Berry curvature induced thermopower in type-I and type-II Weyl Semimetals
BBerry curvature induced thermopower in type-I and type-II Weyl semimetals
Kamal Das ∗ and Amit Agarwal † Department of Physics, Indian Institute of Technology Kanpur, Kanpur 208016, India
Berry curvature acts analogously to a magnetic field in the momentum-space, and it modifiesthe flow of charge carriers and entropy. This induces several intriguing magnetoelectric and mag-netothermal transport phenomena in Weyl semimetals. Here, we explore the impact of the Berrycurvature and orbital magnetization on the thermopower in tilted type-I and type-II Weyl semimet-als, using semiclassical Boltzmann transport formalism. We analytically calculate the full magne-toconductivity matrix and use it to obtain the thermopower matrix for different orientations of themagnetic field ( B ), with respect to the tilt axis. We find that the tilt of the Weyl nodes induces linearmagnetic field terms in the conductivity matrix, as well as in the thermopower matrix. The linear- B term appears in the Seebeck coefficients, when the B -field is applied along the tilt axis. Applyingthe magnetic field in a plane perpendicular to the tilt axis results in a quadratic- B planar Nernsteffect, linear- B out-of-plane Nernst effect and quadratic- B correction in the Seebeck coefficient. I. INTRODUCTION
Weyl semimetals (WSMs) host relativistic masslessfermionic quasiparticles in the vicinity of the Weyl nodeswhich always come in pairs of opposite chirality . Theirexistence has been demonstrated in several materials where either time reversal or space inversion symmetryis broken. Unlike their relativistic counterparts, in crys-talline systems the Weyl quasiparticles can also breakLorentz invariance. Consequently, their dispersion canbe tilted in a specific direction . Depending on thedegree of the tilt, these WSMs can be classified as type-Ior type-II. In a type-I WSM, the Fermi surface enclosesonly one kind of carriers, and has a vanishing density ofstates at the Weyl point. In contrast, a type-II WSM hasnon-vanishing density of states at the Weyl point, and theWeyl point appears at the intersection of an electron anda hole pocket.Interestingly, the Weyl nodes act as a source or sink ofBerry curvature (BC), which in turn acts as a fictitiousmagnetic field in the momentum space . This leadsto the possibility of several interesting transport phenom-ena in isotropic and tilted WSMs . Several of thesehave also been experimentally demonstrated . Forinstance, negative magnetoresistivity (MR) has beenobserved in several WSM candidates including the TaAsfamily and in WSMs induced magnetically fromthree dimensional Dirac semimetals . The anomalousHall effect predicted to exist in time reversal symme-try (TRS) broken WSMs has been recently seen inZrTe . The corresponding effect in thermopower, theanomalous Nernst effect in WSM has been demon-strated in Cd As , NbP and Ti MnX . Chiral mag-netic effect, a chiral anomaly induced phenomena has been reported in ZrTe . The BC induced planarHall effect, where the current response is measured inthe plane of electric and magnetic field, has also beenpredicted in WSM and multi-WSM and experi-mentally demonstrated in WSM . More recently, lin-ear magnetic field dependence in both the MR and Hallresponses is predicted to exist in tilted WSMs . Motivated by these recent studies, in this paper weexplore the BC induced magnetothermopower in tiltedtype-I and type-II WSMs: the Seebeck and the Nernstcoefficients (SCs and NCs, respectively). Our analyticalcalculations for the full conductivity and thermopowermatrix are based on the Berry-connected-Boltzmann-transport formalism and include the effect of the orbitalmagnetic moment (OMM) . The Seebeck effect cap-tures the electric response along the temperature gradi-ent while the Nernst effect captures the electric responseperpendicular to the temperature gradient. The conduc-tivity and the thermopower are connected by the Mottrelation [see Eq. (4)] even in the presence of the OMMcorrection . Thus the magnetothermopower broadly fol-lows the magnetoelectric response, leading to the expec-tation of phenomena such as negative Seebeck effect andplanar Nernst effects in WSMs. A similar kind of phe-nomenon is known to exist in ferromagnetic systems where spin dependent scattering induces a transverse ve-locity component in the charge carriers .In this paper, we have calculated the electrical con-ductivity and thermopower matrix to explore the BC-induced magnetotransport in type-I and type-II WSMs.We have explicitly included the previously ignored im-pact of the OMM in all our calculations. Although theelectrical conductivity matrix is well explored (excludingthe OMM), the BC-induced thermopower and the im-pact of the tilt on it is relatively unexplored, and this isthe primary focus of this paper. In particular, we pre-dict the following BC-induced phenomena: (1) linear- B as well as quadratic- B dependent SCs, (2) the existenceof quadratic- B planar Nernst as well as linear- B out-of-plane Nernst response, (3) negative longitudinal (paral-lel electric and magnetic field) MR and positive perpen-dicular (perpendicular electric and magnetic field) MRin WSM. The rest of the paper is organized as follows:In Sec. II we present the phenomenological equation ofcharge current and establish the relation between chargeconductivity and thermopower. This is followed by adetailed discussion of the full magnetoconductivity ma-trix for type-I and type-II WSM in Sec. III, and MR inSec. IV. We discuss various aspects of the thermopower a r X i v : . [ c ond - m a t . m e s - h a ll ] A ug matrix in Secs. V, VI and VII. We summarize our find-ings in Sec. VIII. II. THERMOPOWER IN PRESENCE OFBERRY CURVATURE
Within the linear response theory, the phenomenologi-cal transport equation for the electrical current j e is givenby , j ei = σ ij E j + α ij ( −∇ j T ) . (1)Here, E j and ∇ j T are the external electric field andtemperature gradient applied along the j th direction, σ ij denotes the elements of electrical conductivity ma-trix (˜ σ ) and α ij are the elements of thermoelectric con-ductivity matrix (˜ α ). These transport coefficients arecalculated by doing a Brillouin zone sum over the rel-evant physical quantities, involving only the occupiedstates. In this paper, we use the semiclassical Boltz-mann transport formalism to calculate the magnetocon-ductivity and magnetothermopower. The details of theBerry-connected-Boltzmann-transport formalism are dis-cussed in Appendix A. The general expressions for BC-induced conductivity and thermopower are presented inEqs. (A10), and (A11), respectively.The thermopower for an open circuit system is definedby setting j ei = 0 in Eq. (1). In this scenario, the electricfield generated by a temperature gradient is given by, E i = ν ij ∇ j T, where ˜ ν ≡ ˜ σ − ˜ α . (2)The diagonal elements ν ii denote the SCs whereas theoff-diagonal elements ν ij ( i (cid:54) = j ) are the NCs. It turnsout that in the low temperature limit ( k B T (cid:28) µ ), BC-induced thermopower can also be expressed in terms ofthe electrical conductivity using the Mott relation . TheMott relation yields, α ij = − π k B T e ∂σ ij ∂(cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) = µ . (3)Using Eq. (3) in Eq. (2), the thermopower matrix can beexpressed solely in terms of the electrical conductivitymatrix as ˜ ν = − π k B T e ˜ σ − ∂ ˜ σ∂(cid:15) (cid:12)(cid:12)(cid:12)(cid:12) (cid:15) = µ . (4)Further, to explicitly track the magnetic field depen-dence analytically, we express σ ij = σ (0) ij + σ (1) ij + σ (2) ij + O ( B ), and α ij = α (0) ij + α (1) ij + α (2) ij + O ( B ). Here the su-perscripts denote zeroth order (Drude and anomalous),linear and quadratic magnetic field terms, respectively.In the next section, we calculate the magnetoconductiv-ity matrix for tilted WSM, including the OMM correc-tions. Since the tilt-axis (we consider ˆ z ) breaks the TRS for each node , the two cases in which B is applied paral-lel and perpendicular to ˆ z result in different forms of theconductivity matrix. III. MAGNETOCONDUCTIVITY IN TYPE-IAND TYPE-II WSMS
The low energy Hamiltonian, for each of the chiral nodeof a tilted WSM is given by, H s ( k ) = (cid:126) C s k z + s (cid:126) v F σ · k . (5)Here, s denotes chirality, C s ( v F ) is the tilt (Fermi) ve-locity, and σ = ( σ x , σ y , σ z ), are the Pauli matrices. Inthis paper, we consider a WSM with a pair of oppositelytilted nodes such that C − s = − C s . The degree of the tiltof the Weyl nodes can be quantified by the ratio of thetilt and the Fermi velocities: R s = C s /v F . The abovedispersion corresponds to the type-I class of WSMs inthe regime | R s | < | R s | > Ω λs = − λs k / (2 k ), where λ = +1 ( λ = −
1) denotes theconduction (valence) band. The OMM can be calculatedfrom Eq. (A4) and can be expressed in terms of the BC , m λs = λev F k Ω λs = − sev F k k . (6)Both of the BC and the OMM are independent of thetilt velocity. Furthermore, the OMM, and the resultingvelocity correction are identical for both the bands.The impact of the tilt on the magnetoconductivity intype-I and type-II WSM was explored in Ref. [58]. Here,we generalize those results to include the effect of theOMM. The conductivity matrix can be expressed as sumof the contributions in absence and presence of a mag-netic field: ˜ σ = ˜ σ D + ˜ σ B , such that ˜ σ B vanishes as B → σ D is the Drude conductivity. For the scenario inwhich B ⊥ ˆ z , and the magnetic field is confined in the x - y plane (planar geometry), we find that the conductivitymatrix has this general form for both type-I and type-IIWSMs, ˜ σ B = σ (2) ⊥ + ∆ σ (2) cos φ ∆ σ (2) sin(2 φ ) / σ (1)t cos φ ∆ σ (2) sin(2 φ ) / σ (2) ⊥ + ∆ σ (2) sin φ σ (1)t sin φσ (1)t cos φ σ (1)t sin φ σ (2)z . (7)Here, φ is the angle of the magnetic field with respect tothe x axis. See Appendix B for the details of calculationof ˜ σ D . Here, σ = σ xy denotes the planar Hall conduc-tivity and in addition there are new linear- B terms suchas σ = σ xz and σ = σ yz , which were discussed inRef. [58].For the other case of B (cid:107) ˆ z , the general form of theconductivity matrix has a diagonal form ,˜ σ B = σ (1)l + σ (2)l σ (1)l + σ (2)l
00 0 σ (1)lz + σ (2)lz . (8)Here, the diagonal components have linear- B dependenceinduced by the tilt. The analytical expression of the dif-ferent conductivity components is presented in the sub-sections below where we have presented results includ-ing OMM. However, to explicitly highlight the impactof OMM, we have presented the general expressions interms of γ , in Appendix C. A. Type-I WSMs
For the B ⊥ ˆ z case, the magnetoconductivity is givenby Eq. (7), where the conductivity coefficients σ (2) ⊥ , σ (2)z and ∆ σ (2) are proportional to B . In type-I WSM, in-cluding the OMM, these are explicitly given by∆ σ (2) = (cid:88) s (cid:0) R s (cid:1) σ ; σ (2) ⊥ = − (cid:88) s σ , (9) σ (2)z = − (cid:88) s (cid:0) − R s (cid:1) σ , (10)where we have denoted the quadratic dependence as σ ≡ e τ π (cid:126) v F µ (cid:18) eB (cid:126) (cid:19) . (11)These terms are finite even in the limit of R s → R s . Thus, the contributions from apair of oppositely tilted nodes just adds up. Note theopposite sign of ∆ σ (2) and σ (2) ⊥ and this will manifest inthe perpendicular MR ( φ = π/
2) being positive, and thelongitudinal MR ( φ = 0) being negative, as discussed inthe next section.In addition to these quadratic- B terms, there arelinear- B dependent off-diagonal conductivity compo-nents as well ( σ xz , σ yz ∝ σ (1)t ∝ B ). These terms arisesolely due to the tilt of the Weyl nodes (which breaks theTRS for each node), and vanish in the limit of R s → B components are σ (1) xz = σ (1) yz ( π/ − φ ) = σ (1)t cos φ , where σ (1)t can be expressed as σ (1)t = (cid:88) s sσ R s (cid:2) R s (cid:0) − R s (cid:1) + F δ s (cid:3) . (12)In the above equation, we have defined F ≡ − R s and, σ = e τ (2 π ) πv F (cid:126) eB (cid:126) , and δ s = ln (cid:18) − R s R s (cid:19) . Note that the contributions for the oppositely tiltednodes simply add up and the overall sign of this com-ponent depends on the details of tilt configuration.For the B (cid:107) ˆ z configuration, the conductivity matrixis given by Eq. (8). As the matrix structure shows, inthis case the longitudinal conductivities have a linear- B dependence in addition to the quadratic- B one. Thequadratic- B correction perpendicular to the tilt is σ (2)l = σ (2) ⊥ = − (cid:80) s σ , whereas along the tilt axis, it is givenby σ (2)lz = (cid:88) s (cid:0) R s (cid:1) σ . (13) The linear- B term in σ xx = σ yy and σ zz is given by σ (1)l = − (cid:88) s sσ R s (2 R s + δ s ) ; σ (1)lz = − (cid:88) s sσ (2 R s ) . (14)We emphasize that all the linear- B conductivity dis-cussed in this section is ∝ σ , in which there is no ex-plicit µ dependence. This is primarily a consequence ofΩ k ∝ /k in WSMs. The only µ dependence of σ arisesfrom the energy dependence of the scattering timescale τ . B. Type-II WSMs
The low energy model Hamiltonian of Eq. (5) corre-sponds to the type-II class in the regime | R s | >
1. In thisregime, the Fermi surface of the Weyl node comprises of“unbounded” electron and hole pockets. Hence both thebands take part in transport. And to truncate the “un-bounded sea” of the charge carriers, we need to introducea cutoff in the momentum space along the radial direc-tion (Λ k ). In real materials, this is akin to the bandwidthof the system. For simplicity, we present all the conduc-tivity terms only upto first order in k F / Λ k ≡ / ˜Λ k , andassume µ > σ (2) = 2 (cid:88) s K (cid:0) R s + 35 R s + 50 R s − R s − (cid:1) , (15) σ (2) ⊥ = (cid:88) s K (cid:0) R s − R s + 25 R s − (cid:1) , (16) σ (2)z = 2 (cid:88) s K (cid:0) R s + 65 R s − R s − R s + 4 (cid:1) , (17)where K ≡ σ | R s | . Note that the tilt induced correctionsoccur as even powers of R s , implying the addition of con-tributions from the oppositely tilted nodes. The linear- B correction in the out-of-plane off-diagonal conductivitiescan be written as σ (1) xz = σ (1) yz ( π/ − φ ) = σ (1)t cos φ . Here, σ (1)t = (cid:88) s sσ R s sgn( R s ) (cid:2) R s − R s − F R s δ s (cid:3) , (18)and we have defined δ s = ln( R s −
1) + 2 ln ˜Λ k . (19)Now, we consider a magnetic field along the directionof the tilt ( B (cid:107) ˆ z ). The linear- B correction to the longi-tudinal component in the x − y plane, σ xx = σ yy , is givenby σ (1)l = − (cid:88) s sσ R s sgn( R s ) (cid:2) R s − − δ s R s (cid:3) . (20)The linear- B correction to σ zz is given by σ (1)lz = − (cid:88) s sσ R s sgn( R s ) (cid:0) R s − R s + 1 (cid:1) . (21) φ (degree) − . − . . M R ( a ) xxzz . . . . R − . − . . ( b ) xxzz FIG. 1. a) MR for type-I WSM ( R = 0) as a function of theangle between E and B for the planar geometry. The planarMR (MR xx ( φ )) varies as cos φ (blue lines). The longitudinalMR (MR xx ( φ = 0)) is negative irrespective of OMM correc-tion. The perpendicular MR (MR zz ) becomes positive onincluding the OMM (solid red line). (b) MR as a function oftilt ( R ) for the configuration R − = − R + = R . The longitudi-nal MR remains negative (blue lines) while the perpendicularMR changes sign at a certain critical R value, beyond which itremains negative (red lines). Here, we have used the followingparameters: µ = 0 . v F = 10 m/s and B = 4 T. The quadratic- B correction to σ xx and σ yy is given by σ (2)l = (cid:88) s σ | R s | (cid:0) − R s + 5 R s (cid:1) . (22)The corresponding term for the σ zz component is givenby σ (2)lz = (cid:88) s σ | R s | (cid:0) R s − R s + 5 R s − (cid:1) . (23)Having obtained the full conductivity matrix for tiltedWSM, now we discuss tilt and OMM dependence of theMR – the quantity generally probed in experiments. IV. MAGNETORESISTIVITY
The resistivity matrix is obtained by inverting the con-ductivity matrix. The corresponding MR is given byMR ii = ρ ii ( B ) /ρ ii (0) −
1. Below we discuss the longitu-dinal and perpendicular MR for the two cases of B ⊥ ˆ z and B (cid:107) ˆ z .For the case of planar geometry, using Eq. (7) we ob-tain the planar resistivity to be ρ xx = ρ D − ρ (2) ⊥ + (cid:20)(cid:16) ρ (1)t (cid:17) ρ zD [ ρ D ] − − ∆ ρ (2) (cid:21) cos φ . (24)Here, we have defined the Drude resistivity in the x - y plane as ρ D = 1 /σ D , and along the z axis as ρ zD = 1 /σ z D .Additionally, we have defined the following: ρ (2) ⊥ = σ (2) ⊥ /σ D2 , ρ (1)t = σ (1)t /σ D2 and ∆ ρ (2) = ∆ σ (2) /σ D2 . Itis evident from Eq. (24) that the planar MR [MR xx ( φ )]is anisotropic and varies as cos φ on changing the planar B direction with respect to the x axis. In Fig. 1 we have − − B (Tesla) − . . . M R ( a ) xxzz . . . . R − . − . . . ( b ) xxzz FIG. 2. MR of type-I WSM for the case of B (cid:107) R with tilt con-figuration of Fig. 1. (a) The magnetic field dependence of theMR. Note that MR xx and MR zz have a linear- B contributionfor a finite tilt (here R = . B = 0 line. (b) Variation of MR as afunction of the tilt parameter R at B = 4 T. Note that the in-clusion of OMM correction forces perpendicular MR (MR xx )to be positive (solid blue line), while longitudinal MR (MR zz )remains negative with or without OMM correction (solid ordashed red line, respectively). Here the parameters used areidentical to those of Fig. 1. plotted the MR with φ and tilt factor. We have useddotted lines for conductivities without the contributionof OMM ( γ = 0) in our plots. Note that the longitudinalMR [MR xx ( φ = 0)] is negative irrespective of inclusionor exclusion of the OMM and the degree of the tilt of theWSM. However, the perpendicular MR [MR xx ( φ = π/ γ = 1)for isotropic WSMs shown in Fig. 1(a). For the out-of-plane perpendicular MR (MR zz ), we obtain the resistiv-ity to be ρ zz = ρ zD − ρ (2)z + (cid:16) ρ (1)tz (cid:17) ρ D [ ρ zD ] − . (25)Here, we have defined ρ (2)z = σ (2)z /σ zD2 and correction dueto the linear- B Hall conductivity component as ρ (1)tz = σ (1)t /σ zD2 . The OMM correction forces the out-of-planeperpendicular MR to be positive (solid red line) for WSMwith small tilt – as shown in panels (a) and (b) of Fig. 1.For the other case of B (cid:107) ˆ z , we calculate resistivityalong the tilt direction from Eq. (8), and it is given by ρ zz = ρ zD − ρ (1)lz + (cid:16) ρ (1)lz (cid:17) [ ρ zD ] − − ρ (2)lz . (26)Here we have defined ρ (1)lz = σ (1)lz / ( σ zD ) and ρ (2)lz = σ (2)lz / ( σ zD ) . Evidently, in this case the longitudinal MR,MR zz , will have linear- B contribution for a tilted WSM,while its absolute value depends on the degree of the tilt,starting with a negative value for an isotropic WSM. Thislinear- B part gives rise to an asymmetry in the MR curveas B goes from negative to positive – see Fig. 2(a). Notethat the inclusion of OMM does not change the sign oflongitudinal MR. The expression for ρ xx is given by ρ xx = ρ D − ρ (1)l + (cid:16) ρ (1)l (cid:17) [ ρ D ] − − ρ (2)l . (27)Here, we have defined ρ (1)l = σ (1)l /σ D2 and ρ (2)l = σ (2)l /σ D2 . Similar to the case of ρ zz , ρ xx also has linear- B contributions leading to asymmetric MR curves aroundthe B = 0 line shown in Fig. 2(a). However, unlike thecase of longitudinal MR, the perpendicular MR, MR xx ,changes sign on including the OMM and reverses fromnegative to positive as shown in Fig. 2(b).Our findings for isotropic WSM that the longitudinalMR is negative, while the perpendicular MR is positive,are consistent with the experimental MR results reportedin Dirac semimetals and isotropic WSMs . We em-phasize that the inclusion of OMM is crucial to capturethe correct sign of the perpendicular MR. V. THERMOPOWER IN WEYL SEMIMETAL
In this section we calculate the magnetic field depen-dent thermopower at low temperature using the Mottrelation . Let us first consider the case B ⊥ ˆ z . Since˜ α ∝ ∂ µ ˜ σ , the thermoelectric conductivity matrix retainsthe form of Eq. (7), and it is given by ˜ α B = α (2) ⊥ + ∆ α (2) cos φ ∆ α (2) sin(2 φ ) / α (1)t cos φ ∆ α (2) sin(2 φ ) / α (2) ⊥ + ∆ α (2) sin φ α (1)t sin φα (1)t cos φ α (1)t sin φ α (2)z . (28)The different thermoelectric conductivity elements in thematrix are connected to the corresponding elements inthe conductivity matrix of Eq. (7) via the Mott relation[Eq. (3)]. At a first glance it seems that the out-of-planeHall components ( α xz and α yz ∝ α (1)t ) are zero for type-I WSM as the corresponding elements in the electricalconductivity matrix are independent of the Fermi energy.However, the scattering timescale is generally dependenton the Fermi energy, and this would lead to a finite linear- B term in the thermoelectric conductivity matrix as well.Another possibility is that the deviations from the linearmodel, for example in a lattice model, can also lead tofinite linear- B contribution. Similar physics is seen in thecase of the finite anomalous Nernst response in a tight-binding model of WSMs , even though the anomalousHall coefficient is independent of the Fermi energy in theisotropic low energy model of WSM .The thermopower matrix can now be calculated by us-ing Eqs. (7) and (28) in Eq. (2). The SC in the pla-nar configuration can be expressed in the form ν yy = ν xx ( π/ − φ ), where ν xx − ν D = ν (2) ⊥ + ∆ ν (2) cos φ . (29)Here, we have defined ν D = α D /σ D as the usual Drudecoefficient calculated in Appendix B and the magnetic field dependent coefficients are given by ν (2) ⊥ = σ − (cid:16) σ D α (2) ⊥ − α D σ (2) ⊥ (cid:17) , (30) ∆ ν (2) = 1 σ (cid:34) ( σ D ∆ α (2) − α D ∆ σ (2) + (cid:16) σ (1)t α D − α (1)t σ D (cid:17) σ (1)t σ zD (cid:35) . (31)The out-of-plane SC (along the z axis) can be expressedas ν zz = ν zD + ν (2)z , where ν zD ≡ α zD /σ zD is the Drudecontribution along the tilt axis and the correspondingquadratic- B correction is given by ν (2)z = σ zD ) (cid:34) σ zD α (2)z − α zD σ (2)z + (cid:16) σ (1)t α zD − α (1)t σ zD (cid:17) σ (1)t σ D (cid:35) . (32)For the planar configuration, we obtain the coefficientfor the planar Nernst effect, ν yx = ∆ ν (2) sin φ cos φ . (33)This has an identical angular dependence on the planarangle between E and B to that of the planar Hall effect.In addition to the planar Nernst effect, we find the out-of-plane linear- B NCs, and are given by ν xz = ν (1)t cos φ = ν yz ( π/ − φ ), with ν (1)t = 1 σ zD (cid:32) α (1)t σ zD − σ (1)t α zD σ D (cid:33) . (34)The angular dependence of the planar SC ( ν xx ∝ cos φ ), is shown in Fig. 3(a) for type-I WSMs and inFig. 4(a) for type-II WSMs. The relative phase differ-ence between the two classes is due to the opposite signof Drude conductivity shown in Appendix B. The planarNC ( ν xy ∝ sin 2 φ ) and the out-of-plane NC ( ν xz ∝ cos φ )are highlighted in Figs. 3 (c), and 4 (c), for type-I andtype-II WSMs, respectively. Again we find a relativephase difference in the linear- B NC between the twoclasses. However this is not due to the Drude conduc-tivity but due to the ‘tilted over’ nature of the type-IIWSM. The inclusion of an OMM has a significant impacton the perpendicular SCs (magnetic field perpendicularto temperature gradient). It reverses the sign of the B -induced contribution in the ν zz for the type-I WSM fromnegative to positive upto a critical tilt parameter, beyondwhich it retains its negative value [see Fig. 3(b)]. This isreminiscent of the sign change also seen in the perpen-dicular MR in Fig. 1 (b). Note that the sign reversal of ν zz ( B ) /ν zz (0) − R ≈ . ν zD as shown in Fig. 7.In the presence of BC, the magnetic field suppressesthe longitudinal SC [ ν xx ( φ = 0)] resulting in what istermed as a negative Seebeck effect. At the same time, itenhances the perpendicular SC ( ν zz ) for a type-I WSM,as shown in Fig. 3(d). This kind of negative longitudinalSeebeck effect and positive perpendicular Seebeck effect φ (degree) − . − . . ν xx ( B ) / ν xx ( ) − ( a ) . . . . R − . . . ν zz ( B ) / ν zz ( ) − ( b ) φ (degree) − . . . ν i j ( ν D ) ( c ) xyxz B (Tesla) − . − . . ν ii ( B ) / ν ii ( ) − ( d ) zzxx FIG. 3. Various components of thermopower in the planargeometry for the type-I class with tilt configuration R − = − R + = R . (a) The cos φ dependence of the planar SC in-cluding (solid line) and excluding (dashed line) the OMM cor-rection. (b) The dependence of the out-of-plane SC with thetilt parameter. Note that the inclusion of OMM correction(solid line) changes the sign of the B dependent contributionfrom negative to positive, up to a critical R . (c) The an-gular dependence of the planar NC ( ν xy ∝ sin 2 φ ) and theout-of-plane NC ( ν xz ∝ cos φ ). (d) The B dependence of thelongitudinal and the out-of-plane transverse SC, which resultsin a negative and a positive Seebeck effect, respectively. Wehave used the parameters of Fig. 1 and R = 0 . has been experimentally observed in the magneticallyinduced isotropic WSM phase in Cd As and NbP .Our calculations predict that for a type-II WSM, the signof both the longitudinal and the perpendicular SC changeas compared to the type-I class, as indicated in Fig. 4(d).This is because of the sign change of the correspondingDrude components as discussed in Appendix B.For the case of B (cid:107) ˆ z , the thermoelectric conductivitymatrix can be written as˜ α − ˜ α D = α (1)l + α (2)l α (1)l + α (2)l
00 0 α (1)lz + α (2)lz . (35)Using Eqs. (8) and (35) in Eq. (2) yields the thermopowermatrix. For this configuration, since both ˜ σ and ˜ α arediagonal, the thermopower matrix has no off-diagonalterms i.e., no Nernst response. The diagonal componentsare given by ν xx = ν yy and ν xx = ν D + ν (1)l + ν (2)l , (36) ν zz = ν zD + ν (1)lz + ν (2)lz . (37)Here, we have defined the linear- B correction along the x axis to be ν (1)l = σ D α (1)l − α D σ (1)l σ , (38) φ (degree)0 . . . ν ii ( B ) / ν ii ( ) − ( a ) xx . . . . . R − ( b ) zz φ (degree) − . . . ν i j ( ν D ) ( c ) B (Tesla)0 . . . ν ii ( B ) / ν ii ( ) − ( d ) zzxx FIG. 4. Same as Fig. 3, but for a type-II WSM. (a) The φ dependence of ν xx . (b) The tilt dependence of ν zz hascontributions from electrons as well as holes. Here the signreversal in ν zz from negative to positive arises from the sign ofthe Drude component reversing at large R (see Fig. 7). Thisis a direct consequence of the hole carriers dominating thetransport on increasing the WSM tilt. (c) The φ dependenceof planar ( ν xy - purple curve) and out-of-plane ( ν xz - greencurve) NCs. Note the phase difference of π in the ν xz responsebetween a type-I and a type-II WSM. (d) The B induced partof the SC has opposite signs for the longitudinal ν xx ( φ = 0),and the out-of-plane transverse ν zz components. Here, wehave used the parameters of Fig. 1 and the tilt parameter R = 1 . k = 10. and the quadratic- B correction in Eq. (36) reads as ν (2)l = 1 σ (cid:32) α (2)l σ D − α D σ (2)l + (cid:16) σ (1)l α D − α (1)l σ D (cid:17) σ (1)l σ D (cid:33) . (39)The linear and quadratic- B correction along the z di-rection can be generated from the above two equationssimply by replacing the x component of σ ’s and α ’s bytheir z components.Interestingly, the SCs have a linear- B dependence,arising from TRS breaking tilt. This is reminiscent oflinear- B terms also appearing in MR. The tilt and B dependence of the longitudinal SCs for type-I WSM isshown in Fig. 5, while the same for type-II is shownin Fig. 6. Evidently the OMM plays an importantrole, reversing the sign of the perpendicular SC ( ν xx )in type-I as well as type-II WSMs. Furthermore, in thecase of a type-II WSM the linear- B component of ν xx dominates for small B , and the corresponding curve for ν xx ( B ) /ν xx (0) − VI. LIMITING CASES: R → AND R → In this section, we summarize our results for differentcomponents of thermopower in the asymptotic limit of
TABLE I. The Berry curvature, OMM ( γ = 1) and tilt induced B -linear correction to thermopower. Only nonzero correctionsare listed below. We have defined the dimensionless thermopower, ν (1) ij = ν ˜ ν (1) ij , and ν is defined in Eq. (40). We haveneglected terms of the order of xf ( x ) for the type-III class, and x (cid:48) g ( x (cid:48) ) and g ( x (cid:48) ) for the type-II class to obtain a simpler formof thermopower.˜ ν (1) ij = ˜ ν (1) ji Type-I [ R → | R | ] O ( R ) Type-III [ | R | → − x ] O ( x ) Type-II [ | R | → x (cid:48) ] O ( x (cid:48) )( B (cid:107) ˆ z ) ˜ ν (1)lz ≈ − R ˜ ν (1)l ≈ R ˜ ν (1)lz ≈ − f ( x ) (2 − x )˜ ν (1)l ≈ f ( x ) x ; f ( x ) ≡ log x − ν (1)lz = − g ( x (cid:48) ) ; g ( x (cid:48) ) ≡ (cid:16) x (cid:48) ˜Λ k (cid:17) ˜ ν (1)l ≈ [ g ( x (cid:48) ) − x (cid:48) ( B ⊥ ˆz ) ˜ ν (1)t ≈ − R ˜ ν (1)t ≈ − x ˜ ν t ≈ x (cid:48) − − B (Tesla) − . − . . ν ii ( B ) / ν ii ( ) − ( a ) xxzz . . . . R − . − . . ( b ) xxzz FIG. 5. SCs of type-I WSM for B (cid:107) R with tilt configuration R − = − R + = R . (a) The B dependence of the SCs at R = . B terms in the ν xx and ν zz expressions leadto the asymmetry in the SC curves as B changes from positiveto negative. (b) The tilt dependence of the SCs at B = 4 T.The longitudinal SC ( ν zz ) is negative irrespective of OMMcorrection (red lines). Note that the inclusion of the OMMcorrection has a significant impact on perpendicular SC ( ν xx )as evident from the difference between the dashed (excludingOMM) and the solid blue lines (including OMM). We haveused the parameters of Fig 1. no tilt, R →
0, and critical tilt, R →
1, which is calleda type-III WSM, and serves as the boundary betweentype-I and type-II. To be specific, we work with the tiltconfiguration R − = − R + = R with R >
0, though theresults are similar for the other configuration as well. Wewill consider three specific cases: (a) vanishing tilt, R →
0, (b) tilt tending to R → − + from below, and (c) tilttending to R → + from above. We present linear- B results in Table I and quadratic- B results in Table II.The linear- B correction to the thermopower is of theorder of − π e (cid:0) k B T (cid:1) ν , where we have defined, ν ≡ σ σ α σ = 3 µ (cid:126) v F µ eB (cid:126) . (40)The quadratic- B correction is of the order of − − B (Tesla) − . . . ν ii ( B ) / ν ii ( ) − ( a ) xxzz . . . . . R − ( b ) xxzz FIG. 6. Same as Fig. 5, but for type-II WSM. (a) B depen-dence of the SCs at R = 2. The linear- B terms in ν xx , domi-nates its behavior for small B with a negative slope, as shown(solid blue line – including OMM). (b) The tilt dependenceof the SC at B = 4 T. Note that for a given R , the signs of∆ ν xx ( B ) /ν xx (0) and ∆ ν zz ( B ) /ν zz (0) are opposite. The signreversal in each of them is a consequence of the correspond-ing Drude components flipping sign. This in turn occurs asdifferent carriers start dominating the transport as shown inFig. 7. We have used the parameters of Fig. reffig.44. − π e (cid:0) k B T (cid:1) ν , where ν is defined as ν ≡ σ α − α σ ( σ ) = − µ (cid:18) (cid:126) v F µ eB (cid:126) (cid:19) . (41)Interestingly, we find that all the B -linear terms tabu-lated in Table I, vanish as R → R → VII. EFFECT OF CHIRAL ANOMALY
So far in this paper, we have discussed the ther-mopower due to intranode scattering and the effect of BC.In this section, we estimate the effect of internode scat-tering as the origin of non-trivial thermopower in type-IWSM. Internode scattering stabilizes the chiral anomalyin WSMs leading to different chemical potential in differ-ent Weyl nodes.
For calculating charge conductiv-ity and thermoelectric coefficient due to internode scat-tering of a tilted WSMs, we borrow the formalism fromRef. [56].For the case of B ⊥ ˆ z , we calculate the charge conduc-tivity matrix due to chiral anomaly to be ˜ σ B = σ (2)ca cos φ σ (2)ca sin φ cos φ σ (1)ca cos φσ (2)ca sin φ cos φ σ (2)ca sin φ σ (1)ca sin φ σ (1)ca cos φ σ (1)ca sin φ . (42)Note the difference in the matrix structure in Eq. (42),from the intranode contribution given in Eq. (7). Forthe case of B (cid:107) ˆ z , the only non zero component is σ zz and it is given by σ zz = (cid:18)
14 + 19 (cid:19) σ (1)ca + σ (2)ca . (43)In Eqs. (42) and (43), we have defined (for γ = 1), σ (2)ca = e τ v π (cid:126) e v F µ B , and σ (1)ca = − e τ v π (cid:126) ev F (cid:126) RB . (44)Note that both these coefficients are proportional tothe internode scattering time τ v as expected. Further-more, while the σ (2)ca term solely arises from the chiralanomaly inducing E · B term, the σ (1)ca term primarilyarises from the tilted nature of the WSM and it vanishesas R →
0. The Fermi energy dependence of both thelinear and quadratic terms in Eq. (35), is identical to thecorresponding intranode contributions. As a consistencycheck, we note that if we ignore the OMM correction in σ (2)ca , then its numerical prefactor changes from 1 /
18 to1 /
4, and σ (2)ca becomes identical to Eq. (17) in Ref. [24.]Using these coefficients and the Mott relation [valid inthe limit µ/ ( k B T ) (cid:29) B ⊥ ˆ z ,the thermopower matrix is given by ˜ ν B = ν (2)ca cos φ ν (2)ca sin φ cos φ ν (1)ca cos φν (2)ca sin φ cos φ ν (2)ca sin φ ν (1)ca sin φ ν (1)ca cos φ ν (1)ca sin φ ν (2)ca , z . (45)Similarly, for B (cid:107) ˆ z , the only non zero component ofthermopower is given by ν zz = (cid:18)
14 + 19 (cid:19) ν (1)ca + ν (2)ca , zz (46)The quadratic- B correction to the thermopower matrixdue to internode scattering is given by can be written, inunits of − π e (cid:0) k B T (cid:1) , with ζ ≡ (cid:126) v F µ eB (cid:126) , as ν (2)ca = − µ ζ τ v τ (cid:16) − τ v τ R (cid:17) , ν (2)ca , z = 8 µ ζ (cid:16) τ v τ R (cid:17) . (47) ν (2)ca , zz = − µ ζ τ v τ (cid:16) − τ v τ R (cid:17) . (48) Similarly, the linear- B correction, in units of − π e (cid:0) k B T (cid:1) can be expressed as ν (1)ca = 24 µ ζ τ v τ R . (49)We emphasize that this linear- B correction in ther-mopower, is one of the significant findings of this paper.It primarily arises due to the tilted nature of the WSMand vanishes as R → , generally we have τ v (cid:29) τ . Ad-ditionally, since the internode scattering terms [Eqs. (47)and (49)] are ∼ τ v /τ times than the intranode scatter-ing terms, the contribution of the internode scatteringterms will dominate in the thermopower as well as in theelectrical conductivity. VIII. CONCLUSIONS
The presence of the BC and OMM in WSMs influ-ences the flow of charge carriers as well as entropy in thepresence of a magnetic field. This manifests as several in-teresting magnetoelectric and magnetothermal transportproperties in WSM. Since the Weyl nodes always come inpairs in a WSM, both the intranode and internode scat-tering play an important role in determining the electricalconductivity and thermopower. In this paper, we haveprimarily focused on the impact of the BC and OMM onthe thermopower due to intranode scattering in a tiltedWSM, and briefly discussed the effect of the internodescattering timescale. Our analytical calculations of thefull conductivity and thermopower matrix, are based onthe BC-connected semiclassical Boltzmann transport for-malism, and explicitly include the effects of the OMM.The latter modifies the energy-dispersion of the Blochelectrons which also manifests in the modified velocityof carriers, as well as in the Fermi function. However,the Mott relation connecting the conductivity matrix tothe thermopower matrix remains intact on including theeffects of the OMM.We find that the OMM has a significant impact onthe perpendicular MR in WSMs. Consistent with exper-iments, our calculations show that the longitudinal MR( B (cid:107) E ) in isotropic WSMs is always negative, while theperpendicular MR ( B ⊥ E ) is positive on including theeffect of the OMM. However, in tilted WSMs, the per-pendicular MR can also flip sign to become negative forthe large tilt parameter [see Fig. 1 (b)].In a type-I WSM, for the case of B ⊥ ˆ z , we find thatincreasing the magnetic field reduces the longitudinal SC,giving rise to a negative Seebeck effect in analogy withnegative MR [see Fig. 3(d)]. For the perpendicular SCwe find it to be positive for small tilt parameters, butit reverses sign for large tilt parameters. Analogous tothe planar Hall effect, we also find the existence of aplanar Nernst effect, which has an angular dependence TABLE II. The Berry curvature, OMM, and tilt induced quadratic- B correction to the thermopower. Only nonzero correctionsare listed below. We have defined the dimensionless thermopower, ν (2) ij = ν ˜ ν (2) ij , and ν is defined in Eq. (41). We have neglectedterms of the order of xf ( x ) for type-III class, and x (cid:48) g ( x (cid:48) ) and g ( x (cid:48) ) for type-II class to obtain a simpler form of thermopower.˜ ν (2) ij = ˜ ν (2) ji Type-I [ R → | R | ] O ( R ) Type-III [ | R | → − x ] O ( x ) Type-II [ | R | → x (cid:48) ] O ( x (cid:48) )( B (cid:107) ˆ z ) ˜ ν (2)l = 2; ˜ ν (2)lz = − ν (2)l = xν (2)lz = − f ( x ) (cid:104) f ( x ) ν − ν ν (cid:105) + f ( x ) ν x ˜ ν (2)l ≈ x (cid:48) ˜ ν (2)lz ≈ − g ( x (cid:48) ) ( B ⊥ ˆ z ) ˜ ν (2)z ≈ ν (2) ≈ −
6; ˜ ν (2) ⊥ ≈ ν (2)z ≈ − f ( x ) (cid:16) ν − (cid:104) ν + ν ν (cid:105) x (cid:17) ∆ ν (2) ≈ − (cid:104) ν − ν ν f ( x ) (cid:105) x ; ˜ ν (2) ⊥ ≈ x ˜ ν (2)z ≈ − g ( x (cid:48) ) ∆˜ ν (2) ≈ − x (cid:48) ; ˜ ν (2) ⊥ = x (cid:48) ν xy ∝ sin(2 φ ). Additionally, we also find a linear- B out-of-plane Nernst response in WSMs with a finite tilt. Forthe other case of B (cid:107) ˆ z , we find the conductivity and thethermopower matrix to be diagonal, with tilt inducedlinear- B terms in the longitudinal as well as perpendicu-lar components. This manifests in an asymmetry in theMR and SC curve around the B = 0 line, as shown inFigs. 5 and 6.For the case of a type-II WSM, the scene is a bit mixedup, owing to the contributions of both electron and holecarriers for all energies. We find that even in the ab-sence of a magnetic field, the Drude SC can be positiveor negative depending on the tilt (see Fig. 7). For thecase of B ⊥ ˆ z in a type-II WSM, we find that in con-trast to the case of a type-I WSM, the longitudinal SCis positive while the perpendicular SC is negative. Theangular dependence of the planar ( ν xy ∝ sin 2 φ ) and theout-of-plane Nernst effect ( ν xz ∝ cos φ ) is the same fortype-I and type-II WSMs. For the other case of B (cid:107) ˆ z , wefind that the linear- B terms dominate the ν xx for smallmagnetic fields. We expect similar effects (such as pla-nar Peltier effect and linear- B out-of-plane Peltier effect,among others) to also arise in the diagonal and the off-diagonal coefficients corresponding to the Peltier effect.Additionally, we have also explored the impact of intra-node scattering and chiral anomaly on the electrical con-ductivity and thermopower matrix in tilted WSMs. Re-markably, we find that the intranode scattering and chiralanomaly in tilted WSMs also lead to B -linear terms in theelectrical conductivity as well as in the thermopower ma-trix. Furthermore, as the conductivity and thermopowermatrix ∝ τ v and since τ v (cid:29) τ , the internode contributiondominates. ACKNOWLEDGMENTS
A. A. acknowledges funding support by Dept. of Sci-ence and Technology, Government of India, via DSTgrant no. DST/NM/NS/2018/103(G), and from SERB grant number CRG/2018/002440. K. D. acknowledgesIndian Institute of Technology Kanpur for PhD fellow-ship.
Appendix A: Berry-connected Boltzmann transportformalism
The Boltzmann transport formalism for magnetotrans-port works well for relatively small magnetic fields wherethe effects of Landau quantization can be ignored. Theequations of motion (EOM) approach works well in theregime where several Landau levels are occupied: (cid:126) ω c (cid:28) µ , with µ denoting the chemical potential, and ω c is thecyclotron frequency. In addition, the relaxation time ap-proximation for the non-equilibrium distribution func-tion (NDF) works well in the regime v F τ (cid:28) l , where v F is the Fermi velocity, τ is the relaxation time scale and l ≡ (cid:112) (cid:126) /eB is the magnetic length for cyclotron motion with B as the magnetic field. For WSM, the Fermi ve-locity is found to be in the range of 10 -10 m/s . TheFermi energy and scattering time are found to be of theorder of a few meV and 0 . .
1. Semiclassical transport with Berry curvatureand orbital magnetic moment
The EOM describing the dynamics of the center of thecarrier wave-packet (location at r , and having the Blochwave-vector k ) in a given band is given by ˙ r = D k (cid:104) ˜ v k + e (cid:126) ( E × Ω k ) + e (cid:126) (˜ v k · Ω k ) B (cid:105) , (A1) (cid:126) ˙ k = D k (cid:20) − e E − e (˜ v k × B ) − e (cid:126) ( E · B ) Ω k (cid:21) . (A2)Here − e is the electronic charge and and we have defined D k ≡ [1 + e (cid:126) ( B · Ω k )] − . The band velocity is given by (cid:126) ˜ v k = ∇ k ˜ (cid:15) k , where ˜ (cid:15) k = (cid:15) k − m k · B is the electronicdispersion modified by the intrinsic OMM. The modified0band velocity can now be expressed as ˜ v k = v k − γ v m k ,where v m k = (cid:126) ∇ k ( m k · B ), and the factor of γ = 0 / E × Ω k term gives rise to the intrin-sic anomalous Hall effect , while the (˜ v k · Ω k ) B termgives rise to the chiral magnetic effect in the presenceof non-zero chiral chemical potential . In Eq. (A2), thefirst two terms are the well known Lorentz force, whereasthe third ( E · B ) Ω k term manifests the effect of the chi-ral anomaly leading to negative MR in WSMs. Themodified EOM also changes the phase space volume by afactor D k . To compensate for this, such that the numberof states in the phase-space volume element is preserved,we have d k → d k /D k . This factor needs to be incorpo-rated whenever the wave-vector summation is convertedin an integral over the Brillouin zone in the presence ofthe BC .The three-component BC and the intrinsic OMM canbe obtained from their respective tensors via the relation: A a = ε abc A bc , where ε abc is the anti-symmetric Levi-Civita symbol. The corresponding Berry tensor is givenby Ω abn = − (cid:104) n | ∂ k a H| n (cid:48) (cid:105)(cid:104) n (cid:48) | ∂ k b H| n (cid:105) ]( (cid:15) n − (cid:15) n (cid:48) ) , (A3)where n is the band index with H| n (cid:105) = (cid:15) n | n (cid:105) . Similarly, the OMM tensor is given by m abn = − e (cid:126) Im [ (cid:104) n | ∂ k a H| n (cid:48) (cid:105)(cid:104) n (cid:48) | ∂ k b H| n (cid:105) ] (cid:15) n − (cid:15) n (cid:48) . (A4)The dynamics of the NDF g r , k , is described by theBoltzmann kinetic equation. In the steady state the NDFkinetic equation for each node is given by ˙ r · ∇ r g r , k + ˙ k · ∇ k g r , k = I coll { g r , k } , (A5)where the right hand side is the collision integral.In the relaxation time approximation, I coll { g r , k } = − g r , k − f eq τ k , where f eq ≡ f eq (˜ (cid:15) k , µ, T ) = ( e β (˜ (cid:15) k − µ ) +1) − is the equilibrium Fermi-Dirac distribution function with β − ≡ k B T . The scattering timescale τ k is the ef-fective intra-node relaxation time which we consider tobe constant ( τ k → τ ) for simplicity. Note that in ananisotropic tilted WSM, the scattering timescale shouldbe anisotropic. However, for simplicity, we will con-sider the scattering timescale to be isotropic, and theanisotropy of the band structure will appear only in themodified anisotropic velocities, and the anisotropic Fermisurface.Substituting Eqs. (A1)-and (A2) in Eq. (A5), we ob-tain an approximate steady state NDF, upto first orderin E and ∇ T : g r , k = f eq + (cid:20) D k τ (cid:18) − e E − (˜ (cid:15) k − µ ) T ∇ r T (cid:19) · (cid:18) ˜ v k + e B (˜ v k · Ω k ) (cid:126) (cid:19)(cid:21) (cid:18) − ∂f eq ∂ ˜ (cid:15) k (cid:19) . (A6)Note that in this paper, our primary focus is on the BCconnected conductivity and we have not included the im-pact of the Lorentz force terms in modifying the NDF in Eq. (A6). The Lorentz force contribution to conduc-tivity proportional to eBµ τ v F and its effect is more promi-nent in scenarios when the magnetic field is perpendicularto the transport direction. The corresponding BC con-tribution is proportional to eBµ (cid:126) v F (for intranode scat-tering) in the electrical conductivity, and its impact ismore when electric and magnetic fields are parallel. Adirect comparison between the Lorentz force terms andthe BC induced terms is not feasible, as far as the MR isconcerned. We refer the reader to Ref. [79] for an excel-lent discussion on this issue, and proceed below with thediscussion on the BC induced conductivity.Armed with the equation of motion and the NDF, wenow proceed to calculate current. In the presence of afinite OMM, the total local current can be expressed as j loc = − e (cid:90) [ d k ] D − ˙ r g r , k + ∇ r × (cid:90) [ d k ] D − m k f eq . (A7)Here we have used the shorthand [ d k ] = d k / (2 π ) d , with d being the dimension of the system. The additionalsecond term arises from the intrinsic OMM of individ-ual carriers, and can be physically attributed to the ro-tating dynamics of the finite width Bloch wave-packet.However, the ‘magnetization current’ is not observablein conventional transport measurement. Consequently,the transport current is defined as j tr = j loc − ∇ r × M ( r ) , (A8)where M ( r ) is the total orbital magnetization in realspace. The magnetization for a given chemical poten-tial ( µ ) and T is given by M = − ∂F/∂ B | µ,T , where F isthe grand-canonical potential defined as F = − β (cid:90) [ d k ] (cid:16) e (cid:126) B · Ω k (cid:17) ln[1 + e − β (˜ (cid:15) k − µ ) ] . (A9)Note that in Eq. (A8), the curl in the real space willinvolve temperature gradients, and the second term givesrise to the anomalous thermo-electric Hall effect.1
2. Electric and thermoelectric conductivity
Using Eqs. (A1), (A6), and (A9) in Eq. (A8), yieldsthe following general expression for the BC dependentpart of the electrical conductivity tensor σ total ij = − e (cid:126) (cid:90) [ d k ] (cid:15) ijl Ω l k f eq + e τ (cid:90) [ d k ] D k (cid:20) ˜ v i + eB i (cid:126) (˜ v k · Ω k ) (cid:21)(cid:20) ˜ v j + eB j (cid:126) (˜ v k · Ω k ) (cid:21)(cid:18) − ∂f eq ∂ ˜ (cid:15) k (cid:19) . (A10)Here ˜ v j denotes the j th component of ˜ v k , and (cid:15) ijl is the Levi-Civita antisymmetric tensor. Similarly, the BC dependentpart of the thermoelectric conductivity tensor can be explicitly obtained to be α total ij = k B e (cid:126) (cid:90) [ d k ] (cid:15) ijl Ω l k ξ k − eτ (cid:90) [ d k ] D k (˜ (cid:15) k − µ ) T (cid:20) ˜ v i + eB i (cid:126) (˜ v k · Ω k ) (cid:21)(cid:20) ˜ v j + eB j (cid:126) (˜ v k · Ω k ) (cid:21)(cid:18) − ∂f eq ∂ ˜ (cid:15) k (cid:19) . (A11)In Eq. (A11) we have defined, ξ k = β (˜ (cid:15) k − µ ) f eq + ln[1 + e − β (˜ (cid:15) k − µ ) ] . (A12)While Eq. (A11) can be evaluated separately, in the lowtemperature limit ( k B T (cid:28) µ ) it can also be obtainedfrom Eq. (A10) by using the Mott relations whichalso hold in the presence of BC and the OMM. In factthe validity of the Mott relation including the OMM cor-rection has also been proved recently, in a more generalsetting, in Ref. [59].The first term on the right hand side of Eqs. (A10)and (A11) denote the anomalous Hall effect andthe anomalous thermoelectric effect , respectively.In WSM, the anomalous Hall conductivity, σ Axy hasbeen shown to be linearly proportional to the internodeseparation . The anomalous thermoelectric conductiv-ity α Axy was shown to be zero in a linearized model butfinite for a lattice model . The finite contribution in α Axy in a lattice model originates from band curvature ef- fects beyond the linear dispersion. In the case of a tiltedWSM, described by a linear dispersion, α Axy is finite forboth the type-I and the type-II class of WSMs .In the last term in Eqs. (A10) and (A11), one of theanomalous velocity terms arises from the E · B term inEq. (A2), and the other from the NDF. For parallel elec-tric and magnetic fields, this is what leads to NMR ,which is quadratic in the magnetic field, and is a rela-tively well established transport signature . This termalso leads to the planar Hall effect , in which a Hallvoltage is generated in the plane of the electric and mag-netic fields, as long as they are not parallel or perpendic-ular to each other.Expanding Eqs. (A10) and (A11) in powers of B (ex-pansion of Fermi function ), the zeroth order, linear andquadratic- B components of the transport coefficients canbe expressed as : σ ( o ) ij ≡ L o ) ij and α ( o ) ij ≡ − eT L o ) ij ,where o = { , , } refers to the order of magnetic field.For the first-order terms we find L p (1) ij = e τ (cid:90) [ d k ] (cid:20) ( (cid:15) − µ ) p (cid:18)(cid:104) e (cid:126) ( v i B j + v j B i ) ( v · Ω ) − e (cid:126) Ω · B v i v j (A13) − γ (cid:0) v i v mj + v j v mi (cid:1) (cid:105) ( − f (cid:48) ) − γ v i v j ( m · B ) ( − f (cid:48)(cid:48) ) (cid:19) − δ ( p −
1) ( γ m · B ) p v i v j ( − f (cid:48) ) (cid:21) . Here p = 0 (or 1) for the electric (or thermoelectric) conductivity. Similarly, the quadratic terms can be expressed2as L p (2) ij = e τ (cid:90) [ d k ] (cid:34) ( (cid:15) − µ ) p (cid:18)(cid:104) v i v j (cid:16) e (cid:126) Ω · B (cid:17) − e (cid:126) Ω · B (cid:110) e (cid:126) ( v i B j + v j B i ) v · Ω − γ (cid:0) v i v mj + v j v mi (cid:1)(cid:111) + (cid:18) eB i (cid:126) eB j (cid:126) ( v · Ω ) − γ (cid:110) e (cid:126) (cid:0) v mi B j + v mj B i (cid:1) ( v · Ω ) − e (cid:126) ( v i B j + v j B i ) ( v m · Ω ) + v mi v mj (cid:111)(cid:19) (cid:105) ( − f (cid:48) ) − γ (cid:104) e (cid:126) ( v i B j + v j B i ) ( v · Ω ) − (cid:0) v i v mj + v j v mi (cid:1) − v i v j e (cid:126) Ω · B (cid:105) ( m · B ) ( − f (cid:48)(cid:48) ) + γ v i v j ( m · B ) ( − f (cid:48)(cid:48)(cid:48) ) (cid:19) − δ ( p −
1) ( γ m · B ) p (cid:16)(cid:104) e (cid:126) ( v i B j + v j B i ) ( v · Ω ) − (cid:0) v i v mj + v j v mi (cid:1) − v i v j e (cid:126) Ω · B (cid:105) ( − f (cid:48) ) − v i v j ( m · B ) ( − f (cid:48)(cid:48) ) (cid:17) (cid:35) . (A14)The last term in both Eqs. (A13) and (A14) only con-tributes to α ij ( p = 1). As an additional consistencycheck, it is straight forward to derive the Mott rela-tions separately for linear- B and quadratic- B terms usingEqs. (A13) and (A14). Appendix B: Drude conductivities
In this section we calculate Drude conductivities oftilted WSMs . The diagonal components of conduc-tivity in the absence of a magnetic field are called theDrude conductivities. For the type-I class, Drude con-ductivity is given by σ (0) xx = σ (0) yy , where σ (0) xx = (cid:88) s σ R s (cid:20) R s (1 − R s ) + ln (cid:18) − R s R s (cid:19)(cid:21) . (B1)Drude conductivity along the z direction is given by σ (0) zz = (cid:88) s σ R s (cid:20) − R s − ln (cid:18) − R s R s (cid:19)(cid:21) . (B2)In the limit R s → σ = 4 π e h µ τh v F . (B3)For the type-II class a finite cutoff in momentum space(Λ k ) is unavoidable to calculate the conductivities. Thisdetermines the Drude conductivity along the x directionas σ (0) xx = (cid:88) s σ | R s | (cid:20) − R s R s − (cid:0) R s − (cid:1) ˜Λ k − δ s (cid:21) . (B4)The same along the z direction is given by σ (0) zz = (cid:88) s σ | R s | (cid:104) − R s + (cid:0) R s − (cid:1) ˜Λ k + δ s (cid:105) . (B5)These expressions of Drude conductivities are exact asthere are no approximations due to large cutoff. . . . R − . . . ν xx ( a ) condvaltotal . . . R . . . ν zz ( b ) condvaltotal . . . R − / ν xx ( c ) . . . R − / ν zz ( d ) FIG. 7. (a) The Drude ( B = 0) component of the SC ( ν xx ),showing the contribution of the conduction and the valancebands separately. It reverses sign from positive to negative, asthe contribution from the holes (valance band states) startsto dominate. (b) The Drude component of the SC ( ν zz ) high-lighting the contribution from the different bands. All thecomponents are scaled by the isotropic Drude counterpart, ν D . (c) and (d) show the flip in sign of the inverse of theDrude components 1 /ν xx , and 1 /ν xx , which is also reflectedin Fig. 6(b). Now we discuss the Drude thermopower. The Drudethermopower for isotropic WSM, using Eqs. (4) and(B3), calculated to be ν = − π k B e k B Tµ . (B6)Note that for a constant relaxation time the Drude SC isscattering time independent. It is evident that for µ > µ < xx com-ponents, the valence band dominates in flow of entropy[see Fig. 7(a)] whereas for zz components the conductionband dominates [see Fig. 7(b)]. Appendix C: Expressions with γ Type-I WSM.–
For B ⊥ ˆ z , the conductivities for thetype-I class are given by σ (2) ⊥ = (cid:88) s (1 − γ ) σ , (C1)∆ σ (2) = (cid:88) s (cid:2) R s − γ (cid:0) R s (cid:1)(cid:3) σ , (C2) σ (2)z = (cid:88) s (cid:2) R s − γ (cid:0) − R s (cid:1)(cid:3) σ . (C3)Here, the factor γ = 1 (0) explicitly keeps track of theterms arising from the presence (absence) of the OMM .The corrections due to the OMM (the γ dependent terms)tend to suppress the conductivities. Most importantlythe inclusion of the OMM in the conductivity changesthe sign of σ (2) ⊥ . The linear- B correction to the transverseconductivity is given by σ (1)t = (cid:88) s sσ R s (cid:2) R s (cid:8) (3 − R s )( γ −
1) + 3 R s (1 − R s ) (cid:9) +3 F δ s ( γ − R s ) (cid:3) . (C4)For B (cid:107) ˆ z , the quadratic corrections are given by σ (2)l = (cid:88) s (1 − γ ) σ ; σ (2)lz = (cid:88) s (cid:2) γ (cid:0) R s − (cid:1)(cid:3) σ . (C5)The linear- B term in σ xx = σ yy is given by σ (1)l = (cid:88) s sσ R s (cid:104) R s (cid:8)(cid:0) − R s (cid:1) ( γ − − γR s (cid:9) − δ s (cid:8)(cid:0) R s − (cid:1) ( γ −
1) + 2 γR s (cid:9) (cid:105) . (C6)For σ zz , the linear- B correction is given by σ (1)lz = (cid:88) s sσ R s (cid:104) R s (cid:8)(cid:0) − R s (cid:1) (1 − γ ) − R s (cid:9) +3 F δ s (1 − γ ) (cid:105) . (C7) Type-II WSM.–
First, we will consider the planar ge-ometry ( B ⊥ ˆ z ). In this case the form of the conductivitymatrix is given by Eq. (7), and the elements of the con- ductivity matrix are given by∆ σ (2) = (cid:88) s K ( A R − γ A M ) , (C8) σ (2) ⊥ = (cid:88) s K ( B R − γ B M ) , (C9) σ (2)z = 2 (cid:88) s K ( D R − γ D M ) . (C10)where K ≡ σ | R s | . For the planar components (re-sponses in the x - y plane) of the conductivity, we havedefined the following polynomials of R s : A R = 2 (cid:0) − R s + 5 R s + 125 R s + 30 R s (cid:1) , (C11) A M = 2 (cid:0) R s − R s + 90 R s (cid:1) , (C12) B R = (cid:0) − R s + 15 R s + 5 R s (cid:1) , (C13) B M = 3 (cid:0) − R s + 25 R s (cid:1) . (C14)For the σ zz component, we have defined the followingpolynomials of R s : D R = (cid:0) − R s − R s + 65 R s + 15 R s (cid:1) , (C15) D M = 2 (cid:0) − R s + 5 R s (cid:1) . (C16)The linear- B correction in the out-of-plane off-diagonalconductivities can be written as σ (1) xz = σ (1) yz ( π/ − φ ) = σ (1)t cos φ . Here, σ (1)t = (cid:88) s sσ R s sgn( R s ) (cid:104) (cid:0) − R s + 21 R s (cid:1) ( γ − γ (cid:0) R s − R s (cid:1) − F δ s ( γ − R s ) (cid:105) . (C17)Now, we consider a magnetic field along the direction ofthe tilt ( B (cid:107) ˆ z ). The linear- B correction to the longitu-dinal component in the x/y plane, σ xx = σ yy , is givenby σ (1)l = (cid:88) s sσ R s sgn( R s ) (cid:104) (11 − R s )( γ − − γ (3 R s − − δ s { ( R s − − γ ) − γR s } (cid:105) . (C18)The linear- B correction to σ zz is given by σ (1)lz = (cid:88) s sσ R s sgn( R s ) (cid:104) (cid:0) − R s + 6 R s (cid:1) (1 − γ ) − − R s + 2 R s ) + 3 δ s F ( γ − (cid:105) . (C19)The quadratic- B correction to σ xx and σ yy is given by σ (2)l = (cid:88) s σ | R s | (cid:2)(cid:0) − R s + 5 R s (cid:1) + 6 γ (cid:0) − R s (cid:1)(cid:3) . (C20)The corresponding term for the σ zz component is givenby σ (2)lz = (cid:88) s σ | R s | (cid:104) (cid:0) − R s + 15 R s + 5 R s (cid:1) (C21)+ γ (cid:0) − R s − R s + 15 R s (cid:1) (cid:105) . (C22)4 ∗ [email protected] † [email protected] H. Nielsen and M. Ninomiya, Physics Letters B , 219(1981) N. P. Armitage, E. J. Mele, and A. Vishwanath, Rev.Mod. Phys. , 015001 (2018) A. Bansil, H. Lin, and T. Das, Rev. Mod. Phys. , 021004(2016) B. Yan and C. Felser, Annual Review of Condensed MatterPhysics , 337 (2017) H.-J. Kim, K.-S. Kim, J.-F. Wang, M. Sasaki, N. Satoh,A. Ohnishi, M. Kitaura, M. Yang, and L. Li, Phys. Rev.Lett. , 246603 (2013) S.-Y. Xu, I. Belopolski, N. Alidoust, M. Neupane, G. Bian,C. Zhang, R. Sankar, G. Chang, Z. Yuan, C.-C. Lee, S.-M. Huang, H. Zheng, J. Ma, D. S. Sanchez, B. Wang,A. Bansil, F. Chou, P. P. Shibayev, H. Lin, S. Jia, andM. Z. Hasan, Science , 613 (2015) J. Xiong, S. K. Kushwaha, T. Liang, J. W. Krizan,M. Hirschberger, W. Wang, R. J. Cava, and N. P. Ong,Science , 413 (2015) B. Q. Lv, H. M. Weng, B. B. Fu, X. P. Wang, H. Miao,J. Ma, P. Richard, X. C. Huang, L. X. Zhao, G. F. Chen,Z. Fang, X. Dai, T. Qian, and H. Ding, Phys. Rev. X ,031013 (2015) S.-Y. Xu, N. Alidoust, I. Belopolski, Z. Yuan, G. Bian,T.-R. Chang, H. Zheng, V. N. Strocov, D. S. Sanchez,G. Chang, C. Zhang, D. Mou, Y. Wu, L. Huang, C.-C. Lee,S.-M. Huang, B. Wang, A. Bansil, H.-T. Jeng, T. Neupert,A. Kaminski, H. Lin, S. Jia, and M. Zahid Hasan, NaturePhysics , 748 (2015) A. A. Soluyanov, D. Gresch, Z. Wang, Q. Wu, M. Troyer,X. Dai, and B. A. Bernevig, Nature , 495 (2015) G. Aut`es, D. Gresch, M. Troyer, A. A. Soluyanov, andO. V. Yazyev, Phys. Rev. Lett. , 066402 (2016) T.-R. Chang, S.-Y. Xu, G. Chang, C.-C. Lee, S.-M. Huang,B. Wang, G. Bian, H. Zheng, D. S. Sanchez, I. Belopolski,N. Alidoust, M. Neupane, A. Bansil, H.-T. Jeng, H. Lin,and M. Zahid Hasan, Nature Communications , 10639(2016) P. Li, Y. Wen, X. He, Q. Zhang, C. Xia, Z.-M. Yu, S. A.Yang, Z. Zhu, H. N. Alshareef, and X.-X. Zhang, NatureCommunications , 2150 (2017) J. Jiang, Z. K. Liu, Y. Sun, H. F. Yang, C. R. Rajamathi,Y. P. Qi, L. X. Yang, C. Chen, H. Peng, C.-C. Hwang,S. Z. Sun, S.-K. Mo, I. Vobornik, J. Fujii, S. S. P. Parkin,C. Felser, B. H. Yan, and Y. L. Chen, Nature Communi-cations , 13973 (2017) S.-Y. Xu, N. Alidoust, G. Chang, H. Lu, B. Singh, I. Be-lopolski, D. S. Sanchez, X. Zhang, G. Bian, H. Zheng,M.-A. Husanu, Y. Bian, S.-M. Huang, C.-H. Hsu, T.-R.Chang, H.-T. Jeng, A. Bansil, T. Neupert, V. N. Strocov,H. Lin, S. Jia, and M. Z. Hasan, Science Advances ,1603266 (2017) A. Pal, M. Chinotti, L. Degiorgi, W. Ren, and C. Petrovic,Physica B: Condensed Matter , 64 (2018) M. Zhang, Z. Yang, and G. Wang, The Journal of PhysicalChemistry C , 3533 (2018) G. Chang, B. Singh, S.-Y. Xu, G. Bian, S.-M. Huang, C.-H.Hsu, I. Belopolski, N. Alidoust, D. S. Sanchez, H. Zheng,H. Lu, X. Zhang, Y. Bian, T.-R. Chang, H.-T. Jeng, A. Bansil, H. Hsu, S. Jia, T. Neupert, H. Lin, and M. Z.Hasan, Phys. Rev. B , 041104 (2018) M. V. Berry, Proceedings of the Royal Society of Lon-don. Series A, Mathematical and Physical Sciences ,45 (1984) D. Xiao, M.-C. Chang, and Q. Niu, Rev. Mod. Phys. ,1959 (2010) H. B. Nielsen and M. Ninomiya, Physics Letters B ,389 (1983) D. T. Son and N. Yamamoto, Phys. Rev. Lett. , 181602(2012) A. A. Zyuzin and A. A. Burkov, Phys. Rev. B , 115133(2012) D. T. Son and B. Z. Spivak, Phys. Rev. B , 104412 (2013) P. Hosur and X. Qi, Comptes Rendus Physique , 857(2013) K.-S. Kim, H.-J. Kim, and M. Sasaki, Phys. Rev. B ,195137 (2014) R. Lundgren, P. Laurell, and G. A. Fiete, Phys. Rev. B , 165115 (2014) A. A. Zyuzin and R. P. Tiwari, JETP Letters , 717(2016) G. Sharma, P. Goswami, and S. Tewari, Phys. Rev. B ,035116 (2016) E. C. I. van der Wurff and H. T. C. Stoof, Phys. Rev. B , 121116 (2017) T. M. McCormick, R. C. McKay, and N. Trivedi, Phys.Rev. B , 235116 (2017) A. Sekine, D. Culcer, and A. H. MacDonald, Phys. Rev.B , 235134 (2017) Y. Ferreiros, A. A. Zyuzin, and J. H. Bardarson, Phys.Rev. B , 115202 (2017) J. F. Steiner, A. V. Andreev, and D. A. Pesin, Phys. Rev.Lett. , 036601 (2017) S. J. Watzman, T. M. McCormick, C. Shekhar, S.-C. Wu,Y. Sun, A. Prakash, C. Felser, N. Trivedi, and J. P. Here-mans, Phys. Rev. B , 161404 (2018) W. Desrat, C. Consejo, F. Teppe, S. Contreras,M. Marcinkiewicz, W. Knap, A. Nateprov, andE. Arushanov, Journal of Physics: Conference Series ,012064 (2015) X. Huang, L. Zhao, Y. Long, P. Wang, D. Chen, Z. Yang,H. Liang, M. Xue, H. Weng, Z. Fang, X. Dai, and G. Chen,Phys. Rev. X , 031023 (2015) J. Hu, J. Y. Liu, D. Graf, S. M. A. Radmanesh, D. J.Adams, A. Chuang, Y. Wang, I. Chiorescu, J. Wei,L. Spinu, and Z. Q. Mao, Scientific Reports , 18674(2016) F. Arnold, C. Shekhar, S.-C. Wu, Y. Sun, R. D. dos Reis,N. Kumar, M. Naumann, M. O. Ajeesh, M. Schmidt, A. G.Grushin, J. H. Bardarson, M. Baenitz, D. Sokolov, H. Bor-rmann, M. Nicklas, C. Felser, E. Hassinger, and B. Yan,Nature Communications , 11615 (2016) H. Li, H. He, H.-Z. Lu, H. Zhang, H. Liu, R. Ma, Z. Fan, S.-Q. Shen, and J. Wang, Nature Communications , 10301(2016) Y. Li, Z. Wang, P. Li, X. Yang, Z. Shen, F. Sheng, X. Li,Y. Lu, Y. Zheng, and Z.-A. Xu, Frontiers of Physics ,127205 (2017) T. Liang, J. Lin, Q. Gibson, T. Gao, M. Hirschberger,M. Liu, R. J. Cava, and N. P. Ong, Phys. Rev. Lett. , Sudesh, P. Kumar, P. Neha, T. Das, and S. Patnaik, Sci-entific Reports , 46062 (2017) A. C. Niemann, J. Gooth, S.-C. Wu, S. B¨aßler,P. Sergelius, R. H¨uhne, B. Rellinghaus, C. Shekhar, V. S¨uß,M. Schmidt, C. Felser, B. Yan, and K. Nielsch, ScientificReports , 43394 (2017) T. Liang, J. Lin, Q. Gibson, S. Kushwaha, M. Liu,W. Wang, H. Xiong, J. A. Sobota, M. Hashimoto, P. S.Kirchmann, Z.-X. Shen, R. J. Cava, and N. P. Ong, Na-ture Physics , 451 (2018) N. Kumar, S. N. Guin, C. Felser, and C. Shekhar, Phys.Rev. B , 041103 (2018) J. Noky, J. Gayles, C. Felser, and Y. Sun, Phys. Rev. B , 220405 (2018) J. Yang, W. L. Zhen, D. D. Liang, Y. J. Wang, X. Yan,S. R. Weng, J. R. Wang, W. Tong, L. Pi, W. K. Zhu, andC. J. Zhang, Phys. Rev. Materials , 014201 (2019) F. D. M. Haldane, Phys. Rev. Lett. , 206602 (2004) A. A. Burkov, Phys. Rev. Lett. , 187202 (2014) Q. Li, D. E. Kharzeev, C. Zhang, Y. Huang, I. Pletikosic,A. . V. Fedorov, R. . D. Zhong, J. . A. Schneeloch, G. . D.Gu, and T. Valla, Nature Physics , 550 EP (2016) S. Nandy, G. Sharma, A. Taraphder, and S. Tewari, Phys.Rev. Lett. , 176804 (2017) A. A. Burkov, Phys. Rev. B , 041110 (2017) R. M. A. Dantas, F. Pe˜na-Benitez, B. Roy, andP. Sur´owka, Journal of High Energy Physics , 69(2018) T. Nag and S. Nandy, arXiv e-prints , arXiv:1812.08322(2018), arXiv:1812.08322 V. A. Zyuzin, Phys. Rev. B , 245128 (2017) G. Sharma, P. Goswami, and S. Tewari, Phys. Rev. B ,045112 (2017) K. Das and A. Agarwal, Phys. Rev. B , 085405 (2019) L. Dong, C. Xiao, and Q. Niu, arXiv e-prints ,arXiv:1812.11721 (2018), arXiv:1812.11721 C. T. Bui and F. Rivadulla, Phys. Rev. B , 100403 (2014) C. T. Bui, C. A. C. Garcia, N. T. Tu, M. Tanaka, andP. N. Hai, Journal of Applied Physics , 175102 (2018) D. Wesenberg, A. Hojem, R. K. Bennet, and B. L. Zink,Journal of Physics D: Applied Physics , 244005 (2018) N. Ashcroft and N. Mermin,
Solid State Physics , HRW international editions (Holt, Rinehart and Winston, 1976) D. Xiao, Y. Yao, Z. Fang, and Q. Niu, Phys. Rev. Lett. , 026603 (2006) J. P. Carbotte, Phys. Rev. B , 165111 (2016) T. Hayata, Y. Kikuchi, and Y. Tanizaki, Phys. Rev. B ,085112 (2017) Z. Jia, C. Li, X. Li, J. Shi, Z. Liao, D. Yu, and X. Wu,Nature Communications , 13013 (2016) U. Stockert, R. D. dos Reis, M. O. Ajeesh, S. J. Watz-man, M. Schmidt, C. Shekhar, J. P. Heremans, C. Felser,M. Baenitz, and M. Nicklas, Journal of Physics: Con-densed Matter , 325701 (2017) A. Thakur, K. Sadhukhan, and A. Agarwal, Phys. Rev. B , 035403 (2018) C.-L. Zhang, Z. Yuan, Q.-D. Jiang, B. Tong, C. Zhang,X. C. Xie, and S. Jia, Phys. Rev. B , 085202 (2017) T. Morimoto, S. Zhong, J. Orenstein, and J. E. Moore,Phys. Rev. B , 245121 (2016) M. Marder,
Condensed matter physics (Wiley, Hoboken,N.J, 2010) N. A. Sinitsyn, Journal of Physics: Condensed Matter ,023201 (2008) D. Xiao, J. Shi, and Q. Niu, Phys. Rev. Lett. , 137204(2005) C. Duval, Z. Horvath, P. A. Horvathy, L. Martina, andP. C. Stichel, Modern Physics Letters B , 373 (2006) M.-C. Chang and Q. Niu, Phys. Rev. B , 7010 (1996) X. Dai, Z. Z. Du, and H.-Z. Lu, Phys. Rev. Lett. ,166601 (2017) C. Jacoboni,
Theory of electron transport in semiconduc-tors : a pathway from elementary physics to nonequilibriumgreen functions (Springer, Berlin, 2010) M. Imran and S. Hershfield, Phys. Rev. B , 205139(2018) N. R. Cooper, B. I. Halperin, and I. M. Ruzin, Phys. Rev.B , 2344 (1997) P. Goswami and S. Tewari, Phys. Rev. B , 245107 (2013) E. V. Gorbar, V. A. Miransky, I. A. Shovkovy, and P. O.Sukhachov, Phys. Rev. B , 155138 (2017) F. M. D. Pellegrino, M. I. Katsnelson, and M. Polini, Phys.Rev. B92