holes, we observe that for the Ca3/4Cd1/4MnGe2O6 cell, and by comparing to the pristine system, m^*_h slightly increases along the AΓ→Γ with m^*_h=45.259 m_0, although showing similar values along AΓ→ A with m^*_h=27.875 m_0.

With respect to the larger supercell (CaMn7/8Cd1/8Ge2O6), we observe a very high increase with respect to the pristine system, with m^*_h=67.562 m_0, along the same high-symmetry segment. 

On the other hand, when the Cd atom was placed at the Mn site, we found that the dopant affected the valence band edge even further, leading to a further flattening of the VBM; however, and once again, the m^*_e remained very similar to the pure unit-cell. A significant increase of the hole effective mass was found for the CaMn3/4Cd1/4Ge2O6 
structure (V_2 →Γ direction), with  m^*_h=118.325 m_0. 
Moreover, for the more diluted system we observed that along 
AΓ→ A the effective mass decreases slightly (m^*_h=28.65 m_0), showing an increase of m^*_h in the direction of AΓ→ A (m^*_h=68.128 m_0).



Regarding the pristine SrMnGe2O6 structure, we found a similar value for the electron effective mass with m^*_e=1.023 m_0. However, a significantly more parabolic dispersion relation at the VBM when compared to CaMnGe2O6, leading to a hole effective mass of m^*_h=9.380 m_0, when considering the Γ→ D segment. We must however state that since the valence band dispersion is quite asymmetric, the interpolation with a quadratic function did not yield a very accurate fit to the dispersion relation; therefore the value of the m^*_h should be understood more as an order of magnitude rather than a quantitative value. 
Once again, we observe that with the Cd substitution, there is only a mild variation of m^*_e, leading to a slightly increased value of m^*_e=1.028 m_0 for Sr7/8Cd1/8MnGe2O6, and a more pronounced increase for the 
Sr3/4Cd1/4MnGe2O6 compound with m^*_e=1.106 m_0. As for the SrMn3/4Cd1/4Ge2O6 and SrMn7/8Cd1/8Ge2O6 systems, a slight increase of the effective masses is observed to m^*_e=1.068 m_0 and m^*_e=1.057 m_0, respectively. 
With respect to the m^*_h, we again note that due to the flat-shape of the bands, a parabolic interpolation does not provide a dependable quantitative value, so we can only state with confidence that this value should remain of the same order as that of the pristine SrMnGe2O6 compound. We must state that such observed differences of the masses between the charge carriers is quite interesting because of the increased possibility to enhance the separation of the electron–hole pairs, and even of evidencing large electron and
hole drift lengths<cit.>, thus affecting the opto-electronic properties of the studied clinopyroxenes <cit.>.
The band gaps were also determined for all systems from the bandstructure calculations, and are summarised in Table <ref>. As already observed for the PDOS, the band gaps for all systems lie between 1.70 eV and 1.82 eV for the pristine systems, SrMnGe2O6 and CaMnGe2O6, respectively; whereas the doped structures show narrower band gaps. All the doped structures show indirect band gap character, even for the Sr-based strucutures, where the pristine counterpart evidences a direct one (figure <ref> and figures Sr2Cd2Mn2Ge4O12_bands_DOS,SrCdMnGe2O6_bands_DOS,Sr2Mn2Cd2Ge4O12_bands_DOS,SrMnCdGe2O6_bands_DOS,Ca2Cd2Mn2Ge4O12,CaCdMnGe2O6,Ca2Mn2Cd2Ge4O12,Ca2Mn2Ge4O12AFM_bands,CaMnCdGe2O6 in Appendix <ref>).





  §.§.§ Ab Initio Electric Field Gradients


The DFT  EFGs calculations 
were performed for the pristine unit-cells, along with the EFGs for the unit-cells and 1 × 1 × 2 supercells with Cd substitution, either at the A- or the Mn-sites. Results were compared with those obtained for the experimental EFG fractions and are summarised in Table <ref>, whilst a more comprehensive list of computed values can be found in appendix <ref>, Table <ref>. It should be noted that in the cases where Cd was placed at a Mn site, the systems were treated as ferromagnetic, for the sake of computational simplicity. An antiferromagnetic calculation for the CaMn7/8Cd1/8Ge2O6 system was nevertheless performed to check how much of an effect the magnetic ordering might have on the computed EFG. Very similar results were obtained in both the ferromagnetic and antiferromagnetic calculations, with V_zz=12 V/Å^2 and η=0.21, in the former case, 
and V_zz=13 V/Å^2 and η=0.37, in the latter, 
thus showing that this approximation is reliable. 





Comparing the EFG parameters for the experimental EFGSMGO1 fraction of SrMnGe2O6  with those computed for the supercells with Cd substitution at the Sr-site, where the 1/4 and 1/8 Cd dilution was considered (unit-cell and 1 × 1 × 2 supercell, respectively), we may observe a very good agreement between V_zz and η. From the PAC measurements we obtain for EFGSMGO1the values of V_zz=120 V/Å^2 and η=0.28; for the Cd probe nucleus for Sr3/4Cd1/4MnGe2O6 they are V_zz=103 V/Å^2  and η=0.14 and for Sr7/8Cd1/8MnGe2O6 these result as V_zz=108 V/Å^2 and η=0.14. Likewise, good agreement is also observed for the EFGSMGO2 fraction of SrMnGe2O6 for the case of the Cd substitution at the Mn-site. We obtain for the EFGSMGO2 fraction the values of V_zz=20 V/Å^2 and η=0.81. From the DFT results we get V_zz=22 V/Å^2 and η=0.67, and V_zz=21 V/Å^2 and η=0.72, for the SrMn3/4Cd1/4Ge2O6 and SrMn7/8Cd1/8Ge2O6 cells, respectively. Based on the obtained theoretical results, we may conclude that the supercells are the systems that simulate quite well the experimental data when the Cd probe is considered, with the EFGs being in good agreement to the EFGSMGO1 and EFGSMGO2 experimental fractions, which correspond to the Cd probe replacing the Sr-sites and the Mn-sites, respectively, in the crystalline structure. 
 Similarly, this same correspondence seems to hold true for the CaMnGe2O6 compound. 
For the EFGCMGO1 experimental fraction, we observe the values of V_zz=130 V/Å^2 and η=0.15, and for the DFT data we obtain values of V_zz=108 (109)  V/Å^2 and η=0.01 (0.02) for  Cd concentration of 1/4 (1/8)  in the Ca atom position.  
The obtained theoretical EFGs are slightly lower than those of the experimental fraction; such is mostly evident for the asymmetry parameter, η. As for the EFGCMGO2 fraction from the PAC experiments, the EFG  result in V_zz=10 V/Å^2 and an indeterminate η, whereas for the DFT diluted supercell of CaMn7/8Cd1/8Ge2O6 we have V_zz=12 V/Å^2 and η=0.21, similar to those obtained for CaMn3/4Cd1/4Ge2O6. These results indicate that first-principles calculations combined with PAC spectroscopy provide a valuable tool to explain the observed two distinct local environments, indicating that Cd probes are distributed between the Sr- or Ca-sites and the Mn-sites in the  SrMnGe2O6 and CaMnGe2O6 basic compounds, respectively. 





§ CONCLUSIONS

We have undertaken DFT calculations to obtain the structural and electronic properties of the AMnGe2O6 clinopyroxene systems, with A=Be, Mg, Ca, Sr. We found, through enthalpy of formation calculations, that the Be- and Mg-based systems
are not thermodynamically stable against dissociation into the stable constituent oxides BeGeO3 or MgGeO3 and MnGeO3. On the other hand, the Ca- and Sr-based systems are thermodynamically stable against dissociation and, accordingly,  have been realized experimentally. From the PDOS, we found that the strongly correlated Mn 3d-states form the narrow band close to the VBM together with the O p-states.  The CBM is composed mainly of low density hybridized p-states of O and Ge and, for higher energies, the density of states significantly increases
forming a narrow resonant band  that is mainly composed of the anti-bound unnocupied Mn 3d-states. The theoretical band gaps of CaMnGe2O6 and SrMnGe2O6 compounds, computed using the PBE+U approximation, 
are 1.82 eV and 1.70 eV, respectively. The calculated effective masses of the charge carriers of the two studied pristine compounds show considerable differences, which implies that the separation of the electron–hole pairs can be enhanced, thus improving the photoelectric efficiency. By considering the Cd dopants, we do not observe much variation on the value of the charge carrier masses, whereas a decrease of the band-gap widths is observed.

The temperature behavior of the experimental EFG V_zz fractions, labeled  EFGSMGO1 and EFGCMGO1, respectively related to the Cd probe on the SrMnGe2O6 and CaMnGe2O6 compounds,   fits precisely, according to the Bayer-Kushida model <cit.>,  with a single phonon frequency, producing frequencies of 8.01(7) × 10^12 and 10.0(2) × 10^12 respectively, suggesting that their behavior can be attributed to a phonon contribution to V_zz.  


The hyperfine parameters were computed to obtain the EFGs and these were compared to PAC measurement results. Data between both sets are consistent with the two obtained experimental fractions, EFGSMGO1 and EFGSMGO2, when considering a supercell approach with a Cd ion replacing  Sr and Mn sites, respectively. 
We also find that this same correspondence seems to be true for the  CaMnGe2O6 compound, where the  EFGCMGO1 and EFGCMGO2 experimental fractions can be atributed, respectively,  to Cd replacing  Ca and  Mn ions.   
Furthermore, through the combination of theoretical ab-initio calculations and PAC results, we could infer that the Cd probe in the AMnGe2O6 clinopyroxene compounds, with A=Ca, Sr, can replace 
the ion located at the A-site as well as the Mn ion, offering new perspectives for new systems synthesis with possibly improved funcionalities. 



The authors acknowledge the support of the technical teams at ISOLDE for their exceptional work in delivering high-quality beams for the presented PAC measurements. The authors also acknowledge project NECL under NORTE-01-0145-FEDER-022096, and FCT projects UIDP/04968/2020 (doi.org/10.54499/UIDP/04968/2020), UIDB/04968/2020 (doi. org/10.54499/UIDB/04968/2020), LaP-MET LA/P/0095/2020, POCI-01-0145-FEDER-029454, POCI-01-0145-FEDER-032527 and CERN/FIS-TEC/0003/2021. Also, BMBF through grants 05K16PGA and 05K22PGA, EU Horizon Europe Framework research and innovation programme under grant agreement no. 101057511 (EURO-LABS) for supporting IS679 ISOLDE-CERN experiment, and PRACE project with reference 2021240118, with access to the Irene Skylake computer. RPM acknowledges support from the Project HPC-EUROPA3 (INFRAIA-2016-1-730897), with the support of the EC Research Innovation Action under the H2020 Programme, with reference HPC171030W; in particular, the support of Dr. A. Stroppa and the hospitality provided by CNR-SPIN at the Department of Physical and Chemical Science of the University of L'Aquila (Italy) during the visit from 15/10/2021 
to 20/12/2021 as well as computer resources and technical support provided by CINECA. RPM also acknowledges computer resources and technical support from Minho Advanced Computing Center through project CPCA/A1/460622/2021, financed by FCT. LVCA (Project 314884/2021-1) and HMP (Project 308438/2022-1) acknowledge funding from CNPq and support from FAPESP (Project 2022/10095-8). AMLL acknowledge the FCT 2021.04084.CEECIND (doi.org/­10.54499/­2021.04084.CEECIND/­CP1655/­CT0002) and ELdS   the 2022.00082.CEECIND (doi.org/­10.54499/­2022.00082.CEECIND/­CP1719/­CT0001) grants. ELdS further acknowledges the High Performance Computing Chair - a R&D infrastructure (based at the University of Évora; PI: M. Avillez), endorsed by Hewlett Packard Enterprise, and involving a consortium of higher education institutions, research centers, enterprises, and public/private organizations. 











 
   






§ CALCULATION OF FORMATION ENERGIES





The formation energies for the studied oxides of the  AMnGe2O6 (A = Be, Mg, Ca, Sr) clinopyroxene series,  were calculated according to the following equation:

    E_form = E_AMnGe2O6 - E_AGeO3 - E_MnGeO3,

where E_AMnGe2O6, E_AGeO3, E_MnGeO3 are the total electronic energies obtained from DFT+U calculations, per formula unit (f.u.), which are summarised in  table <ref>.





§ ELECTRIC FIELD GRADIENT PARAMETERS




 
The calculated EFG parameters are presented in table <ref>. From these results we can observe that the EFGs at the Ge nuclei  were only slightly affected by the Cd substitution, as evidenced by the fact that the symmetry of the Ge sites remained quite similar to SrMnGe2O6 and CaMnGe2O6 basic compounds after Cd substitution. The same cannot be said for the Sr/Ca and Mn sites, which in many cases deviated substantially.




§ ELECTRONIC BAND DISPERSION OF THE CD DOPED SYSTEMS





The bandstructure and the density of states for the AMnGe2O6 (A = Ca, Sr) clinopyroxene compounds with Cd substitution in  A- and Mn-sites are  presented in figures Sr2Cd2Mn2Ge4O12_bands_DOS,SrCdMnGe2O6_bands_DOS,Sr2Mn2Cd2Ge4O12_bands_DOS,SrMnCdGe2O6_bands_DOS,Ca2Cd2Mn2Ge4O12,CaCdMnGe2O6,Ca2Mn2Cd2Ge4O12,Ca2Mn2Ge4O12AFM_bands,CaMnCdGe2O6.



















48
natexlab#1#1bibnamefont#1#1bibfnamefont#1#1citenamefont#1#1url<#>1urlprefixURL 


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 Paper Title: Acceptor and donor impurity levels in hexagonal-diamond silicon.




Sample title]Acceptor and donor impurity levels in hexagonal-diamond silicon


Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France

Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France

Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France
michele.amato@universite-paris-saclay.fr


             

§ ABSTRACT 
Recent advances in the characterization of hexagonal-diamond silicon (2H-Si) have shown that this material possesses remarkably different structural, electronic and optical properties as compared to the common cubic-diamond (3C) polytype. Interestingly, despite the wide range of physical properties analyzed, to date no study has investigated impurity energy levels in 2H-Si. Here, we present results of ab initio DFT simulations to describe the effect of p- and n-type substitutional doping on the structural and electronic properties of hexagonal-diamond Si (2H-Si). We first provide a detailed analysis on how a given impurity can assume a different local symmetry depending on the host crystal phase. Then, by studying neutral and charged dopants, we carefully estimate donors and acceptors transition energy levels in 2H-Si and compare them with the cubic-diamond (3C) case. In the case of acceptors, the formation energy is always lower in 2H-Si and is associated to a shallower charge transition level with respect to 3C-Si. On the other hand, donors prefer the cubic phase and have transition energies smaller with respect to 2H-Si. Finally, by employing a simple model based on the 2H/3C band offset diagram, we prove the physical validity of our findings and we show how holes can be used to stabilize the 2H-Si phase.


[
    M. Amato^*
    August 5, 2024
==================





§ INTRODUCTION

Recent developments and advances in semiconductor growth methods demonstrated that the hexagonal-diamond polytype of Si, Ge and SiGe alloys (2H in the Rasmdell notation <cit.>) can be fabricated in the form of nanowires <cit.>. This crystal structure presents a C_3v symmetry (P6_3/mmc space group) with three equal length bonds in the c-plane (0001) and a larger one along the [0001] direction of the hexagonal crystal cell. Compared to the 3C cubic-diamond phase (Fd-3m group symmetry) where four equal length bonds induce a T_d symmetry, in the 2H phase the atomic ordering along the ⟨ 111 ⟩ cubic direction changes from an ABC to an ABAB stacking. 

The main reason for studying these novel group IV polytypes lies in the possibility to observe unexplored electronic and optical properties with respect to those of their 3C counterparts. To give an example, Ge, which in its cubic form is an indirect semiconductor, has a direct band gap in its hexagonal polytype <cit.>. Yet, hexagonal SiGe alloys <cit.> and quantum wells <cit.> have recently attracted a great attention because of their direct band gap and the experimentally observed light emission that can be tailored via the Ge concentration and quantum confinement effects <cit.>. In the case of Si, while the cubic phase has an indirect band gap of 1.12 eV with a conduction band minimum close to the X point, the 2H structure presents an indirect band gap of about 0.95 eV and a conduction band minimum at the L point <cit.>. The direct band gap is also reduced from 3.40 eV to 1.63 eV <cit.>. Indeed, 2H-Si presents a larger optical absorption spectra in the visible range than 3C-Si <cit.>, making it a good candidate for solar panel materials. Furthermore, 3C/2H Si junctions are also of interest for photovoltaics due to their intrinsic tendency to promote charge carrier separation <cit.>. 

It is well known that most technologies based on carrier transport in semiconductors (e.g. electronics, embedded systems, light emitting devices and solar cells) become functional only if the material can be appropriately doped. Impurities play a crucial role as they allow to modulate the carrier concentration around the Fermi level. Furthermore, in different type of semiconductor compounds, doping can be employed to stabilize one polytype with respect to the others <cit.>. For instance, Cai et al. <cit.> recently demonstrated that the direct band gap 2H-Ge phase can be stabilized by incorporating electrons into the 3C-Ge crystal. Finally, most of the experimental grown nanowires can contain significant concentration of extrinsic dopants due to the substrate atomic contamination. This is, for instance, the case of 2H-Ge and 2H-SiGe alloys <cit.> fabricated with the crystal transfer methods which usually present a high concentration of As atoms ( ∼ 10^18 cm^-3).  

Although the behaviour of impurities in 3C-Si has been widely discussed, to date no study has investigated impurity energy levels in 2H-Si. The experimental characterization of dopants in this polytype still remains challenging because of the difficulties in its fabrications methods. Ab-initio theoretical approaches can be hence an efficient way to investigate this question. A few studies focused on the analysis of the stability of single-point and neutral substitutional defects in 2H-Si <cit.> but they were essentially limited to the structural stability of defects. 
Here, we present a comprehensive study carried out via ab initio DFT simulations on the effect of p- and n-type substitutional defects on the structural and electronic properties of 2H-Si. We first provide a detailed analysis on how a given impurity can assume a different local symmetry depending on the host crystal phase. Then, by studying neutral and charged dopants, we carefully estimate donors and acceptors thermodynamic transition levels in 2H-Si and compare them with the 3C-Si case. Finally, by employing a simple model based on the 2H/3C band offset diagram, we show how the carrier concentration can be used to stabilize one phase with respect to the other.



§ METHODOLOGY

All simulations were performed by spin-polarized ab initio DFT as implemented in the SIESTA code <cit.>, using the generalized gradient approximation (GGA) with the Perdew, Burke, Ernzerhof (PBE) functional <cit.>. Since the diameter of the 2H nanowires grown experimentally is usually larger than 100 nm, we considered bulk supercells as an accurate and reliable approximation for such systems. Indeed at such large diameters, quantum confinement effect on the band gap can be neglected <cit.>. Norm-conserving pseudopotentials of Trouillier-Martins type were chosen to replace the core electrons. Valence electrons were instead represented using a double-ζ (DZ) polarized basis set for Si, and double-ζ plus two polarization orbitals (DZP) basis sets for impurities. We considered p-type (B, Al, Ga, In) and n-type (P, Sb, As) as common substitutional dopants in 2H-Si. Lattice parameters and band gap of pristine bulk 3C- and 2H-Si were calculated by sampling the Brillouin zone using a 8×8×8 Monkhorst-Pack k-points grid <cit.>. Table <ref> presents the converged results of the lattice parameter and the band gap calculations for both the phases. As expected, both quantities are affected by the error associated to the GGA exchange-correlation functional. We used a denser grid of 25×25×25 k-points to calculate the dielectric constant of both phases. This quantity was needed for the charge correction scheme described below <cit.>. The values of the in-plane and out-of-plane components, reported in Table <ref>, are in perfect agreement with other DFT calculations <cit.>. For the doped systems, a supercell of 512 (a 4×4×4 replica of the 8-atoms conventional supercell with orthogonal cell vectors) and of 500 atoms (a 5×5×5 replica of the primitive cell) were used for the 3C and 2H phases, respectively. This choice has guaranteed the convergence of the formation energy value for both neutral and charged impurities as it eliminated the spurious interaction between defect periodical images. Such doped supercells correspond to a dopant concentration between 10^19 cm^-3 and 10^20 cm^-3, a value that is one order of magnitude larger with respect to the impurity concentration usually measured in 2H-samples <cit.>. A 2×2×2 k-grid and a real space grid cut-off of 500 Ry were considered to ensure the proper convergence of the results. Self-consistency field cycles were converged with a tolerance of 10^-5 eV. Atomic positions were optimized until the forces were less than 10^-2 eV/Ang and the stress lower than 10^-1 GPa. Relaxation was performed adopting a conjugate gradient algorithm that calculates DFT forces through the Hellman-Feynman theorem. Charged structures were relaxed starting from the neutral defect optimized structure. A compensating jellium background was added to avoid divergences due to the defect-defect long-range electrostatic interaction between charged periodic cells. 

For all the considered impurities, we calculated their formation energy, E_form, in a given crystal phase (2H or 3C). This quantity is defined as the energy required to replace a host atom with a neutral or charged defect. In the Zhang and Nortrup formalism <cit.>, it reads as follows:

    E_form = E_total^D - E_total^prist + μ_0 - μ_D + q (ϵ_VBM+μ_e) + E_corr,
 
where E_total^D is the total energy of the doped system, E_total^prist is the total energy of the pristine system, μ_0 is the chemical potential of Si in a given phase while μ_D is that of the defect. μ_0 is here defined as the total ground energy per atom of a pristine Si cell whereas μ_D is calculated as the total energy of the isolated impurity atom. However, it is worth to note that there is no univocal criteria to calculate the chemical potential of an impurity as this quantity strongly depends on the synthesis conditions. As a consequence, the calculated value of the formation energy can assume a wide range of values depending on the way μ_D is defined. Nevertheless, since we are interested in the relative stability of a dopant among the two phases, the chemical potential μ_D erases if we refer to the difference between formation energies, Δ E^2H-3C_form = E_form^2H - E_form^3C rather than in E_form value in itself (this also holds because the difference of μ_0 among the two phases can be neglected). Yet, referring back to Eq. (<ref>), the term q (ϵ_VBM+μ_e) is the charge multiplied by the electron chemical potential, μ_e, using as energy reference the valence band maximum, ϵ_VBM. μ_e varies from 0 to the electronic band gap. Finally, the term E_corr corresponds to the Freysoldt-Neugebauer-Van de Walle's finite-size correction <cit.> that cancels artificial interactions between charged defects and that has been proven to give very accurate results for different type of defects <cit.>. As other proposed schemes <cit.>, this approach implies the solution of the Poisson equation for a model system and the alignment of the electrostatic potentials to derive the formation energy correction. In particular, solving the Poisson equation for the periodic model potential needs the calculation of the dielectric tensor profile of the material <cit.>, whose static components are presented in Table <ref>. 

By calculating formation energies, we can also estimate the thermodynamic transition level between two charge states, ϵ(q_1/q_2), which corresponds to the specific value of the electron chemical potential for which the two states have the same value of E_form. This quantity can be calculated as:

    ϵ(q_1/q_2) = E^D_total(q_1) + E_corr(q_1)- E^D_total(q_2) - E_corr(q_2)q_2-q_1
        - ϵ_VBM.
 


Transition levels ϵ(q_1/q_2) correspond to thermal ionization energies that can be measured by deep level transient spectroscopy (DLTS) <cit.>. We employed Eq. (<ref>) to calculate both acceptors, ϵ_A(q_1/q_2) and donor transition levels ϵ_D(q_1/q_2). It is worth to note that transition levels are not affected by the dopant chemical potential which assumes the same value for the charged and neutral impurity and hence cancels out in Eq. (<ref>). 

In the following analysis, Eq. (<ref>), which depends on the valence band maximum of each phase taken separately, allows to compare the doping with a given impurity of the 2H and the 3C phase of Si. On the other hand, to give an accurate quantitative estimation of transition levels, in particular in the case of the phase coexistence <cit.>, we align ϵ(q_1/q_2) to a common reference level, i.e. the vacuum level Φ (see SM for the details of the calculations). Indeed, many works have demonstrated that, even if within a semi-local approach, energy levels of dopants generally remain preserved if they are referred to a common well defined potential <cit.>. This choice improves the quantitative comparison among the two phases. On the basis of this consideration, Eq. (<ref>) can be modified as:


    ϵ(q_1/q_2) = E^D_total(q_1) + E_corr(q_1)- E^D_total(q_2) - E_corr(q_2)q_2-q_1
        - (ϵ_VBM - Φ) = ϵ(q_1/q_2) - ϵ_VBM.
 

where the charge transition level ϵ(q_1/q_2) and the ϵ_VBM are referred to the vacuum potential. 



§ STRUCTURAL PROPERTIES

We start our discussion by a detailed structural analysis of the relaxed doped structures. Table <ref> and <ref> report the first-neighbors bond lengths of group III and group V impurities in the cubic and hexagonal crystals. In the 3C-Si, both neutral donors and acceptors maintain the tetrahedral symmetry (T_d) forming four first-neighbor bonds of equal length. Variations in bond lengths compared with the pristine system are uniform and depend on the mismatch between the covalent radius of the impurity and that of the host element, as already reported by Amato et al. <cit.>. The bond contractions and expansions discussed here agree with previous results. As for smaller defects, such as B and P, the first neighbors impurity bonds are compressed in respect to those of the pristine cell (see first block of Table <ref> and  <ref>) while for all the other reported defects (Al, Ga, In, Sb and As in Table <ref> and  <ref>) they tend to expand since their covalent radius is larger than those of Si. More in general, we can say that impurities in the 3C-Si phase induce an expansion (contraction) of their first neighbors bonds and, consequently, there is a noticeable contraction (expansion) between the first and second neighbors atomic layers that gradually fades within the crystal. The charged impurities also preserve the cubic T_d symmetry. As for B^-, the extra electron does not induce any appreciable change in the bond distances (see first block of Table <ref>). Al^-, Ga^-, and As^- first neighbors distances are only slightly reduced compared to the neutral case (see Table <ref>). These tiny bond length variations are proportional to the impurity size, being more pronounced for larger dopants. Therefore, adding an extra electron to large radius acceptors (Al, Ga, In) in the 3C-Si host lattice induces a larger contraction compared to the neutral case, presumably because the lattice has fewer degrees of freedom to adjust to the extra electron configuration, resulting in a more significant contraction. The analysis of donors shows that, when positive charges are added, bonds tend to contract as well, but without any clear general trend (see Table <ref>).


Compared to the 3C-Si case, the 2H-Si host lattice presents a lower symmetry due to the anisotropy along the [0001] hexagonal direction (C_3v symmetry). In the pristine structure, for a given Si atom, three equal length first-neighbor bonds in the c-(0001) plane can be defined. The fourth bond is longer and perpendicular to the plane, breaking the tetrahedral symmetry. This arrangement of the atoms increases the degrees of freedom for the structural relaxation around the defect. As shown in the first block of Table <ref>, the local symmetry of the B^0 impurity is C_3v: there exist three equivalent bonds in the plane and a longer one along the [0001] crystal direction. This is a consequence of three valence electrons of the acceptor trying to keep the three-fold coordination. The difference between the average of the three in-plane bonds and the out-of-plane one is larger than that of the 2H pristine case. As for B^-, we observe the same symmetry but slightly more pronounced: an equivalence of the three bonds in the plane and a slight expansion of the perpendicular one with respect to the neutral case. Looking at the second block of Table <ref>, we can recognize a similar behavior in the case of Al:  both Al^0 and Al^- keep a clear C_3v symmetry with three equal length bonds in the plane and a longer one along the c hexagonal direction. As for Ga^0 and In^0 (third and fourth blocks of Table <ref>), though it is not very clear, it seems that the local symmetry is closer to the T_d cubic one. When a charge is added, this tendency is even more pronounced, especially for In (fourth block of Table <ref>). 
On the other hand, donors in 2H-Si induce similar contraction and expansion of the local structure, but with a different impurity symmetry. As reported in the first block of the Table <ref>, P^0 has a C_3v symmetry with a difference between the in-plane and the out-of-the-plane bond lengths larger than those of the pure 2H-Si. When a charge is added, the out-of-plane bond length is slightly reduced, while all the in-plane bonds maintain the same value. There is almost no change in the symmetry between neutral and charged cases, keeping a pronounced C_3v symmetry. The same behavior can be recognized also for As and Sb (second and third block of Table <ref>). It is worth to note that the difference between the in-plane and the out-of-plane bond length average is strictly related to the impurity ionic radius being lower for Sb and larger for P.




§ FORMATION ENERGY

The local structural analysis cannot provide alone a full picture of the impurity stability in different phases. Formation energy calculations of both neutral and charged dopants are required to analyze this question in depth. Figures <ref> and <ref> report the formation energy, E_form, calculated with Eq. (<ref>), as a function of the chemical potential relative to the maximum of the valence band for p- and n-type dopants, respectively. Numerical values associated to the graphical plots of Figures <ref> and <ref> are provided in Table <ref> and Table <ref>.

Looking at Figure <ref>, one can evince, as expected, that negatively charged acceptors are more stable than neutral impurities because of their tendency to attract an electron from the host crystal to create a hole in the valence band. Interestingly, for both the neutral and charged cases, their formation energy is always lower in the 2H phase with respect to the 3C one (see Figure <ref>). This is even clearer if one looks at the first column of Table <ref> where the difference in formation energy, Δ E^2H-3C for a given impurity in the 3C and in the 2H phase is reported for both the charged and the neutral cases. The more Δ E^2H-3C is negative, the more the impurity will prefer the 2H phase with respect to the 3C-one.  

This behavior can be ascribed to the clear preference for acceptors to be in the 2H phase as a consequence of its stronger tendency to induce a local C_3v symmetry and a three-fold coordination around the impurity (see Section <ref>). The same explanation still holds when charged impurities are considered as this phase preference does not change (see Figure <ref>).  

Similarly to the case of acceptors, Figure <ref> shows that positively charged donors are more stable than neutral impurities because the additional donor electron, occupying an antibonding state, is only weakly bonded to the impurity and tends to be delocalized in the crystal. Furthermore, while neutral donors do not present any clear phase preference as Δ E^2H-3C is close to zero, charged donors reveal a marked tendency to be more stable in the 3C-phase (see Figure <ref> and the first two columns of Table <ref>). So for charged donors the 3C-host lattice is the ideal crystal environment to form four equal length covalent bonds with a marked T_d symmetry.

Our results clearly show that, close to the 2H/3C homo-junction <cit.>, the Δ E_form^2H-3C of charged donors is positive while for charged acceptors it is negative. To align the Fermi levels of the two phases a space charge will be generated and thus the dopants will be necessarly charged at the interface. It is hence expected that a gradient in the dopant concentration at the interface between the two phases can be build up with donors accumulating on the 3C-Si side and acceptors in 2H-Si.




§ THERMODYNAMIC TRANSITION LEVELS

The calculation of the formation energies of neutral and charged impurities allows to employ Eq. (<ref>) to calculate thermodynamic transition levels, ϵ(q_1/q_2), which are essential for the experimental characterization of the defect. These charge transition energies correspond to the crossing point where the charged and the neutral defect formation energies coincide, as observed in Figures <ref> and <ref>. It is clear from the figures that, for acceptor dopants, the difference in the defects formation energy between the two phases changes in the order of a few meV with the state of charge of the impurities (Δϵ_A^2H-3C also reported in the sixth column of Table <ref>). More negative values of Δϵ_A^2H-3C correspond to a transition level that is shallower in the 2H phase. For B impurities the transition energy is positive, those of Ga and Al are practically zero (less than 2 meV), while that of In is negative. This means that by increasing the size of the defect, the difference between the transition energies between 2H and 3C will be more negative and hence the defect level will be shallower in 2H-Si. In the case of smaller radius impurities, Δϵ_A^2H-3C is positive and the defect level is shallower in 3C-Si. 
In summary, if one looks at the second and sixth column of Table <ref>, it is possible to conclude that charged acceptors are always more stable in the 2H phase and present charge transition levels shallower with respect to the 3C-Si. The magnitude of these energy shift depends from the ionic radius of the dopant.

A similar analysis can be performed for donors. The difference of the transition energies between the two phases is about hundreths of meV (see sixth column of Table <ref> and Fig. <ref>) and are always negative. This means that the donor dopant levels will be always shallower in 3C-Si with respect to 2H-Si. As in the case of acceptors, even here we can recognize a relation with the ionic radius of the dopant. Indeed, Δϵ_D^2H-3C decreases as a function of the radius mismatch. These findings agree with the hydrogenic model of donors that considers four covalent bonds and an unpaired one whose energy relative to the bottom of the conduction band is directly proportional to the effective mass and inversely proportional to the dielectric function. As reported in Table <ref>, the in-plane component of the dielectric function is lower for 2H- than for 3C-Si. This means that the transition level of donors in 2H-Si will be deeper with respect to those of 3C-Si. The discussed behavior, with the appropriate differences, was already observed in polytypes of GaAs, GaP, and InP <cit.>.

Nevertheless a note of caution is needed when looking at the transition levels shown in Figure <ref> and Figure <ref>. Indeed, some of these levels are resonant with the crystal host band states. This is directly related to the use of total energies obtained from semi-local functionals which cannot provide accurate ionization energies for shallow dopants <cit.> but can only qualitatively reproduce the experimental chemical trend among different type of dopants <cit.>. More advanced methods, such as quasi-particle G_0W_0 calculations, are known to provide accurate results but they are associated with a very high computational cost. Although in our case the absolute values of the transition energies for each phase taken separately may not be correct, the calculated semi-quantitative variation between the two phases is still valid and insightful.
To further improve the comparison of the transition energy levels between the two phases, we calculated them with respect to a common reference point, i.e. the vacuum level, Φ, calculated from a Si slab model, as described by Eq. (<ref>) (see SM for more details about the procedure). Indeed, even when the theoretical description is improved with beyond-DFT methods, band edges in semiconductors undergo significant shifts but charge transition levels of defects remain practically unaffected with respect to their semi-local energy positions <cit.>. Thus the calculated Δϵ_A^2H-3C and Δϵ_D^2H-3C presented in Table <ref> and Table <ref> and graphically shown in Fig. <ref> are a valid reference for future experiments aiming to measure the variation in the ionization energy among the two phases. In agreement with other studies <cit.>, we found that the 3C/2H Si junction presents a type-II alignment where the difference between hexagonal and cubic valence bands maximum (VBM), Δ VBM^2H-3C≈ 0.154 eV and between hexagonal and cubic conduction bands minimum (CBM), Δ CBM^2H-3C≈ 0.036 eV. Thus, the 3C-VBM lies below the 2H-VBM with a difference that is of a similar order of magnitude of the calculated |Δ E^2H-3C_form,q|. In particular, when the two phases are compared, donors transition levels of both materials (referred to the vacuum level) are located almost at the same energy position due to the difference in the band offset. On the other hand, acceptors in the 3C-Si will be lower compared to 2H-Si. 



§ BAND-OFFSET MODEL OF DOPING

In order to further confirm our DFT findings and to depict a full physical picture of the dopant stability in 2H and 3C-Si, we build up a simple model following the arguments employed by Dalpian et al. <cit.> to explain the stability of III-V polytypes and recently adopted by Cai et al. <cit.> to study 2H-Ge. Ideal doping can be represented by simply adding electrons and holes to the pristine supercell of 2H- and 3C-Si taking into account total energy correction due to finite size effects <cit.>. Figure <ref> reports the difference in the total energy between the two phases, Δ E_total^2H-3C per atom as a function of the carrier concentration. The results show that introducing electrons tends to stabilize the 3C phase while introducing holes is a way to make the 2H phase more stable. These results are a consequence of the type II band offset between the two phases: since 2H-Si has a higher valence band maximum than 3C (see Fig. <ref>), it will cost less energy to create holes in the 2H-phase than in the 3C-one. On the other hand, since 3C-Si has a lower conduction band minimum than 2H-Si (see Fig. <ref>), it will cost less energy to add electrons in the 3C-Si than in 2H-Si. Moreover, as the 2H/3C conduction band offset is very tiny, the slope of the electron doping curve (red line of Fig. <ref>) is found to be small while that of the hole incorporation is larger (green line of Fig. <ref>). Interestingly, this fully confirms our DFT results for impurity doped supercells which show that i) acceptors are more stable in the 2H phase and have shallower transition energy levels if compared to 3C-Si, ii) donors prefer instead to occupy cubic crystal site with shallower transition energies if compared to 2H-Si.  



§ CONCLUSION

In conclusion, this work presents DFT calculations of the structural and electronic properties of p- and n-type impurities in 2H-Si, comparing them with the 3C-Si case. We find that acceptors tend to be more stable in the hexagonal phase because of its stronger tendency to induce a local C_3v symmetry and three-fold coordination around the impurity. This behaviour is valid for all the considered acceptors (B, Al, Ga and In). In particular, charged acceptors are always more stable in the 2H phase and present a shallower charge transition level with respect to the 3C-Si crystal host.

On the other hand, we find that neutral donors (P, As, Sb) have no phase preference while positively charged donors have lower formation energies in the cubic phase than in the hexagonal one. This can be ascribed to the fact that for charged donors the 3C-host lattice is the ideal crystal environment to form four equal length covalent bonds with a marked T_d symmetry. Furthermore, donors levels are shallowers in 3C-Si than in 2H-Si. 

The physical validity of the doped supercell calculations is reinforced by a simple model based on the 2H/3C band offset diagram through which we show how holes can be used to stabilize the 2H-Si phase while electrons prefer the 3C-Si crystal phase.

All these results assume a particular importance for future experiments. Being accurate theoretical calculations of transition energy levels in 2H-Si, these findings represent a valid theoretical reference for future measurements of the variation in the ionization energy of a given dopant among the two phases. Moreover, in the case of 2H/3C homo-junctions <cit.>, because of the different stability of charged dopants, a gradient in dopant concentration at the interface between the two phases may be experimentally detected with donors accumulating on the 3C-Si side and acceptors in 2H-Si.


All the authors acknowledge the ANR AMPHORE project (ANR-21-CE09-0007). Part of the high-performance computing resources for this project were granted by the Institut du développement et des ressources en informatique scientiﬁque (IDRIS) under the allocations AD010914974 and AD010915077 via GENCI (Grand Equipement National de Calcul Intensif). The authors also acknowledge the use of the CERES high performance computer cluster at the Laboratoire de Physique des Solides of the Université Paris-Saclay (Orsay, France).


apsrev4-1























 


















 Paper Title: Ultrastrong Terahertz Coupling in a van der Waals Heterostructure.


  
[
    Francesco Restuccia
    August 5, 2024
=======================



§ ABSTRACT 
The field of strongly coupled photon-matter systems aims to create novel hybrid phases that enhance or modify material properties. A key challenge has been developing versatile platforms for exploring cavity quantum electrodynamics (QED) of quantum materials. Here, we demonstrate and probe ultrastrong light-matter coupling in atomically-thin van der Waals materials. We developed a broadband time-domain microscope that integrates exfoliated, dual-gated, two-dimensional quantum materials with sub-wavelength photon confinement at terahertz frequencies. Using bilayer graphene as a model system, we perform spectroscopic studies across a wide range of frequencies and gate voltages. For the first time, we measure the field-tunable band gap of bilayer graphene at terahertz frequencies while simultaneously observing clear signatures of ultrastrong coupling, with interaction strengths exceeding Ω/ω≈ 40%. Our work bridges the gap between theoretical predictions and experimental implementation of cavity QED in van der Waals materials, paving the way for further studies of exotic phases and novel light-matter states.


The control of quantum materials with light has emerged as an intriguing direction in materials science and condensed matter physics <cit.>. At the forefront of modern research is the vision that ground state properties of materials could be modified through interaction with cavity-confined electromagnetic fields <cit.>. Even in the absence of external stimuli, matter can couple to quantum fluctuations of the vacuum electromagnetic field, leading to the formation of hybrid light-matter states known as cavity polaritons <cit.>. In the regime of ultrastrong light-matter coupling, where the interaction strength becomes comparable to the energy of material excitations, these novel hybrid phases are predicted to exhibit strongly modified macroscopic properties compared to the uncoupled system, with ground states that contain virtual light and matter excitations <cit.>. Recent theoretical studies have proposed a range of fascinating phenomena in hybrid systems. These include the emergence of superconductivity due to electron-photon interactions <cit.>, the potential for cavity-induced ferroelectric phase transitions <cit.>, as well as stabilization of superradiant excitonic insulators <cit.> and cavity-mediated control of intertwined orders <cit.>. These predictions highlight the transformative potential of strong light-matter coupling in quantum materials. 
Experiments have also shown impressive results demonstrating how confined electromagnetic fields can influence various systems. In the context of polariton chemistry <cit.>, cavity-coupling to molecular vibrations has been shown to affect chemical reactivity <cit.>. In solid-state systems, the critical temperature in a metal-to-insulator transition has been controlled by suppressing the radiative heat flow to the material <cit.>. A recent study has demonstrated cavity-induced breakdown of topological protection at integer filling in the quantum Hall transport of two-dimensional (2D) electron gases <cit.>. 
However, especially in condensed matter systems, clear evidence of cavity coupling to quantum phases and modification of equilibrium properties due to vacuum fields remains scarce.


To address this gap, we have developed a platform specifically designed to explore the physics of strongly coupled light-matter quantum materials, with a particular emphasis on two-dimensional (2D) systems. Heterostructures of van der Waals materials serve as an excellent experimental framework, owing to their diverse array of quantum and correlated phases, ranging from superconductivity <cit.> and Wigner-Mott states <cit.> to exotic magnetism <cit.>. Often, these phases can be explored within a single device by electrostatic gating, making van der Waals materials particularly versatile in the exploration of light-matter phases.


However, integrating 2D materials with suitable cavities presents significant challenges. Strong light-matter coupling occurs in the proximity of collective excitations, which for quantum materials lie within the terahertz photon range. Ultrastrong light-matter coupling in this frequency range is usually achieved by confining the electromagnetic field within a cavity mode noticeably smaller than the photon wavelength, which spans from tens to thousands of microns. While sub-wavelength localization of terahertz fields can be efficiently achieved using micro-structured metallic resonators, conventional approaches rely on metamaterials of repeating resonators placed atop the material of interest <cit.>. This method is incompatible with exfoliated 2D materials, which are usually limited to in-plane dimensions of just a few microns.


Recent advancements have enabled spectroscopy of individual metallic resonators by adapting solid immersion lenses <cit.> to the terahertz range <cit.>. Building on this concept, we have developed a method that simultaneously achieves the broadband frequency operation necessary for quantum materials spectroscopy <cit.> and strong field confinement, while maintaining compatibility with gate-tunable van der Waals devices. Our approach addresses the limitations of conventional techniques in studying micro-structured devices, which often suffer from bandwidth constraints or incompatibility with dual gating. While on-chip terahertz techniques have proven highly effective in probing the ultra-low energy electrodynamics of 2D materials <cit.>, they face inherent bandwidth limitations <cit.>, typically not exceeding 1.5 THz or 6 meV. Notably, much of the physics of quantum <cit.> and van der Waals materials is expected or has been observed at energies of several to tens of meVs. This frequency range encompasses collective modes such as phonons <cit.>, Mott gaps in Moiré systems <cit.>, binding energies <cit.> as well as Higgs and Bardasis-Schrieffer modes in excitonic insulators <cit.>, and Wigner gap and excitations of Wigner crystals <cit.>, among others. With a bandwidth exceeding 20 meV, our technique outperforms existing methods and bridges the gap between on-chip terahertz and photoconductivity measurements <cit.>. Importantly, our approach is versatile, being adaptable to various terahertz generation techniques and cavity designs. This flexibility makes it scalable across a wide range of frequency domains.





§ EXPERIMENTAL SETUP AND MODEL SYSTEM

In our setup (Fig. 1A), a free-space terahertz beam is tightly focused using a high-refractive-index, transparent hyper-hemispherical silicon lens. The beam impinges on a high-resistivity silicon substrate coated with gold, except for a small complementary bow-tie resonator designed to function for a broadband range of terahertz frequencies. This resonator steers the field onto a spot smaller than 2 microns, as represented by the antenna mode in Fig. 1C. The transmitted terahertz field (Fig.1B), with a bandwidth ranging from 1 to 6 THz (4 to 24 meV), is then recollimated by a secondary hemispherical silicon lens and detected using conventional far-field methods, such as electro-optic sampling within a nonlinear crystal. Crucially, only the field that traverses the micron-sized resonator focus is transmitted, enabling sub-wavelength, broadband terahertz spectroscopy of micron-sized samples positioned beneath the antenna focus (Fig. 1D).


We chose Bernal-stacked bilayer graphene (BLG) as our material system due to its ideal characteristics for studying light-matter interactions at their most extreme. While the material is intrinsically a semimetal with quasi-parabolic bands at the K and K' points, a unique "Mexican hat" band gap can be opened and controlled by application of a displacement field perpendicular to the graphene layers <cit.>. As a consequence, BLG offers strong optical excitations such as band-to-band transitions <cit.> and excitons <cit.> that can be continuously tuned from the mid-infrared to the terahertz range, perfectly aligning with our setup capabilities.





In our device (Fig. 1D), BLG is encapsulated between two hexagonal boron nitride (hBN) layers and is fully tunable with two thin graphite gates, allowing independent control of the charge density n = ϵ_hBNϵ_0/e ( V_TG/d_TG + V_BG/d_BG ) and the out-of-plane displacement field D = ϵ_hBNϵ_0/2 ( V_TG/d_TG - V_BG/d_BG ) , where V_TG (BG)[Note that, for simplicity, we include the shift of charge neutrality from 0V in the definition of V_BG and V_TG.], d_TG (BG), e, ϵ_hBN, and ϵ_0 are the top gate and bottom gate voltages, the hBN layer thicknesses, the elemtary charge, the hBN dielectric constant (≈3.4 <cit.>) and vacuum permittivity, respectively. Two Cr/Au etch contacts to BLG <cit.> allow us to probe the transport properties of the device. The stack is completed by the terahertz resonator, which is electrically insulated from the heterostructure and the electrical connections via a 140 nm layer of alumina grown by atomic layer deposition.




§ TRANSPORT CHARACTERISTICS AND TERAHERTZ TRANSMISSION

We first characterized the gate-voltage dependent two-point conductance of our device at 5K (Fig. 2A), identifying the typical transport regimes of bilayer graphene. For V_TG = -V_BGd_TG/d_BG, the out-of-plane displacement field D breaks inversion symmetry between the two graphene layers, opening a band gap that results in a minimum in conductance typical of an insulating state. This line corresponds to the diagonal charge neutrality region in Fig. 2A, where n=0. For V_TG = V_BGd_TG/d_BG, we increase the carrier concentration without opening a band gap, thus making the material metallic and increasing the conductivity. At the Charge Neutrality Point (CNP), where both n=0 and D=0, the material behaves like a semimetal (purple dot in Fig. 2A).



In Fig. 2C, we show the terahertz pulse transmitted through the resonator, the stack and the silicon lenses when the device is biased at the CNP. For non-zero displacement (D ≠ 0) and zero carrier density (n = 0), the opening of the band gap induces changes of a few percent in the terahertz amplitude (inset in Fig. 2C).


The gate-voltage dependence of the terahertz transmission at a fixed time delay (indicated by the black arrow in the Fig. 2C inset) is shown in Figure 2B. This transmission data bears a striking resemblance to the transport measurement results. A distinct charge neutrality line emerges, mirroring the insulating region observed in transport, with a nearly identical slope and intercept, albeit with a significantly larger width.
Notably, the CNP location exhibits a slight shift along the charge neutrality line between the terahertz and transport measurements, as indicated by the purple and green dots in Figures 2A and 2B, respectively. We attribute this displacement to the probing of different sample locations in the two measurement types.
Note that the single-time delay measurement does not provide spectral information. 

Nevertheless, the measurement in Fig. 2B clearly demonstrates that the terahertz signal is sensitive to the electronic state of BLG while remaining unaffected by other elements of the stack, including the gates.




§ DOPING-INDUCED CAVITY MODE SCREENING

To gain insight into the optical properties of the coupled BLG/cavity system, we first measured the spectrally resolved terahertz transmission when BLG is doped. Fig. 3A shows the terahertz spectra measured for different charge densities with n≠0 and D = 0. Each spectrum is referenced to the transmission taken for BLG in an insulating state (D=-0.35V/nm and n=0; red dot in Fig. 2A), where the band gap is larger than the terahertz bandwidth of our setup and the material becomes transparent for THz radiation. Effectively, the reference spectrum corresponds to the transmission of the bare cavity. 
It is crucial to note that changes in the normalized transmission spectra may originate from changes in either BLG or the cavity transmission properties. Here, we observe that for increasing electron densities, a pronounced dip in the terahertz transmission develops around 2.3THz. The dip coincides with the peak position of the simulated cavity transmission spectrum (Fig. 3B), consistent with observations in ref. <cit.>. As the resonance frequency and linewidth remain unchanged (line cuts in Fig. 3B), we conclude that the primary effect is the screening of the cavity mode from the incoupling field by the injected electrons, which results in reduced cavity transmission. The coupled system acts as a terahertz attenuator with modulation depths of around 30% in power for relatively small dopings. 


We qualitatively reproduce this behavior by simulating the interaction between a linear terahertz cavity and a material characterized by a Drude model with varying electron densities (Fig. 3C).









§ OBSERVATION OF ULTRASTRONG COUPLING AND POLARITON FORMATION

The transmission spectra as a function of displacement field D ≠ 0 for n=0 are shown in Figures 4A and B. At large negative fields, the material exhibits an insulating state with an electronic gap exceeding our detection window, resulting in a featureless response (pink curve at -0.33 V/nm in Fig. 4A). As the field decreases, an absorption edge emerges and shifts from higher to lower frequencies (dashed black line in Fig. 4A). This behavior is consistent with the field-tunable band gap of BLG, measured here for the first time in this frequency range. The calculated field dependence following the self-consistent Hartree model for screening of ref. <cit.> is shown in Figure 4B (dark blue line), in excellent agreement with the measured edge.


As the gap approaches the cavity frequency with further decreasing fields, the spectrum undergoes dramatic changes. Two dispersive peaks develop: a strong, narrow peak below and a weak, broad peak above the cavity frequency. These modes correspond to the lower (LP) and upper (UP) polaritons, forming around a deepening polariton gap (the blue dip in Fig. 4B), characteristic of a strongly coupled light-matter system.
The estimated coupling strength, defined by the ratio between the vacuum Rabi (Ω) and the cavity (ω) frequencies, is η = Ω/ω≈1THz/2.3THz=43%, which places our system well into the ultrastrong coupling regime (η>10% <cit.>). The large coupling in our system aligns with the predictions of ref. <cit.>. We qualitatively capture these observations by simulating a linear terahertz cavity coupled to a material characterized by a Lorentz oscillator with a tunable resonance frequency. Importantly, the Lorentz model can be used even in the apparent absence of bound states <cit.>. This model effectively reproduces the key experimental findings: the formation of asymmetric polariton branches and the emergence of a gap in the terahertz spectrum (Fig. 4C).






§ TEMPERATURE DEPENDENCE

Figure 5 presents the temperature dependence of the polariton spectra at 5 K (A), 20 K (B), 35 K (C), and 50 K (D). As temperature increases, we observe a decrease in the contrast of all spectral features, with the lower polariton and the polariton gap weakening (Fig. 5E). Notably, the upper polariton strength increases with temperature (Fig. 5F).
We attribute this behavior to non-radiative decay processes that become dominant at elevated temperatures, leading to line broadening. Interestingly, we do not observe a significant closing of the polariton gap at higher temperatures. This observation aligns with theoretical predictions that features of ultrastrong coupling persist even in the presence of strong dissipation <cit.>.




§ CONCLUSION

Our work demonstrates the successful integration of van der Waals materials with sub-wave-length terahertz cavities, achieving ultrastrong coupling with a remarkable strength of ≈40%. This accomplishment is twofold: first, it enables broadband terahertz spectroscopy of dual-gated and micron-sized materials, which was previously unobtainable with existing techniques. Second, it introduces a platform for creating ultrastrongly coupled light-matter quantum materials while taking advantage of the unique tunability and versatility of van der Waals heterostructures.


The observed coupling strength firmly places our system in the ultrastrong coupling regime, potentially leading to the realization of novel quantum phases and the ability to control material properties through cavity engineering. Future work could explore the impact of ultrastrong coupling on various quantum phenomena in van der Waals heterostructures, such as superconductivity, magnetism, and correlated states.


unsrt



 Paper Title: Transfer Learning for Multi-material Classification of Transition Metal Dichalcogenides with Atomic Force Microscopy.




An iterative transversal CNOT decoder
    Winfried K. Hensinger
    
=====================================




§ ABSTRACT 
Deep learning models are widely used for the data-driven design of materials based on atomic force microscopy (AFM) and other scanning probe microscopy. These tools enhance efficiency in inverse design and characterization of materials. However, limited and imbalanced experimental materials data typically available is a major challenge. Also important is the need to interpret trained models, which have typically been complex enough to be uninterpretable by humans. Here, we present a systemic evaluation of transfer learning strategies to accommodate low-data scenarios in materials synthesis and a model latent feature analysis to draw connections to the human-interpretable characteristics of the samples. Our models show accurate predictions in five classes of transition metal dichalcogenides (TMDs) (MoS2, WS2, WSe2,MoSe2, and Mo-WSe2) with up to 89% accuracy on held-out test samples. Analysis of the latent features reveals a correlation with physical characteristics such as grain density, DoG blob, and local variation. The transfer learning optimization modality and the exploration of the correlation between the latent and physical features provide important frameworks that can be applied to other classes of materials beyond TMDs to enhance the models' performance and explainability which can accelerate the inverse design of materials for technological applications.





  
Keywords: Transition metal dichalcogenides, atomic force microscopy, Computer vision, Model explanation, Transfer learning.



§ INTRODUCTION

Transition metal dichalcogenides (TMDs) are an important class of two-dimensional materials with interesting properties that promise great potential for applications in technological applications.<cit.> The materials are being explored for applications in areas including, energy,<cit.> optoelectronics,<cit.> and sensing.<cit.> 
Different characterization techniques are used to determine the quality and properties of the grown TMDs. Some of these focus on the determination of the chemical signature and characteristics (such as X-ray photoelectron spectroscopy (XPS), mass spectrometry, and Auger Electron Spectroscopy (AES)),<cit.> while others are related to the structural properties of the materials (such as X-ray diffraction (XRD), Raman spectroscopy, electron microscopy, and Atomic Force Microscopy (AFM)),<cit.> and still others on the physical properties (such as electrical measurements, mechanical testing, and optical spectroscopy).<cit.> Irrespective of the characterization method, the materials are a function of their chemical composition and the growth and after-growth conditions.<cit.> AFM, a scanning probe microscopy technique, provides topographic and various surface properties of materials at the nanoscale.<cit.> The AFM images, containing a high-dimension representation of the materials, are analyzed to determine the quality and properties of the grown materials. 


In data-driven design and exploration of materials, deep learning models (DL), including computer vision, have been widely used to develop models for the analysis and classification of different classes of materials based on AFM and other scanning probe microscopies.<cit.> DL and machine learning (ML) minimize the amount of expert time requires for materials characterization, analysis, detections, and classifications. Their deployment eradicates biases inherent in manual postprocessing of materials characterization data. They also provide tools that enhance efficiency and an avenue for high throughput growth and characterization of materials with predefined properties for technological applications.<cit.>

A major challenge in deploying DL in materials science is the large amount of data typically required in training the models.<cit.> While simulated data are being used in some studies,<cit.> the experimentally grown materials data are limited and pose a significant bottleneck to the data-driven design and exploration of materials, as the former at best is only an approximation to the latter. Besides the challenge of insufficient data, we also have the associated limitations of imbalanced training data. Materials with desirable properties are more likely to be grown in larger quantities than those without (Table <ref>). Also, it is common to have a material grown with parameters that result in the quality and property of interest, while less favorable parameters are rarely used.<cit.> 

Transfer learning is the use of knowledge gained in training a source, usually larger, data set in target data to improve the generalization of the model trained on the latter.<cit.> The weights of the model trained on the source data, pre-trained weights, are used, either by fine-tuning them on the target data or by using them to extract the features of the target data, and then training shallow or unsupervised models. It has been a key tool in addressing the challenge of limited data in DL for data-driven materials characterization.<cit.> For instance, VGG16 pre-trained on ImageNet has been used to extract features to classify microstructure images<cit.> and TEM images of carbon nanomaterials.<cit.> Similarly, semantic segmentation has been performed with diverse data sets using several computer vision architectures pre-trained on ImageNet and MicroNet.<cit.> More recently, we used the ResNet18 model pre-trained on ImageNet and another on MicroNet to perform feature extraction and fine-tuning methods for the classification of MoS2 AFM images based on their growth temperature.<cit.> Also, we trained segmentation and regression models based on ImageNet and MicroNet pre-trained weights to predict the mono-layer crystal coverage and crystal-substrate pixel-wise classification of WSe2 AFM samples.<cit.> Instead of random initialization of millions of weights that need to be trained on limited target data, models pre-trained on larger data that share some generic features with the target data can be deployed in transfer learning. A divergence between the source and target data, where common generic features are limited, could result in negative transfer learning.<cit.> The optimization of the transfer learning for best performance using the TMD-AFM data as a demonstration will significantly guide their use in materials science.

Beyond taking measures that ensure trained models perform optimally and can generalize to new data, the models' interpretation is an important consideration associated with DL in materials science. To use DL models to accelerate materials discovery through high-throughput materials design and characterization, it is important to derive as much utility from the trained models, as possible. It is therefore important to go beyond obtaining the predictions provided by the models to explore the basis and the hows of the model's decisions. For instance, images go through different layers of representation in convolutional neural network (CNN) models. Some features of a few dimensions could be obtained from the later layers, usually with fully connected nodes. A major utility is derivable if such features, known as the latent features, could be interrelated with the physical feature of the image. The growth parameters, for instance, could be related to the latent features and hence to the physical features which provide information for an inverse design of materials.

A successful classification of the TMDs based on their AFM images will require identifying latent features peculiar to the different classes, even if they are grown under the same conditions. In addition to the class of the TMDs,  image features related to the chemical constituent are of interest as the high throughput growth of the materials for on-demand technological application is sought. While models are trained to classify materials, a crucial utility that arises is the enablement to represent the image in a few latent dimensions. Latent space from the machine learning models enables a representation of the image in a very reduced dimension compared to the original multidimensional image. An important application derivable from such representation is an easy explanation of the AFM image in terms of the specific material properties, chemical composition, and the material's growth conditions. Additionally,  an interrelationship of the latent and the physical features and the performance metrics of the materials could be analyzed.

We present deep learning models trained on AFM images of five classes of metal-organic chemical deposition (MOCVD) grown TMDs. The materials were grown at the Penn State 2D Crystal Consortium (2DCC)<cit.> and hosted in the Lifetime Sample Tracking (LiST) database.<cit.> The TMDs included in our data are MoS2, WS2, WSe2, MoSe2, and Mo-WSe2. Two classification approaches based on target representation, multiclass (MtC) and multilabel (MtL), and different modes of transfer learning, were investigated to propose what is suitable for such analysis. We also proposed a modality for maximizing transfer learning utility in cases of imbalanced data and when models are trained on a few classes and an update is required to account for more classes than were previously included in the original models. A significant contribution in this study also includes the determination of the discriminative latent features used by the ML models in classifying materials and their correlation with physical features. Also, the processed 1026 AFM image data (termed TMD-AFM)<cit.> is publicly available to the scientific community.




§ METHODS




 §.§ Data Preparation




We have used AFM images of TMDs grown using MOCVD at the 2DCC of Penn State. The data is stored in the LiST database of the 2DCC. The data we have used consists of five different TMDs, MoS2, WS2, WSe2, MoSe2, and Mo-WSe2. The raw SPM files of the data were retrieved and processed to obtain the AFM images. The TMDs were grown with different parameters, including temperature, pressure, time, precursor flux, etc. Additionally, diverse imaging conditions were used to obtain the data. For instance, images were taken from the center and edges of the wafer and at varied resolutions. This results in multiple images per sample of the TMD grown. For consistency in the imaging conditions, we have mostly used images taken from the center of the wafer. Exceptions are the MoSe2 and Mo-WSe2 which have fewer samples, hence their images taken from the edges of the wafer were included. For all the TMDs, only 1  m× 1  m images taken at the center of the 5  m× 5  m AFM scans were included in our data.

The total data, consisting of 1026 AFM images of TMDs, were split into train, validation, and test sets (Table <ref>). Sample images of the consistent TMDs are shown in Figure <ref>. It was ensured that a given sample was assigned to only one of these splits, even though multiple images of each sample were present (i.e., splitting by sample ID). For instance, images from a given sample represented in the train set cannot be in the test set. This ensures unique samples are present in each data set to avoid data leakage. About 10% of the data was held out for testing, while 5-fold cross-validation was carried out using the remaining data.



 §.§ Supervised learning




The award-winning ResNet architectures <cit.> were used in our study since they provide a residual learning framework that eases the training of very deep networks. We used the implementation in  <cit.>, part of the  <cit.> software ecosystem. The ImageNet classifier was replaced with two fully connected layers. The number of nodes for the first layer is 2048 for ResNet50, 101, and 152, while ResNet18 and 34 have 512 nodes. The second layer for all the architectures has 100 nodes, followed by an output layer. A ReLU activation function and a dropout are between the two fully connected layers. The dropout, learning rate, and batch size were optimized using grid search. 

We first explore different modes of transfer learning, the fine-tuning where the whole pre-trained model was fine-tuned on our data, and feature extraction where features were extracted from the pre-trained model and used to train a multilayer perceptron (MLP), were explored. The smallest ResNet architecture, ResNet18, pre-trained on the ImageNet data was examined for the transfer learning. Additionally, as a baseline, the ResNet18 architecture was trained from scratch with randomly initialized weights. A  <cit.> implementation weighted random sampler was used to mitigate the imbalance in the training data.
The data were normalized using the mean and the standard deviation of the ImageNet data, while vertical and horizontal flips at 50% were applied to the training data on the fly. 

An interesting aspect of transfer learning that has not been fully explored in the materials science context is continual learning when introducing novel materials.
When considering material synthesis, wherein collections of elemental constituents can be combined in different combinations, the question naturally arises of whether new models should be created for each new material, or whether some prior knowledge can be leveraged and applied to the new material.
This leads to algorithmic decisions with both the learning itself as well as labeling schemes to use.
The obvious choices are multiclass (MtC), wherein each material is represented by a unique class label, and multilabel (MtL), wherein each material is represented by a collection of elemental labels.
These two strategies are illustrated in Figure <ref>.
We test these two strategies with the idea that MtC is more discerning between materials while potentially leveraging less of the prior knowledge from other materials, whereas MtL uses a common labeling scheme across a series of similar elemental compositions but may learn or retain spurious information as elements combine to form different crystal structures.

The loss functions used for the MtC and MtL classifications are CrossEntropy and BCE, respectively, while the Adam optimizer was used in both cases. 
To quantify the performance of our models while training, the F1 score, and the MSE were used as metrics to determine what model to save while training. After an epoch of training, the previously saved model (if any) is replaced with the current model if and only if the F1 score and MSE value on the validation improve compared to the previously saved model. The model was trained for 60 epochs. However, the best model saved is usually obtained before 40 epochs are reached, as no further improvement in the validation score is recorded beyond that.

To minimize bias in assessing the model performance, a 5-fold cross-validation was used in the training. In 5-fold cross-validation, the training set (train and val set in Table <ref>) is split into 5 folds. A fold is used for validation and the remaining 4 folds for training the model. Therefore, 5 models (with the same hyperparameters), with a unique fold serving as a validation set while the remaining for training, were trained and evaluated. 



 §.§ Model interpretation


To explain the trained model, class activation maps (CAM) of the different TMD classes are obtained. The CAM implementation by Zhou, et al<cit.> was used. After the pre-trained ResNet model is fine-tuned on our data, the feature maps of the last convolutional layer of the fine-tuned model are summed up and then normalized by dividing by the maximum value to obtain a heatmap with the same dimensions as the layer. The heatmaps are obtained for sample images from the different classes of the TMDs. The dark red region on the CAM has the highest activation for the class and, hence, is most significant for the classification.



 §.§ Latent space analysis


The interrelationship of the latent and physical image features was analyzed using Pearson's correlation coefficient and linear regressions implemented in the  library version 1.2.2<cit.>. In analyzing the latent space and examining how they are related to the image's physical features, 100 features of the images were extracted using the trained models. Principal component analysis (PCA) was then applied to the extracted features. PCA is a linear dimensionality reduction method that linearly transforms data into a new coordinate system such that the directions (principal components) represent the largest variance in the data.<cit.> The first 10 principal components, which account for 100% of the cumulative explained variance, were obtained. 

The material images were analyzed using several metrics, including microstructural features such as the grain size, grain density,  and Difference of Gaussian (DoG) blob,<cit.> and topographical and defect-related features such as the surface roughness, step edges, defect density, anisotropy, and local variation. The grains were determined using Otsu’s thresholding algorithm<cit.> implemented in  library. The algorithm works by maximizing the variance between two classes of pixels to determine the threshold separating the classes. The surface roughness is measured by determining the root-mean-squared error between the Gaussian smoothed and the original image. To determine the step edges, the Canny edge detection<cit.> was used, and the edges were enhanced using binary dilation and erosion. Defect density is measured using the local maximum of the Gaussian smoothed image. Anisotropy is the standard deviation of the angle of gradients per the mean of the magnitudes of the gradient. Local variation is the mean of the normalized absolute difference between the Gaussian smoothed and actual image, using the Gaussian filter from the Scipy library.<cit.> Others are the Gray-Level Co-occurrence Matrix (GLCM)<cit.> features such as contrast, dissimilarity, homogeneity, energy, correlation, and angular second moment (ASM) implemented in . Unless otherwise stated, the  version 0.23.2 was used for the image processing.<cit.> More details on the image analysis implementation parameters are provided in the supporting information.




§ RESULTS AND DISCUSSION




 §.§ Model Performance


We first observe the performance of our models using the fine-tuning and feature extraction approaches of transfer learning (Table <ref>). The optimum dropout, learning rate, and batch size values are 0.0, 7.5e-5, and 32, respectively, obtained by grid search. The fine-tuning yielded significantly better results than the feature extraction. While both approaches give better results than a random initialization of the weight, underscoring the importance of transfer learning, a major advantage of the fine-tuning over the feature extraction was associated with the on-the-fly augmentation benefited by the former.<cit.> The following analysis is therefore based on fine-tuning.



Different ResNet architectures with the different ImageNet pre-trained weights available (either V1 or V2, depending on the architectures in Table <ref>) were fine-tuned. This allows for comparing the different architectures and identifying the best one for our data.  The models were trained for 60 epochs, and the loss and F1 scores were evaluated after each epoch. The trained models were saved for every improvement in the loss and F1 score as the training proceeded. The final models obtained were those with the lowest loss and highest value of the F1 scores. Among the different architectures, the differences in model validation performances are minimal. Except for the ResNet101 which has the same performance with both the V1 and V2, the V2 pre-trained weights, which were obtained using a better training recipe<cit.> give better performance than the corresponding V1 weights. Interestingly, the difference between our results and the ImageNet validation accuracy is just about 2% on average. The V2 of ResNet152 was observed to give the best validation accuracy and F1 score (Table <ref>), in agreement with the ImageNet accuracy. This model was, therefore, selected for the subsequent analysis.




Having identified the ResNet architecture and pre-trained weights with the best performance, we compare the performance of our models with two different training modalities: MtC label and MtL (Figure <ref>). Similar results were obtained with the MtL having a minimal improvement over the MtC modality (Table <ref>). While the MtC classification gives validation and test accuracies of about 94, and 87%, respectively, the MtL gives 95 and 89%, respectively. The high value of the F1 score established the lack of biases of our models to more represented class(s) in the data. In addition to the better performance observed, the MtL modality has the advantage of learning the constituent atom and not just the class of TMD. Therefore, to use our trained models to examine the features associated with the TMDs constituent atoms, the MtL classification training was adopted. The analyses that follow are based on MtL unless otherwise specified.

The impressive performance of the models in classifying the materials is noteworthy. This excellent performance cuts across the different data sets. In particular, the accuracy and F1 score recorded for the test set, 89% and 0.90 respectively, which were never used in training nor in selecting the models, attests to the potential of transfer learning in general and to the specific approaches we have adopted in training the models. To gain insight into how the models perform for the different classes of TMDs, the confusion matrix is shown in Figure <ref>. The average number of TMDs in different classes in the held-out test set using the 5 models from the 5-fold cross-validation are shown.
Interestingly, the few misclassifications observed are between MoS2 and WS2. This phenomenon is observed in both the MtL and MtC classifications. A possible reason could be the presence of S in both materials and hence the similarity in the morphological features. The chalcogen in the other three TMDs is Se, with virtually non-existent misclassification among them and neither with the S-containing TMDs.



 §.§ Expanding Models for New TMDs


As observed in our data, a balanced data set where all the classes of materials are equally represented is a rare occurrence in materials science (Table <ref>). The weighted random sampling we have adopted has significantly mitigated the effect of the imbalanced data. However, the imbalance in the data could be more pronounced than we have in our current data set. It is therefore important to explore how best to use transfer learning in such instances. Additionally, it is important to understand how transfer learning could be best adopted when models have to be trained to accommodate additional materials. For instance, if we have a model trained on MoS2 and WS2, and additional TMD, say WSe2, are grown. One option is to train different models from scratch or use the pre-trained weights of the model previously trained on the first two materials. Therefore, we explore the two options to determine which to adopt in such a scenario.

For 3, 4, and 5 TMDs classifications, we observe the effect of training our model by fine-tuning the ImageNet weight every time (ImageNet) or using sequential training where the weight from the previous model obtained from fewer materials classifications (Sequential) is fine-tuned (Figure <ref>). The use of Sequential is better in generalization to the held-out data or is similar to ImageNet in some instances (Figure  <ref>). This shows that, with the additional class of TMDs, the weights from the fewer classes (Sequential) maximize the benefit of transfer learning in multi-material classifications. The difference could be very significant and holds for larger training volumes such as 124 in Figure <ref>((a)) where test set F1 score for ImageNet and Sequential is 0.82 and 0.89, respectively, and smaller training volumes such as 8 in Figure <ref>((b)) where test set F1 score for ImageNet and Sequential is 0.77 and 0.86, respectively.

The Sequential learning is examined for the MtC, and MtL target representations (Figure <ref> ((d), (e), and (f))). These give statistically indistinguishable results in most instances. However, for more materials like the five in Figure <ref>(f), MtL gives significantly better results with F1 scores of 0.92 versus 0.81 for MtC, for the 16 training volumes. This implies that MtL will likely perform better than MtC with increasing materials, particularly for smaller data volumes, which is useful in the context of new materials discovery.



 §.§ Model Explanation






The high accuracy on the out-of-sample test set data underscores the significance of the discriminative latent features that the models associate with the different classes of TMDs. We therefore want to understand how the model learns in the different transfer learning modes, ImageNet and Sequential. Additionally, we want to gain insight into the visually observable image features that correspond to the latent discriminative features that the models associated with the different classes of TMDs. Using the second fully connected layer of the trained model, 100 features were extracted from the images. The models trained on 2, 3, 4, and 5 classes of TMDs were used to extract the image features. The principal component analysis, PCA, was applied to the features to obtain the first few components accounting for most feature variance.  The data's latent space (first two components) is shown in Figure <ref>.

In the Sequential training mode, it is interesting to observe that the different classes of the TMDs occupy relatively the same region even as more classes of materials are added. For instance, in the 4 and 5 TMDs classifications (Figures <ref>(f) and (g)). The relative regions occupied by the MoS2, WS2, WSe2, and MoSe2 in both the models classifying the 4 TMDs and that for 5, including Mo-WSe2, are the same. The added fifth TMD in Figure <ref>(g) only slid in between the MoS2 and WSe2. The maintenance of the relative position by the different classes of the TMD, when more classes are added in Sequential, could indicate that the model retains the weights peculiar to classifying a set of TMDs even as it learns new weights for the added materials. This can lead to better generalization when more material classes are added.

It is also worth noting that the plots show distinct positions occupied by the different TMDs in the latent space. Interestingly, tremendous information in the AFM images could be captured in the 2D space so much that the materials classes could occupy a distinct position. Beyond the materials classes, it is observed that the models can associate a given atom with unique latent features irrespective of its present TMD. For instance, in Figure <ref>(d), a horizontal line could be drawn to separate the region containing TMDs with only S as the constituent chalcogen from the rest, with the former occupying the lower part. Similarly, a slightly rotated diagonal would separate the region with TMDs containing only Mo as the transition metal from others.


Beyond associating the different chemical compositions and TMDs to the unique regions of the 2D space, it is important to uncover potential physically meaningful features of the image that models use in distinguishing the classes of materials. To this end, the images are embedded in the latent space to show how the morphological features vary across the components and the materials (Figure <ref>). Small domain sizes can be associated with the images located in the first quadrant of the space mainly consisting of MoSe2. In contrast, the images in the third and fourth quadrants mostly have bigger sizes of domains. However, it is difficult to visually determine how the image features vary within the classes in the space. Similarly, class activation maps (CAM) show the domain boundary and multilayer region being activated for the TMD classes (Supporting Information). However, the activations are not as discriminative as similar regions are activated irrespective of class (Figure S1 to S10).




 §.§ Latent and Physical Features






Following the challenge of visually discerning the discriminative image properties that the models use in classifying the TMDs, a detailed search of a relationship between the latent features and quantified physically meaningful image features (physical features) was carried out. The material images were analyzed using several metrics, including microstructural features such as the grain size, grain density,  and Difference of Gaussian (DoG) blob, topographical and defect-related features such as the surface roughness, step edges, defect density, anisotropy, and local variation and the Gray-Level Co-occurrence Matrix (GLCM) features such as the contrast, dissimilarity, homogeneity, energy, correlation, and angular second moment (ASM) (Table <ref>). The magnitude of each physical feature was normalized by dividing them by the maximum values such that they all fall into the 0 to 1 range.

After quantifying the physical features, the Pearson correlation coefficient between each quantity and the dimensions of the image's latent features (PCA components of the features extracted using the trained model) was obtained. We determined which dimension of the latent feature has the greatest magnitude of the correlation coefficient with the physical features. The analysis shows the physical features with the highest correlation with any dimension of the latent features are grain density, local variation, and dissimilarity with positive correlation coefficients of 0.48, 0.50, and 0.48, respectively. The seventh dimension of the latent features correlates with the three physical features while minimal negative correlation is observed between the fifth dimension and grain density, the fourth and local variation, and the third and dissimilarity (Table <ref> and Figure <ref>).
For the quantities with the highest correlation, sample images with the high magnitude (75th percentile and upward) and those with the low values (25th percentile and less) are shown in Figure <ref>.

To determine how these properties vary across the classes of TMDs in the latent space, a scatter plot of the data points in dimension with the negative correlation and that with the positive correlations were obtained (Figure <ref>). Additionally, the quantities of the physical features were embedded in the latent space. As expected, unlike the first two components (shown in Figure <ref>), these components do not show a distinctive clustering of the different classes of the TMDs because of their relatively low explained variance. However, we could notice a group of MoS2 and WS2 have the highest values of the local variation and dissimilarity, and MoS2 with the highest grain density. 

To further explain how the latent and physical features are related, we went beyond the correlation to do a regression analysis between the features. As expected, this results in a correlation coefficient of about 0.7, up from the maximum value of 0.5 obtained from the Pearson correlation (Figure <ref>). Features with high correlation with the latent features include the DoG blob, local variation, and dissimilarity. 

The relationship among the regression direction of physical features is analyzed using the Cosine similarity (Figure <ref>). This analysis shows diverse directions and some groupings among the features. For instance, using local variation as the standard, properties such as grain density, DoG blob, and dissimilarity have a Cosine similarity of about 0.9, revealing almost identical regression direction. In contrast, properties such as step edges, anisotropy, and homogeneity show high dissimilarity (opposite direction) with the local variation. The embedding of our data in the representative regression directions similar and dissimilar to the local variation (dissimilarity and homogeneity) show features, for instance, local variation, dissimilarity, and grain density, with high and low values on the right upper, and left lower region, respectively (Figure S12). 
It is interesting to note that despite most of these properties exhibiting high correlation values with the latent features, their regression directions are diverse. This indicates that the model's discriminative latent features could be related to a fusion of several physical features.






§ CONCLUSION


We present deep learning models trained on AFM images of five classes of metal-organic chemical deposition (MOCVD) grown TMDs; MoS2, WS2, WSe2, MoSe2, and Mo-WSe2.
Classification based on two labeling schemes, multiclass (MtC) and multilabel (MtL), were investigated in the context of several transfer learning strategies.
This is especially salient in the case of continual learning, such as new materials being grown within a facility.
We analyzed transfer learning performance in cases of imbalanced data and when models are trained on a few classes, and an update is required to account for more classes than were previously included in the original models.
Finally, we also analyzed the discriminative latent features used by the ML models in classifying materials and their correlation with physical features. 

The fine-tuning and feature extraction methods of transfer learning were explored with the former yielding significantly better performance than the latter, mainly due to the benefit of the on-the-fly data augmentation. The two approaches give better results than a random initialization of the weights, underscoring the importance of transfer learning. We also examined ImageNet and Sequential transfer learning. In ImageNet, training entails the fine-tuning of the ImageNet weights, while Sequential training entails a sequential fine-tuning of the models trained on fewer classes. For instance, 3-material classification is based on fine-tuning the 2-material classification model (obtained by fine-tuning ImageNet's weights), and so on. The Sequential was observed to be better in generalization to the held-out data or is similar to ImageNet in some instances. Our analysis shows that, with the additional class of TMDs, the weights from the fewer classes (using a Sequential scheme) maximize the benefit of transfer learning in multi-material classifications.

Accurate out-of-sample predictions of the trained models are worth noting. The MtC and MtL both perform excellently, with the MtL showing minimal improvement over the MtC labeling. While the MtC classification gives validation and test accuracies of about 94 and 87%, respectively, the MtL gives 95 and 89%, respectively. The high value of the macro F1 score recorded and the confusion matrix analysis established the lack of biases of our models to more represented class(s) in the data.
This is an encouraging result, since each labeling scheme has advantages in different contexts, and our results suggest that either can be used effectively.

In explaining the trained models, we observed that distinct positions are occupied by the different TMDs in the 2D latent space. Interestingly, tremendous information in the AFM images could be captured in the 2D space so much that the materials classes could occupy a distinct position. Beyond the materials classes, it is observed that the models can associate a given atom with unique latent features, irrespective of its present TMD. Further analysis shows the physical features with the highest correlation with any dimension of the latent features are grain density, local variation, and dissimilarity, with positive correlation coefficients of 0.48, 0.50, and 0.48, respectively, and complementary negative correlations. Furthermore, regression analysis between the latent and physical features was also carried out. This results in a correlation coefficient of up to 0.66. The models' explanation analyses suggest that their latent features used to distinguish one material from others are interrelated to a fusion of several physical features. The transfer learning optimization modality and the exploration of the correlation between the models' latent features and the image's physical features provide important frameworks that can be applied to other classes of materials beyond TMDs to enhance the models' performance and explainability which can accelerate the inverse design of materials for technological applications.



§ ACKNOWLEDGMENTS

This study is based on research conducted at The Pennsylvania State University Two-Dimensional Crystal Consortium – Materials Innovation Platform (2DCC-MIP) supported by NSF cooperative agreement DMR-2039351.



§ DATA AVAILABILITY

The raw data required to reproduce these findings are available to download from Ref.
The processed data required to reproduce these findings are available to download from Ref.
The codes used to generate the reported results are available at https://github.com/reinhart-group/Transfer-Learning-for-Multi-material-Classification-of-Transition-Metal-Dichalcogenides.





 Paper Title: Synchrotron x-ray diffraction and DFT study of non-centrosymmetric EuRhGe3 under high pressure.






nsdhami@ifs.hr
Institute of Physics, Bijenička cesta 46, 10000, Zagreb, Croatia


Université Paris-Saclay, CNRS, Laboratoire de Physique des Solides, 91405 Orsay, France


Department of Physics, Faculty of Science, University of Zagreb, Bijenička 32, 10000 Zagreb, Croatia


Institute of Low Temperature and Structure Research, Polish Academy of Sciences, Okólna 2, 50-422 Wrocław, Poland


Institute of Low Temperature and Structure Research, Polish Academy of Sciences, Okólna 2, 50-422 Wrocław, Poland


Synchrotron SOLEIL, L’Orme des Merisiers,  91190 Saint-Aubin, France


Department of Earth Sciences, Department of Physics and Astronomy, University of Western Ontario, London, Ontario N6A-5B7, Canada


The Henryk Niewodniczański, Institute of Nuclear Physics, Polish Academy of Sciences, ul. Radzikowskiego 152, 31-342 Kraków, Poland


yutsumi@ifs.hr
Institute of Physics, Bijenička cesta 46, 10000, Zagreb, Croatia





§ ABSTRACT 
Antiferromagnetic intermetallic compound EuRhGe_3 crystalizes in a non-centrosymmetric BaNiSn_3-type (I4mm) structure. We studied its pressure-dependent crystal structure by using synchrotron powder x-ray diffraction at room temperature. Our results show a smooth contraction of the unit cell volume by applying pressure while preserving I4mm symmetry. No structural transition was observed up to 35 GPa. By the equation of state fitting analysis, the bulk modulus and its pressure derivative were determined to be 73 (1) GPa and 5.5 (2), respectively. Furthermore, similar to the isostructural EuCoGe_3, an anisotropic compression of a and c lattice parameters was observed. Our experimental results show a good agreement with the pressure-dependent structural evolution expected from theoretical calculations below 13 GPa. Reflecting a strong deviation from integer Eu valence, the experimental volume data appear to be smaller than those of DFT calculated values at higher pressures.


Synchrotron x-ray diffraction and DFT study of non-centrosymmetric EuRhGe_3 under high pressure
    Y. Utsumi
    August 5, 2024
===============================================================================================










§ INTRODUCTION


Rare-earth compounds in non-centrosymmetric BaNiSn_3-type structures have attracted considerable attention for their rich magnetic properties and pressure-induced superconductivity <cit.>. A series of Eu-based silicides/germanides EuTX_3 (T= transition metal, X=Si or Ge) have been discovered with the BaNiSn_3-type structure. In the crystallographic unit cell of EuTX_3 systems, Eu atoms occupy the 2a Wyckoff site, silicon/germanium atoms are located at two different Wyckoff positions 2a and 4b, while transition metal atoms occupy the 2a site <cit.>. Due to the lack of inversion symmetry in the crystal structure, the Dzyaloshinskii-Moriya antisymmetric spin interaction plays an active role and results in complex magnetic structures among the EuTX_3 series <cit.>.

EuRhGe_3 possesses magnetic Eu^2+ (4f^7, J=7/2) ions and exhibits antiferromagnetic ordering in the tetragonal ab-plane below T_ N= 11.3 K <cit.>. Temperature-dependent electrical resistivity of EuRhGe_3 shows a linear increase of T_ N by applying pressure up to 8 GPa <cit.>. Since the energy difference between Eu^2+ and nonmagnetic Eu^3+ (4f^6, J=0) is not very large <cit.>, and the ionic radius of Eu^3+ is ∼ 10% smaller than Eu^2+ ion, a pressure-induced Eu valence transition accompanied by a collapse of antiferromagnetic ordering and a volume change was expected. This phenomenon has been observed in ternary Eu-compounds with the ThCr_2Si_2-type (I4/mmm) structure <cit.>. However, the antiferromagnetic ordering and divalent Eu were revealed to be stable against pressure in EuRhGe_3. Recently, we performed high energy resolution fluorescence detected (HERFD) near-edge x-ray absorption spectroscopy (XAS) on EuRhGe_3 as a function of pressure. In the Eu L_3-edge XAS spectra, a prominent Eu^2+ peak was observed at ambient pressure. By increasing pressure, the spectral intensity shifted from Eu^2+ to Eu^3+ peaks. The obtained mean Eu valence from the Eu L_3 XAS spectrum exhibited a continuous increase from ∼ 2.1 at ambient pressure to ∼ 2.4 at 40 GPa without a first-order valence transition <cit.>. 

Pressure-dependent Eu valence and crystal structure were also studied in isostructural antiferromagnets EuCoGe_3 and EuNiGe_3. Their antiferromagnetic phases are stable against pressure <cit.>. Both exhibited a continuous contraction of their unit cell volume by applying pressure without any symmetry changes <cit.>. The mean Eu valence in EuCoGe_3 only changes from 2.2 at 2 GPa to ∼ 2.3 even around 50 GPa <cit.>. In EuNiGe_3, the mean Eu valence changes from 2.13 at 1 GPa to 2.43 at 48 GPa at 8 K <cit.>. As well as EuRhGe_3, the Eu valence does not reach Eu^3+ in both EuNiGe_3 and EuCoGe_3, even under such high pressure. These pressure-dependent changes of Eu valence in the EuTGe_3 series are rather small compared to ternary Eu-compounds with the ThCr_2Si_2-type structure (Eu122-systems), which often show a drastic pressure-induced Eu valence transition below 10 GPa <cit.>. For example, antiferromagnet EuRh_2Si_2 exhibits a pressure-induced valence transition at room temperature. By applying pressure, the mean Eu valence changes from Eu^2.2+ at ambient pressure to almost Eu^3+ at ∼8 GPa <cit.>. Such pressure-induced valence transition is also reported in antiferromagnet EuNi_2Ge_2, where the Eu valence changes from 2.2 at ambient pressure to almost Eu^3+ above 5 GPa at room temperature <cit.>. In the Eu122-systems, a correlation between the Eu valence and its unit cell volume tends to appear as a lattice volume collapse simultaneously happening with the Eu valence transition since the size of the Eu^3+ ion is about 10% smaller than the Eu^2+ ion <cit.>. 



The different pressure behavior in the EuTGe_3 series from the Eu122-systems raises questions about whether the variation of the Eu valence in the EuTGe_3 series correlates with the pressure evolution as well as their crystal structural changes. Unlike the intensively investigated Eu122-systems, pressure-dependent crystal structural studies of the EuTGe_3 series remain scarce. To elucidate a trend of pressure-dependent structural changes in the EuTGe_3 series and its relation to the Eu valence evolutions, a systematic study is needed. Here we performed synchrotron powder x-ray diffraction (XRD) as a function of pressure to study the pressure-dependent crystal structural change in EuRhGe_3, together with the theoretical predictions by density functional theory (DFT) method. The results are discussed in relation to the pressure evolution of the mean Eu valence obtained by HERFD near-edge XAS <cit.>.



§ EXPERIMENT

Single crystals of EuRhGe_3 were synthesized by the metal-flux method using liquid indium as solvent. The crystals were taken from the same batch reported in Ref.<cit.>. The grown crystals were characterized by XRD, electrical resistivity, and magnetic susceptibility measurement <cit.>.

The high-resolution XRD was performed at the PSICHE beamline of SOLEIL synchrotron with a photon energy of 33 keV (λ = 0.3738 Å). The single crystal of EuRhGe_3 was ground into a powder using mortar and pestle with ethanol. Fine grains of EuRhGe_3 afloat on the ethanol were collected by using a syringe and transferred onto a clean glass slide. After drying the powdered sample was loaded in a membrane diamond anvil cell (DAC). Diamonds of 300 μm diameter culet size were used. A rhenium gasket was pre-indented to about 28 μm thickness, and then a hole of 150 μm was drilled through it to serve as a sample chamber. Gold powder was also loaded in the DAC as a pressure reference material. Helium was used as a pressure-transmitting medium. Pressure was controlled by a membrane on the DAC and was determined by the Au equation of state <cit.>.

The pressure evolution of the crystal structure of EuRhGe_3
was also studied theoretically by using the Quantum ESPRESSO DFT package <cit.>. In DFT calculations, we have used pseudopotentials from pslibrary 1.0.0 <cit.>, with the Perdew–Burke–Ernzerhof exchange-correlation functional appropriate for solids <cit.>. The kinetic energy cutoff for wavefunctions was 150 Ry, while for the charge density and potential, it was 700 Ry.
The Brillouin sampling was 16x16x8 (no offset), with the Marzari–Vanderbilt Fermi surface smearing <cit.>. To take into account the antiferromagnetic ordering on Eu,
we have used the simplified formulation of DFT+U proposed by Dudarev <cit.>.
The Hubbard interaction U for Eu was assumed to be 3.8 eV to match the Eu 4f peak in the valence band spectrum by photoelectron spectroscopy measurement <cit.>. This U value was kept constant for all pressure calculations.



§ RESULTS AND DISCUSSION




We performed powder XRD on EuRhGe_3 as a function of pressure at room temperature up to 35 GPa. The contour map of diffraction intensities in the pressure range from 1 to 35 GPa is presented in Fig.<ref>. In order to highlight the pressure evolution of the peak positions, the contour map is plotted in the 2θ range from 5 to 19 . The pressure evolution of Bragg peak positions contains the main phase EuRhGe_3 and gold as pressure reference. When pressure exceeds 25 GPa, new peaks that belong to neither EuRhGe_3 nor gold emerge in the diffraction pattern. These new peaks were identified as rhenium from the gasket. 



We integrated the diffraction images by using Dioptas software <cit.> and performed Rietveld refinement for three phases by using the Profex program <cit.>. The Rietveld refinement fits at selected pressures are presented in Fig. <ref>. The widening of diffraction peaks with increasing pressure might reflect the presence of non-hydrostaticity. Another plausible reason for the peak widening is the bridging of the powder sample between the anvils. Although the DAC was prepared not to have an excessive amount of the sample in the gasket hole, a possible occurrence of bridging cannot be eliminated. Note that, helium pressure medium has the highest hydrostatic limit among other rare gases <cit.>, a development of uniaxial stress is reported above 30 GPa that largely depends on experiments <cit.>. The obtained lattice parameters and refinement parameters at selected pressures are shown in Table <ref>. The detailed atomic coordinates as a function of pressure are presented in Appendix B. Our analysis confirms that EuRhGe_3 maintains the same crystal symmetry (I4mm) up to 35 GPa. Note that the unit cell parameters a = 4.4129 Å, c = 10.0906 Å,  and unit cell volume V_0 = 196.50 Å^3 at ambient pressure are taken from Ref. <cit.>. 
 



Figure <ref> (a) shows the evolution of lattice parameters a- and c as a function of pressure. The lattice parameter a shows greater pressure change compared to c. Similar high-pressure behavior is also observed in EuCoGe_3 <cit.> and EuNiGe_3 <cit.>. We also performed high-pressure powder XRD of EuRhGe_3 by using a neon pressure transmitting medium . The pressure-dependent changes of a and c lattice parameters showed the same tendency (see supporting material), which suggests an intrinsic pressure behavior to the EuTGe_3 series and is independent from the pressure medium. Figure <ref> (b) shows the change of axial ratio (c/a) as a function of pressure. Pressure evolution of the c/a can be divided into two regions: low-pressure (LP) region 13 GPa and high-pressure (HP) region above 13 GPa. The linear fitting of the LP region provides 2.28+0.003P, while it provides 2.31+0.001P in the HP region. In the LP region, the value of the c/a linearly increases with pressure with a rate of 0.003 GPa^-1, then the rate decreases to 0.001 GPa^-1 in the HP region. This change in the c/a increase rate is ascribed to a change in compressibility along the a- and c-axis. The linear modulus of the c-axis increases considerably in the HP region, which makes the c-axis less compressible and changes the slope of c/a (see Appendix A). Among Eu122-systems, the change of c/a slope as a function of pressure was reported as a consequence of pressure-induced isostructural transition. Across such isostructural transition from tetragonal to so-called collapsed tetragonal phase, a-, c-lattice parameters and the unit cell volume greatly change along with the c/a slope <cit.>. Compared to the large structural changes in Eu122-systems, no discontinuous changes are observed in lattice parameters at the pressure range of the c/a slope change in EuRhGe_3. Moreover, the c/a slope change does not coincide with the pressure change of Eu valence in EuRhGe_3. The isostructural phase transitions in the Eu122-systems are accompanied by pressure-induced Eu valence transition from almost Eu^2+ to Eu^3+. However, in EuRhGe_3, the mean Eu valence linearly increases from ∼ 2.1 at ambient pressure to ∼ 2.4 around 25 GPa, then the increase rate becomes smaller and deviates from the linear behavior above 25 GPa <cit.>. Hence, we eliminate the possibility of pressure-induced isostructural transition in EuRhGe_3. It is worth mentioning that the change of the c/a slope as a function of pressure was also observed in EuCoGe_3 <cit.> and EuNiGe_3 <cit.> without any isostructural transitions.







Figure <ref> shows the pressure evolution of the unit cell volume of EuRhGe_3. It exhibits a smooth contraction with increasing  pressure. A slight change in the slope can be seen around 30 GPa (also in the lattice parameters of Fig.3 (a)). This slope change of the unit cell volume coincides with the slope change of the mean Eu valence as a function of pressure. The mean Eu valence of EuRhGe_3 obtained by HERFD XAS exhibited a linear increase by applying pressure up to 25 GPa and exhibited a smaller change at higher pressure <cit.>. Pressure-dependent unit cell volumes obtained by the DFT calculation are also plotted in Fig. 4. The theoretically calculated unit cell volumes exhibit a good agreement with experimental results in the LP region. However, the deviation from the experimental results increases in the HP region. The DFT-calculated unit cell volumes in the HP region tend to be larger than the experimental values. This is due to the change of the c-axis compressibility in the HP region, which is not captured by the DFT calculation. Since the DFT calculations cannot take into account the valence fluctuation, the Eu valence was considered to be Eu^2+ in the whole pressure range. In the LP range, the average Eu valence in EuRhGe_3 stayed close to Eu^2+ (2.1-2.2) <cit.> resulting in a reasonable agreement between the experiment and the DFT calculation. However, the mean Eu valence increases toward 2.4 by further increasing pressure. Considering the different ionic radius between Eu^2+ and Eu^3+ ( 10% smaller), it is somewhat understandable that the experimental unit cell volume appears to be smaller than the DFT-calculated values. In order to study the elastic properties of EuRhGe_3, we performed equation of state (EOS) fitting by using EOSFit7c software <cit.>. We used the 3^rd order Birch Murnaghan equation below for EOS fitting <cit.> :



    P(V)=  3B_0/2 [ (V_0/V )^7/3 -  (V_0/V )^5/3 ] {1+3/4(B_0'-4) [ (V_0/V )^2/3 -1 ] }



Here, B_0 and B_0^' denote the bulk modulus at 0 GPa and its first pressure derivative, respectively. We also performed EOS fitting on the lattice volume of EuRhGe_3 obtained by the DFT calculation. The obtained values from the EOS fitting are listed in the table <ref> and compared with the reported values of EuCoGe_3 and EuNiGe_3, as well as those obtained in the theoretical unit cell volume of EuRhGe_3.






§ CONCLUSION

We performed pressure-dependent synchrotron x-ray diffraction measurements on polycrystalline non-centrosymmetric EuRhGe_3 at room temperature. It revealed a smooth contraction of the lattice constants without any structural transition within the investigated pressure range. An anisotropic compressibility between the a- and c-axis was observed. The lattice parameter along the a-axis exhibits more sensitivity to pressure and greater contraction in comparison to that of the c-axis. The change of slope in the pressure-dependent c/a ratio was observed around 13 GPa, which does not seem to have a direct correspondence to the pressure evolution of the mean Eu valence in EuRhGe_3. By performing the EOS fitting on the obtained unit cell volume, we determined the bulk modulus and its pressure derivative of EuRhGe_3. Our experimental results show a good agreement with the pressure-dependent structural changes expected from the DFT calculations at P13 GPa, and slightly deviates in the HP region. The experimental unit cell volume tends to be smaller than the theoretical values in the HP region as a consequence of the increased mean Eu valence in the HP region. 




§ ACKNOWLEDGMENTS

The synchrotron powder XRD experiments under pressure were performed at the PSICHE beamline station (proposal Nos. 20191645 and 20220361). This work has been supported by the Croatian Science Foundation under project No. UIP-2019-04-2154, and in part under project No. IP-2020-02-9666. N.S.D. acknowledges financing from the Croatian Science Foundation under the “Young Researchers’ Career Development Project:” Project No. DOK-2018-09-9906. N.S.D. and Y.U. acknowledge support of the project from the Cryogenic Centre at the Institute of Physics—KaCIF (Grant No. KK.01.1.1.02.0012), co-financed by the Croatian Government and the European Union through the European Regional Development Fund—Competitiveness and Cohesion Operational Programme. Y. U. thanks Kai Chen for the helpful discussion and for sharing the experimental data of EuNiGe_3 high-pressure XRD.




apsrev
                

APPENDIX
	

§ APPENDIX

	
	




§ LINEAR EOS FITTING

As mentioned in the main text, the pressure evolution of the c/a ratio can be divided into the low-pressure (LP) region < 13 GPa and the high-pressure (HP) region above 13 GPa. We also performed a linear EOS fitting by using the EOSFit7c software <cit.>. The linear modulus is defined as: 

    M_i = -x_i  (∂ P/∂ x)_T
 Thus the relation between the linear modulus and the bulk modulus can be described as follows: 
    B = ( 1/M_1 +1/M_2+1/M_3)^-1
 

Here, M_1, M_2, and M_3 are linear moduli along the a-axis, b-axis, and c-axis respectively. For tetragonal symmetry M_1 and M_2 are the same. The lattice parameters along the a- and c-axis at 0 GPa (a = 4.4129 Å, c = 10.0906 Å) used for the linear EOS fitting were taken from Ref.<cit.>. The linear EOS fitting was performed separately in the LP and the HP regions. The obtained linear moduli along the a- and the c-axis both increase in the HP region. The linear modulus in the HP region especially increases  along the c-axis (See Table III).
The bulk modulus calculated by using the aforementioned formula is consistent with the bulk modulus extracted from the pressure-dependent unit cell volume in the main text. 





§ DFT-CALCULATED AND EXPERIMENTAL ATOMIC COORDINATES

Table IV presents the DFT-calculated atomic coordinates, and table V shows the atomic coordinates extracted from the Rietveld refinement of the XRD data. 













