Phonon dichroism in proximitized graphene

Wen-Yu Shan(单文语)

Department of Physics,School of Physics and Materials Science,Guangzhou University,Guangzhou 510006,China

Keywords: phonon dichroism,spin-orbit coupling,proximitized graphene,electron-phonon interaction

1.Introduction

Recent years have witnessed a surge of interest in studying topological states in two-dimensional materials,[1]such as quantum anomalous Hall effect[2]and quantum spin Hall effect in graphene.[3,4]Since the occurrence of these topological states does not rely on the existence of strong magnetic field as in quantum Hall effect, they provide potential applications in dissipationless electronics and spintronics.

To realize these states,large spin-orbit coupling(SOC)or external magnetization is usually required.However, in pristine graphene,the intrinsic SOC is very weak,yielding a spinorbit splitting～µeV,[5,6]which is unable to drive a topological phase transition.When placing graphene on transitionmetal dichalcogenides, e.g., MoS2or WS2, the spin-orbit splitting can be largely enhanced up to the meV scale.[7-18]The form of SOC,according to the original proposal by Kane and Mele,[4]is uniform on A and B sublattices of graphene.However,recent spin relaxation anisotropy experiments[19-21]suggest that in heterostructures of graphene and transition metal dichalcogenides, the SOC exhibits a staggered form with opposite signs on A and B sublattices.Such a new form of SOC is proposed to support topologically protected pseudohelical edge states inZ2-trivial proximitized graphene.[22,23]

On the other hand, magnetization can be introduced in graphene by proximity to magnetic substrates with uniform(ferromagnetic) exchange coupling.[24-34]A natural question is whether these topological effects can arise from the staggered (antiferromagnetic) exchange coupling, in which case the average magnetization vanishes.Up to now, this question has received rare attention[23,38,39]since it is difficult to realize the N´eel-order antiferromagnetism (AFM) in monolayer graphene.An earlier proposal is to place the graphene on an antiferromagnetic, perovskite BiFeO3,[38]which, however, leads to ferromagnetic exchange in graphene.Fortunately,recent first-principles calculations[23,39-41]suggest that graphene on an Ising AFM MnPSe3[42,43]would be a realistic platform for the staggered exchange coupling.This provides opportunities to investigate new physics in graphene with proximitized AFM.

Similar to electrons, phonons in topological materials may bring rich physics due to the coupling between electrons and phonons.Through the electron-phonon coupling,the topological or geometric information of electrons can be inherited by phonons,giving rise to unusual phononic behaviors, e.g., phonon Hall viscosity,[44]anomalous phonon magnetic moment,[45-49]phonon magnetochiral effect,[50,51]etc.Particularly, when left- or right-handed circularly polarized phonons are propagating in conducting materials,they may be absorbed differently by electrons, a phenomenon named circular phonon dichroism(CPD).[52]Such phenomenon can be regarded as a phononic version of Hall absorption, which requires a breaking of time-reversal symmetry by external magnetic field or magnetization.Up to now, CPD has only been theoretically studied in Weyl semimetals[52]and monolayer transition metal dichalcogenides MoTe2.[53]For MoTe2,ferromagnetic substrate EuO is adopted,which introduces uniform ferromagnetic exchange and a mixture of uniform and staggered SOC in the system.Nevertheless, the individual roles of uniform and staggered SOC in CPD are unclear.Moreover,when replacing the substrate EuO with AFM MnPSe3, will CPD effect still survive? What are the roles of uniform and staggered SOC in that case? These issues are worth further investigation to understand better how the topology affects the phonon physics.

In this work, we study the linear and circular phonon dichroism in proximitized graphene with a combination of uniform (staggered) intrinsic SOC and ferromagnetic (antiferromagnetic) exchange coupling.We find that in each of these situations phonon dichroism can occur with unique features,particularly at the transition points.These phenomena help us better understand the phonon dynamics in proximitized antiferromagnetic graphene on the MnPSe3substrate,and provide opportunities to use phonon dichroism to identify the dominant mechanisms of SOC and exchange coupling in realistic proximitized graphene.

This paper is organized as follows.In Section 2,we introduce model and methods.The results of phonon dichroism for proximitized graphene are given in Section 3,including ferromagnetic and antiferromagnetic exchange couplings.Finally,conclusions and discussions are made in Section 4.

2.Model and methods

We start with the effective model of proximity modified graphene in the vicinity of Dirac points(±Kvalley)He(k)=H0+HI+HR+HZ, where the nearest-neighbor hopping and staggered potential read

Hereσiacts on sublattice(A,B)andsiacts on spin(↑,↓),withi=0,x,y,z.vFis the Fermi velocity andkx,yare wave vectors.Staggered potentialΔbreaks the inversion symmetry between sublattices A and B,which opens the band gap of Dirac cones.The valley indexη=±1 for±K.The intrinsic SOC is given byHI=HUI=ηλUσ0szorHI=HSI=ηλSσzsz,which corresponds to the uniform and staggered intrinsic SOC, respectively.The intrinsic SOC preserves the mirror symmetry alongzaxis,whereas the Rashba SOCbreaks the mirror symmetry and mixes spin↑and↓.These terms constitute the experimentally relevant model for nonmagnetic proximitized graphene.[8,22,54,55]In the presence of magnetic substrates with out-of-plane magnetization,the Zeeman term is given byHZ=mFσ0szfor ferromagnetic exchange coupling,whereasHZ=mAσzszfor AFM exchange coupling.

When phonons are propagating in the graphene plane,they can be absorbed by electrons through the electronphonon couplingHe-ph=u(q)·T(q).Hereu(q)=(ux,uy)is the displacement of ions in the momentum space andT(q)is the effective force exerted by electrons.We mainly focus on the long-wavelength acoustic phonons,which,at each valley, have the form[56,57]Tη=-1(q)= ig[q·σ,(q×σ)z] andTη=1(q)=K[Tη=-1(-q)].Kis the complex conjugation operator.Flexural modes, which correspond to the out-of-plane displacement,are neglected due to the weak electron-phonon coupling[58]and strong suppression by the substrates.[59-61]

The absorption coefficients are given byγαβ(q,ω) =α,β=x,y, where the valley-resolved coefficients read[52,53]

where we only need to consider the intraband transition of electrons, i.e.,m=n.For convenience, theγmatrix can be rewritten as

whereγD(γ¯D) refers to the symmetric (anti-symmetric) longitudinal absorption of phonons, andγA(γ¯A) refers to the symmetric(anti-symmetric)transverse absorption.When linearly polarized longitudinal or transverse phonons are considered, that is,|ul(q)〉 = [cosφq,sinφq]Tand|ut(q)〉 =[-sinφq,cosφq]T, the absorption coefficients followγl/t=γD±cos2φqγ¯D±sin2φqγ¯A.φqis the azimuthal angle ofq.This means that nonzeroγ¯Dorγ¯Adefines the linear phonon dichroism.On the other hand, when circular phonons are injected,i.e.,,the absorption coefficients readγL/R=γD ∓γA.This suggests that nonzeroγAdefines the circular phonon dichroism.These absorption coefficients can be evaluated numerically(see Appendix A).

3.Results

3.1.Ferromagnetic exchange coupling

In this section, we show numerical results of phonon dichroism in graphene with ferromagnetic exchange coupling(mF/=0).For simplicity, we first consider the case without intrinsic SOC,that is,λU=λS=0.If Rashba SOC is also absent,i.e.,λR=0,we find thatγA=0 due to the opposite contributions from valley±K.This means that circular phonon dichroism vanishes although time-reversal symmetry is broken by ferromagnetic order.

When the Rashba SOC is present, i.e.,λR/=0, numerical results are plotted in Fig.1.According to Fig.1(a), band dispersions at valley±Kare asymmetric, suggesting thatγAis non-vanishing.The Fermi level is assumed to be in the conduction band, andkFis the Fermi wave vector at valleyK.Phonons are assumed to propagate along thexdirection,q=qxˆx,andγD/¯D/Aas functions of phonon wave vectorqxare shown in Figs.1(b)-1(d).Hereγ¯A=0 sinceγ¯Ais proportional to sin4φq,[53]whereφq=0 is the azimuthal angle ofq.Results for single valleyK,-Kand both valleys are plotted in green,blue and yellow,respectively.We can see that there are two peaks forγD/¯D/A,labeled byqc1andqc2.The occurrence of peak values is becauseγD/¯D/A∝1/sinθ(see Appendix A)andθ ≈0 atq=qc1orqc2.Hereθis the angle between the phonon wave vectorqand electron wave vectork,andq ≈2katq=qc1orqc2.

Fig.1.(a) Electronic band structure and (b)-(d) phonon absorption coefficients of proximitized graphene with ferromagnetic exchange coupling mF = 0.2 eV in the absence of intrinsic SOC.In (b)-(d), green, blue and yellow lines correspond to valley K, -K and both valleys, respectively. kF is the Fermi wave vector at valley K.Parameters: Δ = 0.3 eV, ¯hvF =6.59 eV·°A, EF =0.33 eV, kF =0.061 °A-1, λR =0.225 eV, g=1.5 eV,[57]ρ = 3.8×10-7 kg/m2, γ0 = ¯hg2k4F/(16πρE2F) = 1.59×107 s-1.Phonon dispersion ω = ¯h¯c|q|, with the average sound velocity ¯c = (cl+ct)/2 =1.66×104 m/s.[57]

The sudden jumps after peaks originate from the dynamical factorFηmn(k,q)in Eq.(3).Whenqxbecomes larger thanqc1orqc2,the conditionEη,n,k-q=ω+EFfromFηmn(k,q)is no longer satisfied, thus excluding the contribution from valleyKor-Kand giving rise to the sudden jumps.The results for valley±Khave the same sign forγD/¯D,whereas they have opposite signs forγA.This originates from the opposite Berry curvature of electrons in valley±K.Since the band dispersion is particle-hole symmetric,we also check the case for the Fermi level lying in the valence band.We find thatγD/¯Dremains the same,whereasγAflips the sign.Such behaviors are similar to that of Berry curvature.This suggests that the circular phonon dichroismγAmay provide additional features to reveal the quantum geometric properties of electronic bands.

When the uniform intrinsic SOC is further included, numerical results are plotted in Fig.2.The Fermi wave vectorkFat valleyKbecomes larger than valley-K.As a result, the transition pointqc1(qc2) occurs when the contribution from valley-K(K) suddenly vanishes.Compared with Fig.1, a prominent feature in Fig.2 is that the contribution from valley-Kis strongly suppressed at the transition pointqx=qc1.Such suppression results from a combination of ferromagnetism,uniform intrinsic and Rashba SOC,which leads to vanishing transition matrix elements between wave vectorkandk-q,i.e.≈0 and≈0.By contrast,the result for valleyKis monotonic and shows sharp peaks at the transition pointqx=qc2.The different behaviors atqx=qc1andqc2originate from the different matrix structures of effective HamiltonianHe(k)at valleysKand-K.

Fig.2.(a) Electronic band structure and (b)-(d) phonon absorption coefficients of proximitized graphene with ferromagnetic exchange coupling mF =0.2 eV and uniform intrinsic SOC λU =0.2 eV.Insets of(b)-(d)show the details of the highlighted region.In (b)-(d), green, blue and yellow lines correspond to valley K,-K and both valleys,respectively.Parameters:kF=0.096 °A-1,γ0=0.99×108 s-1.Other parameters are the same as Fig.1.

Fig.3.(a) Electronic band structure and (b)-(d) phonon absorption coefficients of proximitized graphene with ferromagnetic exchange coupling mF = 0.2 eV and staggered intrinsic SOC λS = 0.2 eV.Insets of (b) and(c) show the details of the highlighted region.Parameters: kF =0.075 °A-1,γ0=3.7×107 s-1.Other parameters are the same as Fig.1.

When the uniform intrinsic SOC is replaced by the staggered one,numerical results are shown in Fig.3.We find thatγD/¯D/Aare strongly enhanced and have peak values at the transition pointqx=qc1,in marked contrast to the case with uniform intrinsic SOC.This is because in the presence of staggered SOC, the transition matrix elements [Tη=-1x]nnk,k-qand[Tη=-1y]nnk,k-qare nonzero,and the presence of factor 1/sinθgives rise to the peak values ofγD/¯D/Aat the transition point(θ ≈0).The above studies suggest that in the presence of ferromagnetic exchange coupling, the uniform and staggered intrinsic SOC have distinguishable features in the linear or circular phonon dichroism.

3.2.Antiferromagnetic exchange coupling

In this section, we show numerical results of phonon dichroism in graphene with antiferromagnetic exchange coupling (mA/= 0).When both uniform intrinsic and Rashba SOC are taken into account, numerical results are plotted in Fig.4.We find that at the transition pointqx=qc1and for valley-K,γD/¯Dhave significant peaks whileγAis strongly suppressed and exhibits non-monotonic behaviors when increasingqx.The difference betweenγD/¯DandγAoriginates from the finite transition[Tη=-1y]nnk,k-qand vanishingly small transition [Tη=-1x]nnk,k-q.Note that there is a correspondence between AFM with uniform intrinsic SOC and ferromagnetism with staggered intrinsic SOC at valleyK, thus they have the sameγD/¯D/Afor valleyKas shown in Figs.3 and 4(denoted by green curves).Nevertheless,such correspondence fails for valley-K,which gives rise to the distinction between the two situations.

Fig.4.(a)Electronic band structure and(b)-(d)phonon absorption coefficients of proximitized graphene with AFM exchange coupling mA=0.2 eV and uniform intrinsic SOC λU =0.2 eV.Insets of (b) and (c)show the details of the highlighted region.Parameters:kF=0.075 °A-1,γ0=3.7×107 s-1.Other parameters are the same as Fig.1.

When the uniform intrinsic SOC is replaced by the staggered one, numerical results are given in Fig.5.At the transition pointqx=qc1and for valley-K,γD/¯D/Aare again strongly suppressed.This is due to the fact that[Tη=-1x]nnk,k-q ≈0 and [Tη=-1y]nnk,k-q ≈0 at the transition point.Such behaviors are quite similar to the situation with ferromagnetic exchange coupling and uniform intrinsic SOC as in Fig.2.The reason is that there is a correspondence between AFM with staggered intrinsic SOC and ferromagnetism with uniform intrinsic SOC at valley-K.As a result,γD/¯D/Afor valley-Kare exactly the same between the two situations as shown in Figs.2 and 5(denoted by blue curves).However,there is a major difference in the behavior ofγA.In Fig.2(d),γAfor valleysKand-Khas opposite signs, therefore the totalγAexhibits a sign change by increasingqx.By contrast,in Fig.5(d),γAfor valleysKand-Khas the same sign,thus the totalγAdisplays no sign change.This provides the distinguishable features from all above situations.

Fig.5.(a)Electronic band structure and(b)-(d)phonon absorption coefficients of proximitized graphene with AFM exchange coupling mA=0.2 eV and staggered intrinsic SOC λS =0.2 eV.Insets of (b) and (c)show the details of the highlighted region.Parameters:kF=0.045 °A-1,γ0=0.5×107 s-1.Other parameters are the same as Fig.1.

4.Conclusions

To summarize, we systematically study the phonon dichroism in proximitized graphene with external magnetization.We find that any combination of uniform(staggered)intrinsic SOC and ferromagnetic (antiferromagnetic) exchange coupling can lead to nonvanishing phonon dichroism.Qualitatively different behaviors of linear and circular phonon dichroism among these situations are compared, particularly at the transition point.This provides opportunities to apply phonon dichroism effect to detect the form of SOC and magnetic exchange coupling in proximitized graphene.

Note that phonon dichroism has no direct relation to the topological properties of the graphene system.Actually, the situation in Figs.2,3,4 and 5 corresponds to Chern phase with Chern numberC=2,metallic phase,Chern phase withC=1 and metallic phase,respectively.[23]Phonon dichroism is more likely a band geometric quantity, which connects electronic states with wave vectorskandk-q.In the long-wavelength limitq →0, circular phonon dichroismγAis related to the interband Berry connection.[62]That is the reason whyγAreverses the sign when the Fermi level is tuned from the conduction to valence band in Fig.1.

In contrast to Weyl semimetals,[52]phonon dichroism in graphene has several distinct features.First, circularly polarized phonons are constructed by non-degenerate longitudinal and transverse acoustic modes in graphene, rather than two degenerate transverse modes in Weyl semimetals.Such non-degeneracy substantially suppresses the outof-plane phonon angular momentum and the screening effect of electrons.[52,63]Second,the absorption of phonons is much stronger in graphene due to the smaller 2D mass density and larger electron-phonon coupling strength.This suggests that the experimental observation of circular phonon dichroism is more feasible in graphene.

Phonon absorption is qualitatively different from photon absorption.For photon absorption, transitions of electrons are allowed between conduction and valence bands,therefore the optical conductivity is contributed by the band geometric quantities of both conduction and valence bands.For absorption of acoustic phonons, transitions of electrons are within valence(conduction)bands with different wave vectors since acoustic phonon modes are unable to induce the interband transitions.As a result,phonon absorption coefficients depend on the information of band geometries of purely valence(conduction)bands.

The proposed CPD effect can be detected by the pulseecho technique[64,65]based on different absorption coefficients between left- and right-handed circularly polarized phonons.An alternative detection is the Raman spectroscopy analysis[66,67]of phonon polarization by injecting linearly polarized acoustic waves.

Appendix A:Numerical method

To numerically calculate the absorption coefficientsγD/¯D/A/¯A, we can change the integral variable fromktok=|k|,k＇=|k-q|,[53]and obtain

l=D, ¯D,A, ¯A,where

andY=δφk=φq+θ+δφk=φq-θ.φk=tan-1(ky/kx)andφq=tan-1(qy/qx).θis the angle betweenqandk,satisfying

By integrating out the delta function,we have

Eη,m,k/k＇is eigen-dispersion of electronic HamiltonianHe(k).Θ(·)is the Heaviside function.The derivatives can be evaluated numerically by

where

with

Acknowledgements

I am grateful to Yurong Weng for many valuable discussions and comments.This work is supported by the National Natural Science Foundation of China (Grant No.11904062),the Starting Research Fund from Guangzhou University(Grant No.RQ2020076), and Guangzhou Basic Research Program, jointed funded by Guangzhou University (Grant No.202201020186).