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




These authors contributed equally to this work

These authors contributed equally to this work

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


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


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



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

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

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





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

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

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

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

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

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

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

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



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

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


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

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

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

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



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

    Γ_D=-2Dγ/μ_s(S·e_y)∇S,

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

    d=a_0Δ/αarctan[sinh(t/Q)],

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



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

    Γ_h=γ he_z×S.

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

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

    h_W=3πα D/8Δμ_s,

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

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

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



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






 Paper Title: Singlet-Triplet Kondo Effect in Blatter Radical Molecular Junctions: Zero-bias Anomalies and Magnetoresistance.








Singlet-Triplet Kondo Effect in Blatter Radical Molecular Junctions: Zero-bias Anomalies and Magnetoresistance 









 
Gautam Mitra^1,2, Jueting Zheng^1,3, Karen Schaefer^4, Michael Deffner^4,5, Jonathan Z. Low^6, Luis M. Campos^6, Carmen Herrmann^4,5, Theo A. Costi^7 and Elke Scheer^1





^1 Department of Physics, University of Konstanz, 78464 Konstanz, Germany

^2 Institute of Condensed Matter and Nanosciences, Université Catholique de Louvain (UCLouvain), 1348 Louvain-la-Neuve, Belgium

^3 State Key Laboratory of Physical Chemistry of Solid Surfaces, College of Chemistry
and Chemical Engineering, Xiamen University, 361005 Xiamen, China

^4 Department of Chemistry, University of Hamburg, 22761 Hamburg, Germany

^5The Hamburg Centre for Ultrafast Imaging, Hamburg 22761, Germany


^6 Department of Chemistry, Columbia University,  New York, NY 10027, USA

^7 Peter Grünberg Institut, Forschungszentrum Jülich, 52425 Jülich, Germany








§ ABSTRACT 
The Blatter radical has been suggested as a building block in future molecular spintronic devices due to its radical character and expected long-spin lifetime. However, whether and how the radical character manifests itself
in the charge transport and magnetotransport properties seems to depend on the environment or has not yet been studied. 
 Here, we investigate single-molecule junctions of the Blatter radical molecule in a mechanically controlled break junction device at low temperature. Differential conductance spectroscopy on individual junctions shows two types of zero-bias anomalies attributed to Kondo resonances revealing the ability to retain the open-shell nature of the radical molecule in a two-terminal device. Additionally, a high negative magnetoresistance is also observed in junctions without showing a zero-bias peak. We posit that the high negative magnetoresistance is due to the effect of a singlet-triplet Kondo effect under magnetic field originating from a double-quantum-dot system consisting of a Blatter radical molecule with a strong correlation to a second side-coupled molecule. Our findings not only provide the possibility of using the Blatter radical in a two-terminal system under cryogenic conditions but also reveal the magnetotransport properties emerging from different configurations of the molecule inside a junction. 








§ INTRODUCTION


Stable organic radical molecules are emerging to become key components for spin-based molecular electronic materials and devices <cit.>. They are open-shell systems which do not readily decompose or chemically react 
under standard conditions and whose stability can be engineered by virtue of chemical design <cit.>. The 1,2,4-benzotriazin-4-yl radical commonly known as the Blatter radical is one of such species, first reported in 1968 <cit.>, gaining interest owing to its many exciting physical and chemical properties <cit.>.  The stability of the Blatter radical under various circumstances enables to incorporate it into manifold electronic devices <cit.>, controlled polymerization setups<cit.>, photodetectors <cit.> and thermoelectric devices<cit.>. In addition, the presence of an unpaired electron in this  molecule can lead to interesting magnetic properties,  

when interfaced with metallic substrates and electrodes <cit.>.





Blatter radicals typically have the unpaired electron delocalized on the molecule (Figure <ref>a). This spatial delocalization of the unpaired electron gives rise to the exceptionally high stability of the radical. Recently, these systems have been subject to studies under different environments of metal-molecule interfaces and have shown to retain the open-shell nature in thin films <cit.>, and upon adsorption on the Au(111) surface under high vacuum <cit.>. Typically, Kondo spectroscopy is used as an evidence to interrogate the  molecular spin of the radical <cit.>. Previously, using scanning tunneling microscopy (STM) techniques under ultrahigh vacuum conditions, a Kondo resonance arising from the interaction between the unpaired electron of the Blatter radical with the conduction electrons was revealed by Repp and coworkers <cit.>. The singly occupied molecular orbital (SOMO) of the Blatter radical lies close to the Fermi level of Au and was suggested to be very sensitive to the environment around the molecule <cit.>. Little emphasis has been given to the study of Blatter junctions in a two-terminal metal-molecule-metal system mainly due to the suspected loss of the open-shell character upon oxidation, as reported in solution-based STM-break junction experiments <cit.>. Later, this issue was tackled by functionalizing the radical by adding an electron-withdrawing group, which lowers the energy levels inside the molecule giving more stability and preventing oxidation <cit.>. Recently, Jiang et al. explored this problem theoretically using first-principles quantum transport calculations <cit.>. Their calculations suggest that the Blatter radical can successfully retain its open-shell character in a junction between Au electrodes irrespective of the environmental conditions since the dative bonding at the molecule electrode interface does not alter the spin states of the radical. 




We posit that stable organic radical junctions can also manifest unconventional magnetotransport behavior in combination with spectroscopic evidence for the Kondo effect and that both can be tuned by the coupling strength to the electrodes. To the best of our knowledge there are no experimental magnetoresistance (MR) studies to date that focus on radicals with a delocalized unpaired electron in the current pathway using in-situ tunable, yet sufficiently stable electrodes. To better understand the intrinsic properties of open shell systems, here we carry out the first systematic transport study of the Blatter radical in a mechanically controlled break junction (MCBJ) operated at low temperature. We were able to produce Blatter molecule junctions with increased stability in a two-terminal device and the signature of the open-shell nature of the radical is confirmed using dI/dV spectroscopy.  The Blatter radical was synthesized with two methylthioether units (–SCH_3) to enable coupling with the Au electrodes <cit.>. We have deposited the molecule by using a dropcasting technique and carried out the electronic transport measurements of the individual junctions in cryogenic vacuum at a temperature of 4.2 K and under magnetic fields. Kondo resonances and magnetoresistance (MR) originating from stable organic radical molecule junctions have been studied with the MCBJ technique before <cit.>. But in all previous cases, the electronic lone-pair was located in a specific site of the molecule and was sterically protected.  We have found two types of Kondo resonances, presumably originating from  different configurations of the individual junctions of the Blatter radical molecule. One type of the observed Kondo resonances features a Kondo temperature (T_ K) of 30 - 35 K with a broad and asymmetric line shape and the other type has a T_ K of 11 - 17 K with a more symmetric and narrow line shape. We have confirmed the Kondo nature with the evolution of the zero bias peak as a function of the magnetic field. These exhibit only a very weak negative finite-bias MR, as predicted by the usual Kondo effect at finite-bias (see Figure S1 in SI). However, we have also observed a significant negative finite-bias MR in certain junctions without having a zero-bias peak at the measurement temperature of 4.2 K. We find a maximum change of 21% in resistance with an applied magnetic field of ± 8 T and a subsequent saturation of MR for higher fields. These results highlight the stability of the unpaired electron in the Blatter radical molecule junctions in low temperature measurements with a MCBJ device. Finally, the negative MR at finite bias is explained in the framework of a singlet-triplet Kondo model. We have illustrated that the presence of an extra side-coupled molecule in the Blatter junction with an antiferromagnetic exchange interaction between the unpaired spins on the two radicals results in a possible scenario of negative MR at finite bias voltage and at a temperature similar to the experiment. A summary of our results from different types of junction formation is schematically shown in Figure <ref>b. 




§ RESULTS AND DISCUSSIONS




 §.§ Statistical investigations


The Blatter radical molecule with two methylthioether units was synthesized and characterized as reported earlier <cit.>. The freshly prepared solution of the molecule is prepared in tetrahydrofuran (THF) and dropcast onto a gold (Au) MCBJ sample at room temperature in a nitrogen atmosphere and properly dried for 45 min. The sample is then characterized using a dipstick equipped with a MCBJ mechanism pumped to high vacuum and cooled down to 4.2 K inside a liquid-He bath cryostat. More details of sample preparation and measurement techniques are described in the Methods section.





First, we measure and analyze the opening traces of the Blatter radical molecule with an applied bias of 100 mV at 4.2 K. Figure <ref>a shows typical examples of these opening traces as a function of electrode displacement. Au single-atom contacts with 1G_0 are observed in all the opening curves, where G_0 = 2e^2/h is the conductance quantum. Below 1G_0, upon further stretching of the electrodes, conductance plateaus are formed corresponding to the single-molecule junctions of the Blatter radical molecule bridged using methylthioether anchoring units. These plateaus shows a maximum displacement up to 0.6 nm before breaking down to the tunneling regime, with conductance G below 10^-6G_0. In Figure <ref>b, we have constructed a conductance histogram with logarithmic binning of 595 opening traces without any data selection. We observed major conductance peaks between 10^-2 and 10^-3G_0. We attribute multiple peaks observed in this conductance range to different configurations of molecule junctions formed between the Au electrodes. Flat plateaus observed in the opening traces correspond to fully stretched junctions of individual molecules whilst partially connected junctions show fluctuations in conductance as the electrode displacement is increased. This behavior is also clearly observed in the two-dimensional conductance-distance histogram as shown in Figure <ref>c. We note that there is also the possibility of dimers with two parallel or side-coupled molecules which are supported by our quantum chemical calculations using density functional theory (DFT, see the Supporting Information (SI)), the main results of which can be summarized as follows: 
First, the  Blatter radicals investigated here bind substantially to each other, as reported earlier for similar species <cit.>. Second, the simulated zero-bias conductance values are in the range of 10^-2 to 10^-3 G_0, with the monomers (denoted as M) at the lower end of this range (in line with <cit.>, see SI). These values agree well with the experimentally observed range. Knowing that DFT has the tendency to overestimate the conductance by up to around an order of magnitude <cit.>, it appears likely that the structures giving rise to higher conductance values in the simulations, hence dimers (denoted as D), correspond to the ones realized in the experiments. This interpretation is in line with the relatively strong binding between the radicals. We will come back to this aspect further below.



The observed conductance peak in the histogram is also in close agreement with the previous studies of the same molecule on a solution-based scanning electron microscopy break junction (STM-BJ) experiment <cit.> and STM studies on thin films of the Blatter radical on a Au surface at room temperature <cit.>. The small conductance obtained from the Blatter radical molecule junctions were very recently tackled theoretically by Jiang et al. using first-principles quantum transport calculations <cit.>. They attribute their unexpectedly low conductance to singly occupied frontier orbitals (SOMO and SUMO, respectively) being weakly electronically coupled to the Au electrodes. Hence, the Blatter radical may retain its unpaired spin upon contacting with Au electrodes in more instances than previously thought. In order to verify this hypothesis, we have carried out several magnetotransport measurements on individual junctions as explained below. 





 §.§ Transport spectroscopy in individual junctions: Kondo resonances


Differential conductance spectroscopy is performed in order to understand the properties of individual Blatter radical molecule junctions in detail.  We discuss the dI/dV characteristics observed in different junctions which are measured using a lock-in amplifier technique 
with 0.1 mV AC amplitude.
These measurements reveal a clear zero-bias anomaly indicating an electronic resonance formed near the Fermi level for about 20% of single-molecule junctions measured in our study. We have observed that the line width and the line shape of this resonance varies among different junctions under the same measurement conditions. Figures <ref>a and <ref>b show examples of two junctions exhibiting different dI/dV spectra. Junction M1 (<ref>a) shows a broad and asymmetric line shape and junction M2 (<ref>b) shows a narrower and symmetric resonance. We attribute the origin of these zero-bias anomalies to Kondo resonances due to the presence of an unpaired electron in the current pathway, hence indicating that the open-shell radical nature of the molecule within the two-terminal device is maintained at low temperature. Furthermore we argue that these junctions are monomer junctions, as we will discuss below.





For the junction M1, in Fig. <ref>a the zero-bias peak resembles a Fano resonance line shape as observed in many previous mesoscopic and nanoelectronics devices at low temperature <cit.>. This type of spectrum was also previously reported with STM-based studies of the Blatter radical molecule <cit.>. A Fano line shape typically occurs when there is an interference between resonant transport channels and non-resonant direct tunneling pathways through the device under study <cit.>. Here, these are the transport assisted through the SOMO due to an unpaired spin, interfering with the continuum tunneling channels. The first pathway is responsible for the Kondo resonance while the latter corresponds to the broad conductance background. 
This can be described by the commonly used equation <ref>, 


    dI/dV = α(q+ϵ)^2/1+ϵ^2 + β; ϵ = eV-E_ K/Γ_ K


where the Fano parameter, q, defines the ratio of transmission amplitudes through the resonant and non-resonant tunneling channels and hence, the peak asymmetry. E_ K is the energy of the resonant state and Γ_ K gives the halfwidth of the resonance, which is a function of temperature <cit.>. Since the experiment is at k_ BT≪Γ_ K (verified a posteriori by our fits), Γ_ K is the zero-temperature halfwidth of the resonance. The constants α and β correspond to the peak and background amplitudes, V is the bias voltage, and e is the elementary charge. The above, however, assumes a Lorentzian lineshape for the Kondo resonance. In reality, the Kondo resonance is known from accurate numerical renormalization group (NRG) calculations to have a non-Lorentzian lineshape, and can only be approximated by a Lorentzian at very low energies |eV|≪ k_ BT_ K <cit.>. In particular, the Kondo resonance has logarithmic tails decaying as 1/ln^2(|eV|/k_ BT_ K) at |eV|≫ k_ BT_ K<cit.>. It is possible to use the essentially exact numerical NRG spectra, together with the Fano effect, to accurately describe Kondo resonance lineshapes in dI/dV <cit.>. Here, we use an approximate analytic approach, the modified
Frota function <cit.>, which simulates well the non-Lorentzian lineshape of the Kondo resonance in the NRG spectra, while including also the Fano effect,

    dI/dV     = b -c  Im[i e^iϕ√(iΓ_ F/E-E_ K+iΓ_ F)],
 
where Γ_ F=Γ_ K/2.542 and ϕ=ϕ(q) is related to the Fano parameter q (see SI and Fig. S10). The constants b and c describe the background and Kondo peak amplitude, respectively. Note that both the Fano and modified Frota functions can describe any type of measured resonance, ranging from a symmetric, asymmetric, antisymmetric or even an antiresonance, depending on the Fano parameter q (or ϕ(q)), see SI. A comparison between the Lorentzian based procedure in Eq. (<ref>) and the more accurate procedure in Eq. (<ref>) for Kondo resonances is given in the SI, together with a table of all parameters extracted from fitting junctions M1 and M2 to Eq. (<ref>).


For the junction shown in Fig. <ref>a we find the energy width Γ_ K = 4.5 mV. We can extract the Kondo temperature, using the result k_ BT_ K=2 Γ_ K/π <cit.>, and found it to be approximately 34 K. This is in good agreement with the values obtained in previous STM-based studies <cit.>. For junction M2, with Γ_ K= 2.2 mV, we find T_ K  to be approximately 17 K.


 The Kondo conductance and the lineshape depends on the strength of the coupling between the molecule and the electrodes and hence on the bonding sites of the molecule with the electrodes <cit.>. Hence, we argue that the resonances in cases M1 and M2 originate from different configurations of the molecular junction. Since we have observed both flat and slanted opening traces in our study, the formation of single-molecule junctions with different configurations giving rise to different T_ K is a likely scenario. For Blatter radicals, it is known that the spin density is mostly centered around the fused benzene ring, benzotriazinyl core and is partially delocalized to one of the phenyl rings attached with the thiomethyl groups <cit.>. The third aromatic ring is spin-isolated. Hence, the difference in electronic coupling with both ends of the electrodes can lead to varying coupling strengths and transport pathways. Consequently, the junctions with metal electrodes having a direct interaction with one of the higher spin density sites can display strong zero-bias resonances. We analyzed many individual junctions with conductances between 10^-2 and 10^-3 G_0 but found no clear correlation with T_ K and G while 70% of them show asymmetric resonance with a T_ K of more than 30 K (type M1). An in-depth theoretical calculation is necessary to distinguish between different Kondo states and junction structures.
 


To further validate the presence of the Kondo resonance and the extracted 
Γ_ K (and T_ K=2Γ_ K/π k_ B) in these junctions, we have recorded the evolution of dI/dV spectra at various magnetic fields from 0 to 10 T at 4.2 K.  A magnetic field typically splits the Kondo resonance due to the Zeeman effect. Figures <ref>c and <ref>d show the corresponding spectra for junction M1 and junction M2, respectively. We have observed no splitting or suppression of the zero-bias peak in junction M1 while a clear splitting of the peak is visible for junction M2. Theoretical predictions suggest that this splitting follows the relation, gμ_BB_c ≈Γ_ K/2 <cit.>. For the junction M2, the zero-bias peak shows a clear splitting beyond the splitting field B_c of ≈ 7 T compared to the theoretical prediction 9.9 T using the extracted Γ_ K=2.29 mV. We do not observe any changes with increasing magnetic field until 10 T for junction M1, in agreement with the estimation from theory (using the extracted Γ_ K=4.57 mV) that the splitting in this case should occur only after the ≈ 20 T which is above our experimental limit. Further examples for the evolution of dI/dV spectra under magnetic field for both types of junctions is given in the SI. We have found that the experimental field dependence of the dI/dV for all the measured junctions shows a remarkable agreement with the theoretical predictions <cit.>, showing a splitting close to the predicted B_c when this lies in the experimentally accessible field range and no splitting when B_c is predicted to lie outside this range. 






 §.§ Negative magnetoresistance and singlet-triplet Kondo effect

  It was reported earlier that for single-radical molecular junctions without showing any Kondo features, a very pronounced MR can be observed at larger bias <cit.>. MR is defined here as the relative change in resistance of a molecule junction at an applied magnetic field with respect to zero field. In order to study the magnetotransport on molecular junctions which do not show any zero-bias peak in our measurements (denoted as type D junctions), we have studied the evolution of the resistance under magnetic field applied perpendicular to the sample plane from -10 to 10 T (red) and 10 to -10 T (black) at a rate of 50 mT/min and measured with a constant bias voltage of 30 mV at 4.2 K.  Figure <ref> shows MR curves obtained from three such type D junctions.
  

We have observed a strong negative MR (i.e. decrease of resistance with increasing field strength) with a maximum change of ≈ 21%. Besides a small hysteresis we observed no clear change in MR amplitude with the sweep direction. MR curves tend to saturate after approximately ± 8 T. We note that in previous studies of radical molecule junctions, this kind of saturation or maxima/minima of the resistance were also observed, but at different field value. We also found that all measured Blatter radical single-molecule junctions show exclusively negative MR in our study. 
In addition, MR measured on type M1 or M2 junctions show only a very small change in resistance of less than 2%, as given in the SI, hence revealing an anti-correlation between Kondo behavior and strong MR. This behavior was first explored in junctions of the perchlortrityl radical molecule <cit.> and is now confirmed for Blatter junctions. 

  
  

  
  The origin and mechanism of large MR in molecule junctions are still under debate. Various mechanisms have been reported based on the studies of different organic radical molecules with regard to the exact structure of the molecule and anchoring groups. Most of these mechanisms rely heavily on the influence of molecule-electrode interface scattering and changes in the electronic coupling strength <cit.>. In our study, we used methylthioether anchoring units to couple the Blatter radical to the Au electrodes. The previous study on the perchlortrityl radical molecule using the same anchoring groups yielded both positive and negative MR that was assigned to spin-dependent scattering at the molecule-electrode interfaces and resulting in a quantum interference effect <cit.>. In the case of Blatter molecule junctions, we have observed exclusively negative MR and the theoretically reported weak electronic coupling of SOMO and SUMO orbitals with gold electrodes suggests the above MR mechanisms may not be prominent. 

  




  
  Hence, we propose a possible explanation of the significant negative MR of type D junctions assuming the junctions actually consist of a double quantum dot system: a Blatter molecule in the junction side-coupled to a second 
  quantum dot, also having a S = 1/2, resulting in an antiferromagnetic exchange interaction of strength J between the two (see Eq. (<ref>) in Theoretical Methods and the SI). The second quantum dot could be, e.g., a gold atom or a cluster with unpaired spin, a molecular fragment, or a second Blatter radical. As mentioned above, the existence of  structures in which a second Blatter radical couples with the first is supported by our quantum chemical calculations (see SI). Furthermore, we found most such pairs to preferentially couple antiferromagnetically, with a coupling constant J' (for a Hamiltonian of the form H = -2 J' S_a S_b) in the range of around -90 to -140 cm^-1, which translates in our Kondo dimer model notation to antiferromagnetic couplings J=-2J' of order 20-40 meV, i.e., 200-400 K.  Examples of the optimized structures of the possible dimer (D) junctions, the resulting transmission functions and the information about energies and spin population are given in the SI. 
  Weak ferromagnetic coupling could also occur for some junctions based on our DFT calculations, but could be pushed to more antiferromagnetic by binding to the electrodes. The latter possibility is also interesting as the D system would form a S=1 which would result in an underscreened Kondo effect <cit.>.


For antiferromagnetic couplings J, the double-dot system would exhibit the so called singlet-triplet Kondo effect, which would provide a mechanism for understanding the observed negative MR in the absence of a zero-bias peak in dI/dV. 

Organic radical dimers, exhibiting a field-induced singlet-triplet Kondo effect have been investigated previously, however, the MR was not discussed <cit.>. Within this picture, see Fig. <ref>a, the ground state of the isolated dimer at B = 0 is a singlet S = 0 state with a triplet S = 1 state lying at an energy J above it. Upon coupling the dimer to the leads, the Kondo effect at low energies, temperatures and voltages (E, k_ BT, eV ≪ J) is therefore suppressed and a zero-bias peak in dI/dV is absent for eV < J. Instead, the differential conductance exhibits a gap of order 2J, which at high temperatures k_ BT>J could be smeared out (see Fig. S7a in SI). Despite the non-magnetic ground state, the system is still strongly affected by a magnetic field via the effect of the latter on the polarizable triplet state: the magnetic field splits the triplet state and decreases the splitting δ(B) = E_S=1,|S_z|=1 - E_S=0,S_z=0 between the lowest component of the triplet state (S_z = +1 for B > 0 or S_z = -1 for B < 0) and the singlet state as shown in Figure <ref>a. This splitting eventually vanishes at a sufficiently large magnetic field B = B^* where B^* is of the order of J/gμ_B (B^* = J/gμ_B exactly when the molecule plus side-coupled quantum dot is detached from the leads, but is slightly reduced when it is attached to the leads <cit.>). Fluctuations between the degenerate states at B = B^* lead to a fully developed singlet-triplet Kondo effect via the hybridization of these states to the leads <cit.>. In this scenario, increasing B from zero towards B^* induces the singlet-triplet Kondo effect and thereby enhances the conductance G(B)=dI/dV, implying that G(B = 0) - G(B) is negative (i.e., positive magnetoconductance), resulting in a negative MR for the field range -J < gμ_BB < +J (see Fig. <ref>b and Fig. S8). This is opposite to the usual S = 1/2 Kondo effect where a magnetic field reduces the conductance (resulting in a positive MR, see Fig. S9 in the SI).  

  

  We now illustrate the generic MR behavior due to the singlet-triplet Kondo effect via numerical renormalization group calculations (see Theoretical Methods and SI). Two energy scales determine the quantitative aspects of the 
  singlet-triplet Kondo effect, the antiferromagnetic dimer coupling J and the Kondo scale T_ K
  in the absence of the side-coupled quantum dot in (<ref>). The former (J) is determined by the coupling strength between the two Blatter molecules, while the latter (T_ K) depends on the tunnel couplings of the Blatter molecule in the junction to the electrodes. 
  However, even with reasonable estimates for J, the Kondo scale T_ K remains largely unknown for the experimental system: while theoretically, T_ K is easily obtained by switching off J in our model, this is clearly not possible for the experimental system (one cannot remove the side-coupled Blatter molecule in order to extract the T_ K for the monomer).  Thus, we consider a wide range of J/k_ BT_ K = 100, 30, and 10 to illustrate the overall qualitative results for the MR. This uncertainty in T_ K
  for the dimer systems, also means that T/T_ K is unknown. We shall use T/T_ K=0.4 with T_ K=10 K for an experimental temperature T=4 K as a typical value, but it should be noted that the MR depends sensitively on the precise value of T/T_ K (see SI). The bias-voltage dependence of the MR will also be calculated. 

  
  Figure <ref>b shows the calculated MR at several dimensionless voltages eV/k_BT_ K. For all voltages, the MR exhibits a similar dependence on the magnetic field as in the experiment. It is largest at V = 0. Finite V introduces decoherence, leading to a suppression of the MR. Nevertheless, even at eV/k_ BT_ K = 30, the MR reaches 10% at gμ_BB/k_ BT_ K = 10. The magnitude of the MR is sensitive to the precise J/k_ BT_ K and T/T_ K (see Fig. S8 for J/k_ BT_ K=30 and T/T_ K=10). The effect of a spin-exchange anisotropy (J_∥/J_⊥ = 2 and J_∥/J_⊥ = 1/2) is shown for the largest bias voltage (eV/k_ KT_ K = 30) and found to be small. Its effect is even smaller for smaller bias voltages. In general, for eV < J its effect is to enhance the magnitude of the MR. The effect of the Dzyaloshinskii-Moriya (DM) interaction, shown in Fig. <ref>b for two bias voltages (dotted lines) and a large D/J=0.5, is also to enhance the magnitude of the MR, while not affecting its overall B-dependence (for smaller D/J=0.2, the enhancement is much smaller, see Fig. S8a). The enhancement in MR ≈ 1-G(V, 0)/G(V, B) is due to the reduction of G(V, B) in the presence of a finite Dzyaloshinskii vector (D), relative to its value for D = 0 (since, a finite D enhances the pseudo-magnetic field in the singlet-triplet Kondo effect, thereby reducing G as discussed in <cit.>).  For comparison, we briefly also show the results for the temperature and bias-voltage dependence of a single Kondo-correlated quantum dot described by the Anderson impurity model in the SI, which is also in clear agreement with the previously reported studies <cit.>. 
  
  
  

 §.§ Conclusions

 To summarize, we have developed single-molecule junctions of the Blatter radical molecule using a mechanically controlled break junction setup at low temperature in a cryogenic vacuum environment. 
 
 Our detailed magnetotransport measurements suggest that the open-shell nature of the Blatter junction remains intact in a two-terminal junction.
 Differential-conductance spectroscopy studies on individual contacts reveal two different types of zero-bias anomalies originating from the different configurations of the metal-molecule-metal junction. They are attributed to a Kondo resonance arising from the delocalized unpaired electron orbital situated in the current pathway. In addition, Blatter radical molecular junctions without a zero-bias peak showed strong negative magnetoresistance (MR) with a maximum change of 21% in resistance measured at high bias voltage. We modeled the origin of these high negative MR towards junctions involving a Blatter molecule with strong correlation with a side-coupled quantum dot configuration which results in a singlet-triplet Kondo effect. The detailed quantum chemical calculations show qualitative agreement between the experiment and theoretical predictions. Our findings reconcile earlier, seemingly contradictory findings on Blatter radical molecular junctions studied in different environments and will open up several new experimental and theoretical avenues to revisit the transport and magnetotransport properties of other radical molecular junctions. They also pave the way for studying various fundamental aspects of the Blatter radical and its derivatives to tailor their properties for spintronic applications.
  
  
  

 §.§ Methods





 §.§ Experimental Methods


Device Fabrication. 
The break junction devices were fabricated on a 500 μm thick, polished insulating Cirlex (Kapton laminate) substrate.  2 μm polyimide sacrificial layer was spin-coated on the substrate, baked at 130°C for 5 min, then hard-baked at 430°C for 90 min under vacuum. Next, a double-layer resist consisting of methylmetacrylat-methacrylat acida/polymethylmethacrylat (MMA-MAA/PMMA) is spincoated on top and baked in an oven at 170^∘C for 30 min. Electron beam lithography was performed on these substrates of size 3 × 18 mm^2 in a Zeiss Cross Beam machine at 10 kV acceleration voltage and are developed using 1:3 methyl isobutylketon: isopropylalcohol (MIBK:IPA) solution for 30 s and rinsed with pure IPA. 80 nm of Au were later deposited using the electron beam evaporation. The samples were then etched using anisotropic reactive ion etching in a mixture of oxygen and sulfur hexafluoride at 1 mbar with a power of 50 W for 30 min. This process removes about 600 nm of the polyimide layer below the electrodes.  We prepared a 0.2 mM solution of Blatter molecules in tetrahydrofuran (THF) and then dropcast it onto a freshly prepared MCBJ substrate in nitrogen atmosphere at room temperature and dried for ≈ 1 hour. 


Transport Measurements. All experimental measurements were carried out using a custom made cryogenic vacuum dipstick equipped with a MCBJ at 4.2 K in a He bath cryostat. The sample is shielded using a copper cap to prevent stray electromagnetic fields. The electrode displacement is calibrated by analyzing the opening curves of a bare Au break junction using the tunneling conductance expression, G(δ x) ∝ exp (-2 r δ x √(2mϕ)/ħ) where ϕ is the work function of Au approximated as 5.0 eV, m is the electron mass, δ x is the pushing rod movement of the break junction setup, and the reduction ratio r has the relation with electrode displacement, δ d = r δ x <cit.>. We found a reduction ratio r = 3.6 × 10^-5 which we use to calculate the electrode displacement. At base temperature (4.2 K), sample is repeatedly opened and closed to form atomically sharp Au electrodes and this method allows to form molecular junctions at low temperature <cit.>.  Electronic transport measurements were carried out using a Yokogawa 7651 as a DC voltage source. The wiring is composed of homemade coaxial cables and SMA connectors. Current and voltage across the sample are recorded using Femto DLCPA and Femto DLPVA low noise amplifiers and measured with the help of Agilent 34410A multimeters. dI/dV measurements were carried out using HF2LI Zurich Instruments lock-in amplifiers by applying an AC modulation voltage of 0.1 mV  at 317 Hz. Magnetotransport measurements were performed with an applied magnetic field up to ± 10 T perpendicular to the sample plane produced from a superconducting magnet inside the liquid-He dewar.  All measurements and devices were remotely controlled using a custom Python program. Control experiments utilizing helium exchange gas for thermalization confirmed that eddy-current heating effects were negligible in our experiments. 
  



 §.§ Theoretical Methods
  
Singlet-triplet model.
  We model the  singlet-triplet Kondo effect by the Anderson impurity model side-coupled to a second S=1/2 quantum dot via an antiferromagnetic exchange interaction of strength J. The Hamiltonian is given by

    H    =H_ dot+H_ leads+ H_ tunneling+H_ J+H_ B.

The first term, H_ dot=∑_σε_0d_σ^†d_σ +Un_d↑n_↓, describes the dot Hamiltonian, representing the
S=1/2 degrees of freedom of the Blatter molecule,
where ε_0 is the level energy, measured relative to  the Fermi level E_ F, n_dσ is the occupation number for spin σ=↑,↓ electrons on the dot, and U is the local Coulomb repulsion 
on the dot. The second term describes the Hamiltonian of the leads and is given by H_ leads=∑_kασϵ_kc_kασ^†c_kασ, where α=L,R labels the two leads, and ϵ_kσ is the kinetic energy of the lead electrons.  The third term, H_ tunneling=∑_kασt_α(c_kασ^†d_σ +H.c.), describes the tunneling of electrons from the leads onto and off the dot with tunneling amplitudes t_α and tunneling rates Γ_α=πρ_ Ft_α^2, with ρ_ F the lead electron density of states at the Fermi level. The term H_ J=J  s_d· S describes an antiferromagnetic (J>0) coupling between the spin s_d of the Blatter molecule and the spin S of the side-coupled quantum dot. Finally, H_ B=-gμ_ BBS_z^ tot describes a magnetic field acting on the combined spin of the quantum dot and side-coupled quantum dot via the total z-component S_z^ tot=s_z,d+S_z. The above is a minimal model for the singlet-triplet Kondo effect <cit.>. It can be generalized to include the effects of anisotropic spin exchange J  s_d· S→J_⊥/2(s_d^+S^-+s_d^-S^+)+J_∥s_d,zS_z, and spin-orbit coupling via a Dzyaloshinskii-Moriya (DM) term H_DM= D· s_d× S. The former (anisotropic spin exchange) does not significantly alter the MR predictions of the isotropic model. The main effect of the latter (DM) is to lift the degeneracy of singlet and triplet states at B=B^*, resulting in an avoided crossing at this field value <cit.>, see Fig. <ref>(a), and therefore a reduced conductance for magnetic fields close to B=B^* <cit.>. Hence, the  largest effect of the DM term will be in the magnitude of the MR in the vicinity of B=B^*≈ J.  For D/J≪ 1, the overall qualitative dependence of the MR on field is close to that of the pure singlet-triplet Kondo effect (isotropic J and D=0), see Fig. <ref>(b).


The total dot-lead tunneling rate is denoted by Γ=Γ_ L+Γ_ R. For a strongly correlated quantum dot we have that U/Γ≫ 1. We shall use U/Γ=24 for the calculations, a typical value expected for molecular junctions. For example, estimates of Γ from the experiment yielded for the Blatter molecular junction 50-60 meV. For U/Γ=24 this implies that U=1.2-1.4 eV, which is a reasonable value for a single-molecule device. The relevant energy scales of the above model are J and k_ BT_ K, where T_ K is the Kondo temperature for the Anderson impurity model (J=D=0). More precisely, our T_ K is defined as k_ BT_ K= (gμ_ B)^2/4χ(0), where χ(0) is the zero-temperature spin susceptibility of the Anderson model. This T_ K is close to that of the well known Haldane expression T_ K^ H=√(Γ U/2)exp(πε_0(ε_0+U)/2Γ U) <cit.>. See SI for other definitions of Kondo scales used in the literature and their interrelationships.
Calculations will be presented for several values of the dimensionless ratio J/k_ BT_ K ranging from 1 to 100.


Numerical renormalization group method.
The model (<ref>) is diagonalized to obtain its many-body eigenstates and eigenvalues via the numerical renormalization group method <cit.>. Briefly stated, this approach consists of three steps: (i) the conduction electron kinetic energy of the leads ∑_kασϵ_k c_kασ^†c_kασ is logarithmically discretized about the Fermi level E_ F, taken as zero of energy, ϵ_k →ϵ_k_n=± DΛ^-n, n= 0,1,…, with D the half-bandwidth of the leads and Λ>1 the logarithmic discretization parameter, (ii), the discretized leads are then transformed to a (tridiagonal) linear chain form using a Lanczos procedure, and, (iii), the resulting linear chain form of (<ref>) is iteratively diagonalized on successively lower energy scales to obtain the eigenvalues E_p=1,2,…^n and eigenstates |p⟩_n on all energy scales ξ_n =DΛ^-n/2,n=0,1,…. Knowledge of the eigenstates and eigenvalues on all energy scales then allows physical properties such as thermodynamics and Green functions to be calculated. Within this approach, with refinements <cit.> described in the SI, we calculate the magnetic field, bias voltage and temperature dependence of the differential conductance G(V,T,B) = dI/dV via

    G(V,T,B)     = ∫_-∞^+∞-∂ f(E-eV,T)/∂ E  A(E,T,B) dE,

where  f is the Fermi function and A(E,T,B) is the spectral function of the impurity level. The general dependence of the spectral function and differential conductance on field, temperature and energy E=eV is shown in Figs. S5-S7. From G(V,T,B) we extract the magnetoresistance (MR) value at bias voltage V and temperature T via
   
    MR[%]    = 100 × (R(B)-R(0))/R(0),
        = 100 × (G(V,T,B=0)-G(V,T,B))/G(V,T,B).


First Principles Calculations.

All molecular structures were optimized using Kohn-Sham density functional theory (KS-DFT) as implemented in the program package Turbomole 6.6 <cit.>. The B3LYP <cit.> exchange–correlation functional was used, along with Ahlrichs' def2-TZVP <cit.> single-particle atom-centered basis set, and Grimme's empirical dispersion corrections (DFT-D3) <cit.> with Becke–Johnson damping <cit.>. Convergence criteria of 10^-7 a.u. for the energy in the self-consistent-field (SCF) algorithm and 10^-4 for the gradient in the molecular structure optimizations were applied, in combination with m4 integration grids. 
The coupling constants J are obtained using the Yamaguchi formula <cit.> based on the energy difference between an open-shell singlet and open-shell triplet (broken-symmetry) determinant <cit.>.


The conductance of the junctions was evaluated assuming coherent tunneling as the dominant transport mechanism (Landauer regime). The transmissions are modeled via a Green's function approach combined with DFT calculations for zero-bias electronic structures, employing the wide-band limit for the self-energies of the electrodes, as described, e.g., in Refs.<cit.>. A local density of states (LDOS) of 0.036 eV^-1 was used. Transmission functions were evaluated with Artaios <cit.>, using the Fock and overlap matrices from the KS-DFT electronic structure calculations on the molecular junctions. The value of the Fermi energy is not straightforward to predict in first-principles calculations on molecular junctions, as it will be affected by factors such as the irregular atomistic shapes of the electrodes and the adsorption of non-bridging molecules on the electrodes. We, therefore, evaluate zero-bias conductances for a range of reasonable Fermi energies (-5 eV, -4.5 eV, -4 eV) and check whether our conclusions are robust with respect to this choice. 
For further information please refer to the SI.
  
  

 §.§ Acknowledgements

 We thank the staff of the nano.lab at the University of Konstanz for technical support. G. M. and E.S. acknowledge funding by the Deutsche Forschungsgemeinschaft  (DFG) through  the Collaborative Research Center SFB 767 “Controlled Nanosystems” (funding ID 32152442).  K. S. and C. H. acknowledge funding via the Graduate School 2536 “Nanohybrid” of the Deutsche Forschungsgemeinschaft (DFG), funding ID 408076438. M. D. and C. H. acknowledge support by the Cluster of Excellence “CUI: Advanced Imaging of Matter” of the Deutsche Forschungsgemeinschaft (DFG) (EXC 2056, funding ID 390715994). T. A. C. gratefully acknowledges computing time on the supercomputer JURECA <cit.> at Forschungszentrum Jülich under grant no. JIFF23.
  
  

 

 Paper Title: Nonlinearity-induced dynamical self-organized twisted-bilayer lattices in Bose-Einstein condensates.





























#15pt#1

    GThese authors contribute equally to this work.Ministry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, ChinaThese authors contribute equally to this work.Ministry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, ChinaMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, ChinaMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, ChinaMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, Chinalishuai0999@stu.xjtu.edu.cnMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, Chinaliubophy@gmail.comMinistry of Education Key Laboratory for Nonequilibrium Synthesis and Modulation of Condensed Matter,Shaanxi Province Key Laboratory of Quantum Information and Quantum Optoelectronic Devices, School of Physics, Xi'an Jiaotong University, Xi'an 710049, China§ ABSTRACT 
        Creating crystal bilayers twisted with respect to each other would lead to large periodic supercell structures, which can support a wide range of novel electron correlated phenomena, where the full understanding is still under debate.
        Here, we propose a new scheme to realize a nonlinearity-induced dynamical self-organized twisted-bilayer lattice in an atomic Bose-Einstein condensate (BEC).
        The key idea here is to utilize the nonlinear effect from the intrinsic atomic interactions to couple different layers and induce a dynamical self-organized supercell structure, dramatically distinct from the conventional wisdom to achieve the static twisted-bilayer lattices. To illustrate that, we study the dynamics of a two-component BEC and show that the nonlinear interaction effect naturally emerged in the Gross-Pitaevskii equation of interacting bosonic ultracold atoms can dynamically induce both periodic (commensurable) and aperiodic (incommensurable) moiré structures. One of the interesting moiré phenomena, i.e., the flat-band physics, is shown through investigating the dynamics of the wave packet of BEC. Our proposal can be implemented using available state-of-the-art experimental techniques and reveal a profound connection between the nonlinearity and twistronics in cold atom quantum simulators.
        Nonlinearity-induced dynamical self-organized twisted-bilayer lattices in Bose-Einstein condensates
            Bo Liu
            August 5, 2024
        ===================================================================================================
        
        
        
        § INTRODUCTION
        
        Superimposed crystal bilayers twisted with respect to each other provide a new knob to unveil a wide
        range of novel electron correlated phenomena <cit.>.
        In particular, moiré patterns associated  with small twist angles result in various novel physical properties <cit.>, including the flat band <cit.>, fractional Chern insulators <cit.>, quantum Hall effect <cit.>, moiré ferroelectricity <cit.>, exotic magnetism <cit.>, moiré excitons <cit.>, non-Abelian gauge potentials <cit.>, and unconventional superconductivity <cit.>.
        In parallel to these developments in solid state materials, examples of twisted-bilayer are also studied in other artificial quantum systems, such as in optics <cit.>, acoustics <cit.> and cold atoms  <cit.>.
        In photonic lattices, the moiré pattern and its induced phenomena, such as the localization of light and flat band physics, have been studied in the photorefractive medium <cit.> and nanostructured photonic moiré superlattices <cit.>.
        
        Ultracold atoms in optical lattices are another one of the most promising platform to
        investigate the remarkable physical properties of twisted artificial structures <cit.>.
        It thus has stimulated tremendous amount of efforts,  including theoretically predicting moiré physics in ultracold fermions <cit.> and bosons  <cit.>. In particular,  experimental breakthrough observation of the spatial moiré pattern and momentum diffraction of a Bose-Einstein condensate (BEC) in twisted bilayer optical lattices has recently been achieved <cit.>.
        However, the conventional wisdom in past studies focuses on achieving the static twisted-bilayer lattices <cit.>.
        How to realize such twisted lattice structures in a dynamical way has so far eluded efforts.
        
        
        
        In this work, we propose a new scheme to dynamically achieve twisted-bilayer lattices that builds on the concept of nonlinearity. Our scheme relies on the  nonlinear interaction effect, which naturally emerges in the Gross-Pitaevskii equations of interacting bosonic ultracold atoms. We find that such nonlinear interaction effect amazingly results in a dynamical self-organized twisted-bilayer lattice. This new scheme frees out the usual requirement of adding external single-particle type coupling between different layers and only relies on the nonlinear effect from the intrinsic atomic interactions,  dramatically distinguished from previous schemes  <cit.>. We show that both dynamical self-organized periodic (commensurable) and aperiodic (incommensurable) twisted-bilayer lattice structures can be realized through our proposed scheme. In particular, for the aperiodic case, we find the localization of the Gaussian BEC wave packet, resulting from the emergence of the flat bands in the nonlinearity-induced self-organized twisted-bilayer lattices. It indicates that our proposed nonlinear interaction effect can dynamically produce the tunable moiré lattices in BEC.
        
        
        
        § EFFECTIVE MODEL
        
        The system considered here is schematically illustrated in Fig.<ref>. A two-component BEC, for instance ^87Rb atoms, is used,  where a spin-dependent harmonic potential is applied <cit.>. Atoms in component 1 are trapped in a tight harmonic trap and the atoms in component 2 are loaded in a square optical lattice confined by a very shallow harmonic trapping potential. After the system reaches equilibrium, the tight harmonic trap for component 1 atoms is tuned to be very shallow and a square optical lattice is added. At the same time, the inter-species interaction is turned on. Such a system can be described by the following model Hamiltonian
        
            𝐇   =   ∫dxdy ∑_α=1,2ψ_α^†[ 𝐤^2/2m+V_trap,α(x,y)+V_α(x,y) ] ψ_α
               +   ∫dxdy(g_11n̂_1^2+g_22n̂_2^2+2g_12n̂_1n̂_2),
        
        where ψ_α(α=1,2) denote the Bose field operators for two-component bosons, respectively. The corresponding density operators  are defined as n̂_α=ψ^†_αψ_α. V_trap,α denotes the harmonic trap. The spin-dependent optical  lattices are  V_1(x,y)=V_1[sin^2(cosθ/2k_Lx-sinθ/2k_Ly)+sin^2(sinθ/2k_Lx+cosθ/2k_Ly)] and  V_2(x,y)=V_2[sin^2(cosθ/2k_Lx+sinθ/2k_Ly)+sin^2(-sinθ/2k_Lx+cosθ/2k_Ly)], respectively, with θ being the twist angle. k_L=2π/λ is the wavevector
        of the laser field with the corresponding lattice constant defined as a_L=π/k_L, and λ is the wavelength of the laser. V_α captures the corresponding lattice depth.
        g_11 and g_22 characterize the intra-species interaction strength for component 1 and  component 2 atoms, respectively. The inter-species interaction strength is captured by g_12.
        
        From Eq. (1), the following
        dimensionless energy functional can be constructed under the Gross-Pitaevskii (GP) mean-field theory
        
            ε   =   ∫dx'dy' ∑_α=1,2ψ_α^* [ -∇^2/2+V'_trap,α(x',y')+V'_α(x',y') ] 
               ×   ψ_α +∫dx'dy'(g'_11|ψ_1|^4+g'_22|ψ_2|^4+2g'_12|ψ_1|^2|ψ_2|^2),
        
        where dimensionless spatial coordinates are defined as x'=x/λ and y'=y/λ, respectively. V'_trap,α=V_trap,α/(ħ^2/mλ^2) represents the harmonic trapping potential. V'_α=V_α/(ħ^2/mλ^2) labels the dimensionless lattice depth. g'_11(12,22)=g_11(12,22)/(ħ^2/m) are the corresponding dimensionless interaction strength.
        The wavefunction is normalized as ∬ |ψ_α|^2dx'dy'=N_α with N_α being the particle number of the two-component atoms, respectively.
        Based on Eq. (2), the dynamics of the system can be well described by the following coupled GP equations
        
            i∂ψ_1/∂ t' =    [ -1/2∇^2+V'_trap,1(x',y')+V'_1(x',y')+g'_11|ψ_1|^2 
               +g'_12|ψ_2|^2 ] ψ_1  
            
            			i∂ψ_2/∂ t' =   [ -1/2∇^2+V'_trap,2(x',y')+V'_2(x',y')+g'_12|ψ_1|^2 
               +g'_22|ψ_2|^2 ] ψ_2,
        
        where t'=t/(mλ^2/ħ) denotes the  dimensionless evolution time.
        
        
        § MOIRÉ LOCALIZATION INDUCED BY THE  NONLINEAR INTERACTION EFFECT
        
        In the following, we will demonstrate that distinguished from previous studies <cit.>, even without
        external single-particle type interlayer coupling,  the existence of nonlinear interaction term in GP equation, i.e., w=g'_11|ψ_1|^2
        +g'_12|ψ_2|^2, can dynamically result in the twisted-bilayer lattices and  lead to one of the interesting moiré phenomena.
        To show that, we monitor the dynamics of BEC for different twist angles θ. Here the initial state for component 1 atoms is the ground state of the harmonic trap, which is chosen as a Gaussian wave packet (left part in Fig. <ref>(a)). While for component 2 atoms, the initial state is chosen as the ground state (right part in Fig.<ref>(a)) determined by the effective Hamiltonian in the right hand-side of Eq. (<ref>) with g'_12=0.
        
        Interestingly, as shown in Fig. <ref>, when tuning the twist angle θ, in the presence of nonlinear term w, even without external single-particle type coupling between two layers, one of the interesting moiré physics, i.e., both the localized and delocalized behavior of component 1 atoms, can be achieved.
        For instance, as shown in Fig. <ref>(b) and (c), when θ is chosen as a Pythagorean angle, for instance, θ= arctan3/4 (Fig. <ref>(b)), the Gaussion wave packet for component 1 atoms displays the delocalized behavior.
        While considering θ being the non-Pythagorean twisted angle, for instance, θ = π/6 (Fig. <ref>(c)), the wave packet for component 1 atoms turns out to be localized.
        Note that here we consider the case with large number imbalance between two components, i.e., N_1 ≪ N_2 and lager intra-species interaction, i.e., g'_22 > g'_12. The effect of inter-species interaction on the dynamics of component 2 atoms is highly suppressed and the density profile of component 2 atoms almost does not change during the evolution, as shown in the right column of Fig. <ref>.
        
        
        
        To quantitatively distinguish between the  delocalized and localized behavior of the wave packet of component 1 atoms, we analyze the following mean square displacement (MSD) <cit.>
            σ(t)=[∬dxdy (x^2+y^2)|ψ_1(t)|^2/∬dxdy|ψ_1(t)|^2]^1/2,
        
        which can characterize the width of the wave packet of component 1
        atoms. The different dynamical behaviors of the wave packet can be
        quantified as σ(t)∝ t^κ with distinct dynamical index κ.  As shown in Fig. <ref>,  when the twist angle θ is chosen as a Pythagorean angle, for instance, θ= arctan3/4, σ(t) tends to linearly increase with respect to the evolution time, i.e., σ(t) ∝ t.  The dynamical index is 1,  indicating
        its delocalized behavior. While θ being a non-Pythagorean angle, for instance, θ = π/6, σ(t) will reach a stable value in a fairly short time, indicating the dynamical index κ being 0. It thus characterizes the localized behavior of the wave packet.  Therefore, through tuning the twist angle θ, one of the moiré physics, i.e., both the localized and delocalized behavior of the wave packet of component 1 atoms, can be achieved.
        
        
        
        § NONLINEARITY-INDUCED DYNAMICAL SELF-ORGANIZED TWISTED-BILAYER LATTICES
        
        In the following, we will explain the reason for the appearance of the moiré physics in our proposed system. Let us focus on Eq. (<ref>) in the GP equation describing the two-component atomic BEC. In the presence of the nonlinear term w,  component 1 atoms suffer from a  dynamical self-organized effective potential V_eff≡ V'_trap,1+V'_1+w=V'_trap,1+V'_1+g'_11|ψ_1|^2+g'_12|ψ_2|^2.
        As shown in Fig. <ref>(a), when the twist angle θ is chosen as a Pythagorean
        angle, for instance, θ =arctan 3/4, the effective potential V_eff possess a spatially periodic structure. While considering θ being a non-Pythagorean twist angle, for instance,  θ=π/6, as shown in Fig. <ref>(b), V_eff behaves as a twisted-bilayer lattice with a spatial aperiodic structure. Therefore, distinguished from previous studies <cit.>, even without external single-particle type interlayer coupling, the nonlinear effect resulting from the intrinsic atomic interactions provides a new way to dynamically induce the twisted-bilayer lattices. Through tuning the twist angle θ, we find that both periodic (commensurable) and aperiodic (incommensurable) moiré structures can be created in such a dynamical twisted-bilayer lattice.
        
        
        
        
        To further understand the distinct dynamical behaviors of the wave packet of component 1 atoms, we rely on the analysis of the appearance of the flat band physics in our proposed nonlinearity-induced self-organized twisted-bilayer lattices. To do that, we calculate the band structure of V_eff for both periodic (commensurable) and aperiodic (incommensurable) moiré structures.
        In particular, when considering the aperiodic moiré lattice, the band structure can be calculated through approximating the non-Pythagorean twist angle by a Pythagorean one. For instance, here we use θ=arctan 120/209 to approximate θ= π/6.
        The band structure of V_eff can thus be obtained by the plane-wave expansion method through introducing the Bloch basis ψ_nk⃗=∑_G⃗ u_nk⃗,G⃗|k⃗+G⃗⟩ with the Bloch vector k⃗ and the reciprocal lattice vector G⃗. Here n labels the band index. Then, the energy spectra can be obtained through solving the eigen-problem via the following relation
        
            (k⃗+G⃗)^2λ^2/2u_nk⃗,G⃗ + ∑_G⃗'⃗⟨k⃗+G⃗ | V_eff| k⃗+G⃗'⃗⟩ u_nk⃗,G⃗'⃗   
            
            		= E'_nk⃗ u_nk⃗,G⃗,
        
        where E'_nk⃗=E_nk⃗/(ħ^2/mλ^2) is the dimensionless energy spectra.
        
        
        
        
        As shown in Fig. <ref>, the band structures of the corresponding
        twisted-bilayer lattices V_eff  demonstrated in Fig. <ref> are obtained. We find that when changing the twist angle from a Pythagorean angle to a non-Pythagorean angle, more dispersive lower bands become extremely flat, accompanying with the lattice structure changing from the periodic (commensurable) case to the aperiodic (incommensurable) one. In particular, the appearance of the lowest flat band will lead to the localization of the wave packet of component 1 atoms, since such a BEC wave packet will mainly concentrate at the lowest band after a long enough time evolution. Furthermore, it is known that the flat band can support quasi-nondiffracting localized modes. Therefore, the wave packet of component 1 atoms loaded in such a  nonlinearity induced self-organized aperiodic (incommensurable) twisted-bilayer lattice will tend to be localized.
        
        
        
        § CONCLUSION
        
        In summary, we have demonstrated a new approach to
        dynamically achieve twisted-bilayer lattices in an atomic BEC.
        To emphasize a remarkable difference, our scheme
        does not require external single-particle type
        interlayer coupling. Instead, we only require the nonlinear effect
        arising from the intrinsic atomic interactions.
        We further show that both periodic and aperiodic  moire lattices, i.e.,  nonlinearity-induced dynamical self-organized twisted-bilayer lattices, can be designed through tuning the twist angle. One of the interesting phenomena in twisted-bilayer lattices, i.e., the flat-band physics, is also shown through investigating the dynamics of the wave packet of BEC. Such nonlinearity-induced approach is rather generic to other quantum systems, such as in photonics and electrical circuits, than restricted to the setup considered in this work, since the nonlinear effects, such as  Kerr nonlinearity, is common in these systems. Our results reveal a profound connection between
        the nonlinearity and twistronics.
        
        
        § ACKNOWLEDGMENT
        
        We acknowledge helpful discussions with L. Chen. This work is supported by the National Key R&D Program of China (2021YFA1401700), NSFC (Grants No. 12074305, 12147137, 11774282), the Fundamental Research Funds for the Central Universities (Grant No. xtr052023002), the Shaanxi Fundamental Science Research Project for Mathematics and Physics (Grant No. 23JSZ003) and  Xiaomi Young Scholar Program. We also thank the HPC platform of Xi'an Jiaotong University, where our numerical calculations was performed.
        
        
        apsrev


 Paper Title: Andreev Reflection in the Quantum Hall Regime at an Al/InAs Junction on a Cleaved Edge.




takafumi.akiho@ntt.com





NTT Basic Research Laboratories, NTT Corporation, 3-1 Morinosato-Wakamiya, Atsugi 243-0198, Japan

PACS number


§ ABSTRACT 
We have fabricated a superconductor/semiconductor junction composed of Al and InAs using cleaved edge overgrowth (CEO). By exploiting the unique geometry with a thin Al/Pt/Al trilayer formed on the side surface of an in-situ cleaved heterostructure wafer containing an InAs quantum well (QW), we achieve a superconducting critical field of ∼ 5 T, allowing superconductivity and quantum Hall (QH) effects to coexist down to filling factor ν = 3. Andreev reflection at zero magnetic field shows a conductance enhancement that is limited solely by the Fermi velocity mismatch, demonstrating a virtually barrier-free, high-quality S/Sm junction. Bias spectroscopy in the QH regime reveals a clear bias dependence indicating the superconducting gap, with the reduced downstream resistance demonstrating that the electron-hole Andreev conversion probability consistently exceeds 50%. Our results, obtained in a new experimental regime characterized by a clean edge-contacted junction, a much smaller superconducting gap, and a narrow superconducting electrode less than the coherence length, open a new avenue for both theoretical and experimental studies of the interplay between superconductivity and QH effects and the engineering of exotic quasiparticles.
	

Andreev Reflection in the Quantum Hall Regime at an Al/InAs Junction on a Cleaved Edge
    Koji Muraki
    August 5, 2024
======================================================================================

There have been ongoing efforts to realize topological superconductivity, which is expected to support exotic quasiparticles that obey non-Abelian statistics, a crucial building block for fault-tolerant quantum computation <cit.>.
Among the various proposals to engineer topological superconductivity by exploiting the proximity coupling with s-wave superconductors, those based on quantum Hall (QH) systems <cit.> are unique in that they hold the promise of otherwise unavailable universal quantum computation using parafermions predicted in the fractional QH regime.
This has led to a renewed interest and experimental efforts <cit.> in superconductor/semiconductor (S/Sm) junctions in the QH regime, despite the experimental challenges posed by the requirements of junction transparency and the coexistence of high magnetic fields and superconductivity, which are generally incompatible with each other.

Superconducting correlation is induced in QH edge channels by Andreev processes, where electrons entering the junction exit as holes, transmitting Cooper pairs into the superconductor.
Experiments on NbN/graphene junctions in the integer <cit.> and fractional <cit.> QH regimes have demonstrated crossed Andreev reflection, in which electron-hole conversion takes place between counterpropagating edge channels through a narrow strip of superconductor in between.
There, the negative resistance in the downstream of the junction (R_d) provided evidence of electron-hole conversion.
In contrast, for a 600-nm-wide MoRe/graphene junction along a single edge, R_d was observed to oscillate or fluctuate between positive and negative as the magnetic field or gate voltage was swept <cit.>.
It was then argued that these oscillations arise from the interference of chiral Andreev edge states (CAESs), hybrid eigenmodes of electron- and hole-like edge states formed along the proximitized region <cit.>.
However, subsequent theoretical investigations have raised interfacial disorder <cit.>, vortices <cit.>, disorder <cit.>, and supercurrent <cit.> in the superconductor, and the geometry of the junction <cit.> as necessary ingredients to explain the experimental observations.
On the other hand, an experiment for a 150-μm-wide NbTiN/InAs junction found negative R_d without oscillations <cit.>.
These clearly show the need for further experiments with a controlled setup to understand even the most fundamental process of Andreev conversion.

Common to the reported superconducting junctions in the QH regime <cit.> is that they use type-II superconductors to sustain superconductivity at high fields and etching to fabricate the junction, both of which make it challenging to form a pristine and uniform junction free of interfacial disorder or in-gap states.
Nonuniform interfaces would result in an inhomogeneous induced gap, leading to a soft gap <cit.>, where in-gap states act as a source of unwanted quasiparticle tunneling into the superconductor.
On the other hand, successful formation of pristine S/Sm junctions by in-situ epitaxial growth of a superconductor on the surface of a semiconductor has been reported <cit.>.
In particular, epitaxial junctions of Al and InAs nanowires <cit.> or two-dimensional electron systems (2DESs) <cit.> have been shown to have a hard superconducting gap free of in-gap states.
Although Al is a type-I superconductor, in thin films it can maintain superconductivity up to a few tesla when the field is in the plane of the film, a property that has been exploited for hybrid nanowires.
Furthermore, by adding heavy elements with strong spin-orbit coupling such as Pt, the in-plane critical field can be increased to a several tesla <cit.>.
However, despite their superior properties, epitaxial Al/InAs junctions could not be used for QH systems, because the field must be applied perpendicular to the 2DEG, i.e., to the superconducting film.

Here, we overcome this limitation by using a novel geometry in which the superconducting thin film is grown on the side surface of a semiconductor wafer, and realize an Al/InAs epitaxial junction that operates in the QH regime.
Using the cleaved edge overgrowth (CEO) technique <cit.>,
we deposited an Al/Pt/Al trilayer on an in-situ cleaved edge surface of a wafer containing an InAs quantum well (QW).
Analysis of the Andreev reflection at zero field indicates that the conductance enhancement is limited only by the Fermi velocity mismatch, demonstrating a high-quality junction that is virtually free of an interface barrier.
At high fields, we observe that R_d is reduced near zero bias in the plateau regions for filling factors from 10 to 3, indicating that the probability of electron-hole conversion consistently exceeds 50%, without oscillatory behavior.
The geometry of our experiment also makes it unique: 1) the field direction, being parallel to the plane of the thin superconductor, eliminates vortices and 2) the 2DES is edge contacted by the superconductor with no spatial overlap along the field direction.
The observation of Andreev conversion with a clear bias dependence in the unique geometry provides new insights into the underlying processes.
In addition, our approach to fabricating etch-free, high-quality superconducting junctions will open a new venue for the study of engineered topological superconductivity.

Our CEO device was fabricated from a heterostructure wafer grown on a GaSb(001) substrate. (See Supporting Information (SI) for the layer structure <cit.>.)
The wafer, cut into a 14 mm × 15 mm piece, was thinned to 300 μm by polishing to facilitate cleaving and thus obtain a flat cleaved edge surface.
The wafer was mounted at 90^∘ on a Mo sample holder and was cleaved in UHV (< 5.0 × 10^-10 mbar) using a wobble stick.
The wafer was then transferred under UHV to the deposition chamber, where a superconducting electrode was deposited on the cleaved edge surface, with the sample holder cooled by direct contact with a liquid nitrogen cold plate, a crucial step for obtaining epitaxial Al layers <cit.>.
We used photolithography to process the sample into a gate-defined Hall bar with normal Ti/Au electrodes and an S/Sm junction at the cleaved edge.
(See SI for fabrication details <cit.>.)
Measurements were conducted in a ^3He cryostat equipped with a superconducting vector magnet.
The 2DES density and mobility are n = 3.0 × 10^15m^-2 and μ = 63 m^2/Vs, respectively.

Figure <ref>(a) shows a schematic of our CEO device.
The Hall bar is defined electrostatically by applying a negative voltage V_FG (≲ -1.2 V) to the front gate and depleting the 2DES beneath [This prevents non-topological edge conduction in the QH regime caused by electron accumulation at the edge, which can occur in etch-defined InAs channels <cit.>. Additionally, it allows the junction ends to be defined electrostatically, avoiding contamination or damage from direct processing.].
The front gate has an opening above the cleaved edge, defining a narrow junction of width L, nominally 3-μm wide, between the superconductor (#8) and the 2DES.
The scanning electron microscope (SEM) image taken near the narrow junction of a dummy sample confirms the formation of a split gate [Fig. <ref>(b)].
The superconductor (#8) was electrically accessed through the wide S/Sm junctions on either side, using and the 2DES regions isolated from the Hall bar by the front gate as leads and the Ti/Au contacts #6 and #7 as current and voltage probes, respectively [This design was employed to access the superconductor while avoiding unintended breaks at steps on the cleaved edge and shorting to the conducting GaSb substrate.].
We applied AC and DC bias to the source electrode #1 and measured the AC current (I) from the drain electrode #6.



Figure <ref>(c) presents the longitudinal (R_xx) and Hall (R_xy) resistances of the device, measured in a perpendicular field B_z using normal contacts, showing well-developed QH effects at Landau-level filling factor ν (= nh/eB_z) of 3 to 10, where h is Planck's constant and e is the elementary charge.
Figure <ref>(d) shows the superconducting property of the Al/Pt/Al film of a control sample fabricated on the cleaved edge of a semi-insulating GaAs(100) substrate.
The two-terminal resistance R of an Al/Pt/Al trilayer of thicknesses 6, 1, and 2 nm from bottom to top is plotted as a function of the field B_z applied perpendicular to the substrate (i.e., parallel to the superconducting thin film).
The critical in-plane field B_c^∥ is ∼5 T for the 6/1/2-nm trilayer, which decreased as the bottom Al thickness increased <cit.>.
Based on this result, we employed the 6/1/2-nm trilayer structure for the S/Sm junction.
The results in Figs. <ref>(c) and (d) together confirm that, below B_z = 5 T, superconductivity and QH effects can coexist in our device employing an Al superconductor.



Now we characterize the S/Sm junctions by measuring the differential resistance at B_z = 0T using the setup shown in Fig. <ref>(a).
Here, we focus on the results for R^0_d = dV_d/dI obtained with contacts #4 and #7, while those for R^0_u = dV_u/dI obtained with contacts #3 and #7 are provided in SI <cit.>.
Figures <ref>(a) show the DC bias (V_dc) dependence of R^0_d at different temperatures [Throughout this paper, V_dc is the measured DC component of V_u, the volatage between contacts #3 and #7.].
A pronounced zero-bias minimum is observed at low temperatures, demonstrating Andreev reflection, i.e., reflection of an incident electron from the 2DES as a hole, thereby enhancing the conductance <cit.>.
From the temperature at which the zero-bias minimum evolves into a broad maximum, the critical temperature T_c is determined to be 1070 mK, with the corresponding BCS gap Δ = 1.77k_BT_c of 163 μeV, where k_B is the Boltzmann constant.

To characterize the S/Sm junction more quantitatively, we evaluate the Andreev reflection probability.
To this end, we derived the differential conductance of the junction, G = (R^0_d - R_s)^-1, by subtracting the series resistance R_s from R^0_d.
We estimated R_s = 1.01kΩ by noting that the voltage drop across the S/Sm junction (V_jct) equals Δ/e at the DC bias (V_Δ) for which the Fermi level in the normal region coincides with the superconducting gap edge at base temperature.
Figure <ref>(b) plots the normalized differential conductance G/G_n as a function of V_jct [≡ (Δ/e)V_dc/V_Δ], where G_n is the differential conductance at a finite bias outside the superconducting gap.
The solid line represents a fit based on the modified Blonder-Tinkham-Klapwijk (BTK) model described by <cit.>,

    G/G_n = C∫_-∞^∞ f(E-eV) - f(E)  (1 + A - B) dE.

Here, C is a constant, E is the energy, f is the Fermi distribution function, and A = A(E, Δ, Z, Γ) and B = B(E, Δ, Z, Γ) are the Andreev and normal reflection probabilities, respectively
[
A = a^∗ a and B = b^∗ b, where a = u_0 v_0 /γ, b = -(u_0^2 -  v_0^2) /γ, and γ = u_0^2 +  Z^2(u_0^2 - v_0^2) with u_0^2 = 1/2[ 1 + √((E + i Γ)^2 - Δ^2)/E + i Γ] and v_0^2 = 1/2[ 1 - √((E + i Γ)^2 - Δ^2)/E + i Γ]
].
Z is a dimensionless parameter characterizing the effective barrier at the S/Sm interface, and Γ, which enters A and B as the imaginary part of the energy, describes the lifetime broadening.

As shown in Fig. <ref>(b), Z = 0.42 and Γ = 0.17Δ reproduce the measured bias dependence, including the conductance enhancement factor G/G_n of 1.28 at zero bias.
Both finite Z and Γ reduce the Andreev reflection probability, resulting in a conductance enhancement factor less than 2, the value expected for an ideal junction.
Note, however, that a mismatch in the Fermi velocity at the junction leads to a finite normal reflection probability even in the absence of a barrier.
The resulting I-V characteristics can be described by replacing Z by an effective value Z_eff=√(Z^2+(1-r)^2/4r), where r is the Fermi velocity ratio <cit.>.
In fact, for our InAs 2DES with v_F = 5 × 10^5 m/s, the Fermi velocity mismatch alone is sufficient to make Z_eff≥ 0.42, assuming bulk Al <cit.>.
Including the two-dimensionality of the 2DES, which is not considered in the BTK model, would further increase Z_eff <cit.>.
These observations suggest that our junction is virtually barrier-free.
Interestingly, the obtained Γ = 0.17Δ corresponds to a quasiparticle lifetime τ_qp (= h/2 πΓ) of 24 ps.
This is more than 10 times longer than that for the NbN/graphene device <cit.>, indicating that a high-quality superconducting electrode has been obtained by CEO.

We also measured Andreev reflection with the magnetic field B_x applied perpendicular to the superconducting thin film [inset of Fig. <ref>(b)].
From the critical field of B_c^⊥ = 0.39 T, the effective coherence length is estimated to be ξ = 29nm.
Although this value is much shorter than the typical values for Al (> 1 μm), possibly due to the spin-orbit interaction induced by Pt <cit.>, it is important to note that the thickness (10 nm) of our superconducting film is even smaller.



We next study the Andreev reflection in the QH regime by applying a magnetic field B_z perpendicular to the 2DES (i.e., parallel to the superconducting thin film).
Figure <ref>(a) plots R_d, the resistance measured on the edge channels downstream of the S/Sm junction, as a function of V_dc and B_z.
The plot reveals a region of reduced R_d around V_dc = 0 V, which narrows above 4 T but remains open up to 4.8 T.
We associate this with the superconducting gap [The width of the V_dc range corresponding to the superconducting gap becomes narrower at low fields due to the increasing bulk conductivity near the S/Sm junction.].
Figure <ref>(b) displays horizontal line cuts of Fig. <ref>(a) at several integer values of ν, showing reduced R_d inside the superconducting gap.
We emphasize that such a clear bias dependence, indicating a superconducting gap in the QH regime, has not been previously reported for junctions along a single edge [The only reports are for the setups of crossed Andreev reflection <cit.>.].
We first noticed that although R_d is clearly reduced around zero bias, it remains positive.
This contrasts with the sign-oscillating R_d reported for MoRe/graphene junctions <cit.>, where negative R_d indicates electron-hole conversion.
We note that in our data R_d is finite outside the superconducting gap.
This indicates that the coupling between the edge channels and the Al electrode is imperfect; that is, even in the absence of superconductivity, some fraction of electrons in the upstream channels are transferred to the downstream channels.
As we show below, this effect can be accounted for using a Landauer-Büttiker-type model, from which the probability P_h of electron-hole conversion by the Andreev process can be derived.

We construct a Landauer-Büttiker-type model appropriate for our device layout involving a superconducting electrode.
The current-voltage relation can be expressed as 𝐈 = ν G_0𝐌𝐕, where G_0 = e^2/h is the conductance quantum, and 𝐈 = (..., I_i, ...)^T and 𝐕 = (..., V_i, ...)^T with I_i (V_i) the current (voltage) of the ith contact (i = 1, 3, 4, 6, 7, 8).
(Contacts #2 and #5, which are not used in the R_d(u) measurements, are excluded.)
𝐌 is the normalized conductance matrix.
The effect of imperfect edge coupling is accounted for by introducing a parameter T_t.
As shown schematically in the inset of Fig. <ref>, a portion of the electrons in the incoming ν edge channels are reflected before reaching the superconducting electrode #8 with probability 1-T_t and fed into the outgoing channels.
Definition of P_h is such that the injection of an electron into the junction results in the emission of an average P_h hole into the outgoing channels.
That is, P_h = 1 and 0 correspond to 100% electron-hole conversion (i.e., perfect Cooper-pair transmission) and perfect normal reflection (i.e., no current through the junction), respectively.
P_h = 0.5 means that on average these two events occur with equal probability.
Thus, the matrix 𝐌 is given as

    𝐌 =
    [      1      0     -1      0      0      0;     -1      1      0      0      0      0;      0    η-1      1      0      0     -η;      0      0      0    η^'      0   -η^';      0      0      0      0    η^'   -η^';      0     -η      0   -η^'   -η^' η+2η^';        ],

where η≡ 2T_tP_h and η^' is the counterpart of η for the wide junctions connected to contacts #6 and #7.
By solving 𝐈 = ν G_0𝐌𝐕 for 𝐕 with 𝐈=(-I,0,0,I,0,0)^T, we obtain [The consistency of the model can be checked by seeing that T_t = 1 and P_h = 0.5 give R_d = 0 and R_u = 1/ν G_0 as expected for the conventional QH effects with normal contacts, while T_t = 1 and P_h = 1 give R_d = -1/2 ν G_0 and R_u = 1/2 ν G_0 as expected for perfect Andreev reflection.]

    R_d   ≡V_4-V_7/I= ( 1/2T_tP_h - 1 ) 1/ν G_0,
    
    	R_u   ≡V_3-V_7/I = 1/2T_tP_h1/ν G_0.

Note that for T_t<1, R_d becomes negative only when P_h > 1/2T_t, even if P_h > 0.5.



We deduced T_t at each ν from the measured R_d and R_u outside the superconducting gap (V_dc = -10 mV) by setting P_h= 0.5 in Eqs. (3) and (4).
As shown in the upper panel of Fig. <ref>, R_d and R_u give consistent results, both showing that T_t gradually increases from 0.8 to 0.96 as ν decreases from 10 to 3.
We used these T_t values to derive P_h from R_d and R_u at V_dc = 0 mV.
As shown in the lower panel of Fig. <ref>, R_d and R_u give similar results, showing that P_h consistently exceeds 0.5.

Two main aspects of our results need to be discussed first: the absence of oscillatory behavior in R_d and the preferential reflection of holes, indicated by P_h>0.5.
Theories predict that, for an ideal junction without disorder, P_h should oscillate as a function of junction width L <cit.> and bias <cit.>, due to the interference between CAESs, reflecting their momentum difference.
On the other hand, it is also argued that the oscillations will only be visible at specific magnetic fields per Landau level; away from these fields, the oscillation amplitude will rapidly decay, causing P_h to average out to 0.5 <cit.>.
The absence of such field dependence in the reported experiment <cit.> is a key aspect that triggered the subsequent theoretical investigations <cit.>.
In our data, R_d (and hence P_h) remains nearly constant within each QH plateau, maintaining P_h >0.5, without  oscillations or sharp field dependence.
One could speculate on the reasons for the absence of oscillations, such as the involvement of multiple edge channels or factors like temperature, momentum difference, or junction width being inadequate for resolving the interference.
However, in either case, the question arises as to why P_h does not average out to 0.5.
The absence of sharp field dependence also remains unexplained.

Turning to the result in Fig. <ref>(b), P_h gradually increases with increasing ν (i.e., decreasing B_z).
No clear difference in P_h is observed between the odd and even ν, where the incoming edge channels are partially spin-polarized and unpolarized, respectively.
This suggests the presence of a spin-flip mechanism, likely due to the spin-orbit interaction in the Pt layer.

Another notable feature in our data is the absence of discontinuous jumps in R_d during magnetic field sweeps, which were reported for MoRe/graphene junctions and attributed to a phase shift in Δ resulting from vortices hopping in and out of the superconductor <cit.>.
In our experiment, the superconductor is thinner than the coherence length, and we utilize a two-axis vector magnet to align the magnetic field parallel to the film. This minimizes the possibility of vortices playing any role, consistent with the absence of discontinuous jumps.
In an attempt to account for the P_d > 0.5 observed in their 150-μm-wide NbTiN/InAs junction, the authors of Ref. <cit.> speculated that electrons may have a higher probability than holes to tunnel into the superconductor.
Although the origin of such asymmetric tunneling, if present, is unclear, the absence of vortices, which are necessary for electron/hole tunneling into the superconductor, makes this scenario unlikely in our CEO device.
While the effects of vortices have been studied from different theoretical perspectives <cit.>, the irrelevance of vortices in our CEO device is expected to help elucidate the mechanism of Andreev conversion in the QH regime.
The much smaller Δ compared to type-II superconductors, along with the narrowness of the superconducting electrode below the coherence length, places our experiment in a unique regime distinct from other experiments.
The small Δ makes the velocity of CAESs (v_c) and their momentum difference small, the latter being determined by v_c and the induced superconducting gap (≤Δ).
Our observation of P_d > 0.5 along with a clear bias dependence in such a regime will stimulate further theoretical investigations.

In summary, using cleaved edge overgrowth, we have successfully fabricated a clean Al/InAs junction that is virtually barrier-free.
Bias spectroscopy in the QH regime reveals a clear bias dependence indicating the superconducting gap up to 4.8 T, a behavior not previously observed in junctions along a single edge.
The reduced downstream resistance inside the superconducting gap indicates an electron-hole conversion probability that consistently exceeds 50%.
Our results, obtained in a unique regime characterized by a clean junction, a much smaller superconducting gap, and a narrow superconducting electrode (less than the coherence length), excluding vortices, open a new avenue for both theoretical and experimental studies toward engineered topological superconductivity.

The authors thank H. Murofushi for device processing, M. Imai for drawing support, H. Kamata for measurement advice, and T. Wakamura for discussions on Andreev reflection data.










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 Paper Title: Persistent nodal magnon-photon polariton in ferromagnetic heterostructures.







School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China


School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China


Department of Materials, ETH Zurich, Zurich 8093, Switzerland



School of Integrated Circuit Science and Engineering, Beihang University, Beijing 100191, China


taoyuphy@hust.edu.cn
School of Physics, Huazhong University of Science and Technology, Wuhan 430074, China




§ ABSTRACT 
 Exceptional points with coalescence of eigenvalues and eigenvectors are spectral singularities in the parameter space, achieving which often needs fine-tuning of parameters in quantum systems.  We predict a persistent realization of nodal magnon-photon polariton, i.e., a polariton without any gap splitting in a thin ferromagnetic insulator film sandwiched by two normal metals, which persistently exists when the ferromagnet is sufficiently thick ∼ 100 nm. We perform the model calculation beyond the perturbation theory using a classical approach, develop a quantum scheme able to account for the Ohmic dissipation, and find ultrastrong coupling with coupling strength comparable to the bare magnon frequency. Via revealing a simple conversion relation we extend this formalism to superconductors and predict the gap opened by the ultrastrong coupling strongly depends on the direction of polariton propagation.  Our findings may help search for robust non-Hermitian topological phases in magnonic and spintronic devices.




Persistent nodal magnon-photon polariton in ferromagnetic heterostructures
    Tao Yu
    August 5, 2024
==========================================================================






§ INTRODUCTION


Ferromagnetic heterostructures composed of ferromagnets (FMs) and normal metals (NMs) are common devices intensively explored and exploited in spintronics over decades, tracing back to the discovery of giant magnetoresistance <cit.>. In these heterostructures, NM|FM, FM|NM|FM, or NM|FM|NM could be basic unit cells. Due to the Ohmic dissipation in NMs, these heterostructures are intrinsically dissipative and non-Hermitian, thus providing platforms for searching for useful non-Hermitian topological phases with magnetism <cit.>. For example,  Liu
et al. experimentally observed the exceptional points (EPs) of magnons, a state with both eigenstates and eigenvalues coalescing at specific parameters, in FM|NM|FM heterostructure in which the NM layer mediates a RKKY exchange interaction <cit.>. This realization needs fine-tuning of parameters <cit.>, including the gain, loss, and RKKY exchange interaction, which is device-dependent and, thereby, difficult to achieve universally.


It has been revealed that the NM affects the excitations of magnetization in FMs, i.e., spin waves or magnons as their quanta. They act as promising information carriers to transport spin information and perform logic operations in “magnonics" <cit.>. The dipolar fields emitted by spin waves drive the diamagnetic eddy currents in the NMs, which are dissipated according to Ohm’s Law and thereby cause an additional dissipation channel of magnons <cit.>.  Moreover, the chirality of the stray field translates into non-local chiral damping, i.e., it only affects spin waves that obey the right-hand rule, rendering non-Hermitian skin effect <cit.> and unidirectional transmission across potential barriers <cit.>. However, despite being useful for spintronic and magnonic devices, the studies addressing collective modes in the NM|FM|NM systems are still limited, in which the magnon-photon interaction could be much enhanced due to the “cavity effect".   


Different functionalities appear when the NMs become superconducting at low temperatures since the Ohmic dissipation disappears. The induced eddy currents in superconductors (SCs) also generate Oersted magnetic fields that blue-shift the magnon frequencies <cit.>. Superconducting gates, therefore, act as non-dissipative repulsive potential barriers for propagating spin waves <cit.>. Recently, ultrastrong coupling <cit.> between magnon and Swihart photon modes of SCs <cit.> was reported in SC|FM|SC Josephson junctions <cit.>. Recent experiments <cit.> addressed a large anticrossing between Kittel magnon and so-called Swihart photon in SC-FM multilayers, thereby inspiring the on-chip realization of ultrastrong coupling with magnons.  Silaev <cit.> predicted ultrastrong coupling between propagating magnon with Swihart photon in SC|FM|SC heterostructures, useful for quantum squeezed vacuum states, but focusing on the bulk volume configuration with wave vectors aligned with the saturation magnetization. Ultrastrong coupling comparable to the bare magnon frequency goes beyond the conventional devices with a millimeter-sized magnetic sphere loaded into a metallic microwave cavity <cit.>, in which case similar coupling strength requires much larger magnet/cavity volume ratio <cit.>.  


In this work, we investigate collective modes in the ferromagnetic insulator (FI) sandwiched by two NMs, as illustrated in Fig. <ref>(a). In such NM|FI|NM heterostructure, we predict the coupling between the magnon and photon is ultrastrong with the coupling strength comparable to the bare magnon frequency due to the strong confinement of microwave magnetic field across the heterostructure [Fig. <ref>(b)]. We develop a non-Hermitian quantization scheme with non-Hermitian operators that allows for accounting the Ohmic dissipation, different from the scheme for the LRC circuit (e.g., Ref. <cit.>). By going beyond the perturbation theory, we find unexpectedly that the eddy-current-induced damping favors the EPs with the coalescence of both eigenstates and eigenvalues at a specific wave vector <cit.>. Surprisingly, such EPs in this heterostructure persistently exist in the wave-vector space when the thickness of the FI is thicker than a “critical" one. Recently, EPs have been experimentally demonstrated in magnonic systems <cit.>, but the realization of such singularity lies in the perturbation theory and needs fine-tuning of the parameters. Finally, we extend the formalism to the SC|FI|SC Josephson junction by revealing a close connection between the two systems: the formalisms with NMs connect with those with SCs by simply replacing the skin depth δ with London's penetration depth λ. We find strong anisotropy in the coupling: the coupling is ultrastrong in the bulk volume configuration (the wave vector k∥ẑ in Fig. <ref>) but vanishes in the  Damon-Eshbach case (k⊥ẑ) due to the chirality of magnons. 



This article is organized as follows. In Sec. <ref>, we introduce the model and the basic equations for solving the collective modes in the ferromagnetic heterostructures. We focus on the NM|FI|NM heterostructure in Sec. <ref> and predict the robust non-Hermitian nodal polariton. Section <ref> shows the anisotropy in the magnon-photon ultrastrong coupling when the NMs become superconducting at low temperatures. We summarize our results and give an outlook in Sec. <ref>.
 




§ MODEL AND FORMALISM

 

We consider a FI of thickness 2d_F sandwiched by two thick NMs or SCs of thickness ≫λ, where λ∼ O(100  nm) is the penetration depth of microwaves in the NMs or SCs. The interfacial exchange interaction that induces additional damping of magnons due to the spin pumping effect <cit.> can be suppressed by inserted thin insulators without affecting the long-range dipolar interaction. The in-plane magnetic field H_0ẑ along the ẑ-direction biases the saturation magnetization M_s, as shown in Fig. <ref>(a).  For a thin ferromagnetic film, the magnetization fluctuation of long wavelength is nearly uniform across the film normal x̂-direction. In the linear response regime, we focus on the transverse fluctuation M_x(y)( r,t)= M_x(y) e^i  k·ρ-iω t with amplitude M_x(y), wave vector k=k_yŷ+k_zẑ, and frequency ω, where ρ=yŷ+zẑ.  The transverse magnetization fluctuation radiates the electric and magnetic fields, captured by Maxwell's equations <cit.>. 



The electric field obeys Maxwell's equations <cit.>


    ∇× E( r, t)=-∂ B( r, t)/∂ t,
       ∇× H( r, t)= J( r, t)+ϵ_r∂ E( r, t)∂ t,


where ϵ_r is the dielectric constant depending on the materials: ϵ_r=ϵ_ FI in the FI, while in the SC and NM, ϵ_r=ϵ_0 is the dielectric constant in the vacuum. 
J( r, t) is the electric current.  In the FI, J( r, t)=0, so taking the curl of Eq. (<ref>) and substituting it into (<ref>) yields 

    in FI:   ∇^2 𝐄( r, t)+k^2_0 𝐄( r, t)
    =-iωμ_0  ∇× M( r,t),
 
where μ_0 is the  vacuum permeability and k_0≡ω√(μ_0ϵ_ FI). Equation (<ref>) implies that the “magnetization current" ∇× M radiates the electric field.
In the NM, the electric field drives the electric current according to Ohm's Law J( r, t)=σ_c  E( r, t),
where σ_c is the electric conductivity.
The electric field in the NM is governed by 

    in NM:    ∇^2 𝐄( r, t)+k_n^2𝐄( r, t)=0,

 where k_n=√(ω^2μ_0ϵ_0+iωμ_0σ_c). 
 
 
 When the NM enters into the superconducting states at low temperatures, the electric field, governed by London's equation ∂ J( r,t)/∂ t= E( r,t)/(μ_0λ^2), drives the “supercurrent" J( r,t)=σ_s E( r,t) that is out of phase of the electric field since the “conductivity" σ_s=i/(ωμ_0λ^2) of SCs is purely imaginary when far below the superconducting transition temperature T_c, such that the electric field inside the SCs obeys
 
    in SC:    ∇^2 𝐄( r, t)+k_s^2𝐄( r, t)=0,

 where k_s=√(ω^2μ_0ϵ_0-1/λ^2) depends on the London's penetration depth λ. 



With the spin waves M_x(y)= M_x(y)e^i k·ρ-iω t, we assume the formal solution 
 E( r,t)=Ẽ(x)e^i k·ρ-iω t, in which the amplitudes 
 

    in FI:  Ẽ(x)= U + E_0e^i  A_k x+ E_0'e^-i  A_k x,
       in NM/SC “1" (x>d_F):  Ẽ(x)= E_1e^i  B_k x,
       in NM/SC “2" (x<-d_F):  Ẽ(x)= E_2e^-i  B_k x,


where A_k=√(k_0^2-k^2) and in the NM (SC) B_k=√(k_n(s)^2-k^2). In Eq. (<ref>), U=ωμ_0(-k_z  M_yx̂+k_z M_xŷ -k_y  M_xẑ)/ A_k^2 is the special solution of Eq. (<ref>). 



The oscillating electric field converts to the magnetic field according to Faraday's Law, i.e., iωμ_0[ H( r,t)+ M( r,t)]=∇× E( r,t), leading to 

     
    H_y=1/(i ωμ_0)(∂_z E_x-∂_x E_z)-M_y,
       H_z=1/(i ωμ_0)(∂_x E_y-∂_y E_x),
       H_x=1/(i ωμ_0)(∂_y E_z-∂_z E_y)-M_x.

 




The coefficients in Ẽ(x) are then solved by the boundary conditions of the electric and magnetic fields at material interfaces. The in-plane component E_∥ is continuous at the interfaces. On the other hand, for the “good" NM conductors and SCs, the characteristic relaxation time of electrons τ≪1/ω, implying no charge accumulation is built inside the conductors and hence ∇· E( r,t)=0 therein. Then by the continuity of E_∥,  ∂_xE_x( r,t) is continuous at the interfaces <cit.>.  For the magnetic field, at the interfaces B_⊥ and H_∥ are continuous <cit.>.


To solve E_x( r,t), we need another boundary condition, however. The x-component of Eq. (<ref>) obeys
iωμ_0(ik_yH_z-ik_zH_y)=(iωμ_0σ_c(s)+ω^2μ_0ϵ_r)E_x.
Then the continuity of H_y and H_z implies that the current  (iωμ_0σ_c(s)+ω^2μ_0ϵ_r)E_x is continuous at the interfaces.
Accordingly, we find inside the FI   

    E_0x= E_0x'=R_kU_x(1-R_k)e^i A_k d_F-(1+R_k)e^-i A_k d_F,

where R_k=[( A_k^2+k^2) B_k]/[( B_k^2+k^2) A_k] depend on the surroundings. 



For E_y, from the continuity of E_y and H_z, we find

    E_0y   =k_y(Ẽ_x|_d_F^--Ẽ_x|_d_F^+)( B_k+ A_k)e^-i  A_k d_F-( B_k- A_k)e^i  A_k d_F
       - B_k U_y( B_k+ A_k)e^-i  A_k d_F+( B_k- A_k)e^i  A_k d_F,
     E_0y'   =- B_k U_y( B_k+ A_k)e^-i  A_k d_F+( B_k- A_k)e^i  A_k d_F
       -k_y(Ẽ_x|_d_F^--Ẽ_x|_d_F^+)( B_k+ A_k)e^-i  A_k d_F-( B_k- A_k)e^i  A_k d_F,

in which,  according to Eq. (<ref>) and the continuity of H_y and H_z, Ẽ_x|_d_F^-=U_x+ E_0xe^i  A_k d_F+ E'_0xe^-i  A_k d_F and Ẽ_x|_d_F^+=(iωμ_0σ_c(s)+ω^2μ_0ϵ_0) /(ω^2μ_0ϵ_ FI)Ẽ_x|_d_F^-. These components depend explicitly on k_y and are thereby anisotropic. 
Similarly, we solve the amplitudes in E_z as

    E_0z   =-1k_z[ A_k R_k U_x(1-R_k)e^i A_k d_F-(1+R_k)e^-i  A_k d_F.
       +k_y^2(Ẽ_x|_d_F^--Ẽ_x|_d_F^+)( B_k+ A_k)e^-i  A_k d_F-( B_k- A_k)e^i  A_k d_F
       .-k_y B_k U_y( B_k+ A_k)e^-i  A_k d_F+( B_k- A_k)e^i  A_k d_F],
     E_0z'   =1k_z[
     A_k R_k U_x(1-R_k)e^i A_k d_F-(1+R_k)e^-i  A_k d_F.
       +k_y^2(Ẽ_x|_d_F^-
        -Ẽ_x|_d_F^+)( B_k+ A_k)e^-i  A_k d_F-( B_k- A_k)e^i  A_k d_F
       +.k_y B_k U_y( B_k+ A_k)e^-i  A_k d_F+( B_k- A_k)e^i  A_k d_F].



With the solution of electric fields Eqs. (<ref>), (<ref>), and (<ref>), we find the magnetic field via Eq. (<ref>).
For the thin FI, the radiated magnetic field has a long wavelength and is, therefore, nearly uniform across the thickness. We thereby take its value at x=0 as the averaged one, i.e., H_x,y=H_x,y|_x=0= H_x,ye^i k·ρ-iω t, where H_x,y are the amplitudes. According to Eqs. (<ref>) and (<ref>), the amplitudes

    H_y   =- M_y+(k_z/ωμ_0)(U_x+2 E_0x)
       +2 A_k/ωμ_0k_z[ A_k R_k U_x/(1-R_k)e^i A_kd_F-(1+R_k)e^-i A_kd_F.
       .+k_y^2(1-k_0^2/k_n(s)^2)(U_x+ E_0x(e^i A_kd_F+e^-i A_kd_F))/( B_k+ A_k)e^-i A_kd_F-( B_k- A_k)e^i A_kd_F],
     H_x   =- M_x+(k_y/ωμ_0)
       ×[U_z+2k_y B_kU_y/k_z(( B_k+ A_k)e^-i A_kd_F+( B_k- A_k)e^i A_kd_F)]
       -k_z/ωμ_0[U_y-2 B_kU_y/( B_k+ A_k)e^-i A_kd_F+( B_k- A_k)e^i A_kd_F],

in which the amplitudes of magnetization M_x,y is solved by combining the Landau-Lifshitz equation. 




§ NM|FI|NM HETEROSTRUCTURE

 



 §.§ Classical approach


We can significantly simplify the amplitudes of the magnetic field  (<ref>) when the frequency ω of the collective modes in the NM|FI|NM heterostructure is low, and their wavelength 2π/k is long.
When k<10^4 rad/m and ω≪ω_c, where ω_c≡σ_c/ϵ_0 is the plasma frequency in the NMs <cit.>, B_k=√(ω^2μ_0ϵ_0+iωμ_0σ_c-k^2)≈√(iωμ_0σ_c)= i/δ, where δ=(1+i)/√(2ωμ_0σ_c) is the penetration depth of microwaves in NMs that contains both the real and imaginary components. For typical conductivity of copper σ_c=5.96× 10^7 Ω^-1· m^-1, the approximation is suitable as long as  ω≪ 10^9 GHz. Furthermore, when k<10^4 rad/m and d_F≲ 1 μm, | A_k|=|√(k_0^2-k^2)|<k and | A_kd_F|<k d_F≪ 1.
Therefore, for thin FI layers, we are allowed to expand e^± i A_kd_F  to the leading order 1± i A_kd_F. 
The amplitudes of magnetic fields (<ref>) are then reduced to

    H_x=- M_x1-k_0^2d_Fδ1- A_k^2d_Fδ≈ - M_x,
        H_y=- M_yd_Fk_0^2δ+ A_k^2d_F(k_0^2-d_F/δ+d_Fk_y^2),

such that H_x is out of phase π to M_x, while H_y contains the component out of phase π/2 to M_y due to the imaginary component of δ. 


By substitution of Eq. (<ref>) into the (linearized) Landau-Lifshitz equation
    
    -iω M_x+μ_0γ H_0M_y-μ_0γ M_sH_y    =0,
    μ_0γ H_0 M_x-μ_0γ M_sH_x+iω M_y   =0,

 we arrive at the coupled equation 

    ([         0 -μ_0γ M_s       -iω  μ_0γ H_0; -μ_0γ M_s         0  μ_0γ H_0        iω;         1         0         1         0;         0       α_k         0      β_ k;           ])
    ([ H_x; H_y; M_x; M_y ])=0,

where α_k=k_0^2(δ+d_F)-k^2d_F and β_ k=k_0^2d_F-d^2_F/(δ+d_F)k_y^2 are both complex numbers. It leads to the characteristic equation for solving the frequency of the collective modes  

    k^2+K_ωd_F/d_F +δk_y^2=k_0^2(1+K_ω+δ/d_F),

where K_ω=(μ_0^2γ^2 M_s(M_s+H_0))/(μ_0^2γ^2 H_0(M_s+H_0)-ω^2).


We resort to the numerical calculation to solve Eq. (<ref>).
Since the penetration depth δ of microwaves in the NMs is a complex quantity, the solution of (<ref>) for the collective-mode dispersion is also a complex number. The complex component of ω represents the inverse lifetime of collective modes.  In our system, the dissipation is introduced by the Ohmic dissipation of the eddy current in the NM layers. 





 §.§ Quantum scheme with Ohmic dissipation


In the NM|FI|NM heterostructure, the photon modes are well confined between two NM layers, acting as an ultrathin microwave waveguide. Magnon modes in the FI are also renormalized since the stray magnetic field is changed due to the back-and-forth reflection by the NMs according to Eq. (<ref>). 
The interaction between magnon and photon in the ferromagnetic heterostructure can be conveniently understood using the quantum approach. However, the Ohmic dissipation in the  NM|FI|NM heterostructure presents a challenge that requires a different quantum scheme from conventions.  


In the long wavelength limit, the eigenfrequency of the magnon modes is captured by Kittel's formula <cit.>  

    ω_N=μ_0γ√((H_0+N_xxM_s)(H_0+N_yyM_s)),

where N_xx and N_yy are the “demagnetization factors". Without the NMs, N_xx=1 and N_yy=0 are real numbers for magnetic films. Nevertheless, the NMs reflect and dissipate the magnetic stray field, rendering the demagnetization factors to be complex numbers. According to Eq. (<ref>), when k→0, 

    N_xx=1,   N_yy=d_F/(δ+d_F).

So N_xx is a constant, while N_yy depends on the collective-mode frequency ω via δ. 
By substitution to Kittel's formula (<ref>), the eigenfrequency of the magnon mode in the NM|FI|NM is strongly renormalized to be  
 
    ω_N=μ_0γ√((H_0+M_s)(H_0+d_F/δ+d_FM_s)),

 which is a complex number and cannot be described by the Gilbert phenomenology, different from that of the NM|FM bilayer <cit.>. Such ferromagnetic resonance (FMR) with significant shift and broadening can be conveniently measured in microwave absorption experiments. 


 
Combining the Landau-Lifshitz equation (<ref>) and the normalization condition ∫ d𝐫(M_xM_y^*-M_x^*M_y )=-i/2 for the magnon modes <cit.>, we find the normalized amplitudes

    M_x=i√(ω_N/8d_Fμ_0γ(H_0+M_s)),
        M_y=-√(μ_0γ(H_0+M_s)/8d_Fω_N),

with which we expand the magnetization operators as  
    M̂_x   =-i∫d k/2π√(ħ M_sω_N/4d_Fμ_0(H_0+M_s))(e^i k·ρm̂_ k-e^-i k·ρm̂_ k^†),
    M̂_y   =∫d k/2π√(ħμ_0γ^2(H_0+M_s)M_s/4d_Fω_N)(e^i k·ρm̂_ k+e^-i k·ρm̂_ k^†),

where  m̂_ k^† and m̂_ k are the creation and annihilation operator of magnons. In this ansatz, M̂_x,y are no longer Hermitian operators since ω_N is not purely real, different from the conventional quantization scheme. 



To derive the bare photon modes in the heterostructure, we construct an NM|non-magnetic-insulator(I)|NM  heterostructure, in which the thickness of the non-magnetic insulator is also set to be 2d_F. The electric fields are governed by

    in I:    ∇^2 𝐄( r, t)+k_0^2 𝐄( r, t)=0,
    in NMs:    ∇^2 𝐄( r, t)+k_n^2 𝐄( r, t)=0.

Given the isotropy in the propagation y-z plane, without losing generality, we consider the wave propagation along the ẑ-direction. We assume the plane-wave solution E=Ẽ(x)e^ikz-iω t, where the amplitudes

    In I,  Ẽ(x)= E_0e^i A_k x+ E'_0e^-i A_k x,
       In NM (x>d_F),  Ẽ(x)= E_1e^i B_k x,
       In NM (x<-d_F),  Ẽ(x)= E_2e^-i B_k x,

in which A_k=√(k_0^2-k^2) and B_k=√(k_n^2-k^2). According to Faraday's Law,  the magnetic field H_y=1/(i ωμ_0)(∂_z E_x-∂_x E_z). E_z and ∂_x E_x at the interfaces are continuous, leading to the secular equation for the components { E_0x, E'_0x}:  

    |
        
        (R_k-1)e^i A_k d_F    (R_k+1)e^-i A_k d_F
    
        (R_k+1)e^-i A_k d_F    (R_k-1)e^i A_k d_F|=0,

i.e., 

    i(R_k^2+1)sin(2 A_kd_F)=2R_kcos(2 A_kd_F).

Since the wavelength 2π/| k| is much larger than the insulator thickness d_F, sin(2 A_kd_F)≈2 A_kd_F and cos(2 A_kd_F)≈1. Further,  when the frequency  ≲ 10 GHz, ωμ_0σ_c≫ω^2μ_0ϵ_0 for the NMs, such that B_k≈ i/δ and 

    R_k=( A_k^2+k^2) B_k/ A_k( B_k^2+k^2)≈-ik_0^2/ A_kδ≪1
        .

Accordingly, Eq. (<ref>) is simplified as 

    k^2=Ω_n^2μ_0ϵ_ I(1+δ/d_F),

where ϵ_ I is the dielectric constant of the insulator, from which 

    Ω_n=√(d_F/(d_F+δ)μ_0ϵ_ I)| k|

is the dispersion of photon modes in the NM|I|NM heterostructure, which is no longer purely real since δ contains the imaginary component. They correspond to the TM modes with E_y=H_x=H_z=0 and only have three components {E_z,E_x,H_y}. 


 When -d_F<x<d_F, the electromagnetic fields derived from Eq. (<ref>) read

    Ẽ_x,k(x)   = E_0xe^i A_k x+ E_0x'e^-i A_k x,
    Ẽ_z,k(x)   = A_k/k( E_0xe^i A_k x- E_0x'e^-i A_k x),
    H̃_y,k(x)   =ωϵ_ I/k( E_0xe^i A_k x+ E_0x'e^-i A_k x),

where E_0x'=(1-R_k)/(1+R_k)e^2i A_kd_F E_0x≈ E_0x, and  with the dispersion of the photon mode,   (<ref>)

    A_k   =√(k_0^2-k^2)=ik√(δ/d_F+δ),
     B_k   =√(k_n^2-k^2)=i√(1/δ^2+δ/d_F+δk^2)≈i/δ.

Accordingly, the fields in the insulator are simplified as 

    Ẽ_x,k(x)   ≈ E_0x(e^i A_k x+e^-i A_k x)≈ 2 E_0x,
    Ẽ_z,k(x)   ≈ A_k/k E_0x(e^i A_k x-e^-i A_k x)≈ i2 A_k^2/k E_0xx,
    H̃_y,k(x)   ≈ωϵ_ I/k E_0x(e^i A_k x+e^-i A_k x)≈2ωϵ_ I/k E_0x.
 
Thereby, H_y≡ H_0y and E_x≡ E_0x are uniform across the thin insulator layer.
On the other hand, when x > d_F and x < -d_F, the electromagnetic fields

    Ẽ_x,k(x)   =ϵ_ I/ϵ_0+iσ_c/ωẼ_x,k(± d_F^∓)e^± i B_k (x∓ d_F),
    Ẽ_z,k(x)   =Ẽ_z,k(± d_F)e^± i B_k (x∓ d_F),
    H̃_y,k(x)   =H̃_y,k(± d_F)e^± i B_k (x∓ d_F),

where “+" and “-" stand for x>d_F and x<-d_F, respectively. Therefore, H_y≈ H_0ye^- sgn(x)(x- sgn(x) d_F)/δ outside of the insulator. 



Figure <ref>(a) addresses the skin effect of the electric field E_z, generated by waves propagating along the ẑ-direction, with a finite leakage into the adjacent NMs. Such electric field drives the eddy currents that exhibit an oscillatory decay over the skin depth |δ| from the NM|I interfaces. In contrast, the x-component of the electric field is discontinuous because of the discontinuity of the dielectric properties across the interfaces, which results in the accumulation of bound charges, as depicted in Fig. <ref>(b).




With the photon dispersion (<ref>), we now quantize the photon magnetic fields in terms of operators.  To this end, we interpret our model by LC circuits, as shown in Fig. <ref>(a). We note again that since the NM|I|NM heterostructure is isotropic in the y-z plane, without losing generality, we focus on the wave propagation along the ẑ-direction. To convert our model system to the equivalent LC circuit representation, we introduce the voltage V across the thickness of the heterostructure and total current I per unit length along the  ŷ-direction. 
From the spatial profile of the electric fields in Fig. <ref>(a) and (b), the voltage is only contributed by the x-component of the electric field, i.e., V=∫ dx E_x=2E_0xd_F. The current I=∮ B· d l follows Ampère's circuital Law, for which we select an integration route traversing the insulator in the x-y plane. The width of this route along the ŷ-direction is set to the unit length and extends to infinite along the x̂-direction. Considering the localization of the magnetic stray field, we arrive at I=H_0y. 


An integration over Maxwell's equation ∇×𝐄 = -∂𝐁/∂ t leads to 

    ∫^+∞_-∞ dx∂_z E_x=-∫^+∞_-∞ d x Ḣ_y,

where from Eq. (<ref>) ∫_-∞^-d_F d x H_y=∫_d_F^+∞ dx H_y=H_0yδ. Then Eq. (<ref>) is simplified to be 

    2∂_z E_0xd_F=-2Ḣ_0y d_F(1+δ/d_F).

Substituting E_0x and H_0y with V and I, Eq. (<ref>) becomes 

    ∂_z V=-2μ_0d_F(1+δ/d_F)İ.

Comparing Eq. (<ref>) with the circuit equation ∂_z V = -L İ, we find the effective inductance 

    L=2μ_0d_F(1+δ/d_F),

which is not purely real since δ is a complex number. Additionally, substituting E_0x and H_0y with V and I to Maxwell's equation ∂_z H_0y=-ϵ_ IĖ_0x yields ∂_z I=-ϵ_ I/(2d_F) V̇. Comparing again with the circuit equation ∂_z I=-C V̇, we find the effective capacitance 

    C=ϵ_ I/(2d_F).
 


With the effective inductance (<ref>) and capacitance (<ref>), we are now allowed to express the Hamiltonian of NM|I|NM heterostructure to the effective one 

    Ĥ=∫ dz 1/2( LÎ^2+C V̂^2).

In the wave vector space, k=kẑ, and 

    Ĥ=∫ dk 1/2(L Î_k Î_-k+CL^2/k^2Î̇_kÎ̇_-k),

where Î_k(t)=∫ dz/√(2π)Î(z,t) e^-ikz.


We then define the canonical conjugate variables:  q̂_k = Î_k represents the generalized coordinate and π̂_k = CL^2Î̇_k/k^2 represents the canonical momenta. With them, we express the effective Hamiltonian (<ref>) of the system as

    Ĥ=∫ dk(m_k/2Ω_n^2 q̂_k q̂_-k+1/2m_kπ̂_k π̂_-k),

where m_k=CL^2/k^2.


We now define the annihilation and creation operators of photon modes as 

    p̂_k=1/√(2m_kħΩ_n)(m_kΩ_n q̂_k + i π̂_-k),
    p̂_k^†=1/√(2m_kħΩ_n)(m_kΩ_n q̂_-k + i π̂_k),

which obey the bosonic commutation relation [p̂_k, p̂_k'^†] = δ(k-k'). Inversely, 

    q̂_k=√(ħ/2m_kΩ_n)(p̂_k+p̂_-k^†),
       π̂_k=i√(m_kħΩ_n/2)(p̂_k^†-p̂_-k).

We note again in this quantization scheme that q̂_k and π̂_k are no longer Hermitian since the photon dispersion Ω_n is not purely real due to the Ohmic dissipation. 
This allows us to express the quantized magnetic field Ĥ regarding the photon creation and annihilation operators, i.e.,

    Ĥ   =Ĥ_yŷ =∫ dk /√(2π)q̂_k e^ikzŷ
       =∫ dk/√(2π)√(ħ/2m_kΩ_n)(p̂_ke^ikz+p̂_k^† e^-ikz)ŷ.

For the waves propagating with arbitrary wave vector k, we express their magnetic field operators by extending Eq. (<ref>) as

    Ĥ_y   =∫d k/2π √(ħ/2m_kΩ_n)cosθ_ k( e^i k·ρp̂_ k+e^-i k·ρp̂_ k^†),
    Ĥ_z   =-∫d k/2π √(ħ/2m_kΩ_n)sinθ_ k(e^i k·ρp̂_ k+e^-i k·ρp̂_ k^†).




In terms of the magnetization operator M̂ (<ref>) and the magnetic-field operator Ĥ (<ref>), 
the Zeeman interaction describes the interaction between the magnon and photon 

    Ĥ_ int   =-μ_0∫ d rM̂·Ĥ=-μ_0∫ d rM̂_yĤ_y
       =∫ d k g_ k(m̂_ kp̂_ k^†+m̂_ k^†p̂_ k-m̂_ k^†p̂_ -k^†-m̂_ kp̂_ -k),

where the coupling constant

    g_ k/ħ=cosθ_ kħγ/2√(μ_0(H_0+M_s)M_s/2m_kΩ_nω_Nd_F)

strongly depends on the propagation direction of the collective modes. 
The total Hamiltonian of the system

    Ĥ_ tot   =∫ d k (ħω_N(m̂_ k^†m̂_ k+1/2)+ħΩ_n(p̂_ k^†p̂_ k+1/2).
       .+g_ k(m̂_ kp̂_ k^†+m̂_ k^†p̂_ k-m̂_ k^†p̂_ -k^†-m̂_ kp̂_ -k))

is non-Hermitian since g_ k g_ k^*. This Hamiltonian goes beyond the perturbation theory without invoking the rotating-wave approximation. 
The 4× 4 matrix Hamiltonian (<ref>) under the ultrastrong coupling regime <cit.> leads to the (formal) dispersion relation

    ω_±^2=1/2(Ω_n^2+ω_N^2±√((Ω_n^2-ω_N^2)^2+16Ω_nω_Ng_ k^2/ħ^2)).
  
By employing the Bogoliubov transformation, we can recover the dispersion relation for our system 

    ω^4-ω^2(ω_N^2+Ω_n^2)+ω_N^2Ω_n^2-4ω_NΩ_ng_ k^2/ħ^2=0,

which is exactly the same as Eq. (<ref>) by the classical approach. 





 §.§ Nodal magnon-photon polariton


We now turn to address the properties of the collective modes in the NM|FI|NM heterostructure by performing the numerical calculation. We consider the typical magnetic insulator—yttrium iron garnet (YIG)—with the saturation magnetization μ_0 M_s=0.17 T and relative dielectric constant ϵ_ FI=8ϵ_0 <cit.>, sandwiched by two thick copper films with the conductivity σ_c=5.96× 10^-7 Ω^-1· m^-1. The bias magnetic field μ_0 H_0=50 mT is applied along the in-plane ẑ-direction, as shown in Fig. <ref>. Ferromagnetic insulator EuS is also the material of choice that has been exploited in FI|SC|FI heterostructure <cit.>. In the following, we address persistent non-Hermitian topological phenomena that appear in the wave vector space for the coupled magnon and photon modes, driven by the eddy current in the NMs that induces additional anisotropic dissipation Δα( k).  




For the collective modes propagating parallel to the magnetization with θ_ k=0,  we numerically solve the eigenfrequency ω̃_ k=ω_ k-iΓ_ k according to Eq. (<ref>), which contain the real ω_ k and imaginary Γ_ k components.
Such a dispersion has two branches, namely, the upper branch ω^ upper _ k and the lower branch ω^ lower _ k, with the frequency gap δω=min(ω^ upper _ k-ω^ lower _ k) between the two branches minimized at the wavevector k=k_cẑ, as shown in Fig. <ref>(a). The upper branch by the red curves approaches the photon-mode frequency (the dotted curves) for large k and recovers to the Kittel frequency (the dashed curves) when k→0. 
On the other hand, the lower branch by the blue curves reaches the Kittel frequency for large k and recovers to the photon mode when k→ 0.
The frequency gap δω between these two branches is sensitive to the thickness d_F of the FI.
When d_F=50 nm, these two branches are gaped with δω>0; nevertheless, when d_F=30 nm they are gapless with δω=0 at a finite wave vector.  
Analogous to the frequency gap δω, the damping gap δΔα=Δα_k_c^ upper-Δα_k_c^ lower is defined as the difference of the additional damping of the two branches at k=k_cẑ, as plotted in Fig. <ref>(b). In contrast to the real part ω_ k of the frequency, the additional damping  Δα_ k=Γ_ k/ω_ k of the upper and lower branches are gapless with δα=0 when d_F=50 nm but are gapped with δα>0 when d_F=30 nm [Fig. <ref>(b)].





Thereby, we expect a specific FI thickness d_F exists that renders the simultaneous vanished δω and δΔα, implying the collapse of two collective modes with the same real and imaginary eigenvalues. 
 Such a “critical point" implies the existence of the EPs or the exceptional nodal phase since it appears in the wave vector space. To trace the EPs, we plot in Fig. <ref>(c) and (d) for the thickness dependence of ω_k_c and Δα_k_c of the upper and lower branches with d_F∈ [20,70] nm. The EP appears when d_F≈46 nm, which is experimentally feasible.



It appears that the emergence of the EPs needs the fine-tuning of parameters. For particular propagation directions, this is indeed the case, but the EPs can appear at other wave numbers. Although the damping of the bare photon and magnon modes is independent of wave propagation direction θ_ k, the damping of collective modes depends on θ_ k since the magnon-photon coupling g_ k [Eq. (<ref>)] is anisotropic. According to Eq. (<ref>), when θ_ k=π/2, g_ k=0, so the photon and magnon modes decouple; the coupling is maximal when θ_ k=0.
Accordingly, although it appears above that the EPs depend on the specific thickness of the FIs, the anisotropy in the eddy-current-induced damping may render the EPs to appear persistently at the other wave vectors when the collective modes propagate along other directions, as shown below.



In Fig. <ref>(a), we compare the dispersion of the collective modes when the spin waves propagate parallel (θ_ k=0)  or skew (θ_ k=π/6) to the saturation magnetization when  d_F=50 nm. In this configuration, when θ_ k=0, the upper and lower branches of the collective modes are close yet separated by a gap, while when  θ_ k=π/6, the two branches intersect without any gap. Figure <ref>(b) presents the associated additional damping: when θ_ k=0, the additional damping of the two collective modes are the same, while they become different when θ_ k=π/6. These imply the existence of the EPs in the wave vector space since the wave vector continuously changes. 
Indeed, we demonstrate in Fig. <ref>(c) and (d) the existence of closure of the frequency gap and damping gap simultaneously at θ_ k≈π/10. 



The conditions of EPs with ultrastrong coupling go beyond the perturbation theory and differ from those with strong coupling.  
In the context of the 2× 2 matrix Hamiltonian after the rotating-wave approximation under the strong coupling, the condition for the emergence of EPs is given by <cit.>

    (ω_N-Ω_n)^2+4g_ k^2=0.
 
However, when substituting the parameters in our system with which EPs are unexpected to emerge, Eq. (<ref>) is inaccurate in predicting the occurrence of EPs.
Considering 
the Hamiltonian Eq. (<ref>) the EPs appear when 

    (Ω_n^2-ω_N^2)^2+16Ω_nω_Ng_ k^2/ħ^2=0.





Figure <ref> addresses the persistent existence of the EPs in the wave-vector space when the thickness of the FIs exceeds a critical value d_F^(0), as shown in Fig. <ref>(a) and (b). The dashed curve indicates the EPs with both gapless frequency and damping of the collective modes. In other words,  by tuning the parameters {d_F,H_0,θ_ k} of our system, we can discover exceptional lines, as seen in Fig. <ref>. Furthermore, we are allowed to tune the direction of the external bias magnetic field H_0 in our system to persistently discover the position of EPs at a particular propagation direction. 



EPs of magnons have been observed in ferromagnetic multilayers <cit.> as well as in systems combining magnon with cavity photons through the dipolar interaction <cit.>.
The emergence of EPs in the passive magnonic 𝒫𝒯-symmetric device reported in FM|NM|FM heterostructure <cit.> is realized through fine-tuning the gain, dissipation, and the RKKY interaction by changing the NM thickness. 



The coalescence of the two modes is different from the level attraction phenomena <cit.>. The level attraction between two modes was observed in dissipative coupling systems <cit.> when the absolute value of the dissipative coupling strength is larger than the difference of their eigenfrequencies. Within the realm of level attraction, the eigenfrequencies of the two modes become degenerate while their dissipation rates differ. In contrast, the coalescence of two modes is the hallmark of EPs, with both the eigenfrequencies and dissipation rates of the two modes being the same. In this sense, the EPs could be used for the synchronization of two oscillators. 






§ SC|FI|SC JOSEPHSON JUNCTION





 §.§ Classical approach


As an extension of the above formalism, we consider the SC|FI|SC heterostructure. We find most formalism with NMs addressed above can recover that with SCs when replacing the skin depth δ by London's penetration depth λ, although the former is non-Hermitian and the latter is Hermitian. 


Since 1/λ^2≫ k_0^2 when ω≲ 100 GHz, B_k≈ i/λ.  On the other hand, | A_k|∼ k such that for a thin FI,  | A_kd_F|∼ kd_F≪ 1. Accordingly, we expand e^± i A_kd_F=1± i A_kd_F, which simplifies  Eq. (<ref>)  to be 
    
    H_x= -1-k_0^2d_Fλ1- A_k^2d_Fλ M_x≈- M_x,
        H_y= -d_Fk_0^2λ+ A_k^2d_F(k_0^2-d_F/λ+d_Fk_y^2) M_y.

This implies that the out-of-plane B_x is nearly zero in the FI. This is reasonable because the magnetic induction cannot exist inside the SC, and B_x is continuous across the SC|FI interfaces. The SC then mainly modulates the in-plane magnetic field along the ŷ-direction.
On the other hand, the electric field inside the FI according to Eqs. (<ref>), (<ref>), and (<ref>) is reduced to 
    
    Ẽ_x   =-ωμ_0k_zd_F/ A_k^2d_F+k_0^2 λ M_y,
    Ẽ_y   =-ωμ_0λ d_F k_z  M_x-iλ/d_F+λωμ_0 d_F/ A_k^2d_F+k_0^2λk_yk_z M_yx,
    Ẽ_z   =ωμ_0λ d_F k_y  M_x-iωμ_0(k_0^2-d_F/λ+d_Fk_y^2)λk_0^2λ+ A_k^2d_F M_yx.


Combining with the linearized Landau-Lifshitz equation (<ref>),  the dispersion relation of the collective mode is governed by 

    k^2+K_ωd_F/d_F+λk_y^2=k_0^2(1+K_ω+λ/d_F).

We note that replacing λ in Eq. (<ref>) with δ reduces exactly this characteristic equation of the SC|FI|SC Josephson junction to that of the NM|FI|NM heterostructure (<ref>).
From Eq. (<ref>), we obtain two real solutions

    ω^2_u (l)( k)=1/2(Ω_s^2(k)+ω_S^2±√((Ω_s^2(k)-ω_S^2)^2+4Δ_ k^2)),

where 

    Ω_s(k)   =√(d_F/(d_F+λ))ck,
    ω_S   =μ_0γ√((H_0+M_s)(H_0+M_sd_F/(d_F+λ))),

are, respectively, the frequency of the Swihart mode <cit.> between two SCs (refer to Appendix <ref>) and the “renormalized" FMR frequency by the SCs <cit.>, and 

    Δ^2_ k=cos^2θ_ kΩ_s^2(k)μ_0^2γ^2M_s(H_0+M_s)d_F/d_F+λ

relates to the anisotropic coupling between the Swihart and Kittel modes. Here c=1/√(μ_0ϵ_ FI) is the light velocity in the FI. 


The maximal value of Δ_ k^2 appears when cos^2θ_ k=1, i.e., the modes propagating parallel to M_s, resulting in the largest splitting. While when the collective modes propagate perpendicular to the external magnetic field or M_s, cos^2θ_ k=0, the Swihart mode and Kittel mode decouples with  

    ω_u={   ω_S,      k_y≤ k_c
       Ω_s(k_y), k_y>k_c
    ., ω_l={   Ω_s(k_y), k_y≤ k_c
       ω_S,       k_y>k_c
    .,

where k_c=√(1+λ/d_F)ω_S/c is the crossing point of the two modes. 
These features in the coupling between 
the two modes can be understood from the Zeeman interaction -μ_0M_yH_y.
When 𝐤 is perpendicular to M_s, substituting Ω_s into Eq. (<ref>) yields H_y = 0. Consequently, the Zeeman interaction between the magnon and photon modes vanishes. On the other hand, when 𝐤 is parallel to M_s, k_y = 0, and H_y reaches its maximum according to Eq. (<ref>). As a result, the Zeeman interaction between the two modes is maximized in the bulk volume configuration.


When k→ 0, the frequencies of the two branches are reduced to

    ω_u   =ω_S,
    ω_l   =√(H_0(d_F+λ)+M_sd_Fsin^2θ_ k/H_0(d_F+λ)+M_sd_Fd_F/d_F+λ)ck→ 0.

Accordingly, ω_u(k→ 0) is the shifted FMR <cit.>, while ω_l(k→ 0) renders the Swihart photon modes anisotropic and reduces the group velocity. 
On the other hand, when k is large, i.e., ck≫μ_0γ(H_0+M_s), 

    ω_u   =√(d_F/d_F+λ(c^2 k^2+μ_0^2γ^2(H_0+M_s)M_scos^2θ_ k))
       ≈√(d_F/d_F+λ)ck=Ω_s(k), 
    ω_l   =μ_0γ√((H_0+M_s)(H_0+d_F/d_F+λM_ssin^2θ_ k)).

Thereby, ω_u is switched to the Swihart mode, while ω_l becomes the shifted FMR. 






 §.§ Quantum approach



In the SC|FI|SC Josephson junction, the photon modes are well confined between two SC layers, acting as an ultrathin microwave waveguide without introducing dissipation. Such photon modes are described by the Swihart mode <cit.>. 
In the long wavelength limit, similar to the NM case (<ref>), the eigenfrequency of magnon is also given by Kittel's  formula <cit.> 

    ω_S=μ_0γ√((H_0+N_xxM_s)(H_0+N_yyM_s)),

but the demagnetization factors are renormalized 
 to be N_xx=1 are  N_yy=d_F/(d_F+λ) by the SCs. The interaction between magnon and Swihart photon is described by the Zeeman interaction

    Ĥ_ int=-μ_0∫ d r M̂·Ĥ_ Sw,

where M̂ is the magnetization operator for the magnon mode and Ĥ_ Sw is the magnetic-field operator for the Swihart mode. 


Combining the Landau-Lifshitz equation and normalization relation ∫ d𝐫(M_xM_y^*-M_x^*M_y )=-i/2 <cit.>, we find the normalized amplitudes

    M_x=i√(ω_S/8d_Fμ_0γ(H_0+M_s)),
        M_y=-√(μ_0γ(H_0+M_s)/8d_Fω_S).

The magnetization is then quantized as  

    M̂_x   =∫d k/2π(-i√(ħ M_sω_S/4d_Fμ_0(H_0+M_s))e^i k·ρm̂_ k+ H.c.),
    M̂_y   =∫d k/2π(√(ħμ_0γ^2(H_0+M_s)M_s/4d_Fω_S)e^i k·ρm̂_ k+ H.c.).

According to Appendix <ref>, the magnetic field is quantized according to  

    Ĥ_ Sw,y   ≈∫d k/2π (Ω_s/2k√(ϵ_ FIħΩ_s/d_F)cosθ_ k e^i k·ρp̂_ k+ H.c.),
    Ĥ_ Sw,z   ≈∫d k/2π (-Ω_s/2k√(ϵ_ FIħΩ_s/d_F)sinθ_ k e^i k·ρp̂_ k+ H.c.),

where p̂_ k is the annihilation operator of Swihart photons. 
These quantizations lead to  

    Ĥ_ int   =-μ_0∫ d rM̂_yĤ_ Sw, y
       =∫ d k g_ k(m̂_ kp̂_ k^†+m̂_ k^†p̂_ k-m̂_ k^†p̂_ -k^†-m̂_ kp̂_ -k),

where the coupling constant

    g_ k/ħ=cosθ_ kμ_0γΩ_s/2| k|√((H_0+M_s)M_sΩ_sμ_0ϵ_ FI/ω_S)

strongly depends on the propagation direction of the collective modes. 
The total Hamiltonian of the system reads 

    Ĥ_ tot   =∫ d k (ħω_S(m̂_ k^†m̂_ k+1/2)+ħΩ_s(k)(p̂_ k^†p̂_ k+1/2).
       .+g_ k(m̂_ kp̂_ k^†+m̂_ k^†p̂_ k-m̂_ k^†p̂_ -k^†-m̂_ kp̂_ -k)),

which goes beyond the perturbation theory. 



After the Bogoliubov transformation on the Hamiltonian (<ref>), the eigenfrequency of the collective modes is governed by the characteristic equation 

    ω^4( k)-ω^2(Ω_s^2(k)+ω_S^2)+Ω_s^2(k)ω_S^2-4Ω_s(k)ω_Sg_ k^2/ħ^2=0.

Substitution of Ω_s(k) (<ref>) and g_ k (<ref>) into Eq. (<ref>) yields 

    ω^4-ω^2(d_F/d_F+λk^2/μ_0ϵ_ FI+ω_S^2)+ω_S^2d_F/d_F+λk^2/μ_0ϵ_ FI
       -μ_0^2γ^2(H_0+M_s)M_s(d_F/d_F+λ)^2k_z^2/μ_0ϵ_ FI=0,

which is exactly the same as Eq. (<ref>) derived by the classical approach. 


The solution of Eq. (<ref>) is 

    ω_u(l)^2=1/2(Ω_s^2(k)+ω_S^2±√((Ω_s^2(k)-ω_S^2)^2+16Ω_s(k)ω_Sg_ k^2/ħ^2)).

When Ω_s(k)=ω_S, the magnon and Swihart modes cross with each other, and the solutions at the crossing point ω^*_u(l)=√(ω_S^2±2ω_Sg_ k/ħ) defines the anti-crossing gap as 

    Δω   =ω^*_u-ω^*_l
       =√(ω_S^2+2ω_Sg_ k/ħ)-√(ω_S^2-2ω_Sg_ k/ħ).

When g_ k/ħ≪ω_S, Δω≈ 2g_ k/ħ is the same as the solution obtained by the perturbation theory <cit.>, which, however, breaks down when g_ k/ħ is comparable to the bare magnon frequency ω_S. We also note that the stable solution only exists when 2g_ k/ħ<ω_S. Because 2g_ k/ħ=cosθ_ kμ_0γ√((H_0+M_s)M_sd_F/(d_F+λ))<μ_0γ√((H_0+M_s)(H_0+M_sd_F/(d_F+λ)))=ω_S, we confirm that the solution of (<ref>) is always stable.


 The gap Δω approaches a constant when d_F≫λ. In this limit the frequency of the Swihart mode Ω_s→ c| k|, the “renormalized" Kittel frequency ω_S→μ_0γ(H_0+M_s), and the coupling constant g_ k/ħ→cosθ_ kμ_0γ√((H_0+M_s)M_s)/2. Accordingly, 
    Δω|_d_F≫λ   →μ_0γ√((H_0+M_s))
       ×(√((H_0+M_s)+√((H_0+M_s)M_s)).
       .-√((H_0+M_s)-√((H_0+M_s)M_s)))

is free of geometry parameters. 




 §.§ Anisotropic ultrastrong coupling


We then address the parameter dependence in the frequencies of the two collective-mode branches ω_u( k) and ω_l( k) [Eq. (<ref>)] in the SC|FI|SC heterostructure,  which behave as an anti-crossing in the wave-vector space. Figure <ref> plots the anisotropic ultrastrong coupling between the magnon mode and Swihart photon mode at different propagation directions and
addresses the wave-vector and thickness dependencies of the coupling strength to find optimal parameters. In the calculation, the thickness of YIG film d_F=100 nm, the saturation magnetization at low temperatures is enhanced to be μ_0 M_s=0.24 T <cit.>, biased by the external magnetic field μ_0 H_0=50 mT, and ε_ FI=8ε_0 <cit.>. We use the superconducting NbN film with London's penetration depth λ=80 nm at T=0.1T_c∼ 1 K <cit.>.




When the collective mode propagates parallel to the magnetization with θ_ k=0, according to Eq. (<ref>) in the classical approach, the two branches show level repulsion with a frequency gap Δω, as shown in Fig. <ref>(a). For large k, the upper branch by the red curve approaches the high-frequency Swihart photon mode, and when k→ 0, it recovers to the renormalized FMR, which is shifted giantly in comparison to the bare Kittel frequency <cit.>.
On the other hand, the lower branch by the blue curve reaches the bare Kittel frequency for large k and recovers to the Swihart photon mode when k→0. The frequency gap Δω=ω_u-ω_l is defined at the crossing point k= k_cẑ of the renormalized FMR and Swihart-mode frequency. In Fig. <ref>(a), the gap Δω∼ 39.7 GHz is even larger than the bare Kittel frequency but comparable to the renormalized Kittel frequency, showing the ultrastrong coupling between the magnon and Swihart-photon modes. 


These features in the bulk-volume configuration were also revealed recently by Silaev in Ref. <cit.>. References <cit.> found the level repulsion in the S|F|S'|I|S” multilayers, which may be interpreted as the Kittel mode with shifted FMR frequency in S|F|S' couples with the Swihart photon mode in S'|I|S”, mediated by the supercurrent in S'.




Reference <cit.> focuses on the specific bulk-volume configuration with θ_ k=0. Here, we go beyond this configuration by considering the arbitrary propagation direction of the collective modes. As shown in Fig. <ref>(b) with θ_ k≠0, the anti-crossing gap, or the level repulsion strongly depends on the propagation direction. When θ_ k=π/4, the frequency gap Δω at | k|=k_c becomes smaller than that in Fig. <ref>(a). Furthermore, when the collective modes propagate normally to the magnetization with θ_ k=π/2, the crossing of the upper and lower branches is not gapped, i.e., the magnon mode and Swihart photon mode decouples since g_ k=0 [Eq. (<ref>)]. Figure <ref>(c) summarizes the dependence of the anti-crossing gap Δω on the propagation direction θ_ k. Since  the coupling constant  2g_k_c/ħ∼cosθ_ k according to Eq. (<ref>), the gap reaches its maximum when θ_ k=0 and becomes zero when θ_ k=π/2 as plotted in Fig. <ref>(c). Furthermore, the gap Δω>2g_k_c/ħ when θ_ k→ 0, exhibiting the feature of ultrastrong coupling, while Δω≈ 2g_k_c/ħ when θ_ k→π/2 such that the perturbation theory applies. 




The thickness of the ferromagnetic film  d_F strongly affects the coupling strength as well when d_F<λ, but becomes saturated to that in Eq. (<ref>) when d_F≫λ, as shown in Fig. <ref>(d) when θ_ k=0. Here, it is natural to expect the coupling strength to vanish when d_F→ 0 since then the FI is absent. For the thick YIG film, d_F≫λ increases the anti-crossing gap up to ∼ 55 GHz, close to reaching the limit [about 56 GHz according to Eq. (<ref>)].  This suggests the optimal thickness of the FI is d_F∼λ, which depends on the temperature since λ=λ(T=0)(1-(T/T_c)^4).  Δω>2g_k_c/ħ suggests again the perturbation theory breaks down under the ultrastrong coupling.


By changing the direction of the magnetic field while fixing the propagation direction of the collective modes, we can either maximize the coupling strength to enter the ultrastrong coupling regime or minimize it to decouple from photons, which is advantageous for broadband microwave filters by the forbidden band <cit.> and exploring quantum effects with high information processing efficiency and tunability <cit.>.





§ DISCUSSION AND CONCLUSION

 

In conclusion, we develop a non-perturbation theory using both classical and quantum approaches to investigate the collective magnon-photon modes in the ferromagnetic heterostructures composed of ferromagnetic insulators sandwiched by either normal metals or superconductors, thereby representing either dissipative or non-dissipative states of electrons. We develop a non-Hermitian quantization scheme with non-Hermitian magnetization and magnetic/electric-field operators to account for the Ohmic dissipation. A simple conversion relation between the two electronic states is revealed by finding most formalisms with normal metals can be extended to those with superconductors by simply replacing the skin depth δ with London's penetration depth λ, although the former is non-Hermitian and the latter is Hermitian.

In the NM|FI|NM heterostructure, we predict the persistent realization of non-Hermitian nodal magnon-photon polariton, where the coupling strength is comparable to the bare magnon frequency. 
Remarkably, the exceptional points at which the eigenvalues and eigenvectors are coalescent persist over a wide range of wave vectors when the ferromagnetic layer is sufficiently thick ∼ 100 nm, without the need for fine-tuning of parameters. The dissipationless nature of the superconductors eliminates the eddy-current-induced damping present in the normal metal case. In the SC|FI|SC Josephson junction, we find that magnon-photon coupling exhibits strong anisotropy. The coupling is ultrastrong when the electromagnetic field propagates along the direction of the saturation magnetization but vanishes when they are normal to each other due to the chirality of the dipolar stray field.  


Exceptional points hold the potential application in the enhancement of sensitivity in magnonic devices, as recently observed in the enhanced magnonic frequency combs <cit.>. Our results suggest that the common ferromagnetic heterostructures can host persistent EPs and anisotropic ultrastrong magnon-photon coupling, which may enable improved sensitivity and performance of the magnonic devices <cit.>. The predicted persistent nodal polariton and strongly anisotropic coupling provide new opportunities to explore the synergies between magnonics and superconducting electronics <cit.>.



This work is financially supported by the National Key Research and Development Program of China under Grant No. 2023YFA1406600, the National Natural Science Foundation of China under Grants No. 12374109 and No. 52201200, as well as the startup grant of Huazhong University of Science and Technology. H.W. acknowledges support from the China Scholarship Council (CSC) under Grant No. 202206020091.






§ SWIHART MODE AND ITS QUANTIZATION



In this appendix, we first review the electromagnetic modes in the SC|I|SC heterostructure. The thickness of the middle non-magnetic insulator is 2d_I. Swihart found that the superconductor strongly modulates the distribution of the electromagnetic field, significantly slowing down the speed of electromagnetic waves, which is now known as the Swihart mode <cit.>.



According to Maxwell's equations <cit.>, in different regions, the electric field obeys 

    in I:      ∇^2 𝐄( r, t)+k^2_0 𝐄( r, t)
    =0,
    in SC:      ∇^2 𝐄( r, t)+k^2_s 𝐄( r, t)
    =0,

where k_0=ω√(μ_0ϵ_ I). Our system is isotropic in the y-z plane, so we consider the electromagnetic waves propagating along the ẑ-direction with wavevector k without losing generality. We assume the formal solution E(x,z,t)=Ẽ(x)e^ikz-iω t+ H.c., where the amplitudes

    in I:  Ẽ(x)= E_0e^i  A_k x+ E_0'e^-i  A_k x,
       in SC “1" (x>d_I):  Ẽ(x)= E_1e^i  B_k x,
       in SC “2" (x<-d_I):  Ẽ(x)= E_2e^-i  B_k x,

where A_k=√(k_0^2-k^2) and B_k=√(ω^2μ_0ϵ_0-1/λ^2-k^2). 
In the non-magnetic insulator, the magnetic field is governed by Faraday's Law, i.e., iμ_0ω H( r,t)=∇× E( r,t), leading to
     
    H_x=1/(i ωμ_0)(∂_y E_z-∂_z E_y),
       H_y=1/(i ωμ_0)(∂_z E_x-∂_x E_z),
       H_z=1/(i ωμ_0)(∂_x E_y-∂_y E_x).

     
The boundary conditions are already described in the main text, with which we find the amplitudes { E_0x,   E_0x',     E_0y,   E_0y'} obey the matrix equation


    (
             (R_k-1)e^i A_k d_I    (R_k+1)e^-i A_k d_I    0     0 
    
             (R_k+1)e^-i A_k d_I    (R_k-1)e^i A_k d_I    0     0 
    
             0     0     ( A_k- B_k)e^i A_k d_I    -( A_k+ B_k)e^-i A_k d_I
    
             0     0     ( A_k+ B_k)e^-i A_kd_I    -( A_k- B_k)e^i A_kd_I
    )(
         E_0x
     E_0x'
     E_0y
     E_0y'
        )=0.


This implies that the x- and y-components of the electric fields decouple, forming two distinct modes. 
The non-zero solutions require the determinant of the coefficient matrix to equal zero. 


The first mode is the TE mode with E_x=E_z=H_y=0. It has three components {H_z,H_x,E_y}. From the secular equation 

    | 
    ( A_k- B_k)e^i A_k d_I    -( A_k+ B_k)e^-i A_k d_I
    
    ( A_k+ B_k)e^-i A_kd_I    -( A_k- B_k)e^i A_kd_I
    |=0,

we find its dispersion relation 

    ω=1/√(μ_0ϵ_ I)√(k^2+1/2(1/λ^2+1/d_Fλ)).

which holds a large cut-off frequency and hence mismatches with the Kittel frequency in the magnetic film. 



The second one is the TM mode with E_y=H_x=H_z=0. This mode has three components {E_z,E_x,H_y}. From the secular equation

    |
        
        (R_k-1)e^i A_k d_I    (R_k+1)e^-i A_k d_I
    
        (R_k+1)e^-i A_k d_I    (R_k-1)e^i A_k d_I|=0,

we find the Swihart-mode frequency <cit.>  

    Ω_s=√(d_I/d_I+λ)1/√(μ_0ϵ_ I)| k|.
  


Next, we quantize the Swihart mode. The Hamiltonian associated with the Swihart mode of the SC|I|SC Josephson junction includes the energy of electromagnetic field as well as the supercurrents <cit.>:

    Ĥ_ Sw=∫ d r(ϵ_r/2Ê^2+μ_0/2Ĥ^2+μ_0/2λ^2Ĵ_ s^2),

where J_ s is the super-current flowing in the superconductor. We expand the electromagnetic field and supercurrent in terms of the photon operators p̂_k as 

    Ê   =∫^+∞_-∞ dk/√(2π)(Ẽ_k(x)e^ikzp̂_k+ H.c.),
    Ĥ   =∫^+∞_-∞ dk/√(2π)(H̃_k(x)e^ikzp̂_k+ H.c.),
    Ĵ_s   =∫^+∞_-∞ dk/√(2π)(J̃_s,k(x)e^ikzp̂_k+ H.c.),

where Ẽ_k(x), H̃_k(x), and J̃_s,k(x) are the amplitudes. The photon creation and annihilation operators obey the commutation relation [p̂_k, p̂^†_k' ]=iħδ(k-k'). 



When -d_I<x<d_I, the electromagnetic field is derived from Eq. (<ref>) and the boundary conditions as

    Ẽ_x,k(x)   = E_0xe^i A_k x+ E_0x'e^-i A_k x,
    Ẽ_z,k(x)   = A_k/k( E_0xe^i A_k x- E_0x'e^-i A_k x),
    H̃_y,k(x)   =ωϵ_ I/k( E_0xe^i A_k x+ E_0x'e^-i A_k x),
    J̃_s,k(x)   =0.

They are incorporated into the electromagnetic field at x>d_I and x<-d_I after applying the continuity conditions for the E_z and H_y at the interfaces:
    
    Ẽ_x,k(x)   =ϵ_ I/ϵ_0+iσ_s/ωẼ_x,k(± d_I^∓)e^± i B_k (x∓ d_I),
    Ẽ_z,k(x)   =Ẽ_z,k(± d_I)e^± i B_k (x∓ d_I),
    H̃_y,k(x)   =H̃_y,k(± d_I)e^± i B_k (x∓ d_I),
    J̃_s,k(x)   =σ_sẼ_z,k(x)=i/ωμ_0λ^2Ẽ_z,k(± d_I)e^± i B_k (x∓ d_I).

Here with the dispersion of the Swihart mode (<ref>)

    A_k   =√(k_0^2-k^2)=ik√(λ/d_I+λ),
     B_k   =√(k_s^2-k^2)=i√(1/λ^2+λ/d_I+λk^2).

When the wavelength of the photon is much larger than the scale of the system, i.e., k≪ 1/λ and k≪ 1/d_I, sin 2 A_k d_I/(2 A_k)≈ d_I, cos 2 A_k d_I ≈ 1, and 

    B_k=i√(1/λ^2+λ/d_I+λk^2)≈i/λ.

Furthermore, the amplitudes

    E_0x'   =( B_k+k^2) A_k-( A_k+k^2) B_k/( B_k^2+k^2) A_k+( A_k^2+k^2) B_ke^2i A_kd_I E_0x
       ≈1+d_I/d_I+λkλ/1-d_I/d_I+λkλ E_0x≈ E_0x.




We normalize E_0x according to 

    1/2∫ dx [ϵ_r(Ẽ_x,kẼ^*_x,k+Ẽ_z,kẼ^*_z,k)
       +μ_0H̃_y,kH̃^*_y,k+μ_0λ^2J̃_s,kJ̃_s,k^*]
        
       ≈ 4ϵ_ I E_0x E_0x^*d_I(1+k_0^2/k^2)+2ϵ_ I E_0x E_0x^*λ
       ×(k_0^2/k^2+1/2k^2d_Iλ(-1/24 A_k^2 d_I^2))
       =8ϵ_ I E_0x E_0x^*d_I=ħΩ_s/2.

Then with relation

    1/2∫ dx [ϵ_r(Ẽ_x,k^(*)Ẽ_x,-k^(*)+Ẽ_z,k^(*)Ẽ_z,-k^(*))
       +μ_0H̃_y,k^(*)H̃_y,-k^(*)+μ_0λ^2J̃_s,k^(*)J̃_s,-k^(*)]
       =4ϵ_ I E_0x^(*) E_0x^(*)d_I(1-k_0^2/k^2(1+λ/d_I))=0

and by substitution of Eqs. (<ref>) and (<ref>) into Eq. (<ref>), we arrive at the Hamiltonian of Swihart mode   
Ĥ_ Sw=(1/2)∫ dk ħΩ_s(p̂_k^†p̂_k+p̂_kp̂_k^†).









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 Paper Title: Femtosecond switching of strong light-matter interactions in microcavities with two-dimensional semiconductors.










	
	

    
    Contributed equally
 	Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
  
  	
    Contributed equally
    Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
    NanoPhotonics Centre, Cavendish Laboratory, Department of Physics, JJ Thompson Ave, University of Cambridge, Cambridge, UK
    
	
	Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy

    
	Department of Physics, University of Exeter, Stocker Road, EX4 4PY, Exeter, UK

   	
    Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
    Department of Mechanical Engineering, Columbia University, New York, NY 10027, USA

    
    Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
    
    
    Dipartimento di Scienze Matematiche, Fisiche e Informatiche, Università di Parma, Parco Area delle Scienze 7/A, 43124 Parma, Italy
    
    
    Department of Physics and Astronomy, University of Sheffield, Hounsfield Road, S3 7RH, Sheffield, UK
    
    
    Department of Physics and Astronomy, University of Sheffield, Hounsfield Road, S3 7RH, Sheffield, UK

  	
	Department of Physics and Astronomy, University of Sheffield, Hounsfield Road, S3 7RH, Sheffield, UK
 
	
	Advanced Materials Laboratory, National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044, Japan
 	
	Advanced Materials Laboratory, National Institute for Materials Science, 1-1 Namiki, Tsukuba, 305-0044, Japan
 
    
	Department of Physics and Astronomy, University of Sheffield, Hounsfield Road, S3 7RH, Sheffield, UK
 
    
	Department of Physics, University of Exeter, Stocker Road, EX4 4PY, Exeter, UK
 
    
    
	Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
 
	
     a.tartakovskii@sheffield.ac.uk
	Department of Physics and Astronomy, University of Sheffield, Hounsfield Road, S3 7RH, Sheffield, UK
 
 	
    giulio.cerullo@polimi.it
	Dipartimento di Fisica, Politecnico di Milano, Piazza Leonardo Da Vinci 32, 20133 Milano, Italy
    CNR-IFN, Piazza Leonardo da Vinci 32, Milano, 20133, Italy
	
	Femtosecond switching of strong light-matter interactions in microcavities with two-dimensional semiconductors
    Giulio Cerullo
    August 5, 2024
==============================================================================================================





Ultrafast all-optical logic devices based on nonlinear light-matter interactions hold the promise to overcome the speed limitations of conventional electronic devices. Strong coupling of excitons and photons inside an optical resonator enhances such interactions and generates new polariton states which give access to unique nonlinear phenomena, such as Bose-Einstein condensation, used for all-optical ultrafast polariton transistors. However, the pulse energies required to pump such devices range from tens to hundreds of pJ, making them not competitive with electronic transistors. Here we introduce a new paradigm for all-optical switching based on the ultrafast transition from the strong to the weak coupling regime in microcavities embedding atomically thin transition metal dichalcogenides. Employing single and double stacks of hBN-encapsulated MoS_2 homobilayers with high optical nonlinearities and fast exciton relaxation times, we observe a collapse of the 55-meV polariton gap and its revival in less than one picosecond, lowering the threshold for optical switching below 4 pJ per pulse, while retaining ultrahigh switching frequencies. 
As an additional degree of freedom, the switching can be triggered pumping either the intra- or the interlayer excitons of the bilayers at different wavelengths, speeding up the polariton dynamics, owing to unique interspecies excitonic interactions. 
Our approach will enable the development of compact ultrafast all-optical logical circuits and neural networks, showcasing a new platform for polaritonic information processing based on manipulating the light-matter coupling.






§ INTRODUCTION


All-optical switches based on nonlinear optical materials have been extensively investigated to overcome the speed limitations of electronic circuits, thanks to their potential to work at much higher frequencies owing to the inherently fast light-matter interactions underlying their operation <cit.>.
Demonstrations of ultrafast all-optical logic gates have been achieved in a plethora of solid state platforms, exploiting optical nonlinearities (χ^(2) and χ^(3)) <cit.> and saturable absorption <cit.>.
Notable examples employed microring resonators <cit.>, plasmonic nanostructures <cit.>, photonic crystals <cit.>, metasurfaces <cit.>, 2D materials <cit.> or even single molecules <cit.>. 
Such devices showcased switching times down to tens of femtoseconds, but usually at the expense of the switching energy or the on/off contrast <cit.>. More recently, an optimal combination of femtosecond switching times and femtojoule operating energies has been obtained in an all-optical nonlinear device based on a lithium niobate waveguide <cit.>, but with millimeter-scale lengths hindering the production of densely on-chip integrated circuits. 
Therefore, combining integration capabilities, full scalability and high performances in all-optical switches is still an open challenge. 


Excitons in semiconducting materials embedded in optical resonators can be used for all-optical switching, as they show a highly nonlinear response when they are in strong coupling (SC) with resonant photons confined in the structure. In such a regime, the rate of coherent energy transfer between the energy-degenerate excitons and photons is higher than the loss rate, and new hybrid light-matter quasiparticles arise, called polaritons <cit.>. The energy splitting of the polariton states (Rabi splitting) is a direct measure of their coupling strength, which is proportional to the quality factor of the resonator and to the exciton absorption cross section.

Harnessing polaritonic nonlinear interactions is of key importance for a broad range of phenomena and applications, such as lasing <cit.>, optical parametric amplification <cit.>, Bose-Einstein condensation (BEC) <cit.>, or for quantum effects (polariton blockade) <cit.>.
A combination of blueshift of the polariton states and gain in photoluminescence intensity occurring above the threshold of BEC has been used as operational principle of all-optical polariton logic devices and neural networks <cit.>. 
Ultrafast optical switching (with switching times ≤ 1 ps) relying on BEC has been demonstrated even at room temperature <cit.>, with the drawback of using high pulse energies to reach the condensation threshold (≈ 30-300 pJ per pulse at least). 

Alternative strategies for ultrafast all-optical switching in the SC regime rely on the modulation of the light-matter coupling strength acting on the exciton absorption, which produces a spectral shift of the polariton states, acting as a gate for the light transmitted/reflected by the device.
Varying strongly the coupling strength would eventually lead to a complete transition from strong to weak coupling regime or viceversa <cit.>. SC can be switched off in strongly coupled optical microcavities comprising GaAs quantum wells through optical saturation of excitons <cit.> or via electrically-tuned charge build-up <cit.>. Alternatively, it can be switched on by optically induced absorption, i.e. for inter sub-band transitions <cit.>. 
However, achieving complete on/off switching cycle below 1 ps acting on the coupling strength in these material platforms is not possible due the long excitons and excited carriers lifetime <cit.>. 

Atomically thin transition metal dichalcogenides (TMDs) are promising nonlinear optical materials <cit.>, where excitons remain stable up to room temperature due to large binding energies and oscillator strength <cit.>. 
Owing to these properties, TMDs can easily enter the SC regime, when integrated in optical resonators <cit.>. Recent studies of TMD polaritons aim to maximise nonlinear interactions going beyond the use of 1s neutral excitons, exploring higher Rydberg excitonic states, charged excitonic complexes, moiré or dipolar excitons <cit.>. Nonlinearities in MoS_2 bilayers have been shown to be particularly pronounced, due to a combination of Coulomb dipole-dipole interactions, phase space filling and interspecies interactions between hybridized intra- and interlayer excitons <cit.>. Moreover, the fast dynamics of TMD excitons <cit.> makes these materials very promising for ultrafast logic gates. While strong exciton nonlinear interactions can quench the Rabi splitting in TMD-based microcavities <cit.>, all-optical ultrafast SC switching in atomically thin TMD systems has not been demonstrated.

Here we introduce a new strategy to perform all-optical switching in microcavities made of hBN-encapsulated MoS_2 bilayers, exploiting optical saturation of excitons and their fast relaxation times to produce an ultrafast collapse and revival of the SC regime. We employ femtosecond transient reflectivity (TR) spectroscopy to demonstrate sub-picosecond switching of the SC regime at pulse energies lower than 4 pJ, in ultra-compact devices comprising an atomically thin active material. We show the full tuneability of this process through different degrees of freedom, such as pumping frequency, pulse power and bilayer configuration. 
The SC switching leads to a strong modulation of the polariton peaks splitting, reducing the initial energy separation from 42 meV to less than their linewidth, making them indistinguishable from a single peak. The Rabi splitting modulation is further enhanced by placing a stack of two bilayers separated by hBN in the cavity, going from 55 meV to a complete collapse, resulting in an effective extinction ratio of about 7.5 dB working in reflection.
We further demonstrate an on/off SC switching frequency as high as 250 GHz, which can be extended up to 1 THz.





§ RESULTS




 §.§ Static and dynamic optical behaviour of MoS_2 bilayers


The TMD structures used in our work are made of monolayers (MLs) and bilayers (BLs) of MoS_2 encapsulated in hBN, placed on distributed Bragg reflectors (DBRs), for the subsequent fabrication of optical microcavities. All the spectroscopy experiments in this work are performed at T = 8K. Figure <ref>a shows the Reflectance Contrast (RC) spectra of ML and BL MoS_2 outside the cavity. The absorption of the intralayer A exciton in the BL (X_A-BL) is higher than in the ML, due to presence of the additional layer. In the BL a new excitonic resonance appears at ≈ 2 eV, which is attributed to dipolar hybridized interlayer excitons (hIX) with a high oscillator strength, resulting from the coherent tunnelling of holes between the valence bands of the two layers (Fig. <ref>a inset) <cit.>.

We study the ultrafast response of MoS_2 excitons by ultrafast TR micro-spectroscopy. We deliver two collinear ≈ 100-fs pulses, a narrow-band pump and a broad-band probe, focused on the sample using a microscope objective (Figure <ref>b). We then vary the delay time τ between them and monitor the changes in the broadband reflectivity spectrum of the probe (see Methods for experimental details). Figure <ref>c shows the differential reflectivity (ΔR/R) map measured as a function of the probe photon energy and pump-probe delay for a BL MoS_2, tuning the energy of the pump pulses at ≈ 1.94 eV, in resonance with the X_A-BL. 

Generally, the shape of the TR spectra in TMD MLs is a result of multiple effects, such as optical saturation (photo-bleaching), line broadening and spectral shift of the exciton peaks, leading to positive and negative TR signals around the exciton energies <cit.>. 
To show more clearly the temporal evolution of the exciton features in our system, we perform an analysis of the transient ΔR/R response based on the Transfer Matrix Method (TMM) to extract the time-dependent RC spectra of the material (see Methods for details) <cit.>. Figure <ref>d shows the MoS_2 BL dynamic RC spectrum at a delay of 0.3 ps (purple curve), compared to the one before the pump pulse (-0.3 ps, orange curve). We fit the dynamic RC with three Lorentzians (solid lines in Fig. <ref>d) to extract the time-varying intensity and energy shift of each excitonic mode. After excitation, the X_A-BL peak is quenched and slightly blue-shifted, as a result of phase-space filling and Coulomb interactions occurring at high exciton densities <cit.>. The small shoulder at 1.91 eV appearing at positive delays is instead related to a photo-induced absorption of the trion (X^*_A-BL) <cit.>. 
Since the excitation pulses are in resonance with X_A-BL (shaded yellow area in Fig. <ref>d), at lower energies compared to hIX, we would expect negligible optical saturation of the latter if the two excitonic species were totally uncoupled. On the contrary, we observe a photo-bleaching of the hIX absorption, although less intense than in the X_A-BL case. This demonstrates their hybridization with intralayer excitons due to the coherent hole tunnelling between the valence bands of the two layers and to the fermionic interactions between holes of X_A-BL and hIX sharing the same valence band (see inset of Fig. <ref>a) <cit.>.

Tracking the RC peak intensity as a function of the delay time, we can extract the ultrafast dynamics of the excitonic species (Fig. <ref>e). For both X_A-BL and hIX, the exciton population rises instantaneously (within the ≈ 100 fs temporal resolution of our setup), then decays exponentially, with about 50% of the initial population already relaxed within 2 ps (a comparison with  MoS_2 ML exciton dynamics is reported in Supplementary Note S1). 
The simultaneous build-up of hIX population is again an indication of strong hybridization and fermionic interactions with intralayer excitons.
The fast exciton decay time, resulting from both radiative and non-radiative relaxation processes of bright excitons <cit.>, the latter being significant in TMD bilayers, is also similar between X_A-BL and hIX. 






 §.§ Femtosecond switching of the strong coupling regime


We exploit the highly nonlinear exciton interactions in MoS_2 BL to drastically modify the light-matter coupling strength in microcavities on ultrafast time-scales. 
The microcavity samples are fabricated by covering the hBN-encapsulated MoS_2 heterostructures placed on DBRs with a transparent polymeric spacer (polymethylmethacrylate, PMMA) and a top silver (Ag) mirror, as illustrated in Fig. <ref>a. 
We perform k-space (Fourier) spectroscopy to image the angular dispersion of the monolithic cavity embedding the MoS_2 BL (Fig. <ref>b). Two distinct anticrossings appear when the cavity mode is in resonance with X_A-BL and hIX energies, a clear signature of the SC regime, resulting in upper, middle and lower polariton branches (UPB, MPB, LPB). Fitting the dispersion with a three coupled oscillators model, we extract Rabi splittings of Ω_A_BL= 42 meV and Ω_hIX= 23 meV for X_A-BL and hIX respectively. The Rabi splitting for strongly coupled intralayer excitons in a ML cavity is reduced to Ω_A_ML= 28 meV, due to the lower oscillator strength (see Supplementary Note S2 for the coupled oscillators model fit of the ML cavity).

We use ultrafast TR spectroscopy to excite the MoS_2 BL-based microcavities with narrowband ultrashort pulses tuned at the energy of X_A-BL. To better visualize the dynamic behaviour of polariton spectrum, we plot directly the reflectance (1-R) spectra measured on the cavity as a function of delay time and probe photon energy (Fig. <ref>c,d), while we include the TR data of the same measurement in Supplementary Note S3. Considering that in the spectral region of interest the reflectance of the cavity without the TMD is close to 1, plotting 1-R as a function of time is equivalent to showing the dynamic RC.
We focus our analysis on incidence angles close to normal, on the anticrossing between the cavity mode and the X_A-BL, resulting in the MPB and LPB. 
In a stark contrast to the out-of-cavity experiments, we do not observe a strong quenching of the absorption, but rather measure huge shifts of the polariton states. At negative delays, MPB and LPB are clearly separated, located at 1.953 eV and 1.911 eV respectively. When the pump and probe pulses are synchronous, the two polariton peaks collapse symmetrically in one broad central peak at ≈ 1.94 eV (purple line in Fig. <ref>c). Already after 2 ps, the two polariton branches start to reappear, while after 100 ps they have almost completely recovered. The 1-R map as a function of the time delay (Fig. <ref>d) shows more clearly the complete collapse and revival of polaritons, which can be only explained as a reversible transition from the strong to the weak coupling regime. We note that this trend follows well the dynamic absorption of the excitonic species measured outside the cavity (Fig. <ref>e), being a direct consequence of density-dependent optical saturation of excitons.   

We performed a quantitative analysis of the ultrafast behaviour of MoS_2 BL polaritons by fitting the experimental 1-R peaks with Gaussian functions. The results are shown in Fig. <ref>e, where the extracted peak energies and linewidths are plotted against the time delay up to 1.5 ps. Within few hundreds of femtoseconds from the zero-delay, the LPB shows a blueshift of about 27 meV, while the MPB redshifts by about 14 meV, merging in a single peak at about 250 fs. Such huge shifts cannot be explained just taking into account the bare exciton energy variations, which are in the order of only 5 meV (Fig. <ref>d). When the energy separation of the polariton states is lower than the linewidth of the cavity mode or the exciton, the anticrossing is not visible anymore and the system falls into the weak coupling regime (red star in Fig. <ref>e). 
Already after ≈ 500 fs, the SC is recovered. 
We note that in the weak coupling, the cavity mode is strongly broadened by the background absorption of the excitons, already broad due to excitation-induced dephasing <cit.>. Such broadening also affects the polariton peaks after the collapse, as shown in the color bars of Fig. <ref>e, which become more discernible only after 2 ps. 
We consider that the SC switching has been reached when the Rabi splitting is equal to the unperturbed exciton linewidth (the FWHM of X_A-BL is ≈ 20 meV in static conditions).
On the other hand, considering as switching threshold the splitting being equal to the broader polariton linewidths would be a less stringent criterion. In the latter case the pump energy to induce a peak splitting collapse would be even lower, also resulting in a faster SC recovery.


We observed a similar SC collapse also in microcavities embedding a ML of MoS_2, but in that case the longer exciton lifetimes led to a much slower SC recovery, while the smaller Rabi splitting worsened the on/off contrast, i.e. the signal intensity ratio between the 1-R spectra of the cavity in the unperturbed SC and weak coupling conditions respectively (see Supplementary Note S4).




The pump pulse energy plays a major role in the SC switching dynamics, as shown in Fig. <ref>a where MoS_2 BL cavity spectra taken at a delay of 250 fs for different excitation pulse energies demonstrate the gradual quenching of the Rabi splitting. The SC collapse in TMD cavities is a direct consequence of exciton nonlinear interactions, which scale proportionally to their density <cit.>. 
We demonstrate this effect by carrying out theoretical simulations of the cavity 1-R spectra using the TMM (Fig. <ref>b), employing the MoS_2 BL optical constants calculated from the exciton nonlinear absorption as a function of the density (see Supplementary Note S5). The match between experiments and simulations proves that the main cause behind the observed femtosecond switching of the SC regime is the optical saturation of MoS_2 BL excitons at high excitation densities, which recovers very rapidly due to the fast radiative and non-radiative exciton relaxation mechanisms in this system. 

Figure <ref>c reports the MPB-LPB energy difference against the time delay, normalized with respect to its value before excitation. In this figure, the blue dots are related to the experiment reported in Fig. <ref>, performed at 3.75 pJ (pump fluence: 212 μJ cm^-2), and show the recovery of SC occurring on two different time-scales, a fast one within 1 ps and a slow one which is concluded after ≈ 100 ps. We ascribe those two recovery steps to the population decay dynamics of the bare excitons (Fig. <ref>e). The dashed horizontal line represents the threshold for the weak coupling regime, when the polariton splitting is smaller than the exciton linewidth. Increasing or decreasing by few picojoules the excitation energy, we can extend the temporal window of weak coupling regime (7.5 pJ, red dots in Fig. <ref>c) or suppress completely the transition (2.5 pJ, yellow dots in Fig. <ref>c). 

We provide a theoretical explanation on the energy dependent dynamic SC switching upon direct excitation of the intralayer excitons, considering several possible contributions to the dynamics. We note that the ≈ 1 ps timescale for the fast recovery coincides with the bare X_A-BL exciton fast decay. 
However, this short timescale cannot account for the later slower recovery (∼100 ps) of the Rabi splitting, indicating that a significant exciton population remains in the sample and nonlinear phase space filling impacts the spectrum. 
One plausible explanation is the existence of long-lived dark excitonic states at lower energies <cit.>. In this case photoexcited excitons can be transferred to such states which interact with light weakly, forming a reservoir that contributes to the nonlinear phase space filling. In a bilayer MoS_2, low-energy states are represented by spin-forbidden states due to spin-orbit coupling, or momentum-forbidden states <cit.> due to indirect bandgap 
<cit.>. These states possess a very long lifetime and can explain the TR dynamics. The corresponding model is summarized in the Methods, while we provide more details about the simulated polariton dynamics in Supplementary Note S6.

Leveraging on the ultrafast recovery times of SC in our samples, we demonstrate the possibility to modulate light-matter interactions at very high frequencies, illuminating the cavity with two subsequent pump pulses at 1.94 eV delayed by only ≈ 4 ps. To produce such pulse pair, we used a birefringent YVO_4 crystal with optical axis rotated by 45^∘ with respect to the polarization of the incoming pump pulse, followed by a linear polarizer (see Fig. <ref>d and Methods for more details). The first pulse energy was tuned to be slightly lower than the experiment in Fig. <ref> in order to get a faster SC recovery, while the second pulse energy was adjusted to take into account the residual exciton population after the first pulse. The resulting transient 1-R map shows two reversible on/off cycles (Fig. <ref>e), proving a very fast switching frequency of ≈ 250 GHz. We note that this value was limited by technical constraints (the fixed delay between the pulse pair determined by the thickness of the available YVO_4 crystal), while the theoretical limit is given by the recovery time of the SC.





 §.§ Ultrafast SC switching by interspecies interactions


We exploit the interspecies exciton interactions specific of MoS_2 BL to generate optical saturation of X_A-BL acting on the hIX, exciting selectively the latter and probing the quenching of the Rabi splitting on X_A-BL (Fig. <ref>a). This process relies on nonlinear fermionic interactions (i.e. involving a single charge carrier constituting the exciton) between the two excitonic species: the X_A-BL valence band is shared with hIX, therefore exciting the latter causes optical quenching of the former, due to Pauli blocking of holes for X_A-BL <cit.>. Figure <ref>b shows the transient 1-R map of the MPB-LPB, pumping the hIX of the BL: the femtosecond switching of SC regime occurs very clearly also in this case. Comparing this result with the previous case of resonant X_A-BL pumping (Fig. <ref>c), the fast SC recovery is even more distinct, with the two polariton peaks being clearly visible and well separated already after 1 ps, as shown in Figure <ref>c.

The dynamics of the nonlinear response when pumping in resonance with the hIX is also consistent with the developed model based on the nonlinear saturation and phase space filling from the long-lived states (see Methods). 
In this case, we considered similar lifetimes for hIX compared to X_A-BL, but we assumed the rate for transferring the pumped hIX to the long-lived reservoir states contributing to the phase space filling  
to be smaller than in the X_A-BL case. 
This may be understood as the result of the hIX’s wavefunction spreading in the out-of-plane direction. This implies that hIX is less 2D than X_A-BL, leading to a weaker scattering effect with disorder and to a smaller transfer rate. Furthermore, thermalization through exciton-phonon scattering is another important mechanism converting the bright states into momentum-dark states which may also gives smaller transfer rate upon hIX pumping. 
These scattering effects yield a faster recovery of SC, in agreement with our experimental observation (see Supplementary Note S6). 

Another effect leading to faster recovery in the hIX pumping scheme is the mitigation of the Pauli blockade. In contrast to X_A-BL pumping, the pumped exciton only shares holes with the probed exciton but not the electrons <cit.>, see Fig. <ref>a, leading to a weaker saturation effect. This can also result in a faster recovery of SC if similar conditions as in X_A-BL case are used (except the pump photon energy). Combining the effects of a smaller bright-to-dark exciton transfer rate and weaker Pauli blockade, the influence of the reservoir long-lived states is less significant exciting the hIX. Therefore, a faster recovery time of SC is easier to achieve in this case.
Such fast recovery would allow to further increase the switching frequency, up to ≈ 1 THz. We underline that to achieve such hIX-induced Rabi quenching we use an excitation energy of 4.37 pJ, only moderately higher compared to the resonant excitation case. Higher pulse energies will increase the recovery time and the weak coupling time window.





 §.§ SC switching in a double BL microcavity


Finally, we fabricated a device comprising two vertically stacked bilayer MoS_2 separated by an hBN spacer of  40 nm. We placed this structure in a microcavity made of the same DBR and silver mirror used in the single BL cavity, with a PMMA spacer between the TMD stack and the top mirror (Fig. <ref>a). Similarly to microcavities with multiple quantum wells <cit.>, the SC is enhanced in this sample, as the Rabi splitting is increased to 55 meV due to the additional BL unit (see Supplementary Note S7 for the coupled oscillators model fit of the strongly coupled cavity dispersion).
Figure <ref>b shows a comparison between the static 1-R spectra of a single BL (blue line) and a double BL (red line) cavity, taken at small angles. It clearly appears that the MPB-LPB peaks are more separated in the double BL cavity compared to the single BL sample, being also redshifted, as an effect of the more negative detuning of the former. 
The negative detuning also leads to a decrease of the LPB linewidth in the double BL cavity, being more cavity-like.

By exciting the double BL cavity with pump pulses resonant with X_A-BL, we observed again an ultrafast collapse of the MPB-LPB polariton peaks into a weakly coupled cavity mode, followed by their later recovery (Fig. <ref>c). We note that the pulse energy used for this experiment (8.25 pJ) was not adjusted to obtain a sub-ps SC recovery, but just to demonstrate the SC switching. 
Figure <ref>d shows the full dynamics of the SC collapse and recovery in the double BL cavity, where the energy separation between MPB and LPB (blue and red lines respectively) drops from ≈ 55 meV to zero immediately after the pump pulse.
Considering the unperturbed SC condition as the on state of the optical switch and the weak coupling as the off state, we calculated the spectral power extinction ratio (ER) from the 1-R spectra before and after the pump pulse, taken at -350 fs and 350 fs respectively, where ER(dB)=10log((1-R_on)/(1-R_off)). 
The ER is increased significantly in the double BL cavity compared to the single BL device, over a broad energy range, as shown in Fig. <ref>e. For both the devices, the maximum ER in absolute value is reached around the energy of the weakly coupled cavity mode, between the LPB and MPB, which transmits (reflects) more during the off (on) state. While for the single BL the ER absolute value reaches 3.2 dB at 1.932 eV, it is enhanced up to 7.5 dB at 1.915 eV in the double BL.
It is worth to mention that the ER is also high in the spectral regions of the LPB and MPB, where it shows opposite sign, meaning that the optical switch can be used in direct or reverse mode just by changing the operational wavelength.
We note that working at high frequencies the effective extinction ratio for a second switching event decreases due to the residual exciton population after the first pulse excitation (e.g. by about four times in the double pump pulse experiment shown in Fig. <ref>). We foresee that this drawback can be mitigated by reducing the long exciton decay component, for example suppressing the exciton scattering to dark states and using a different optical resonator with higher quality factor and smaller mode volume, to induce a strong Purcell effect. 




§ DISCUSSION


In summary, we exploit the transient behaviour of MoS_2 exciton-polaritons to demonstrate ultrafast optical switches, whose operational principle is based on the instantaneous transition from the strong to the weak coupling regime due to optical saturation of excitons, which can recover on the sub-picosecond timescale. The MoS_2 BL system uniquely combines nanometric thickness, large nonlinearities and large Rabi splitting with short lifetimes, with the latter enabling observation of the exceptionally fast recovery. 
Furthermore, we show that by increasing slightly the pump pulse energies above the threshold for the SC collapse, the weak coupling time window can be significantly extended and deterministically controlled, being sensitive to energy variations of hundreds of femtojoules and below, crucially important for sensing and low light applications <cit.>. 
This system, also, offers additional degrees of freedom. It can operate at different excitation energies, for example in resonance with either intra- or interlayer excitons, leveraging on the strong interspecies interactions between these excitonic species, unique to the MoS_2 BLs. We foresee this property to be particularly useful for multiplexed logic operations <cit.>. 
Owing to the fast recovery of SC, we were able to perform subsequent switching events delayed by 4 ps, demonstrating an operational frequency of ≈ 250 GHz. Considering the sub-ps SC switching time, this frequency can be pushed up to 1 THz, surpassing even the fastest electronic transistors demonstrated so far <cit.>. 
We also demonstrate that using a microcavity with two stacked MoS_2 BLs can boost the Rabi splitting and greatly enhance the on/off extinction ratio, reaching a maximum of 7.5 dB in a single switching event. An improvement of the optical resonator quality factor and a shorter exciton lifetime will lead to even greater on/off contrast for high frequency switching. 

Our work highlights atomically thin TMDs as a flexible system with rich physics in which sub-ps all-optical switching can be achieved and finely controlled.   
The insights provided can be pivotal for the development of TMD-based high speed all-optical circuits. Moreover, the developed ultrafast nonlinear switching unit can improve the performance of optical neural networks <cit.> acting as an all-optical nonlinear activation function. 
We believe that further progress in boosting the nonlinear interactions of TMD excitons, for example employing localized excitons in moiré heterostructures, could reduce even more the energy required for the ultrafast switching processes, reaching eventually the quantum regime of operation down to a few photons, eventually leading to single-photon transistor-type devices <cit.>. 
The ultrafast switching operation can be extended in principle up to room temperature using our devices, owing to the large binding energies of TMD excitons. 
Considering also the nanometric thickness of each hBN/BL/hBN stack, a microcavity could be filled by many TMD units, greatly increasing the Rabi splitting and subsequently the spectral shifts when used as ultrafast switches, which will lead to enhanced on/off extinction ratio.
Moreover, the integration of electrical contacts in the microcavity structures <cit.> would enable the fabrication of electro-optical interfaces by tuning the electrostatic doping and electric field, which can provide giant shifts of the hIX energy in MoS_2 BLs <cit.>, also enhancing their nonlinear interactions. 
Finally, highly nonlinear TMD systems combined with novel light-confining nanostructures supporting photonic resonances of high quality factor, such as quasi-bound states in the continuum <cit.>, will allow the development of ultra-compact nanodevices, easy to be integrated in planar photonic circuits architectures.

Developing ultrafast all-optical switches based on the transition from the strong to the weak coupling regime would be crucial also to unveil more exotic physical phenomena. 
The crossing point between the strong and weak coupling regimes, called exceptional point, enfolds exotic physics arising from the non-Hermitian Hamiltonian describing such condition <cit.>. 
Both the Hamiltonian eigenvalues and their corresponding eigenstates collapse in exceptional points, which for systems of coupled exciton and photon states translates in their energy and losses becoming identical, making the system very sensitive to external perturbations.
Changing non-adiabatically the parameters of the coupled system in order to encircle the exceptional point can produce topological effects and chiral mode conversions, where the direction of the encirclement dictates the final state of the system <cit.>. Owing to the ultrafast collapse and recovery of SC, our system could be used to demonstrate for the first time this effect in planar microcavities.





§ METHODS




 §.§ Sample Fabrication

The hBN/MoS_2/hBN heterostructures were assembled using a polydimethylsiloxane (PDMS) polymer stamp method. The PMMA spacer for the monolithic cavity was deposited using a spin-coating technique, while a silver mirror of 45 nm thickness was thermally evaporated on top of it.



 §.§ Optical Measurements

For the transient reflectivity measurements 100-fs pulses from an amplified Ti:Sapphire laser at 2 kHz repetition rate are used. The laser output is split in two beams. A portion of the laser output is utilized to drive a non-collinear optical parametric amplifier (NOPA), which allows tuning the pump wavelength. The rest is used for the generation of the broadband white light probe pulse by focusing the beam on a sapphire plate. The delay between pump and probe pulses is controlled by a mechanical delay line. The pulses are combined collinearly and focused on the sample using a 50x objective, resulting in a spot size of ≈ 1.5 μm. The sample is kept in a helium cryostat at 8 K. The differential reflectivity (Δ R/R) spectra are recorded at various time delays τ to track the changes induced by the pump. Specifically, the reflectivity spectrum of the probe with the pump on, R_PumpOn, is compared at each delay with a reference spectrum obtained when the pump is off, R_PumpOff. These are used to calculate Δ R/R = R_PumpOn - R_PumpOff/R_PumpOff, shown in the TR maps. The dynamic probe maps are extracted from a combination of the measured RC spectrum with the pump off and the Δ R/R maps, as R_PumpOn=R_PumpOff(1+Δ R/R). For the double pump pulses experiments, we use a thick YVO_4 birefringent crystal with optical axis rotated at 45^∘ with respect to the vertical pump polarization, which produces a replica of the pulse with horizontal polarization delayed by ≈ 4 ps. The rotation of a subsequent polarizer is changed to finely adjust the energy of each pulse in order to ensure the SC recovery after each excitation pulse.



 §.§ Transfer Matrix Method Analysis


In order to extract the spectral and temporal evolution of the excitonic optical properties from the Δ R/R maps, we follow a procedure recently reported in refs <cit.>. The transient reflectivity R(ω,τ) of MoS2 BL at each delay time is determined from the equilibrium reflectivity R(ω), which is reconstructed from the TMM fit of the static RC spectrum, and the transient reflectivity Δ R/R(ω,τ), following this relation:



R(ω,τ)=R(ω)(Δ R/R(ω,τ) + 1)



Then, the dynamic RC is obtained by applying the formula: 
RC(ω,τ) = 1-(R(ω,τ)/R_sub),
where the substrate reflectivity R_sub is simulated with the TMM. See Supplementary Note S5 for more details on the TMM simulations.





 §.§ Theoretical Model for the Polariton Dynamics


To gain further insight into the dynamics in the system, we develop a mean-field model that captures the main trends in our experiment over different excitation regimes. The Hamiltonian corresponds to the coupled cavity-photon system, where the X_A-BL mode (being the probed A exciton of a homobilayer) hybridizes with the cavity mode. The Hamiltonian reads


    H=[          E_c+iκ 1/2g(n_X)Ω_A_BL; 1/2g(n_X)Ω_A_BL       E_A_BL+iγ ],


where E_c and κ are the energy and linewidth of the cavity photon, and E_A_BL and γ are the energy and linewidth of the probed exciton. In the Hamiltonian above Ω_A_BL is the Rabi splitting at weak pumping, and g(n_X)=e^-α n_X is the dimensionless nonlinear coupling, with α being the nonlinear phase space filling (saturation) coefficient <cit.>. The magnitude of the Rabi splitting is dependent on the total number of excitons in the system, n_X. In general n_X(t)=n_p(t)+n_R(t) is time-dependent, and includes excitons (electron-hole pairs) from different states. Specifically, we separate the two fractions corresponding to n_p and n_R being the population of the pumped exciton and the long-lived excitons in the reservoir. Crucially, both contribute to the nonlinear saturation effect. The dynamics of excitonic fractions can be described by rate equations defining the transfer and population redistribution, which read

    dn_p/dt=   -γ_p n_p-r n_p+Θ(t),
    dn_R/dt=   -γ_R n_R+r n_p,

where γ_p is the pumped exciton decay rate, γ_R is the decay rate of the long-lived exciton in the reservoir, and r is the rate constant for transferring the pumped excitons into the reservoir. The function Θ(t) depends on the pump laser profile in time. For example, Θ(t) can be a Heaviside step function to model the pump as an on-off switching field. In this model, we assume γ_R≈γ_p/100 for the long-lived states corresponding to 100 ps decay time, and consider the decay timescale being similar to that of spin-forbidden dark states <cit.>. In fact, the decay time may be different, but this does not change our later conclusion in a qualitative way. Furthermore, when considering a spin-conserving process, we let the transfer rate be comparable to the pumped exciton decay rate <cit.>, r≈γ_p, such that this allows the pumped exciton transfer into the reservoir. With this, we find a good qualitative agreement between the theoretical spectrum and the experimental measurement (see Supplementary Note S6). Particularly, the theory demonstrated the excitation pulse energy dependence of the recovery time of SC. 



§ ACKNOWLEDGEMENTS


AG and GC acknowledge support by the European Union Marie Sklodowska-Curie Actions project ENOSIS H2020-MSCA-IF-2020-101029644. CT acknowledges the European Union’s Horizon Europe research and innovation programme under the Marie Skłodowska-Curie PIONEER HORIZON-MSCA-2021-PF-GF grant agreement No 101066108. AG, CT, SDC and GC acknowledge funding from the European Horizon EIC Pathfinder Open programme under grant agreement no. 101130384 (QUONDENSATE). This work reflects only authors’ view and the European Commission is not responsible for any use that may be made of the information it contains.
AIT and SR acknowledge financial support of the EPSRC grants EP/V006975/1, EP/V026496/1 and EP/S030751/1. OK and KWS acknowledge the support from UK EPSRC grant EP/X017222/1.


The authors declare no competing interests.



§ AUTHOR CONTRIBUTIONS


AG, CL, CC and CT carried out the optical spectroscopy experiments with contribution from GDB and GS. CL and SR fabricated the 2D samples. KW and TT synthesized the high quality hBN. PC, RJ and DGL fabricated the microcavities. AG, CL, CC, GDB and GS analyzed the data with contribution from SDC, AIT and GC. AG performed the transfer matrix simulations. KWS and OK developed the theory on dynamic optical saturation in microcavities. AG, CL and GC wrote the manuscript with contribution from all other co-authors. DGL, OK, SDC, AIT and GC managed various aspects of the project. SDC, AIT and GC supervised the project.



§ DATA AVAILABILITY


The data that support the findings of this study are available from the corresponding authors upon reasonable request.



§ CODE AVAILABILITY


The computer codes and algorithms used to process the data included in this study are available from the corresponding authors upon reasonable request.



§ INCLUSION AND ETHICS STATEMENT


All collaborators of this study that have fulfilled the criteria for authorship required by Nature Portfolio journals have been included as authors, as their participation was essential for the design and implementation of the study. Roles and responsibilities were agreed among collaborators ahead of the research. This work includes findings that are locally relevant, which have been determined in collaboration with local partners. This research was not severely restricted or prohibited in the setting of the researchers, and does not result in stigmatization, incrimination, discrimination or personal risk to participants. Local and regional research relevant to our study was taken into account in citations.



§ ADDITIONAL INFORMATION


Supplementary Information is available for this paper.





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






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


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



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


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






§ INTRODUCTION

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

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

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



§ METHODS AND MODEL

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



 §.§ Excitons

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

    |nk⟩=1/√(N)∑_ν,R^ık·Ra_nkν|ν,R⟩,

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

    ∑_ν'ℋ_νν'(k)a_nkν'=ϵ_nka_nkν,

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

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

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

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

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

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

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

and

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

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

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

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





 §.§ Phonon-assisted Auger Decay Rate

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

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


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

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

and the electron-electron interaction

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

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

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

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



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

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

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

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

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





 §.§ Exciton-Phonon Coupling Strength

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

    γ(T)=γ_0+2π g^2 DOS_exc n_ph,

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





§ RESULTS

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



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

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

    1/τ=1+2n_ph/τ_0ρ/ρ_0ϵ_0+1/ϵ+1,

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



§ CONCLUSIONS

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


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






 Paper Title: Dephasing-assisted transport in a tight-binding chain with a linear potential.


1

Dephasing-assisted transport with a linear potential]Dephasing-assisted transport in a tight-binding chain with a linear potential 

] 
 
 





[
    [
    Received ..... .., ....; accepted ......, ....
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§ ABSTRACT 


§ 


An environment interacting with a quantum system can enhance transport through the suppression of quantum effects responsible for localization. In this paper, we study the interplay between bulk dephasing and a linear potential in a boundary-driven tight-binding chain. A linear potential induces Wannier-Stark localization in the absence of noise, while dephasing induces diffusive transport in the absence of a tilt. We derive an approximate expression for the steady-state current as a function of both dephasing and tilt which closely matches the exact solution for a wide range of parameters. From it, we find that the maximum current occurs for a dephasing rate equal to the period of Bloch oscillations in the Wannier-Stark localized system. We also find that the current displays a maximum as a function of the system size, provided that the total potential tilt across the chain remains constant. Our results can be verified in current experimental platforms and represents a step forward in analytical studies of environment-assisted transport.

 

§ KEYWORDS:
 Dephasing-assisted transport, Wannier-Stark localization





§ INTRODUCTION


The quantum features of an open system interacting with a macroscopic environment are inevitably destroyed in a process known as decoherence <cit.>. However, the notion that the environment is always detrimental for quantum processes – such as information processing and transport – has been challenged for more than a decade, prompted by investigations of quantum effects in biological systems <cit.>. It is now understood that the environment can assist energy transport in non-interacting quantum systems <cit.>, an effect which has been experimentally verified in quantum networks of photons <cit.>, trapped ions <cit.> and superconducting circuits <cit.>. Although several mechanisms for environment-assisted quantum transport have been proposed and debated <cit.>, a clear mechanism is at play in localized quantum systems – when destructive interference responsible for localization and transport suppression is destroyed by the environment, quantum transport is enhanced. This is indeed the expected impact of environmental coupling on Anderson and Wannier-Stark localization. In Anderson localized systems, quantum transport is suppressed as a consequence of lattice disorder <cit.>. On the other hand, Wannier-Stark localization occurs in the presence of a linear lattice potential such as an electric field <cit.>; in this case, coherent (Bloch) oscillations take place within the region of localization <cit.>. Despite the fact that Wannier-Stark localized systems subject to noise have been investigated <cit.>, the literature on this subject is still scarce.

Environment-assisted transport can also affect the non-equilibrium transport properties of many-body quantum systems. A common approach to evidence this effect is to drive the quantum system to a non-equilibrium steady state by high temperature reservoirs located at the boundaries <cit.>. The dephasing effect of the environment renders the transport diffusive, which has been shown to enhance the steady-state current in boundary-driven systems with disorder <cit.> or quasi-periodicity <cit.> and to modify the transport properties of systems with long-range hopping <cit.>. Moreover, Markovian boundary-driven non-interacting systems are amenable to analytical treatment even in the presence of dephasing, with exact and approximate solutions of steady-state quantities available for tight-binding chains <cit.>.

In this paper, we study dephasing-assisted transport in a boundary-driven, tight-binding chain with a linear potential. We make use of a steady-state ansatz elaborated in Refs. <cit.> to numerically access the relevant steady-state quantities for up to one thousand chain sites. By performing a minimal approximation, we derive an analytical expression for the steady-state current which matches the exact dynamics for a vast range of parameters. From it, we find that the current is maximized at a dephasing rate equal to the period of Bloch oscillations in a Wannier-Stark localized system. We also find that the current displays a maximum as a function of the system size, provided that the total potential tilt across the chain is kept constant. Our results present a significant contribution in the analytics of environment-assisted transport and can be experimentally verified in several platforms.

The paper is organized as follows. In Sec. <ref> we introduce the model and the steady-state ansatz used in our study. In Sec. <ref> we present and discuss our results for dephasing-assisted transport. In Sec. <ref> we present the conclusion and outlook of our work. Sec. <ref> is the supplementary material, containing technical details referenced throughout the main text. 



§ SETUP





 §.§ The model


We study a non-interacting, one-dimensional lattice of L sites with Hamiltonian given by the XX model

    H = J ∑_j=1^L-1  (σ_j^x σ^x_j+1 + σ_j^y σ^y_j+1) + ∑_j=1^Lϵ_j σ_j^z   .

Here, {σ^x_j, σ^y_j, σ^z_j }_j=1^L are Pauli matrices, J is the hopping term and ϵ_j is a local field which will be specified shortly. In order to induce a non-equilibrium state in the system, we couple the boundary sites to high-temperature reservoirs with
different chemical potentials; in addition, each site is also exposed to its own local reservoir which induces dephasing. All the reservoirs are assumed to be ideal and couple weakly to the system. The dynamics of the system's density operator ρ(t) at time t is given by the Lindblad master equation <cit.>

    d ρ(t)/dt = -i/ħ[H,ρ(t)] + ∑_α = l,r,dℒ^α[ρ(t)]   .

Here the superoperator ℒ^α describes dissipation induced by the reservoirs α = l,r,d corresponding to left boundary, right boundary and dephasing, respectively. It is a sum of local Lindblad jump operators of the form

    ℒ^α  [·] = ∑_j,β L^αβ_j · L_j^αβ† - 1/2{ L_j^αβ† L^αβ_j, ·}  ,

where j labels the site acted on by the reservoir α and β labels different jump operators on that same site. The effect of each boundary reservoir (α = l,r) is represented by two jump operators

    L^l ±_1 = √(Γ(1 ± f)/2)σ_1^± ,  L^r±_L = √(Γ(1 ∓ f)/2)σ_L^±  ,

where Γ is the coupling rate at the boundaries, f is the chemical potential bias and σ_j^± = (σ_j^x ± i σ_j^y)/2. For forward bias (0 ≤ f ≤ 1) excitations are mostly created on the first site and annihilated on the last site; the opposite reasoning applies for reverse bias (-1 ≤ f ≤ 0). In order to describe the effects of dephasing, we consider L jump operators acting on individual sites as if each of them were coupled to its own reservoir

    L_j^d = √(γ/2)σ^z_j   ,  j = 1, 2, ..., L   ,

where γ is the dephasing rate. 

The key observables in the system are the magnetization and its associated current. We denote by ⟨ A ⟩(t) ≡Tr[A ρ(t)] the expectation value of an observable A at time t. Differentiating with respect to time and using Eq. (<ref>) yields a continuity equation (see Sec. <ref> for more details)

    d ⟨σ^z_j ⟩(t)/dt    = ⟨ I_j-1⟩(t) - ⟨ I_j⟩(t) + δ_j,1Γ [f - ⟨σ^z_j ⟩(t)] - δ_j,LΓ [f + ⟨σ^z_j ⟩(t)]    ,

involving  the expectation value of the magnetization current ⟨ I_j ⟩(t)  flowing from j to j+1, where

    I_j = 2J/ħ (σ_j^xσ_j+1^y - σ_j^yσ_j+1^x)

is the corresponding current operator. Note that the dephasing reservoir does not change the average magnetization and therefore does not contribute explicitly to Eq. (<ref>), although it still affects the average magnetization and currents implicitly. Moreover, the boundary reservoirs do not contribute to the expression of the current in the bulk of the chain. 



When the steady state is reached, the left-hand side of Eqs. (<ref>) and (<ref>) vanishes. We denote by ⟨ A ⟩_∞≡Tr[A ρ(∞)] the expectation value of A at the steady state ρ(∞). From Eq. (<ref>) we obtain the following equations for the steady-state current

    ⟨ I_1 ⟩_∞    = Γ (f - ⟨σ^z_1 ⟩_∞)   , 
    ⟨ I_j-1⟩_∞    = ⟨ I_j⟩_∞  ,  j = 2, 3, …, L-1 
    ⟨ I_L-1⟩_∞    = -Γ (f + ⟨σ^z_L ⟩_∞)      .

This current is uniform across the chain, with its value dictated by the average magnetization at the boundaries; from now on, we remove the subscript and denote it simply by ⟨ I ⟩_∞. In the absence of any bias (f=0), the boundary reservoirs induce an infinite temperature (maximally-mixed) steady state ρ(∞) = 𝕀/2^L, in which the values of magnetization and current all vanish.



 §.§ Steady-state ansatz


In order to study the transport properties of the model at the steady state, both numerically and analytically, we make use of a perturbative ansatz characterized by an expansion in terms of potential bias f as <cit.>

    ρ    = 1/2^L[𝕀 + f (H + B) + 𝒪(f^2) ]   , 
    
        H     = ∑_r=1^L∑_j=1^L+1-r h^(r)_j H^(r)_j   , 
     
        B     = ∑_r=2^L∑_j=1^L+1-r b^(r)_j B^(r)_j   ,

where h_j^(r) and b_j^(r) are expansion coefficients, and the corresponding operators are given by

    H^(1)_j    = - σ^z_j , 
    
        H^(r)_j    = σ^x_j Z^(r-2)_j+1σ^x_j+r-1 + σ^y_j Z^(r-2)_j+1σ^y_j+r-1  ,  r ≥ 2 
    
        B^(r)_j     = σ^x_j Z^(r-2)_j+1σ^y_j+r-1 - σ^y_j Z^(r-2)_j+1σ^x_j+r-1  ,  r ≥ 2   .

These operators are thus strings of Pauli matrices starting at site j and having length r, where Z^(r-2)_k+1 = σ^z_k+1σ^z_k+2 ... σ^z_k+r-3σ^z_k+r-2 are strings of Pauli z matrices of length r-2. The defining characteristic of this ansatz is that all operators in the expansion are orthogonal according to the Hilbert-Schmidt inner product ⟨ A, B⟩≡Tr[A^†B]. This implies that the expectation values of H^(r)_j and B^(r)_j are exactly determined within first order in f. In particular, H_j^(1) and B_j^(2) are respectively proportional to the magnetization and current

    ⟨σ^z_j⟩_∞ = -f h^(1)_j  , ⟨ I_j ⟩_∞ = f 4J b^(2)_j/ħ  .

Once we determine the first order expansion coefficients h^(r)_j and b^(r)_j, we have access to the exact current and magnetization profile <cit.>. In order to obtain them, we insert Eq. (<ref>) into Eq. (<ref>) and set the left-hand side to zero as demanded at stationarity (see Refs. <cit.> and Sec. <ref> for more details). This gives rise to a set of equations for the coefficients

    ħΓ(1+h^(1)_1) - 4J b_1^(2) = 0   , 
       ħΓ(1-h^(1)_L) - 4J b_L-1^(2) = 0   , 
        b_j^(2) - b_j-1^(2) = 0   ,  j = 2,...,L-1   ,

for r = 1. Note that Eqs. (<ref>)–(<ref>) just express the fact that the current is uniform in the steady state, with its value dictated by the boundary reservoirs as discussed before. For r ≥ 2 we have

    J [ h_j^(r-1) - h_j+1^(r-1) + h_j^(r+1) - h_j-1^(r+1)] +  (ϵ_j - ϵ_j+r-1)h_j^(r) + ħ[ γ + Γ/4(δ_j,1 + δ_j+r-1,L) ] b^(r)_j = 0   , 
        J [b_j^(r-1) - b_j+1^(r-1)+ b_j^(r+1) - b_j-1^(r+1)] + (ϵ_j - ϵ_j+r-1)b_j^(r) - ħ[ γ + Γ/4(δ_j,1 + δ_j+r-1,L) ] h^(r)_j = 0   .

In total, Eqs. (<ref>)–(<ref>) are a closed set of L^2 coupled equations and will be the starting point of our study. This set of equations is equivalent to a Lyapunov-type equation for the correlation matrix in the presence of dephasing <cit.>. Instead of the exponential scaling with the system size, the quadratic scaling makes the problem computationally tractable. Moreover, as we show below, it also allow us to find analytical expressions for the current under some mild assumptions.



§ DEPHASING-ASSISTED TRANSPORT



We now discuss separately the effect of the tilt and dephasing on the transport properties of the model, while their interplay is addressed in the next section. The bias f of each reservoir at the boundaries induces transport along the chain. Intuitively, a similar effect is expected in the presence of an electric field which increases/decreases at each site at a rate U, or tilt between neighbour sites. If the tilt decreases from left to right, the local potentials are

    ϵ_j = - Uj   .

Setting the bias to f=1 drives excitations in the same direction (see Fig. <ref>). Note that changing the sign of f inverts the sign of the current and magnetization, while changing the sign of U affects neither <cit.>. 

Since the tilt U represents the energy difference between nearest neighbours in the chain, the total tilt V=U(L-1) represents the difference in energy across the whole chain. This quantity diverges in the thermodynamic limit, so that a physically meaningful analysis requires fixing the total tilt V and allowing U to decrease with the system size L. As discussed in Secs. <ref> and <ref>, this procedure introduces a non-trivial dependence of the current with system size in the presence of dephasing: while for fixed U the current is expected to decrease with the system size L, this will no longer be the case for fixed V.



Such a linear potential is known to induce Wannier-Stark localization in the absence of dephasing, with U/ħ being the frequency of Bloch oscillations <cit.>.
In the absence of dephasing γ = 0 and for non-vanishing tilt V>0, the steady state magnetization profile shows the formation of domain walls <cit.> – regions of constant magnetization ⟨σ^z_j ⟩_∞≃± f starting at the boundaries and extending inwards as shown in Fig. <ref>(a). According to Eqs. (<ref>) or (<ref>), this implies a vanishing current, which characterizes the localized system. Indeed we observe in Fig. <ref>(b) that, as V increases, the current displays an exponential decay with the system size ⟨ I ⟩_∞∼ e^-L/L_0 with L_0 being the localization length. Note that L_0 here is different from the localization region of a single particle in a Wannier-Stark localized system given by J/U <cit.>. We did not find any direct relationship between the two.



The results in the absence of tilt U = V = 0 are shown in Figs. <ref> and <ref>(a), respectively for the magnetization and the current. We distinguish two regimes: γ = 0 and γ > 0. The first case corresponds to ballistic transport, characterized by a small (but non-zero) magnetization at the boundaries and zero bulk magnetization (inset of Fig. <ref>), together with a size-independent current (see later Fig. <ref>(a)). The second case corresponds to diffusive transport, characterized by an emerging linear magnetization profile as the dephasing rate increases, accompanied by a decreasing current in accordance with Eqs. (<ref>) or (<ref>). For very large dephasing or system sizes the current scales as ⟨ I ⟩_∞∼ 1/(γ L) signaling diffusive transport. The exact expression for the current in the absence of tilt is given by Eq. (<ref>) (with U=V=0) as we discuss in Sec. <ref>.



 §.§ Exact current for small system sizes


Having shown the degrading effect of the tilt and dephasing separately on the spin transport across the system, we now study their interplay which gives rise to dephasing-enhanced transport. In this section, we focus on transport for small system sizes L=2,3. An exact and compact solution for the current can be found from Eqs. (<ref>)–(<ref>) in these cases. For L = 2 we obtain

    ⟨ I ⟩_∞ = f 16 J^2 Γ (2 γ + Γ)/16 J^2 (2 γ + Γ) + 4 Γ U^2 + ħ^2Γ(2 γ + Γ)^2  .

Dephasing-assisted transport can be characterised through the maximum of Eq. (<ref>) with respect to the dephasing rate γ, yielding an optimal dephasing rate

    γ_max = U/ħ - Γ/2≥ 0   .

The behaviour of Eq. (<ref>) is shown in Fig. <ref>. Note that since the dephasing is always positive, the maximum only occurs if U/ħ≥Γ/2.

For L=3 we have a similar expression, namely

    ⟨ I ⟩_∞ = f 16 J^2 Γ (4 γ + Γ)/16 J^2 (4 γ + Γ) + 16 Γ U^2 + ħ^2Γ(4 γ + Γ)^2   ,

where the maximum occurs at

    γ_max = U/ħ - Γ/4≥ 0   .

The complexity of these expressions increases rapidly for L > 3, so the exact evaluation of the current and its maximum is too involved to be displayed here for larger systems. In the following section, we resort to an approximate solution for the current.



 §.§ Tridiagonal approximation 


In order to find an exact solution for systems of arbitrary size, we perform the tridiagonal approximation which consists of retaining strings of Pauli matrices up to r=2, that is, ignoring higher order coefficients in Eqs. (<ref>)–(<ref>) so that h^(r)_j,b^(r)_j = 0 for r ≥ 3. In the correlation matrix approach, this is equivalent to retaining only the diagonal and first off-diagonal elements <cit.>. Physically, this implies that spin and current correlations can be ignored within first order in f, which allows the derivation of an exact expression for the current as a function of all parameters as we now show. We only have to consider Eqs. (<ref>) and (<ref>) for r=2. Summing Eqs. (<ref>) from j=1 to j=L-1 we obtain

    J(h^(1)_1 - h^(1)_L)     + ∑_j=1^L-1 (ϵ_j - ϵ_j+1) h^(2)_j +  ħ b [ γ(L-1) + Γ/4] = 0   ,

where b ≡ b^(2)_j is the coefficient associated with the current, which is independent of j as a consequence of Eq. (<ref>). First, we can write h^(1)_1 - h^(1)_L as a function of b by using Eqs. (<ref>) and (<ref>), which yields

    h^(1)_1 - h^(1)_L = 8 J b/ħΓ - 2   .

Second, we can also express h^(2)_j as a function of b by using Eq. (<ref>), which leads to

    h^(2)_j =  (ϵ_j - ϵ_j+1) b/ħ[γ + (δ_j,1 + δ_j,L-1)Γ/4]  .

We can now substitute the previous two equations into Eq. (<ref>) and obtain an expression for b, namely

    b = 4J/2 ħγ(L-1) + ħΓ + 16J^2/ħΓ  + 2/ħ∑_j=1^L-1(ϵ_j - ϵ_j+1)^2/γ +(δ_j,1 + δ_j,L-1)Γ/4  .

The current is now obtained from Eq. (<ref>). For the local tilted potential in Eq. (<ref>) it becomes

    ⟨ I ⟩_∞ = f 16J^2 Γ/16J^2 + ħ^2 Γ[Γ + 2 γ(L-1)] + 4 Γ U^2 
        
          1/(2γ + Γ)     L = 2 
    
          (L-3)/(2γ) + 4/(4γ + Γ)     L ≥ 3
          .

We note that this expression is exact for L=2 and L=3, in which case we recover Eqs. (<ref>) and (<ref>), respectively. It is also exact in the absence of tilt U=V=0 <cit.>, which is verified in Fig. <ref>(a). In last three panels of Fig. <ref> we plot the exact solution and the tridiagonal approximation for non-zero values of total tilt V > 0. We observe that the approximation works remarkably well, deviating from the exact solution only for very small dephasing rates and system sizes; when γ = 0, the approximation fails dramatically and only the exact solution is shown. The latter case corresponds to a Wannier-Stark localized system which we already discussed. Moreover, we see that for fixed γ the current can now increase with the system size up to a critical size L_max where it is maximal, and only then decreases. In the same way, we see that for fixed L there is a critical dephasing rate γ_max at which the current is maximal. 





 §.§ Interplay between dephasing and tilt 


Since the validity of Eq. (<ref>) is established, we now use it to determine both L_max and γ_max. The maxima of Eq. (<ref>) as a function of L (for fixed V and γ) and γ (for fixed L and V) can be determined exactly. However, as before, the expressions are too lengthy to be displayed. More physically intuitive expressions are obtained by assuming that L ≫ 1 and L ≫ 8 γ / (4γ + Γ) in the denominator of Eq. (<ref>) which leads to the approximate maxima

    L_max   ≃V/ħγ  , 
    γ_max   ≃V/ħ L = U/ħ  .

These maxima characterize dephasing-assisted transport as we show in Fig. <ref>(a). 

Below the critical length (L_max≫ L ≫ 1), dephasing is not yet effective and an increase in L decreases the local tilt U = V/(L-1) and thus localization, which increases the current according to Eq. (<ref>). Above this length scale (L ≫ L_max≫ 1) dephasing becomes effective and diffusive transport clearly emerges ⟨ I ⟩_∞∼ 1/(γ L). Analogously, below the critical dephasing rate (γ_max≫γ) dephasing jump events happen at a much lower frequency than Bloch oscillations, so the current flowing remains hampered by localization. Increasing the rate of dephasing destroys localization and increases the current. When the dephasing rate passes this threshold (γ≫γ_max)  diffusive behaviour emerges. We illustrate this in Fig. <ref>(b) where we plot the (finite size) diffusion coefficient defined by ⟨ I ⟩_∞≡ f D/L as a function of dephasing rate for the largest system size we reached L = 1000.

The imprint of dephasing-enhanced transport is also evident in the magnetization profile, as shown in Fig. <ref>. Here, varying the dephasing strength towards its critical value (<ref>) decreases the magnetization gradient, rendering the magnetization profile almost uniform. This is in contrast with the results obtained in the absence of tilt, where an increasing dephasing rate tends to increase the magnetization gradient and decrease uniformity as shown in Fig. <ref>(a). Uniformity as a feature of environment-assisted transport has been pointed out for quantum systems subject to on-site disorder <cit.>; our results confirm that the same phenomenon appears in the presence of a linear potential inducing Wannier-Stark localization.





§ CONCLUSION AND OUTLOOK


In this paper, we have exploited an exact ansatz to obtain the steady state of a non-interacting spin chain subject to bulk dephasing, a linear potential and boundary-driving. This description has allowed us to unravel the delicate interplay between Wannier-Stark localization and dephasing-induced diffusive transport. We applied the tridiagonal approximation, where only on-site and nearest-neighbour correlation matrix elements are retained, to derive an approximate expression for the steady-state current as a function of both dephasing and tilt. This expression is found to closely match the exact solution for a wide range of parameters, and reveals that the maximum current occurs for a dephasing rate equal to the period of Bloch oscillations in the Wannier-Stark localized system. Fixing the total tilt across the system then revealed a maximum in the current as a function of system size. This evidences a critical system size beyond which Bloch oscillations are suppressed by dephasing.

Our work motivates future analytical and numerical analysis on the interplay between dephasing and a tilted potential in more complex and richer systems, e.g. those including interactions between neighboring sites <cit.>, time-periodic driving <cit.>, or more realistic (non-Markovian) boundary-driving schemes <cit.>. Furthermore, our results can be readily verified in current photonic <cit.>, ion-trap <cit.> and cold-atom <cit.> experimental platforms.



§ FUNDING


The authors declare financial support was received for the research, authorship, and/or publication of this article. S.L.J. acknowledges the financial support from a Marie Skłodowska-Curie Fellowship (Grant No. 101103884). S.R.C. gratefully acknowledge financial support from UK's Engineering and Physical Sciences Research Council (EPSRC) under grant EP/T028424/1 which is part of an EPSRC-SFI joint project QuamNESS funded also by the SFI under the Frontier For the Future Program. J.G. is supported by a SFI Royal Society University Research Fellowship.



§ ACKNOWLEDGEMENTS


The authors want to thank Gabriel T. Landi for useful discussions.



§ SUPPLEMENTARY MATERIAL





 §.§ Heisenberg equation



For completeness, we show here how to obtain Eq. (<ref>). Differentiating ⟨σ^z_j ⟩(t) ≡Tr[σ_j ρ(t)] with respect to time and using Eq. (<ref>) yields

    d ⟨σ^z_j ⟩(t)/dt = i/ħ Tr[ [H,σ^z_j] ρ(t) ] + ∑_α = l,r,dTr[ σ^z_j ℒ^α[ρ(t)] ]   .

Throughout this section, we make use of several relations for the Pauli matrices, which can all be deduced from their commutation and anti-commutation relations [σ^α,σ^β] = 2i ε_αβγσ^γ and {σ^α,σ^β} = 2 δ_αβ𝕀_2, respectively. Here α, β, γ∈{ x, y, z }, 𝕀_2 is the identity operator for a two-level system and ϵ_αβγ is the Levi-Civita symbol with the Einstein summation notation being implicit. The commutator in Eq. (<ref>) can then be easily computed

    i/ħ[H,σ^z_j]     = iJ/ħ∑_k=1^L-1[ σ_k^x σ^x_k+1 + σ_k^y σ^y_k+1,σ^z_j] 
        =  2J/ħ (σ_j-1^xσ_j^y - σ_j-1^yσ_j^x) - 2J/ħ (σ_j^xσ_j+1^y - σ_j^yσ_j+1^x) 
        = I_j-1 - I_j  ,

In the first line only the kinetic energy part of the Hamiltonian is non-vanishing. In the second line, only the pairs (k,k+1) which overlap with j survive. In the last line, we identified the current operator in Eq. (<ref>). Regarding the last term in Eq. (<ref>), we can make progress by considering the dual superoperator acting on σ^z_j defined by Tr[ σ^z_j ℒ^α[ρ(t)] ] = Tr[ ℒ^α†[σ^z_j] ρ(t) ] with

    ℒ^α†[·] = ∑_j,β L^αβ†_j · L_j^αβ - 1/2{ L_j^αβ† L^αβ_j, ·}  .

The dephasing superoperator is equal to its dual and reads explicitly

    ℒ^d[·] = γ/2∑_j=1^L ( σ_j^z ·σ_j^z - · ) = ℒ^d †[·]   .

It is straightforward to see that its action vanishes when applied to any Pauli z-matrix. Therefore dephasing does not contribute to Eq. (<ref>). For the reservoir at the left boundary, we have the dual superoperator

    ℒ^l †[·]     = Γ(1+f)/2 O_1^l+ †[·] + Γ(1-f)/2 O_1^l- †[·] 
    
        O_1^l±†[·]    ≡σ_1^∓·σ_1^± - 1/2{σ_1^∓σ_1^±,·}  ,

and a similar expression holds for the right reservoir in Eq. (<ref>). Applying these superoperators to the Pauli z-matrix yields after some manipulation

    ℒ^l †[σ^z_j]     = δ_j,1Γ(f - σ^z_1)   , 
    ℒ^r †[σ^z_j]     = - δ_j,LΓ(f + σ^z_L)   .

Using the last two expressions and Eq. (<ref>) in Eq. (<ref>) immediately leads to Eq. (<ref>).



 §.§ Steady-state ansatz



In order to derive Eqs. (<ref>)–(<ref>), we insert Eq. (<ref>) into Eq. (<ref>) and set the left-hand side to zero as demanded at steady state. This gives

    0     = 1/2^L∑_α = l,r,d,0[ ℒ^α[𝕀] + f ( ℒ^α[H] + ℒ^α[B])]   , 
    ℒ^α[H]     = ∑_r=1^L∑_j=1^L+1-r h^(r)_j ℒ^α[H^(r)_j]   , 
    ℒ^α[B]     = ∑_r=2^L∑_j=1^L+1-r b^(r)_j ℒ^α[B^(r)_j]   .

In the last expression we ignore all terms beyond first order in f and defined the superoperator for free evolution

    ℒ^0 [·] ≡ -i/ħ[H, · ]   .




  §.§.§ Action of the dephasing reservoir


We start by computing the action of the dephasing superoperator which reads explicitly

    ℒ^d[·] = γ/2∑_j=1^L ( σ_j^z ·σ_j^z - · ) ≡γ/2∑_j O^d_j[·]   .

It is straightforward to see that the action of this superoperator vanishes when applied to 𝕀 and σ^z. For r ≥ 2, we have to compute terms of the following form

    ℒ^d[σ^α_j Z^(r-2)_j+1σ^β_j+r-1]     = γ/2[O^d_j[σ^α_j] Z^(r-2)_j+1σ_j+r-1^β + σ^α_j Z^(r-2)_j+1 O^d_j+r-1 [σ_j+r-1^β] ]   ,

Note that the right-hand side has only two non-vanishing terms on the boundaries since the action on σ^z vanishes. We then have

    O^d_j[σ^α_j] = σ_j^z σ_j^ασ_j^z - σ_j^α = - 2 σ_j^α  (δ_α,y + δ_α,x)   .

Using this last expression and carrying through simple algebraic manipulation, the action of the dephasing is written as

    ℒ^d[𝕀]     = ℒ^d[H_j^(1)] = 0   , 
    ℒ^d[H_j^(r)]     = -2 γ H_j^(r) ,  r ≥ 2 
    ℒ^d[B_j^(r)]     = -2 γ B_j^(r) ,  r ≥ 2   .




  §.§.§ Action of the boundary reservoirs


We first illustrate the action for the left boundary reservoir. The superoperator is given by

    ℒ^l[·]     = Γ(1+f)/2 O_1^l+[·] + Γ(1-f)/2 O_1^l-[·]   , 
    
        O_1^l±[·]    ≡σ_1^±·σ_1^∓ - 1/2{σ_1^∓σ_1^±,·}  .

In opposition to dephasing, this superoperator does not preserve the identity

    O_1^l±[𝕀] = ±σ_1^z = ∓ H^(1)_1   .

As a consequence, its action on the Pauli matrices σ_j^z depends on j. Namely, we obtain

    O_1^l±[σ_j^z] = 
        
          - σ^z_1   ,     j = 1 
    ±σ_1^zσ^z_j   ,     j = 2,3...L   .

Inserting these results into Eq. (<ref>) we obtain the actions

    ℒ^l[𝕀]     = -Γ f H^(1)_1   , 
    ℒ^l[H^(1)_j]     = 
        
            - Γ H_1^(1)    j = 1 
    
            f Γ H_1^(1) H_j^(1)    j = 2,3,…,L   .

Inspecting the last expression together with Eq. (<ref>), we see that for j=2,3,..L the reservoir creates terms in the expansion which are proportional to f^2 instead of f – they represent higher order spin correlations which do not affect the steady state values of magnetization and current. We henceforth ignore these second order terms by considering the action of the reservoir on Pauli matrices located at the reservoir site. Proceeding with the calculations for r ≥ 2, we have to compute terms of the form

    ℒ^l[σ^α_j Z^(r-2)_j+1   σ^β_j+r-1] = δ_j,1{Γ(1+f)/2 O^l+_1[σ^α_1] + Γ(1-f)/2 O^l-_1[σ^α_1] } Z^(r-2)_j+1σ^β_j+r-1  .

It is easy to verify that

    O^l ±_1[σ^α_1] = -σ_1^α/2 , α = x,y   .

Using the last two expressions and carrying out the algebra, the results for the left reservoir to first order in f are

    ℒ^l[𝕀]     = - Γ f H^(1)_1   , 
    ℒ^l[H^(1)_j]     = -δ_j,1Γ H^(1)_j   , 
    ℒ^l[H^(r)_j]     = - δ_j,1Γ/2 H_j^(r) ,  r ≥ 2   , 
    ℒ^l[B^(r)_j]     = - δ_j,1Γ/2 B_j^(r) ,  r ≥ 2   .

Proceeding the same way for the right boundary reservoir we obtain the result

    ℒ^r[𝕀]     = Γ f H^(1)_L   ,
    ℒ^r[H^(1)_j]     = -δ_j,LΓ H^(1)_j   , 
    ℒ^r[H^(r)_j]     = - δ_j+r-1,LΓ/2 H_j^(r) ,  r ≥ 2   , 
    ℒ^r[B^(r)_j]     = - δ_j+r-1,LΓ/2 B_j^(r) ,  r ≥ 2   .




  §.§.§ Action of Hamiltonian


Regarding the Hamiltonian superoperator in Eq. (<ref>), it is easy to see that it preserves the identity ℒ^0[𝕀] = 0 while ℒ^0[H_j^(1)] yields the well-known result

    ℒ^0[H_j^(1)] = - 2J/ħ(B^(2)_j - B^(2)_j-1)   .

This result was already implicitly used in Eq. (<ref>) and simply expresses that the change in magnetization at a given site is due to differences between incoming and outgoing currents at that site. For r ≥ 2, it is useful to write the superoperator  
as a sum of two contributions ℒ^0≡ℒ^0_1 + ℒ^0_2, where

    ℒ^0_1·   ≡ -i J/ħ∑_j=1^L-1 { [σ_j^x σ_j+1^x,·] + [σ_j^y σ_j+1^y,·] }  , 
    ℒ^0_2·   ≡ -i/ħ∑_j=1^L ϵ_j [σ_j^z,·]   .

We compute only the action of ℒ^0_1 and the one for ℒ^0_2 is obtained in a similar manner. We first evaluate operators for r = 2 which contain no Pauli z-strings (see Eq. (<ref>) for the exact expressions). It involves computing terms of the form

    ∑_j=1^L-1 [σ_j^γσ_j+1^γ, 
     σ_k^ασ_k+1^β]     = σ^γ_k-1[σ_k^γ,σ_k^α]σ_k+1^β+ [σ_k^γσ_k+1^γ,σ_k^ασ_k+1^β] + σ^α_k[σ_k+1^γ,σ_k+1^β]σ_k+2^γ
        = 2i [ε_γαωσ_k-1^γσ_k^ωσ_k+1^β + ε_αβω(δ_γασ_k+1^ω - δ_γβσ_k^ω) + ε_γβωσ_k^ασ_k+1^ωσ_k+2^γ]   .

In the first line, the only non-vanishing terms are for those pairs (j,j+1) that partially or completely overlap with (k,k+1). In the second line, we used the standard properties of the Pauli matrices. The last expression is enough to compute the action of ℒ^0_1 on H^(2)_k and B^(2)_k which reads

    ℒ^0_1[H^(2)_j]     = -2J/ħ(B_j^(3) - B_j-1^(3))   , 
    ℒ^0_1[B^(2)_j]     = -4J/ħ(H_j+1^(1) - H_j^(1))-2J/ħ(H_j^(3) - H_j-1^(3))   .

For r ≥ 3, the operators in Eq. (<ref>) have Pauli-z strings. We need to compute terms of the form

    ∑_j=1^L-1   [σ_j^γσ_j+1^γ,σ_k^α Z^(r-2)_k+1σ_k+r-1^β] = σ_k-1^γ [σ_k^γ,σ_k^α] Z^(r-2)_k+1σ_k+r-1^β + [σ_k^γσ_k+1^γ,σ_k^ασ_k+1^z] Z^(r-2)_k+2σ_k+r-1^β
        + σ_k^α Z^(r-2)_k+1[σ^γ_k+r-1,σ^β_k+r-1]σ^γ_k+r + σ_k^α Z^(r-3)_k+1 [σ_k+r-2^γσ_k+r-1^γ,σ_k+r-2^zσ_k+r-1^β]   , 
        = 2i [ ε_γαωσ_k-1^γσ_k^ω Z^(r-2)_k+1σ_k+r-1^β + ε_γβωσ_k^α Z^(r-2)_k+1σ_k+r-1^ωσ_k+r^γ
        + (ε_γαωδ_γ zσ_k^ω + ε_γ z ωδ_γασ_k+1^ω) Z^(r-3)_k+2σ_k+r-1^β + σ_k^α Z^(r-3)_k+1 (ε_γ z ωδ_γβσ_k+r-2^ω + ε_γβ zδ_γ zσ_k+r-1^ω)]   .

Again, the only non-vanishing terms are for those pairs (j,j+1) that partially or completely overlap with the string (k,k+1,...k+r-1). No commutator appears containing pairs of σ_z since these also vanish. The last expression suffices to compute the action of ℒ^0_1 on H^(r)_k and B^(r)_k for r ≥ 3 and a similar reasoning holds for ℒ^0_2. We summarize the results below:

    ℒ^0[𝕀]     = 0   , 
    ℒ^0[H_j^(1)]     = - 2J/ħ(B^(2)_j - B^(2)_j-1)   , 
    ℒ^0[B_j^(2)]     = 4J/ħ(H_j^(1) - H_j+1^(1))+2J/ħ(H_j^(3) - H_j-1^(3)) + 2(ϵ_j - ϵ_j+1)/ħH_j^(2)  , 
    ℒ^0[H_j^(r)]     = - 2J/ħ[ (B_j^(r-1) - B_j+1^(r-1)) + (B_j^(r+1) - B_j-1^(r+1)) ] - 2(ϵ_j - ϵ_j+r-1)/ħB_j^(r) ,  r ≥ 3   , 
    ℒ^0[B_j^(r)]     = 2J/ħ[ (H_j^(r-1) - H_j+1^(r-1)) + (H_j^(r+1) - H_j-1^(r+1)) ] + 2(ϵ_j - ϵ_j+r-1)/ħH_j^(r) ,  r ≥ 3   .

With the action of all superoperators computed and using Eq. (<ref>) we obtain the set of Eqs. (<ref>)–(<ref>) for the coefficients used in the main text.

Frontiers-Harvard




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


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


 
 
 
 Mirco Nydegger1
       and Ralph Spolenak1,^*
       


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

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

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

]




§ INTRODUCTION




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



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

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

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

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

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

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



§ RESULTS

 


 




 §.§ Voltammogram of electrohydrodynamic redox printing

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

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

 

 

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

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



 §.§ Nucleation and growth of Cu on Au

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

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

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

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



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

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

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

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

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



§ DISCUSSION

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

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

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

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

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



§ CONCLUSIONS AND OUTLOOK

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

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

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

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



§ EXPERIMENTAL



 §.§ Materials

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



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



 §.§ Deposition

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


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



§ AUTHOR CONTRIBUTIONS

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



§ CONFLICTS OF INTEREST

The Authors declare no conflicts of interest.



§ ACKNOWLEDGEMENT

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

 
rsc 



Supplementary Information





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



Mirco Nydegger and Ralph Spolenak






 

 

 




 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: Quantized electrical, thermal, and spin transports of non-Hermitian clean and dirty two-dimensional topological insulators and superconductors.





Department of Physics, Lehigh University, Bethlehem, Pennsylvania, 18015, USA


Department of Physics, Lehigh University, Bethlehem, Pennsylvania, 18015, USA



§ ABSTRACT 
From lattice-regularized models, devoid of any non-Hermitian (NH) skin effects, here we compute the electrical (σ_xy), thermal (κ_xy), and spin (σ^sp_xy) Hall, and the electrical (G_xx) and thermal (G^th_xx) longitudinal conductivities for appropriate NH planar topological insulators and superconductors related to all five non-trivial Altland-Zirbauer symmetry classes in the Hermitian limits. These models feature real eigenvalues over an extended NH parameter regime, only where the associated topological invariants remain quantized. In this regime, the NH quantum anomalous and spin Hall insulators show quantized σ_xy and G_xx, respectively, the NH p+ip (p ± ip) pairing shows half-quantized κ_xy (G^th_xx), while the NH d+id pairing shows quantized κ_xy and σ^sp_xy in the clean and weak disorder (due to random pointlike charge impurities) regimes. We compute these quantities in experimentally realizable suitable six-terminal setups using the Kwant software package. But, in the strong disorder regime, all these topological responses vanish and with the increasing non-Hermiticity in the system this generic phenomenon occurs at weaker disorder.       


Quantized electrical, thermal, and spin transports of non-Hermitian clean and dirty two-dimensional topological insulators and superconductors
    Bitan Roy
    August 5, 2024
==============================================================================================================================================





Introduction. Topological crystals displaying the hallmark bulk-boundary correspondence in terms of robust gapless modes living on their boundaries <cit.> typically encounter the notorious non-Hermitian (NH) skin effect in open systems interacting with a bath or the environment <cit.>. It corresponds to an accumulation of all the right and left eigenvectors at opposite ends of the system with open boundary conditions. Even though topological invariants can be defined in NH systems their signatures on the the boundary modes get masked by the NH skin effect. Theoretically, this obstacle can be bypassed through the bi-orthogonal bulk-boundary correspondence, devoid of any NH skin effect <cit.>. Nevertheless, any direct manifestation of the topological invariants on experimentally measurable quantities even in a NH toy topological model remains illusive <cit.>, leaving aside their actual measurements in open quantum crystals. In this quest, two-dimensional NH topological models stand as cornerstones, since in the Hermitian or closed systems their topological invariants can be computed and measured in six-terminal Hall-bar setups <cit.>.      

    
















Here, we consider skin effect free NH generalization of square lattice models for all five non-trivial Hermitian Altland-Zirnbauer symmetry classes <cit.> that over an extended NH parameter regime feature real eigenvalue spectrum and non-vanishing topological invariants <cit.>. Using numerical Kwant software package <cit.>, we compute their electrical (σ_xy), thermal (κ_xy), and spin (σ^sp_xy) Hall, and electrical (G_xx) and thermal (G^th_xx) longitudinal conductivities in a six-terminal setup (Fig. <ref>), to arrive at the following conclusions. The NH quantum anomlaous Hall insulator (QAHI) [quantum spin Hall insulator (QSHI)] belonging to class A [class AII] in the Hermitian limit, shows quantized σ_xy [G_xx] in units of e^2/h (Fig. <ref>), where e (h) is the electrical charge (Planck's constant). By contrast, the NH p+ip (p ± ip) paired state, belonging to class D (class DIII) in Hermitian systems, supports half-quantized κ_xy (G^th_xx) in units of κ_0=π^2 k^2_ B T/(3h) as the temperature T → 0, where k_ B is the Boltzmann constant (Fig. <ref>). Finally, the NH d+id paired state, falling in class C in the Hermitian limit, accommodates quantized κ_xy (in units of κ_0) and σ^sp_xy (in units of σ^sp_0=ħ/(8π)). See Fig. <ref>. These conclusions hold in the entire topological regime in a clean and weakly disordered (due to pointlike random charge impurities) systems. In the strong disorder regime, all these responses vanish, which with increasing non-Hermiticity in the system occur at weaker disorder (Fig. <ref>). 














Model. For all the gapped topological phases belonging to any one of the five non-trivial Altland-Zirnbauer symmetry classes in two dimensions, the model Hamiltonian on a square lattice take the following universal form 

    H= α∑_k⃗Ψ^†_k⃗( ∑^3_j=1 d_j(k⃗) Γ_j ) Ψ_k⃗≡α∑_k⃗Ψ^†_k⃗ℋ(k⃗) Ψ_k⃗.
 
The internal structure of the spinor Ψ_k⃗ with momentum k⃗=(k_x,k_y) and representation of the Hermitian Γ matrices depend on the symmetry class, while always satisfying the anticommuting Clifford algerba {Γ_j, Γ_k }=2 δ_jk. The third component of the d⃗(k⃗)-vector takes the form d_3(k⃗)= m_0-t_0 [ cos(k_x a) + cos(k_y a) ], and unless otherwise stated d_1 (k⃗) = t_1 sin(k_x a) and d_2 (k⃗) = t_1 sin(k_y a), where a is the lattice spacing. Within the topological regime (|m_0/t_0| <2), for topological insulators (superconductors) d_3(k⃗) features a band inversion (Fermi surface) near the Γ=(0,0) and M=(1,1)π/a points of the Brillouin zone (BZ) when 0<m_0/t_0 <2 and -2<m_0/t_0<0, respectively, with α=1 (α=1/2 due to the Nambu doubling). Throughout, we set t_1=t_0=1.














Note that d_3(k⃗) and Γ_3 do not break any crystal symmetry, and ℋ_⊥(k⃗)=d_1(k⃗) Γ_1 +  d_2(k⃗) Γ_2 is also invariant under all the crystal symmetries. So, ℋ(k⃗) transforms under the trivial A_1g representation. Then the anti-Hermitian operator ℋ_ AH(k⃗)=Γ_3 ℋ_⊥(k⃗)/t_1 does not break any crystallographic symmetry either. Finally, we introduce the desired NH topological operator 

    ℋ_ NH(k⃗)= ℋ(k⃗) + t_2 ℋ_ AH(k⃗)
   
that is symmetric under all crystal symmetries, such as reflections, rotations, and inversion. Hence, it is devoid of any NH skin effect for all parameter values <cit.>. Here, t_2 determines the strength of non-Hermiticity in the system. Eigenvalues of ℋ_ NH(k⃗) are ± E_ NH(k⃗), where

    E_ NH(k⃗) =[ (t^2_1-t^2_2) { d^2_1(k⃗) + d^2_2(k⃗) } + d^2_3(k⃗) ]^1/2.

They are purely real when t_2<t_1. Then ℋ_ NH(k⃗) fosters non-trivial topological invariants, when |m_0/t_0|<2.   














NH QAHI. For Γ_j =τ_j and Ψ^⊤_k⃗=(c_+, c_-)(k⃗), where c_τ(k⃗) is the fermionic annihilation operator on orbital with parity eigenvalue τ=± and momentum k⃗, we find the NH incarnation of the Qi-Wu-Zhang model for QAHI <cit.>. The Pauli matrices {τ_μ} act on the orbital indices. Then the NH operator takes the form ℋ_ NH(k⃗) = ∑^3_j=1d^ NH_j(k⃗) τ_j, with d^ NH_1(k⃗)=t_1 sin(k_x a)-i t_2 sin(k_y a), d^ NH_2(k⃗)=t_1 sin(k_y a)+i t_2 sin(k_x a), and d^ NH_3(k⃗)=d_3(k⃗). The first Chern number of ℋ_ NH(k⃗) reads <cit.>  

    C= ∫_ BZd^2 k⃗/4π[ ∂_k_xd̂⃗̂^ NH(k⃗) ×∂_k_yd̂⃗̂^ NH(k⃗) ] ·d̂⃗̂^ NH(k⃗),

where d̂⃗̂^ NH(k⃗)=d⃗^ NH(k⃗)/√([d⃗^ NH(k⃗)]^2). The momentum integral is performed over the first BZ. For any |t_2|<1, we find C=-1 (+1) for 0<m_0/t_0<2 (-2<m_0/t_0<0). Next, we discuss the ramification of non-trivial C on the quantized σ_xy, computed in a six-terminal setup (Fig. <ref>), detailed in the Supplemental Material <cit.>. 


A voltage gradient is applied between two horizontal leads, yielding a longitudinal electrical current (I_el) between them, with no current flowing between the transverse leads, serving as the voltage probes. The current-voltage relation is then given by I_el=  G_el V, with V^⊤=(-Δ V/2,V_2,V_3,Δ V/2,V_5,V_6) and I^⊤_el=(I_el,0,0,-I_el,0,0). Upon computing the conductance matrix G_el using Kwant <cit.>, containing only the transmission blocks of the scattering matrix, we extract different voltages from the current-voltage relation. Subsequently, we compute the transverse electrical resistance R^el_xy=(V_2+V_3-V_5-V_6)/(2 I_el), and find σ_xy=1/R^el_xy=C (in units of e^2/h). See Fig. <ref>(a) and (b).


NH QSHI. A NH generalization of the Bernevig-Hughes-Zhang model for QSHI is realized with Γ=(Γ_01, Γ_32, Γ_03), where Γ_μν=σ_μτ_ν and the Pauli matrices {σ_μ} operate on the spin indiex <cit.>. The four-component spinor reads as Ψ^⊤_k⃗=(c_+^↑, c_-^↑, c_+^↓, c_-^↓)(k⃗), where c_τ^σ(k⃗) is the fermionic annihilation operator with parity τ=±, spin projection σ=↑, ↓ in the z direction, and momentum k⃗. Following the same steps, highlighted for the NH QAHI model, we compute the Chern number for individual spin components, given by C_↑ and C_↓. Within the topological regime, we find that the total Chern number of the model is C_total=C_↑ + C_↓ =0. Nevertheless, we can define an invariant, the spin Chern number C_sp=|C_↑ - C_↓| which is non-trivial in the entire topological regime (|m_0/t_0|<2 and |t_2|<1), given by C_sp=2.    
    


The non-trivial C_sp leaves its footprint on the quantized G_xx for which we employ the same six-terminal arrangement, previously discussed for the NH QAHI. In this case, σ_xy=0 as C_total=0 therein. But, from the same current-voltage relationship, we now compute the longitudinal electrical resistance R_xx=(V_3-V_2)/I_el=(V_5-V_6)/I_el, in turn yielding G_xx = R^-1_xx = C_sp (in units of e^2/h), as shown in Fig. <ref>(c) and (d). 




NH p+ip pairing. For a NH p + ip paired state among spinless or spin-polarized fermions Ψ^⊤_k⃗=(c_k⃗, c^†_-k⃗), where c_k⃗ (c^†_k⃗) is the fermionic annihilation (creation) operator with momentum k⃗. The Pauli matrices {τ_μ} operate on the Nambu or particle-hole index <cit.>. The effective single-particle NH Bogoliubov de-Gennes operator ℋ_ NH(k⃗) has the topological invariant C [Eq. (<ref>)], which manifests via a half-quantized thermal Hall conductivity (κ_xy). Now, a thermal current (I_th) flows between two horizontal leads, held at fixed but different temperatures (Fig. <ref>). Four vertical or transverse leads serve as the temperature probe, with no thermal current flowing between them. The thermal current-temperature relation is a matrix equation I_th=  A_th T, where I^⊤_th=(I_th,0,0,-I_th,0,0) and T^⊤=(-Δ T/2,T_2,T_3,Δ T/2,T_5,T_6). The elements of A_th are 

    A_th,ij =  ∫_0^∞E^2/T( -∂ f(E, T)/∂ E)[δ_ijμ_j-   Tr( t_ij^† t_ij) ] dE,

where f(E,T)=1/(1+exp[E/(k_ BT)]) is the Fermi-Dirac distribution function, μ_j denotes the number of propagating modes in the jth lead,  t_ij is the transmission part of the scattering matrix between the leads i and j, and the trace (Tr) is taken over the conducting channels. From A_th, we extract temperatures at various leads. The transverse thermal resistance is then given by R^th_xy=(T_2+T_3-T_5-T_6)/(2I_th) <cit.>, from which we compute κ_xy= ( R^th_xy)^-1 at T=0.01, showing k_xy=α  C in units of κ_0 [Fig. <ref>(a) and (b)], where α=1/2 accounts for the Nambu doubling [Eq. (<ref>)]. Thus our proposed model NH p+ip paired state features a half-quantized κ_xy in the entire topological regime.       















NH p ± ip pairing. This paired state occurs among spin-1/2 fermions, with p + ip and p-ip pairing symmetries for opposite spin projections. The associated four-component spinor is Ψ_k⃗=(ψ_k⃗, σ_2 ψ^⋆_-k⃗) with ψ_k⃗=(c_↑, c_↓)(k⃗), where c_σ(k⃗) is the fermionic annihilation operator with spin projection σ=↑, ↓ and momentum k⃗. The involved Γ matrices are Γ_1=σ_0 τ_1, Γ_2=σ_3 τ_2, and Γ_3=σ_0 τ_3. Similar to the situation in NH QSHI, C_total=0 but C_sp=2 for the NH p ± ip paired. We numerically confirm that now κ_xy=0. Nevertheless, from the longitudinal thermal resistance R^th_xx=(T_3 -T_2)/I_th=(T_6 -T_5)/I_th, we find that G^th_xx=(R^th_xx)^-1=α C_spκ_0 with α=1/2, as shown in Fig. <ref>(c) and (d).      




NH d+id pairing. In the same four-component Nambu-doubled spinor basis, a spin-singlet NH d+id paired state is realized with d_1(k⃗)=t_1 [cos(k_x a)-cos(k_y a)], d_2(k⃗)=t_1 sin(k_x a) sin(k_y a), and Γ_j=σ_0 τ_j for j=1,2,3 <cit.>. Within the topological regime (|m_0/t_0|<2 and |t_2|<1), the Chern number of the corresponding NH operator for each spin projection is C=2, and thus its total Chern number is 2C=4. This NH paired state accommodates κ_xy=C (in units of κ_0), as shown in Fig. <ref>(a) and (b). The spin-charge separation allows this NH paired state to harbor quantized spin Hall conductivity (σ^sp_xy), for which a spin current passes through the scattering region due to a difference in the magnetic field bias between the two horizontal leads. Then magnetization (m_j) develops in the vertical leads, which we compute from the spin current-magnetization matrix relation I_sp= G_sp M, where I^⊤_sp=(I_sp,0,0,-I_sp,0,0) and M^⊤=(-Δ H/2, m_2, m_3, Δ H/2, m_5, m_6), and G_sp is the spin conductance matrix, which we extract using Kwant <cit.>. From its solutions, we compute the transverse spin Hall resistance R^sp_xy=(m_2+m_3-m_5-m_6)/(2 I_sp), subsequently yielding σ^sp_xy=( R^sp_xy)^-1=C (in units of σ^sp_0), as shown in Fig. <ref>(c) and (d). 



Disorder. Finally, we unfold the effects of disorder on all the topological responses, discussed so far in clean NH systems. We consider only pointlike charge impurities, the dominant source of elastic scattering in any real material. In the above NH systems, we introduce the terms V(r⃗) τ_0, V(r⃗) σ_0 τ_0, V(r⃗) τ_3, V(r⃗) σ_0 τ_3, and V(r⃗) σ_0 τ_3, respectively, where V(r⃗) is uniformly and randomly distributed within the range [-W/2, W/2] at each site of the scattering region and W denotes the strength of disorder. The results are shown in Fig. <ref>. All the topological transport quantities (disorder averaged) retain robust (half-)quantized values in the weak disorder regime, while they all vanish in the strong disorder regime. Furthermore, with the introduction of non-Hermiticity in the system as t_1 →√(t^2_1-t^2_2) [Eq. (<ref>)], fermions become weakly dispersive and the latter event occurs at weaker disorder, which can be seen by comparing the results for t_2=0 and 0.5. However, even L=2D=180 systems [Fig. <ref>] turn out to be insufficient to compare the results for different finite t_2 values. Numerical simulations in larger systems become too time expensive for us to pursue at this stage. Nevertheless, most of the results from Fig. <ref> strongly suggest that with increasing t_2, the disappearance of the topological responses occurs at weaker disorder. 


Discussions. Here we identify a family of NH operators, devoid of NH skin effects, for planar topological insulators and superconductors that manifests the topological invariant through (half-)quantized electrical, thermal, and spin transport quantities, closely mimicking the ones previously found in Hermitian systems <cit.>. Even though the requisite six-terminal Hall bar arrangement is well-developed by now, controlled synthesis of NH quantum crystals remains far from reality. Nevertheless, optical lattices of neutral atoms constitute a promising testbed for our theoretical predictions, where a plethora of topological band engineering protocols has been proposed <cit.> and realized <cit.>. Simplicity of our construction, in which the NH operators result from nearest-neighbor hopping modulations, makes it achievable on such highly tunable platforms that also harbor superfluids (charge-neutral superconductors). In such systems, the Hall conductivity can be obtained from the “heating effect" <cit.>, for example, which can also be employed to measure quantized longitudinal transports.           


Acknowledgments. S.K.D. was supported by the Startup Grant of B.R. from Lehigh University. B.R. was supported by NSF CAREER Grant No. DMR-2238679. Portions of this research were conducted on Lehigh University's Research Computing infrastructure partially supported by NSF Award No. 2019035. We thank Vladimir Juričić for critical reading of the manuscript.





 


















