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A new decomposition of the Kubo-Bastin formula

A new decomposition of the Kubo-Bastin formula 1 1;2y Varga Bonbien and Aur elien Manchon Physical Science and Engineering Division (PSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia Aix-Marseille Univ, CNRS, CINaM, Marseille, France The Smrcka-Streda version of Kubo's linear response formula is widely used in the literature to compute non-equilibrium transport properties of heterostructures. It is particularly useful for the evaluation of intrinsic transport properties associated with the Berry curvature of the Bloch states, such as anomalous and spin Hall currents as well as the damping-like component of the spin-orbit torque. Here, we demonstrate in a very general way that the widely used decomposition of the Kubo-Bastin formula introduced by Smrcka and Streda contains an overlap, which has lead to widespread confusion in the literature regarding the Fermi surface and Fermi sea contributions. To remedy this pathology, we propose a new decomposition of the Kubo-Bastin formula based on the permutation properties of the correlation function and derive a new set of formulas, without an overlap, that provides direct access to the transport e ects of interest. We apply these new formulas to selected cases and demonstrate that the Fermi sea and Fermi surface contributions can be uniquely addressed with our symmetrized approach. I. INTRODUCTION Bruno presented a detailed and widely cited derivation of the Kubo-Bastin and Smrcka-Streda formulae from the 1,2 Kubo formula. The seminal work of Kubo showed that, in the per- The Smrcka-Streda formula has been widely used to turbative weak- eld limit, transport coecients can be 9,10 compute charge and spin Hall currents as well as spin- expressed as correlation functions of quantum mechani- orbit torques . Whereas a few works use the full Smrcka- cal observable operators. The resulting Kubo formalism 12{15 Streda formula , most theoretical studies exploit a has become a staple of quantum transport theory cal- simpli ed version of it, obtained by assuming constant culations and surged in popularity following the realiza- 16{18 scattering time and in the weak disorder limit tion that applying it to transport phenomena in crystals provides direct access to topological invariants, thereby X ^ ^ 1 Re[hnkjBjmkihmkjAjnki] yielding an explanation for the robustness of the quan- A ! ; (3) surf 2 2 2 2 [(" " ) + ][(" " ) + ] F nk F mk tized Hall e ect . k;n;m While the original Kubo formula is formally satisfy- X ^ ^ Im[hnkjBjmkihmkjAjnki] ing, realistic calculations with it are rather impractical. A ! (f (" ) f (" )): nk mk sea 4 5 (" " ) nk mk Bastin et al. and later Streda and Smrcka used Green's k;n6=m functions to rewrite the Kubo formula and arrived at a (4) result directly applicable to computations in the static limit. Later on, Smrcka and Streda further decomposed Here is the homogeneous broadening and jnki is a the Bastin formula into two terms, Bloch state of the crystal. This simpli ed version readily n o attributes A to intraband transitions, yielding a surf r r a ^ ^ ^ ^ ^ A = d"@ f (")Re tr[AG B(G G )] ; (1) I " 1= dependence, and A to interband transitions, 2 sea n o which are nite in the clean limit ( ! 0). In fact, r r r r ^ ^ ^ ^ ^ ^ ^ ^ A = d"f (")Re tr[AG B@ G A@ G BG ] ; these simpli ed formulae elegantly connect the Fermi II " " sea transport contributions to the Berry curvature of the (2) Bloch states, and, to date, the Berry curvature formula, Eq. (4), has been widely used to characterize the ^ ^ Here A is the operator of the perturbation and B is the 19{21 intrinsic spin Hall e ect of bulk materials . As we r(a) operator of the observable, G (") is the retarded (ad- mentioned already, this formula is only valid in the clean vanced) Green's function of the system and we have sup- limit and does not apply in realistic materials where pressed the energy argument in the formulae for brevity, momentum scattering is important. More speci cally, it f (") is the Fermi-Dirac distribution and @ indicates an becomes invalid when the broadening is comparable energy derivative. Because of their connection to @ f (") to, or larger than the local orbital gaps resulting from and f ("), Eqs. (1) and (2) were wrongly referred to as avoided band crossings, and where Berry curvature is Fermi surface and Fermi sea terms, respectively. These 22{24 maximized. Indeed, further investigations have terms were used by Streda in his famous analysis of 7 addressed the spin Hall e ect of metals using the full the quantized Hall e ect . More recently, Cr epieux and Smrcka-Streda formula, Eqs. (1)-(2), showing evidence that the spin Hall conductivity of 5d transition metals is dominated by A . Similarly, an in uential work [email protected] by Sinitsyn et al. demonstrated that in the case of a [email protected] gapped Dirac cone, spin Hall e ect is entirely due to A arXiv:2005.04678v1 [cond-mat.mes-hall] 10 May 2020 2 in the metallic regime, while it is entirely due to A in function with respect to its energy argument, that we II the gap. These observations, valid for speci c examples, have suppressed in the formula for brevity. Across the led to the confusion that A always dominate in metals. manuscript, ::: ^ denotes an operator and tr(: : : ) is the 12{14 For instance, some investigations (including ours) trace operation. have computed the spin-orbit torque using only A based on Kontani et al. 's argument. However, recent Splitting (5) into two halves, integrating one of them calculations have demonstrated that certain transport by parts and combining it with the other half yields the properties associated with Berry curvature, such as Smrcka-Streda decomposition of the Kubo-Bastin for- I II the dampinglike torque in magnetic heterostructures, mula [Eq. (A10) of Ref. 8] with  =  +  , where kl kl kl 26{28 have contributions from both A and A (see also I II Ref. 29). This suggests that attributing purely Fermi I r a ^ ^ ^ ^ ^ ^ = d" @ f (") tr j G j j G j surface origin to A and purely Fermi sea origin to A " k l l k I II kl is incorrect. (6) r a ^ ^ G G In this paper, we rst show in a very general way that the A -A decomposition of the Kubo-Bastin formula I II and introduced by Smrcka and Streda contains an overlap, and there appears to be widespread confusion regarding this aspect in the literature. This overlap was hinted II r r r r ^ ^ ^ ^ ^ ^ ^ ^ = d" f (") tr j G (")j @ G j @ G j G k l " k " l kl at for the special case of a 2-dimensional Dirac mate- rial by Sinitsyn et al. , but the fact that Smrcka-Streda a a a a ^ ^ ^ ^ ^ ^ ^ ^ + j @ G j G j G j @ G : and many subsequent authors unjusti ably neglected a l " k l k " subtle term relating to position operators in certain ver- (7) sions of the Smrcka-Streda formula responsible for geo- 7,8,31 metric e ects, went unmentioned . This subtlety is Integrating (7) by parts shall not yield any surface unnoticeable for simple models | such as the quadratic terms, thus we might naively conclude that this term magnetic Rashba gas | when A is vanishingly small describes e ects resulting purely from the sea. However, II away from the avoided band crossing, since the neglected this is not the case, since there is signi cant overlap be- I II geometric term exactly cancels out Streda's orbital sea tween  and  . Indeed, manipulating (6) and (7) we kl kl term in A which, due to the overlap, also appears in arrive at (see Appendix) II the A term. However, in the general case, A is non- I II 29,32 negligible and thus, Smrcka and Streda's decompo- I surf ol =  +  ; (8a) kl kl kl sition of the Kubo-Bastin formula into A ;A does not I II II sea ol lend itself to a proper analysis of di erent physical ef- =   ; (8b) kl kl kl fects. To remedy this, we propose a new decomposition where of the Kubo-Bastin formula based on the permutation properties of the correlator and derive a new set of for- mulas without an overlap, that provides direct access to ~ surf r a ^ ^ ^ ^ = d" @ f (") tr j G G j " k l kl the intrinsic geometric e ects. r a ^ ^ G G ; (9) II. THE KUBO-BASTIN FORMULA AND THE SMRCKA-STREDA DECOMPOSITION sea r a ^ ^ ^ ^ = d" f (") tr j @ G + @ G j k " " l kl The Kubo-Bastin formula for the electrical conductiv- ity,  , in the static limit as obtained from the Kubo kl r a ^ ^ ^ ^ j (@ G + @ G j l " " k formula is [Eq. (A9) of Ref. 8] r a ^ ^ Z  G G (10) r a ^ ^ ^ ^ ^ ^ = d" f (") tr j @ G j j @ G j kl k " l l " k and the overlap term r a ^ ^ G G ; (5) ol r a ^ ^ ^ ^ = d" @ f (") tr j G + G j " k l kl ^ ^ where j ; j are electric charge current opera- k l r(a) ^ r a ^ ^ ^ ^ tors in the k; l 2 fx; y; zg directions, G (") = j (G + G j l k lim 1=("H i) is the retarded(advanced) Green's !0 0 function corresponding to the equilibrium Hamiltonian r a ^ ^ G G : (11) r(a) H and @ G (") is the derivative of the Green's 0 " 3 surf surf surf I ol sea Upon closer inspection, we note that  is symmet-  =  +  =  +  . Thus we see that kl kl kl kl kl kl II sea ol for the particular case of a vanishing  term, Streda's ric whereas  along with  are antisymmetric under kl kl orbital sea term (12) is exactly equal to the geometric ^ ^ the exchange of operators j and j . Furthermore, in k l sea surf term  and consequently describes Berry curvature kl the special case of k = l,  can be recognized as the kk e ects. This is an advantage in the zero temperature Kubo-Greenwood formula for the diagonal conductivity. I case, since @ f (") ! (" " ) as T ! 0, meaning that " F The separation of  into symmetric and antisymmetric surf ol we can simply evaluate the Green's functions in (12) at parts yielding  and  is already present in the 32,33 the Fermi energy and there is no need for a complete literature , however it was not realized that the ol energy integration, as would be required for (10) or (13). antisymmetric part  is an overlap and gets exactly cancelled when considering an appropriate separation of II sea ol into  and  , as considered here. sea ol In order to gain some understanding of  and kl kl III. THE PERMUTATION DECOMPOSITION we use the expressions j = ie=~[G ; x ^ ], where x ^ is k k k r(a) r(a) 2 ^ ^ the position operator and @ G = (G ) . In the Once we exclude pathological toy models from our in- r 1 a 1 ^ ^ ^ ^ ^ ^ clean limit (G G ! 1; G G ! 1), the overlap term vestigations, such as the quadratic Rashba gas mentioned ol from (11) becomes kl above, and turn our focus to real materials, the general II 29,32 sea term  is strictly non-vanishing , and so we pro- kl pose not to consider the conventional Smrcka-Streda de- ie ol r a ^ ^ ^ ^ surf ! d" @ f (") tr G G x ^ j x ^ j ; (12) I II ol sea ol " k l l k kl composition  =  + = ( + ) + (  ) kl kl kl kl kl kl 4 kl with the overlap term in any capacity. Rather, we o er sea whereas  from (10) simpli es to a new one, the permutation decomposition: kl sea r a surf ^ ^ sea ! d" f (") tr G G [x ^ ; x ^ ] : (13) k l  =  +  ; (14) kl kl kl kl 2~ surf sea We recognize (12) as Streda's orbital sea term . How- where  and  are expressed in Eqs. (9) and kl kl surf ever, contrary to the original derivation in Ref. 6 as (10) respectively. As brie y mentioned above, kl well as in the re-derivation in Ref. 8 (see also Refs. 31, sea is symmetric whereas  is antisymmetric under the kl II 34, 35), this is not equivalent to  but, as seen in Eq. surf sea kl ^ ^ exchange of j and j . Due to  and  being in k l kl kl (8b), is an overlap term which has no overall e ect since di erent permutation classes they cannot overlap, and it gets cancelled out. Indeed, looking at Appendix A of so they can be derived directly from the Bastin formula Ref. 8, we see that their  ~ from Eq. (A11) is the same in Eq. (5) by decomposing the latter into symmetric and I II as our  in Eq. (6), but their  ~ in Eq. (A12), which antisymmetric terms with respect to the permutation of II ol should be the total  is only our overlap term  . ^ ^ j and j , e ectively foregoing the need to go through k l In other words, something was 'lost' while going from the Smrcka-Streda decomposition and all subsequent II the general term  | the second integral in their Eq. analysis. (A10) and our Eq. (7) | to the 'simpli ed' or orbital sea term that is their Eq. (A12) and what we call the To see the direct derivation explicitly, we rst sym- 'overlap' term in Eq. (12). What was 'lost' is precisely sea metrize (5) , expressed in Eq. (10) and in the clean limit as Eq. (13), due to the fact that the position operators were as- sumed to commute. However, the latter is not necessarily 1 1 true, since the weighting with the Fermi-Dirac distribu- = ( +  ) + (  ): (15) kl kl lk kl lk 2 2 tion projects the total space of states to the lled states, and such terms containing non-commuting position op- It is important to add that although the notation erators are responsible for certain geometric e ects such 36,37 suggests symmetrizing the cartesian indices of the as those stemming from the Berry curvature . conductivity tensor, we are in fact exchanging the Then why is it that, even though Streda's orbital operators themselves. In the given case, these are sea term { what we call the 'overlap' term { from (12) ^ ^ equivalent since the two current operators j ; j only has no overall e ect and the geometric term (13) has k l di er in their direction. The distinction is, however, been neglected in the literature, it is still possible to crucial for other cases, such as the spin response to an obtain proper results including Berry curvature e ects electric eld, where the two operators under consider- for certain cases? In order to answer this question, II II ation are not the same, but are in fact s ^ ; j , where consider the case of a vanishing  term:  = 0, k l kl ^ ^ s ^ is the spin operator in the k direction, instead of j ; j . k k l such as for the 2D metallic Dirac gas, or quadratic 9,38 magnetic Rashba gas . From (8a) and (8b) we II sea ol sea ol have  =   = 0 )  =  giving The symmetric part becomes kl kl kl kl kl 4 Z  Z n o ~ 1 ~ surf r a r a r a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ = d" f (") tr j (@ G @ G )j A = d"@ f (")Re tr[A(G G )B(G G )] ; k " " l surf " kl 2 2 4 (16) (20) r a r a ^ ^ ^ ^ ^ ^ + j (@ G @ G )j G G : l " " k n o r a r a ^ ^ ^ ^ ^ ^ A = d"f (")Re tr[A(G G )B(@ G + @ G )] : sea " " Next, we use the following identity (21) As discussed in the previous section, it is clear that r a ^ ^ ^ ^ d" f (") tr j (@ G @ G )j k " " l A + A = A + A . Now, we would like to show I II surf sea how the new separation can speci cally distinguish be- r a r a ^ ^ ^ ^ ^ ^ tween extrinsic and intrinsic phenomena. To do so, we + j (@ G @ G )j G G l " " k consider non-equilibrium transport (i) in the magnetic Rashba gas, (ii) in a multiorbital tight-binding model of r a r a ^ ^ ^ ^ ^ ^ = d" @ f (") tr j G G j G G ; " k l a ferromagnet/normal metal heterostructure and (iii) in a non-collinear antiferromagnet. (17) that can be shown straightforwardly via integration IV.1. Magnetic Rashba gas by parts and the cyclicity of the trace, leading directly to surf the expression of  in (9). The antisymmetric part kl (10) is obtained directly from the antisymmetrization of Let us rst consider the canonical magnetic Rashba (5) without any intermediate steps. gas regularized on a square lattice and described by the Hamiltonian The terms arrived at in this way carry a physical inter- H =2t(cos k + cos k ) +  ^ (22) x y z pretation. Consider the clean limit ( ! 0). In this case, surf +t ( ^ sin k  ^ sin k ): vanishes as is seen by using j = ie=~[G ; x ^ ] in R x y y x k k kl sea (9) and so is purely extrinsic. On the other hand, kl Here t is the nearest-neighbor hopping, t is the Rashba does not vanish, reduces to (13) and so is an intrinsic parameter, and  is the s-d exchange. This model has contribution. In the general case of a material with im- been central to the investigation of the anomalous Hall purities the intrinsic contribution thus arises purely from 39,40 41,42 e ect and spin-orbit torque . Here, we do not sea , which can be very helpful when trying to extract in- kl consider the vertex correction since our interest is to il- formation from experimental results by comparing them lustrate the superiority of our new permutation decompo- to numerical calculations performed using the permuta- sition of the Kubo-Bastin formula. The Green's function tion decomposition. r(a) 1 ^ ^ is simply given by G (") = (" H  i) , being A further utility of decomposing the Kubo formula into the homogeneous broadening coming from short-range permutation classes is the possibility of dealing with dis- (delta-like) impurities. In this section, we compute the tinct physical e ects arising as higher order responses in non-equilibrium properties induced by the electric eld a straightforward manner. This has been completed this (A = ej ), with particular focus on the longitudinal for second order response and is currently under prepa- conductivity (B = ej ), the transverse conductivity ration. ^ ^ (B = ej ), the eldlike torque (B =  ^ ) and the y y dampinglike torque (B =  ^ ). The conductance of IV. APPLICATION TO HALL EFFECTS, SPIN the two-dimensional electron gas is in and the spin CURRENTS AND SPIN-ORBIT TORQUE torque is expressed in terms of an e ective spin conduc- 1 1 tivity (~=2e) m . Finally, for the parameters we In this section, we compute the transport properties of take t = 2:4t,  = 0:2t and = 0:1t. three illustrative systems using the two di erent decom- Figure 1 reports the (a) longitudinal and (b) trans- positions of the Kubo-Bastin formula, the Smrcka-Streda verse Hall conductivities as well as the torque compo- decomposition, nents, (b) eldlike and (d) dampinglike, as a function of the energy. In this gure and the ones following, the n o r r a ^ ^ ^ ^ ^ A and A contributions of the Smrcka-Streda formula A = d"@ f (")Re tr[AG B(G G )] ; I II I " are represented with red and blue solid lines, while the (18) Fermi surface (A ) and Fermi sea (A ) contributions surf sea n o ~ of our permutation decomposition of the Kubo-Bastin r r r r ^ ^ ^ ^ ^ ^ ^ ^ A = d"f (")Re tr[AG B@ G A@ G BG ] ; II " " formula are represented by black and red dots, respec- tively. The black line represents the sum A + A . In I II (19) the case of transport properties only involving the Fermi and our new permutation decomposition, surface, such as the longitudinal conductivity [Fig. 1(a)] 5 FIG. 1. (Color online) Energy dependence of (a) longitudinal conductivity, (b) eldlike torque, (c) transverse conductivity and (d) dampinglike torque in the two-dimensional magnetic Rashba gas. The solid red (blue) curve refers to the A (A ) I II contribution, whereas the black curve is their sum A +A . FIG. 2. (Color online) Energy dependence of (a) longitudinal I II The black (red) dots refer to A (A ). The inset of (b) conductivity and (b) eldlike torque in the multiorbital tran- surf sea shows the band structure of the magnetic Rashba gas. The sition metal bilayer model. The solid red curve refers to the dashed horizontal line indicates the position of the avoided A and the black dots refer to A . The conductivity is in I surf 1 1 1 1 band crossing and the dotted line stands for the maximum m and the spin conductivity is in (~=2e) m . energy taken in this calculation. The conductivity is in 1 1 1 m and the spin conductivity is in (~=2e) m . Hall e ect, whereas A contains these contributions in sea itself. and the eldlike torque [Fig. 1(b)], A = A = 0 and II sea A = A , so using either the conventional Smrcka- I surf IV.2. Transition metal bilayer Streda decomposition or our permutation decomposition is equivalent. The previous calculation shows that the contribution of The transport properties involving Fermi sea are more A becomes particularly crucial when crossing local at interesting to consider. Indeed, as discussed in the previ- II bands. Nonetheless, one might argue that this sensitivity ous section, it clearly appears that when using the con- is due to the simplicity of the Rashba model that only ventional Smrcka-Streda formula, both A (red) and A I II involves two bands of opposite chirality. To generalize (blue) contributions are equally important. In fact, the variations of A can be readily correlated with the band these results, we now move on to a more complex system, II a metallic bilayer made of two transition metal slabs and structure displayed in the inset of Fig. 1(b). The A II curve exhibits two peaks, one close to the bottom of the modeled using a multiorbital tight-binding model within the Slater-Koster two-center approximation. This model lowest band, where the dispersion is quite at (around has been discussed in detail in Refs. 14 and 43 and -2.5t), and one when the Fermi level lies in the local gap here we only summarize its main features. The struc- corresponding to the avoided crossing of the two bands ture consists of two adjacent transition metal layers with [dashed line in the inset of Fig. 1(b)]. Away from this bcc crystal structure and equal lattice parameter. The local gap, A vanishes. This is an important observa- II 10 d-orbitals are included and the tight-binding param- tion because it indicates that the overlap contribution eters are extracted from Ref. 44. Importantly, we con- of the Smrcka-Streda formula is peaked close to locally sider atomic (Russell-Saunders) spin-orbit coupling, so at bands, irrespective whether it is geometrically trivial (around -2.5t) or non-trivial (around -2t). When sum- that bulk and interfacial spin-orbit coupled transport are modeled in a realistic manner. ming A and A , the complex structure of A close to I II II the bottom of the lowest band compensates A exactly, Figure 2 reports the same transport properties as so that the total contribution A + A = A has a Fig. 1, i.e., (a) longitudinal conductivity (i.e., the two- I II sea much simpler overall structure and is peaked only at the dimensional conductance divided by the thickness of the local (geometrically non-trivial) gap, which illustrates the bilayer), as well as (b) the eldlike torque as a function Berry curvature origin of this contribution. This simple of the energy. Again, we nd that Fermi surface proper- calculation points out the dramatic need to compute both ties are well-described by the surface terms when using A and A contributions to obtain correct Fermi sea either the conventional Smrcka-Streda or our permuta- I II contributions such as dampinglike torque and anomalous tion decomposition of the Kubo-Bastin formula [Fig. 2(a, 6 FIG. 3. (Color online) Energy dependence of (a, b) transverse conductivity and (c, d) dampinglike torque in the multiorbital transition metal bilayer model. The solid red (blue) curve refers to the A (A ) contribution, the black curve is their I II FIG. 4. (Color online) Angular dependence of (a) in-plane and sumA +A and the red dots refer toA . The conductivity I II sea 1 1 1 1 (b) out-of-plane spin Hall e ect in the non-collinear antifer- is in m and the spin conductivity is in (~=2e) m . romagnetic Kagome lattice model. The solid red (blue) curve refers to the A (A ) contribution, the black curve is their I II sum A +A and the black (red) dots refer to A (A ). I II surf sea The inset displays the angle made by the applied electric eld b)]. Nonetheless, the Fermi sea properties displayed on with respect to the crystal axes. The spin conductivity is in Fig. 3 exhibit a much richer behavior. The consider- (~=2e) ably more complex band structure of the multiorbital model (e.g., see Fig. 4 in Ref. 14) possesses a high density of at band regions which results in highly os- cillating A and A contributions, in both transverse I II picted in the inset of Fig. 4. The model is the same as conductivity [Fig. 3(a)], and dampinglike torque [Fig. Ref. 45, and the Hamiltonian reads 3(b)]. These oscillations are partially washed out when X X y y summing both contributions [Fig. 3(c,d)] so that the H = t c ^ c ^ +  c ^  ^  m c ^ : (23) i i j i remaining oscillations are only associated to the local hi ;j i Berry curvature of the band structure. These results agree with our recent work where we demonstrated, us- Here, t is the nearest neighbor hopping, and  is the s-d ing a similar multi-band model for topological insula- exchange. The indices ; refer to the di erent mag- tor/antiferromagnet heterostructures, that both A and I netic sublattices of a magnetic unit cell, and i; j refer to A contributions are necessary to obtain the appropri- II di erent unit cells. In this work, we set  = 1:7t. Such a 17,50 ate magnitude of the damping-torque, particularly in system displays two types of transverse spin currents , the regions displaying avoided band crossing . Fig- even in the absence of spin-orbit coupling: one spin cur- ure 3 clearly shows that both contributions should be rent  possesses a polarization perpendicular to the accounted for when computing dampinglike torque and plane, and the other  has a polarization in-plane and anomalous transport. Taking only A into account like normal to the applied electric eld. We refer to the for- in Refs. 12 and 14 is insucient. mer as perpendicular spin Hall current and the latter is called in-plane spin Hall current. We compute in Fig. 4 the (a) in-plane and (b) perpen- IV.3. Non-collinear antiferromagnet dicular spin conductivities as a function of the angle of the electric eld with respect to the crystal lattice direc- We conclude this investigation by considering one last tions. We obtain that the in-plane spin current is purely a system of interest: a non-collinear antiferromagnet dis- Fermi surface term, corresponding to the "magnetic spin playing anomalous transverse spin currents even in the Hall e ect" predicted by Zelezny  et al. and observed absence of spin-orbit coupling. As a matter of fact, the by Kimata et al. . This spin current strongly depends transport of spin and charge in non-collinear antiferro- on the orientation of the electric eld with respect to the magnets has been the object of intense scrutiny recently, crystallographic axes. In contrast, the perpendicular spin as anomalous Hall as well as magnetic spin Hall e ects current shows a weak angular dependence and is purely 17,45,46 47{49 50 have been predicted and observed . We test given by the Fermi sea contribution . Again, the A II our permutation decomposition on an ideal Kagome lat- contribution is small but non-zero. The reduced magni- tice with 120 magnetic moment con guration, as de- tude of A compared to A is due to the fact that the II I 7 Fermi level is taken away from the avoided band crossing be decomposed into symmetric and antisymmetric parts, in this particular case. which gives direct access to Fermi surface and Fermi sea contributions. The superiority of this new permutation decomposition over Smrcka-Streda's, apart from its ap- V. CONCLUSION parent conceptual clarity, has been illustrated by com- puting the extrinsic and intrinsic transport coecients of three selected systems. This observation has sub- We have shown that the widely used Smrcka-Streda stantial impact on quantum transport calculations, espe- decomposition of the celebrated Kubo-Bastin formula cially when considering Berry curvature induced mech- possesses an overlap that makes it inappropriate to dis- anisms such as Hall conductance and torques, since it tinguish between Fermi sea and Fermi surface contri- provides a neat way of separating the intrinsic part of butions to transport coecients. This is particularly these anomalous transport e ects from Fermi surface re- crucial in multiband systems possessing a high density lated e ects, removing spurious e ects stemming from of locally at bands and avoided band crossings. As local trivial band atness. a matter of fact, whereas intrinsic (Berry-curvature in- duced) transport properties are dominated by geomet- rically non-trivial avoided band crossings, the overlap is ACKNOWLEDGMENTS enhanced close to any (trivial and non-trivial) locally at bands, as illustrated in the case of the magnetic Rashba gas. Therefore, the Smrcka-Streda decomposition of the This work was supported by the King Abdullah Uni- Kubo-Bastin formula can lead to an incorrect estima- versity of Science and Technology (KAUST) through the tion of the intrinsic transport properties. To remedy this award OSR-2017-CRG6-3390 from the Oce of Spon- diculty, we demonstrated that the Kubo formula can sored Research (OSR). Appendix: Derivation of the overlap term The  term (6) can be handled in a simple way by separating it into symmetric and antisymmetric permutations ^ ^ of j and j as follows: k l I r a r a ^ ^ ^ ^ ^ ^ ^ ^ = d" @ f (") tr j G j j G j G G " k l l k kl r a r a r a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ = d" @ f (") tr j (G G )j + j (G G )j G G " k l l k r a r a r a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ + d" @ f (") tr j (G + G )j j (G + G )j G G " k l l k (A.1) r a r a ^ ^ ^ ^ ^ ^ = d" @ f (") tr j (G G )j G G " k l r a r a r a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ + d" @ f (") tr j (G + G )j j (G + G )j G G " k l l k surf ol = +  : kl kl surf ol ^ ^ It is clear that  is symmetric and  is antisymmetric in the exchange of j and j . k l II The  term (7) is more complicated, requiring the following manipulations: II r r r r a a a a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ = d" f (") tr j G j @ G j @ G j G + j @ G j G j G j @ G k l " k " l l " k l k " kl 1 ~ r r r r a a a a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ = d" f (") tr j G j @ G j @ G j G + j @ G j G j G j @ G k l " k " l l " k l k " 2 4 1 ~ r r r r a a a a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ + d" f (") tr j G j @ G j @ G j G + j @ G j G j G j @ G (A.2) k l " k " l l " k l k " 2 4 1 ~ r a r a r a r a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ = d" f (") tr j (G G )j (@ G + @ G ) j (@ G + @ G )j (G G ) k l " " k " " l 2 4 1 ~ r a r a r a r a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ + d" f (") tr j (G + G )j (@ G @ G ) j (@ G @ G )j (G + G ) : k l " " k " " l 2 4 8 Looking at the terms after the last equality in (A.2), we integrate by parts the second term and combine the result with the rst term. 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A new decomposition of the Kubo-Bastin formula

Bonbien, Varga; Manchon, Aurelien
Condensed Matter , Volume 2020 (2005) – May 10, 2020
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ARCH-3331
DOI
10.1103/PhysRevB.102.085113
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Abstract

1 1;2y Varga Bonbien and Aur elien Manchon Physical Science and Engineering Division (PSE), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia Aix-Marseille Univ, CNRS, CINaM, Marseille, France The Smrcka-Streda version of Kubo's linear response formula is widely used in the literature to compute non-equilibrium transport properties of heterostructures. It is particularly useful for the evaluation of intrinsic transport properties associated with the Berry curvature of the Bloch states, such as anomalous and spin Hall currents as well as the damping-like component of the spin-orbit torque. Here, we demonstrate in a very general way that the widely used decomposition of the Kubo-Bastin formula introduced by Smrcka and Streda contains an overlap, which has lead to widespread confusion in the literature regarding the Fermi surface and Fermi sea contributions. To remedy this pathology, we propose a new decomposition of the Kubo-Bastin formula based on the permutation properties of the correlation function and derive a new set of formulas, without an overlap, that provides direct access to the transport e ects of interest. We apply these new formulas to selected cases and demonstrate that the Fermi sea and Fermi surface contributions can be uniquely addressed with our symmetrized approach. I. INTRODUCTION Bruno presented a detailed and widely cited derivation of the Kubo-Bastin and Smrcka-Streda formulae from the 1,2 Kubo formula. The seminal work of Kubo showed that, in the per- The Smrcka-Streda formula has been widely used to turbative weak- eld limit, transport coecients can be 9,10 compute charge and spin Hall currents as well as spin- expressed as correlation functions of quantum mechani- orbit torques . Whereas a few works use the full Smrcka- cal observable operators. The resulting Kubo formalism 12{15 Streda formula , most theoretical studies exploit a has become a staple of quantum transport theory cal- simpli ed version of it, obtained by assuming constant culations and surged in popularity following the realiza- 16{18 scattering time and in the weak disorder limit tion that applying it to transport phenomena in crystals provides direct access to topological invariants, thereby X ^ ^ 1 Re[hnkjBjmkihmkjAjnki] yielding an explanation for the robustness of the quan- A ! ; (3) surf 2 2 2 2 [(" " ) + ][(" " ) + ] F nk F mk tized Hall e ect . k;n;m While the original Kubo formula is formally satisfy- X ^ ^ Im[hnkjBjmkihmkjAjnki] ing, realistic calculations with it are rather impractical. A ! (f (" ) f (" )): nk mk sea 4 5 (" " ) nk mk Bastin et al. and later Streda and Smrcka used Green's k;n6=m functions to rewrite the Kubo formula and arrived at a (4) result directly applicable to computations in the static limit. Later on, Smrcka and Streda further decomposed Here is the homogeneous broadening and jnki is a the Bastin formula into two terms, Bloch state of the crystal. This simpli ed version readily n o attributes A to intraband transitions, yielding a surf r r a ^ ^ ^ ^ ^ A = d"@ f (")Re tr[AG B(G G )] ; (1) I " 1= dependence, and A to interband transitions, 2 sea n o which are nite in the clean limit ( ! 0). In fact, r r r r ^ ^ ^ ^ ^ ^ ^ ^ A = d"f (")Re tr[AG B@ G A@ G BG ] ; these simpli ed formulae elegantly connect the Fermi II " " sea transport contributions to the Berry curvature of the (2) Bloch states, and, to date, the Berry curvature formula, Eq. (4), has been widely used to characterize the ^ ^ Here A is the operator of the perturbation and B is the 19{21 intrinsic spin Hall e ect of bulk materials . As we r(a) operator of the observable, G (") is the retarded (ad- mentioned already, this formula is only valid in the clean vanced) Green's function of the system and we have sup- limit and does not apply in realistic materials where pressed the energy argument in the formulae for brevity, momentum scattering is important. More speci cally, it f (") is the Fermi-Dirac distribution and @ indicates an becomes invalid when the broadening is comparable energy derivative. Because of their connection to @ f (") to, or larger than the local orbital gaps resulting from and f ("), Eqs. (1) and (2) were wrongly referred to as avoided band crossings, and where Berry curvature is Fermi surface and Fermi sea terms, respectively. These 22{24 maximized. Indeed, further investigations have terms were used by Streda in his famous analysis of 7 addressed the spin Hall e ect of metals using the full the quantized Hall e ect . More recently, Cr epieux and Smrcka-Streda formula, Eqs. (1)-(2), showing evidence that the spin Hall conductivity of 5d transition metals is dominated by A . Similarly, an in uential work [email protected] by Sinitsyn et al. demonstrated that in the case of a [email protected] gapped Dirac cone, spin Hall e ect is entirely due to A arXiv:2005.04678v1 [cond-mat.mes-hall] 10 May 2020 2 in the metallic regime, while it is entirely due to A in function with respect to its energy argument, that we II the gap. These observations, valid for speci c examples, have suppressed in the formula for brevity. Across the led to the confusion that A always dominate in metals. manuscript, ::: ^ denotes an operator and tr(: : : ) is the 12{14 For instance, some investigations (including ours) trace operation. have computed the spin-orbit torque using only A based on Kontani et al. 's argument. However, recent Splitting (5) into two halves, integrating one of them calculations have demonstrated that certain transport by parts and combining it with the other half yields the properties associated with Berry curvature, such as Smrcka-Streda decomposition of the Kubo-Bastin for- I II the dampinglike torque in magnetic heterostructures, mula [Eq. (A10) of Ref. 8] with  =  +  , where kl kl kl 26{28 have contributions from both A and A (see also I II Ref. 29). This suggests that attributing purely Fermi I r a ^ ^ ^ ^ ^ ^ = d" @ f (") tr j G j j G j surface origin to A and purely Fermi sea origin to A " k l l k I II kl is incorrect. (6) r a ^ ^ G G In this paper, we rst show in a very general way that the A -A decomposition of the Kubo-Bastin formula I II and introduced by Smrcka and Streda contains an overlap, and there appears to be widespread confusion regarding this aspect in the literature. This overlap was hinted II r r r r ^ ^ ^ ^ ^ ^ ^ ^ = d" f (") tr j G (")j @ G j @ G j G k l " k " l kl at for the special case of a 2-dimensional Dirac mate- rial by Sinitsyn et al. , but the fact that Smrcka-Streda a a a a ^ ^ ^ ^ ^ ^ ^ ^ + j @ G j G j G j @ G : and many subsequent authors unjusti ably neglected a l " k l k " subtle term relating to position operators in certain ver- (7) sions of the Smrcka-Streda formula responsible for geo- 7,8,31 metric e ects, went unmentioned . This subtlety is Integrating (7) by parts shall not yield any surface unnoticeable for simple models | such as the quadratic terms, thus we might naively conclude that this term magnetic Rashba gas | when A is vanishingly small describes e ects resulting purely from the sea. However, II away from the avoided band crossing, since the neglected this is not the case, since there is signi cant overlap be- I II geometric term exactly cancels out Streda's orbital sea tween  and  . Indeed, manipulating (6) and (7) we kl kl term in A which, due to the overlap, also appears in arrive at (see Appendix) II the A term. However, in the general case, A is non- I II 29,32 negligible and thus, Smrcka and Streda's decompo- I surf ol =  +  ; (8a) kl kl kl sition of the Kubo-Bastin formula into A ;A does not I II II sea ol lend itself to a proper analysis of di erent physical ef- =   ; (8b) kl kl kl fects. To remedy this, we propose a new decomposition where of the Kubo-Bastin formula based on the permutation properties of the correlator and derive a new set of for- mulas without an overlap, that provides direct access to ~ surf r a ^ ^ ^ ^ = d" @ f (") tr j G G j " k l kl the intrinsic geometric e ects. r a ^ ^ G G ; (9) II. THE KUBO-BASTIN FORMULA AND THE SMRCKA-STREDA DECOMPOSITION sea r a ^ ^ ^ ^ = d" f (") tr j @ G + @ G j k " " l kl The Kubo-Bastin formula for the electrical conductiv- ity,  , in the static limit as obtained from the Kubo kl r a ^ ^ ^ ^ j (@ G + @ G j l " " k formula is [Eq. (A9) of Ref. 8] r a ^ ^ Z  G G (10) r a ^ ^ ^ ^ ^ ^ = d" f (") tr j @ G j j @ G j kl k " l l " k and the overlap term r a ^ ^ G G ; (5) ol r a ^ ^ ^ ^ = d" @ f (") tr j G + G j " k l kl ^ ^ where j ; j are electric charge current opera- k l r(a) ^ r a ^ ^ ^ ^ tors in the k; l 2 fx; y; zg directions, G (") = j (G + G j l k lim 1=("H i) is the retarded(advanced) Green's !0 0 function corresponding to the equilibrium Hamiltonian r a ^ ^ G G : (11) r(a) H and @ G (") is the derivative of the Green's 0 " 3 surf surf surf I ol sea Upon closer inspection, we note that  is symmet-  =  +  =  +  . Thus we see that kl kl kl kl kl kl II sea ol for the particular case of a vanishing  term, Streda's ric whereas  along with  are antisymmetric under kl kl orbital sea term (12) is exactly equal to the geometric ^ ^ the exchange of operators j and j . Furthermore, in k l sea surf term  and consequently describes Berry curvature kl the special case of k = l,  can be recognized as the kk e ects. This is an advantage in the zero temperature Kubo-Greenwood formula for the diagonal conductivity. I case, since @ f (") ! (" " ) as T ! 0, meaning that " F The separation of  into symmetric and antisymmetric surf ol we can simply evaluate the Green's functions in (12) at parts yielding  and  is already present in the 32,33 the Fermi energy and there is no need for a complete literature , however it was not realized that the ol energy integration, as would be required for (10) or (13). antisymmetric part  is an overlap and gets exactly cancelled when considering an appropriate separation of II sea ol into  and  , as considered here. sea ol In order to gain some understanding of  and kl kl III. THE PERMUTATION DECOMPOSITION we use the expressions j = ie=~[G ; x ^ ], where x ^ is k k k r(a) r(a) 2 ^ ^ the position operator and @ G = (G ) . In the Once we exclude pathological toy models from our in- r 1 a 1 ^ ^ ^ ^ ^ ^ clean limit (G G ! 1; G G ! 1), the overlap term vestigations, such as the quadratic Rashba gas mentioned ol from (11) becomes kl above, and turn our focus to real materials, the general II 29,32 sea term  is strictly non-vanishing , and so we pro- kl pose not to consider the conventional Smrcka-Streda de- ie ol r a ^ ^ ^ ^ surf ! d" @ f (") tr G G x ^ j x ^ j ; (12) I II ol sea ol " k l l k kl composition  =  + = ( + ) + (  ) kl kl kl kl kl kl 4 kl with the overlap term in any capacity. Rather, we o er sea whereas  from (10) simpli es to a new one, the permutation decomposition: kl sea r a surf ^ ^ sea ! d" f (") tr G G [x ^ ; x ^ ] : (13) k l  =  +  ; (14) kl kl kl kl 2~ surf sea We recognize (12) as Streda's orbital sea term . How- where  and  are expressed in Eqs. (9) and kl kl surf ever, contrary to the original derivation in Ref. 6 as (10) respectively. As brie y mentioned above, kl well as in the re-derivation in Ref. 8 (see also Refs. 31, sea is symmetric whereas  is antisymmetric under the kl II 34, 35), this is not equivalent to  but, as seen in Eq. surf sea kl ^ ^ exchange of j and j . Due to  and  being in k l kl kl (8b), is an overlap term which has no overall e ect since di erent permutation classes they cannot overlap, and it gets cancelled out. Indeed, looking at Appendix A of so they can be derived directly from the Bastin formula Ref. 8, we see that their  ~ from Eq. (A11) is the same in Eq. (5) by decomposing the latter into symmetric and I II as our  in Eq. (6), but their  ~ in Eq. (A12), which antisymmetric terms with respect to the permutation of II ol should be the total  is only our overlap term  . ^ ^ j and j , e ectively foregoing the need to go through k l In other words, something was 'lost' while going from the Smrcka-Streda decomposition and all subsequent II the general term  | the second integral in their Eq. analysis. (A10) and our Eq. (7) | to the 'simpli ed' or orbital sea term that is their Eq. (A12) and what we call the To see the direct derivation explicitly, we rst sym- 'overlap' term in Eq. (12). What was 'lost' is precisely sea metrize (5) , expressed in Eq. (10) and in the clean limit as Eq. (13), due to the fact that the position operators were as- sumed to commute. However, the latter is not necessarily 1 1 true, since the weighting with the Fermi-Dirac distribu- = ( +  ) + (  ): (15) kl kl lk kl lk 2 2 tion projects the total space of states to the lled states, and such terms containing non-commuting position op- It is important to add that although the notation erators are responsible for certain geometric e ects such 36,37 suggests symmetrizing the cartesian indices of the as those stemming from the Berry curvature . conductivity tensor, we are in fact exchanging the Then why is it that, even though Streda's orbital operators themselves. In the given case, these are sea term { what we call the 'overlap' term { from (12) ^ ^ equivalent since the two current operators j ; j only has no overall e ect and the geometric term (13) has k l di er in their direction. The distinction is, however, been neglected in the literature, it is still possible to crucial for other cases, such as the spin response to an obtain proper results including Berry curvature e ects electric eld, where the two operators under consider- for certain cases? In order to answer this question, II II ation are not the same, but are in fact s ^ ; j , where consider the case of a vanishing  term:  = 0, k l kl ^ ^ s ^ is the spin operator in the k direction, instead of j ; j . k k l such as for the 2D metallic Dirac gas, or quadratic 9,38 magnetic Rashba gas . From (8a) and (8b) we II sea ol sea ol have  =   = 0 )  =  giving The symmetric part becomes kl kl kl kl kl 4 Z  Z n o ~ 1 ~ surf r a r a r a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ = d" f (") tr j (@ G @ G )j A = d"@ f (")Re tr[A(G G )B(G G )] ; k " " l surf " kl 2 2 4 (16) (20) r a r a ^ ^ ^ ^ ^ ^ + j (@ G @ G )j G G : l " " k n o r a r a ^ ^ ^ ^ ^ ^ A = d"f (")Re tr[A(G G )B(@ G + @ G )] : sea " " Next, we use the following identity (21) As discussed in the previous section, it is clear that r a ^ ^ ^ ^ d" f (") tr j (@ G @ G )j k " " l A + A = A + A . Now, we would like to show I II surf sea how the new separation can speci cally distinguish be- r a r a ^ ^ ^ ^ ^ ^ tween extrinsic and intrinsic phenomena. To do so, we + j (@ G @ G )j G G l " " k consider non-equilibrium transport (i) in the magnetic Rashba gas, (ii) in a multiorbital tight-binding model of r a r a ^ ^ ^ ^ ^ ^ = d" @ f (") tr j G G j G G ; " k l a ferromagnet/normal metal heterostructure and (iii) in a non-collinear antiferromagnet. (17) that can be shown straightforwardly via integration IV.1. Magnetic Rashba gas by parts and the cyclicity of the trace, leading directly to surf the expression of  in (9). The antisymmetric part kl (10) is obtained directly from the antisymmetrization of Let us rst consider the canonical magnetic Rashba (5) without any intermediate steps. gas regularized on a square lattice and described by the Hamiltonian The terms arrived at in this way carry a physical inter- H =2t(cos k + cos k ) +  ^ (22) x y z pretation. Consider the clean limit ( ! 0). In this case, surf +t ( ^ sin k  ^ sin k ): vanishes as is seen by using j = ie=~[G ; x ^ ] in R x y y x k k kl sea (9) and so is purely extrinsic. On the other hand, kl Here t is the nearest-neighbor hopping, t is the Rashba does not vanish, reduces to (13) and so is an intrinsic parameter, and  is the s-d exchange. This model has contribution. In the general case of a material with im- been central to the investigation of the anomalous Hall purities the intrinsic contribution thus arises purely from 39,40 41,42 e ect and spin-orbit torque . Here, we do not sea , which can be very helpful when trying to extract in- kl consider the vertex correction since our interest is to il- formation from experimental results by comparing them lustrate the superiority of our new permutation decompo- to numerical calculations performed using the permuta- sition of the Kubo-Bastin formula. The Green's function tion decomposition. r(a) 1 ^ ^ is simply given by G (") = (" H  i) , being A further utility of decomposing the Kubo formula into the homogeneous broadening coming from short-range permutation classes is the possibility of dealing with dis- (delta-like) impurities. In this section, we compute the tinct physical e ects arising as higher order responses in non-equilibrium properties induced by the electric eld a straightforward manner. This has been completed this (A = ej ), with particular focus on the longitudinal for second order response and is currently under prepa- conductivity (B = ej ), the transverse conductivity ration. ^ ^ (B = ej ), the eldlike torque (B =  ^ ) and the y y dampinglike torque (B =  ^ ). The conductance of IV. APPLICATION TO HALL EFFECTS, SPIN the two-dimensional electron gas is in and the spin CURRENTS AND SPIN-ORBIT TORQUE torque is expressed in terms of an e ective spin conduc- 1 1 tivity (~=2e) m . Finally, for the parameters we In this section, we compute the transport properties of take t = 2:4t,  = 0:2t and = 0:1t. three illustrative systems using the two di erent decom- Figure 1 reports the (a) longitudinal and (b) trans- positions of the Kubo-Bastin formula, the Smrcka-Streda verse Hall conductivities as well as the torque compo- decomposition, nents, (b) eldlike and (d) dampinglike, as a function of the energy. In this gure and the ones following, the n o r r a ^ ^ ^ ^ ^ A and A contributions of the Smrcka-Streda formula A = d"@ f (")Re tr[AG B(G G )] ; I II I " are represented with red and blue solid lines, while the (18) Fermi surface (A ) and Fermi sea (A ) contributions surf sea n o ~ of our permutation decomposition of the Kubo-Bastin r r r r ^ ^ ^ ^ ^ ^ ^ ^ A = d"f (")Re tr[AG B@ G A@ G BG ] ; II " " formula are represented by black and red dots, respec- tively. The black line represents the sum A + A . In I II (19) the case of transport properties only involving the Fermi and our new permutation decomposition, surface, such as the longitudinal conductivity [Fig. 1(a)] 5 FIG. 1. (Color online) Energy dependence of (a) longitudinal conductivity, (b) eldlike torque, (c) transverse conductivity and (d) dampinglike torque in the two-dimensional magnetic Rashba gas. The solid red (blue) curve refers to the A (A ) I II contribution, whereas the black curve is their sum A +A . FIG. 2. (Color online) Energy dependence of (a) longitudinal I II The black (red) dots refer to A (A ). The inset of (b) conductivity and (b) eldlike torque in the multiorbital tran- surf sea shows the band structure of the magnetic Rashba gas. The sition metal bilayer model. The solid red curve refers to the dashed horizontal line indicates the position of the avoided A and the black dots refer to A . The conductivity is in I surf 1 1 1 1 band crossing and the dotted line stands for the maximum m and the spin conductivity is in (~=2e) m . energy taken in this calculation. The conductivity is in 1 1 1 m and the spin conductivity is in (~=2e) m . Hall e ect, whereas A contains these contributions in sea itself. and the eldlike torque [Fig. 1(b)], A = A = 0 and II sea A = A , so using either the conventional Smrcka- I surf IV.2. Transition metal bilayer Streda decomposition or our permutation decomposition is equivalent. The previous calculation shows that the contribution of The transport properties involving Fermi sea are more A becomes particularly crucial when crossing local at interesting to consider. Indeed, as discussed in the previ- II bands. Nonetheless, one might argue that this sensitivity ous section, it clearly appears that when using the con- is due to the simplicity of the Rashba model that only ventional Smrcka-Streda formula, both A (red) and A I II involves two bands of opposite chirality. To generalize (blue) contributions are equally important. In fact, the variations of A can be readily correlated with the band these results, we now move on to a more complex system, II a metallic bilayer made of two transition metal slabs and structure displayed in the inset of Fig. 1(b). The A II curve exhibits two peaks, one close to the bottom of the modeled using a multiorbital tight-binding model within the Slater-Koster two-center approximation. This model lowest band, where the dispersion is quite at (around has been discussed in detail in Refs. 14 and 43 and -2.5t), and one when the Fermi level lies in the local gap here we only summarize its main features. The struc- corresponding to the avoided crossing of the two bands ture consists of two adjacent transition metal layers with [dashed line in the inset of Fig. 1(b)]. Away from this bcc crystal structure and equal lattice parameter. The local gap, A vanishes. This is an important observa- II 10 d-orbitals are included and the tight-binding param- tion because it indicates that the overlap contribution eters are extracted from Ref. 44. Importantly, we con- of the Smrcka-Streda formula is peaked close to locally sider atomic (Russell-Saunders) spin-orbit coupling, so at bands, irrespective whether it is geometrically trivial (around -2.5t) or non-trivial (around -2t). When sum- that bulk and interfacial spin-orbit coupled transport are modeled in a realistic manner. ming A and A , the complex structure of A close to I II II the bottom of the lowest band compensates A exactly, Figure 2 reports the same transport properties as so that the total contribution A + A = A has a Fig. 1, i.e., (a) longitudinal conductivity (i.e., the two- I II sea much simpler overall structure and is peaked only at the dimensional conductance divided by the thickness of the local (geometrically non-trivial) gap, which illustrates the bilayer), as well as (b) the eldlike torque as a function Berry curvature origin of this contribution. This simple of the energy. Again, we nd that Fermi surface proper- calculation points out the dramatic need to compute both ties are well-described by the surface terms when using A and A contributions to obtain correct Fermi sea either the conventional Smrcka-Streda or our permuta- I II contributions such as dampinglike torque and anomalous tion decomposition of the Kubo-Bastin formula [Fig. 2(a, 6 FIG. 3. (Color online) Energy dependence of (a, b) transverse conductivity and (c, d) dampinglike torque in the multiorbital transition metal bilayer model. The solid red (blue) curve refers to the A (A ) contribution, the black curve is their I II FIG. 4. (Color online) Angular dependence of (a) in-plane and sumA +A and the red dots refer toA . The conductivity I II sea 1 1 1 1 (b) out-of-plane spin Hall e ect in the non-collinear antifer- is in m and the spin conductivity is in (~=2e) m . romagnetic Kagome lattice model. The solid red (blue) curve refers to the A (A ) contribution, the black curve is their I II sum A +A and the black (red) dots refer to A (A ). I II surf sea The inset displays the angle made by the applied electric eld b)]. Nonetheless, the Fermi sea properties displayed on with respect to the crystal axes. The spin conductivity is in Fig. 3 exhibit a much richer behavior. The consider- (~=2e) ably more complex band structure of the multiorbital model (e.g., see Fig. 4 in Ref. 14) possesses a high density of at band regions which results in highly os- cillating A and A contributions, in both transverse I II picted in the inset of Fig. 4. The model is the same as conductivity [Fig. 3(a)], and dampinglike torque [Fig. Ref. 45, and the Hamiltonian reads 3(b)]. These oscillations are partially washed out when X X y y summing both contributions [Fig. 3(c,d)] so that the H = t c ^ c ^ +  c ^  ^  m c ^ : (23) i i j i remaining oscillations are only associated to the local hi ;j i Berry curvature of the band structure. These results agree with our recent work where we demonstrated, us- Here, t is the nearest neighbor hopping, and  is the s-d ing a similar multi-band model for topological insula- exchange. The indices ; refer to the di erent mag- tor/antiferromagnet heterostructures, that both A and I netic sublattices of a magnetic unit cell, and i; j refer to A contributions are necessary to obtain the appropri- II di erent unit cells. In this work, we set  = 1:7t. Such a 17,50 ate magnitude of the damping-torque, particularly in system displays two types of transverse spin currents , the regions displaying avoided band crossing . Fig- even in the absence of spin-orbit coupling: one spin cur- ure 3 clearly shows that both contributions should be rent  possesses a polarization perpendicular to the accounted for when computing dampinglike torque and plane, and the other  has a polarization in-plane and anomalous transport. Taking only A into account like normal to the applied electric eld. We refer to the for- in Refs. 12 and 14 is insucient. mer as perpendicular spin Hall current and the latter is called in-plane spin Hall current. We compute in Fig. 4 the (a) in-plane and (b) perpen- IV.3. Non-collinear antiferromagnet dicular spin conductivities as a function of the angle of the electric eld with respect to the crystal lattice direc- We conclude this investigation by considering one last tions. We obtain that the in-plane spin current is purely a system of interest: a non-collinear antiferromagnet dis- Fermi surface term, corresponding to the "magnetic spin playing anomalous transverse spin currents even in the Hall e ect" predicted by Zelezny  et al. and observed absence of spin-orbit coupling. As a matter of fact, the by Kimata et al. . This spin current strongly depends transport of spin and charge in non-collinear antiferro- on the orientation of the electric eld with respect to the magnets has been the object of intense scrutiny recently, crystallographic axes. In contrast, the perpendicular spin as anomalous Hall as well as magnetic spin Hall e ects current shows a weak angular dependence and is purely 17,45,46 47{49 50 have been predicted and observed . We test given by the Fermi sea contribution . Again, the A II our permutation decomposition on an ideal Kagome lat- contribution is small but non-zero. The reduced magni- tice with 120 magnetic moment con guration, as de- tude of A compared to A is due to the fact that the II I 7 Fermi level is taken away from the avoided band crossing be decomposed into symmetric and antisymmetric parts, in this particular case. which gives direct access to Fermi surface and Fermi sea contributions. The superiority of this new permutation decomposition over Smrcka-Streda's, apart from its ap- V. CONCLUSION parent conceptual clarity, has been illustrated by com- puting the extrinsic and intrinsic transport coecients of three selected systems. This observation has sub- We have shown that the widely used Smrcka-Streda stantial impact on quantum transport calculations, espe- decomposition of the celebrated Kubo-Bastin formula cially when considering Berry curvature induced mech- possesses an overlap that makes it inappropriate to dis- anisms such as Hall conductance and torques, since it tinguish between Fermi sea and Fermi surface contri- provides a neat way of separating the intrinsic part of butions to transport coecients. This is particularly these anomalous transport e ects from Fermi surface re- crucial in multiband systems possessing a high density lated e ects, removing spurious e ects stemming from of locally at bands and avoided band crossings. As local trivial band atness. a matter of fact, whereas intrinsic (Berry-curvature in- duced) transport properties are dominated by geomet- rically non-trivial avoided band crossings, the overlap is ACKNOWLEDGMENTS enhanced close to any (trivial and non-trivial) locally at bands, as illustrated in the case of the magnetic Rashba gas. Therefore, the Smrcka-Streda decomposition of the This work was supported by the King Abdullah Uni- Kubo-Bastin formula can lead to an incorrect estima- versity of Science and Technology (KAUST) through the tion of the intrinsic transport properties. To remedy this award OSR-2017-CRG6-3390 from the Oce of Spon- diculty, we demonstrated that the Kubo formula can sored Research (OSR). Appendix: Derivation of the overlap term The  term (6) can be handled in a simple way by separating it into symmetric and antisymmetric permutations ^ ^ of j and j as follows: k l I r a r a ^ ^ ^ ^ ^ ^ ^ ^ = d" @ f (") tr j G j j G j G G " k l l k kl r a r a r a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ = d" @ f (") tr j (G G )j + j (G G )j G G " k l l k r a r a r a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ + d" @ f (") tr j (G + G )j j (G + G )j G G " k l l k (A.1) r a r a ^ ^ ^ ^ ^ ^ = d" @ f (") tr j (G G )j G G " k l r a r a r a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ + d" @ f (") tr j (G + G )j j (G + G )j G G " k l l k surf ol = +  : kl kl surf ol ^ ^ It is clear that  is symmetric and  is antisymmetric in the exchange of j and j . k l II The  term (7) is more complicated, requiring the following manipulations: II r r r r a a a a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ = d" f (") tr j G j @ G j @ G j G + j @ G j G j G j @ G k l " k " l l " k l k " kl 1 ~ r r r r a a a a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ = d" f (") tr j G j @ G j @ G j G + j @ G j G j G j @ G k l " k " l l " k l k " 2 4 1 ~ r r r r a a a a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ + d" f (") tr j G j @ G j @ G j G + j @ G j G j G j @ G (A.2) k l " k " l l " k l k " 2 4 1 ~ r a r a r a r a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ = d" f (") tr j (G G )j (@ G + @ G ) j (@ G + @ G )j (G G ) k l " " k " " l 2 4 1 ~ r a r a r a r a ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ + d" f (") tr j (G + G )j (@ G @ G ) j (@ G @ G )j (G + G ) : k l " " k " " l 2 4 8 Looking at the terms after the last equality in (A.2), we integrate by parts the second term and combine the result with the rst term. 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Published: May 10, 2020

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