Kinetic orbital moments and nonlocal transport in disordered metals with nontrivial geometry
KKinetic orbital moments and nonlocal transport in disordered metals with nontrivialgeometry
J. Rou, C. S¸ahin, J. Ma, D. A. Pesin
Department of Physics and Astronomy, University of Utah, Salt Lake City, UT 84112 USA (Dated: July 14, 2018)We study the effects of spatial dispersion in disordered noncentrosymmetric metals. These includethe kinetic magnetoelectric effect, natural optical activity of metals, as well as the so-called dynamicchiral magnetic effect as a particular case of the latter. These effects are determined by the linearin the wave vector of an electromagnetic perturbation contribution to the conductivity tensor ofa material, and stem from the magnetic moments of quasiparticles near the Fermi surface. Weidentify new disorder-induced contributions to these magnetic moments that come from the skewscattering and side jump processes, familiar from the theory of anomalous Hall effect. We show thatat low frequencies the spatial dispersion of the conductivity tensor comes mainly either from theskew scattering or intrinsic contribution, and there is always a region of frequencies in which theintrinsic mechanism dominates. Our results imply that in clean three-dimensional metals, current-induced magnetization is in general determined by impurity skew scattering, rather than intrinsiccontributions. Intrinsic effects are expected to dominate in cubic enantiomorphic crystals with pointgroups T and O , and in polycrystalline samples, regardless of their mobility. I. INTRODUCTION
When a crystal is subjected to an ac electromagneticperturbation of frequency ω and wave vector q , its linearresponse to the perturbation is fully described by theoptical conductivity tensor, σ ab ( ω, q ).Allowing for a moment for the possibility of time-reversal symmetry breaking in a crystal, denoted as theexistence of magnetization M , the optical conductivitytensor σ ab ( ω, q ; M ) can be written for small M and q as σ ab ( ω, q ; M ) ≈ σ ab ( ω ) + χ abc ( ω ) M c + λ abc ( ω ) q c . (1)In this expression, σ ab ( ω ) is the usual local optical con-ductivity, while the pseudotensor χ abc and tensor λ abc describe optical activity of the crystal , either due to theanomalous Hall effect ( χ abc ), or natural optical activity( λ abc ). These tensors determine the antisymmetric partof the conductivity tensor. Indeed, the form of responseis restricted by the Onsager relations : σ ab ( ω, q ; M ) = σ ba ( ω, − q ; − M ) , (2)which stem from the microscopic reversibility of the lawsof physics, and imply that χ abc and λ abc are antisymmet-ric with respect to the first pair of indices: (cid:20) λ abc χ abc (cid:21) = − (cid:20) λ bac χ bac (cid:21) . (3)In this paper, we focus on the theory of the tensor λ abc ( ω ) in metals, at frequencies that are small com-pared to all band splittings at the Fermi surface. In otherwords, we consider the effects of spatial dispersion in dis-ordered time-reversal invariant noncentrosymmetric met-als. We show that the linear-in- q part of the conductivitytensor of these systems is of geometric origin. The term“geometric origin” is understood in the same way as theorigin of the anomalous Hall effect, which is rooted in Berry phase physics . As explained below, the results ofthis work represent the full theory of the natural opticalactivity in metals, and contain the theory of the dynamicchiral magnetic effect as a particular case.The theory of the anomalous Hall effect (AHE), andthus the tensor χ abc , is now quite mature . Importantly,since the seminal work of Haldane , it has been under-stood that (nonquantized part of) the low-frequency limitof this tensor is a Fermi surface property, originating fromthe Berry curvature in the band structure. Berry phasesat the Fermi surface not only determine the so-called in-trinsic contribution to the anomalous Hall effect (AHE),but also can be shown to be behind the extrinsic – sidejump and skew scattering – mechanisms . By now thecorresponding ideas have been well established, and ex-cellent reviews have been written on the subject We show that mechanisms analogous to those respon-sible for the anomalous Hall effect determine the leadingnonlocal correction to the conductivity tensor in time-reversal invariant disordered noncentrosymmetric met-als. Specifically, we build the full theory of the tensor λ abc that determines the linear in wave vector part of theconductivity tensor (1) in disordered metals. As a result,we identify two disorder-induced corrections to the or-bital magnetic moment of quasiparticles that determinethe extrinsic contribution to λ abc .From a physical point of view, the study is motivatedby a recent revival in the interest in the geometric prop-erties of the Fermi surface coming from two seeminglydisconnected directions. One is the magnetohydrody-namics of the quark-gluon plasma created in heavy-ion collisions in High Energy Physics, the other is therecent advent of Weyl semimetals in Condensed MatterPhysics . Both areas are related by the concept ofthe chirality, be it the intrinsic chirality of fermions inLorentz-invariant quark-gluon plasma, or the chirality ofconduction band electrons imparted by the chirality ofthe crystalline structure. a r X i v : . [ c ond - m a t . m e s - h a ll ] M a y A particular example of a phenomenon connecting thetwo fields is the so-called chiral magnetic effect .One variety of it – the dynamic Chiral Magnetic Ef-fect (dCME) – is defined as the electric current re-sponse to a slowly oscillating magnetic field, j cme ( ω ) = γ ( ω ) B ( ω ), where the chiral magnetic conductivity γ ( ω )is the response coefficient. One can easily see that thedCME is a particular case of Eq. (1). Indeed, it can beshown that in an isotropic clean metal λ abc ∝ (cid:15) abc /ω atlow frequencies, and Faraday’s law q × E = ω B ensuresthat the current response described by such λ abc is ex-actly the dCME, establishing the relation.The rest of the paper is organized as follows: Section IIcontains the main results of the paper. Section III laysout the kinetic equation formalism for the problem. InSection IV we apply the formalism of Section III to cal-culate the nonlocal correction to the conductivity tensorin metals. Section V contains the description of the phys-ical applications of the developed theory: natural opticalactivity and current-induced magnetization phenomenain disordered three-dimensional metals. Finally, in Sec-tion VI we summarize our findings. II. MAIN RESULTS AND QUALITATIVECONSIDERATIONS
In this Section we summarize main results obtainedin the rest the paper, and provide estimates for mag-nitudes of various quantities in a typical Tellurium-likehelical metal. Most of the equations here will be repeatedthroughout the paper, often with more in-depth discus-sion.The theory of the intrinsic contribution to the tensor λ abc in Eq. (1) in crystals was developed in Refs. 22–26,often under the disguise of considering the chiral mag-netic effect at finite frequencies. It was realized that theintrinsic contribution was related to the so-called intrin-sic orbital magnetic moment of quasiparticles, m int p : m int p = i (cid:126) e (cid:104) ∂ p u p | × ( h p − (cid:15) p ) | ∂ p u p (cid:105) , (4)where | u p (cid:105) is the periodic part of the Bloch wave func-tion (band index suppressed, only quasimomentum p shown). This magnetic moment determined λ abc via thegyrotropic tensor g ab dual to the latter, λ abc = (cid:15) abd g dc ,where g ab = e ( ω + iτ ) (cid:90) ( d p )( m int p a ∂ b f p − δ ab m int p · ∂ p f p ) . (5)Here τ is the transport scattering time, f p is the equi-librium distribution function of the electrons, ∂ a is thederivative with respect to the a th component of thequasimomentum, and ( d p ) ≡ d p/ (2 π (cid:126) ) .In this paper, we show that in the presence of disor-der, the tensor λ abc still stems from an appropriately de-fined magnetic moment of quasiparticles, which appears only under nonequilibrium conditions. There are twocontributions to such kinetic magnetic moment: skewscattering from impurities, and the coordinate shiftupon impurity scattering (the “side jump” effect ).First, we present the final expression for these magneticmoment contributions, and then discuss the quantitiesthat enter into them. The skew scattering contributionis given by m sk p = eτ (1 − iωτ ) Q sk p × ∂ p (cid:15) p , (6)and the side jump one is m sj p = eτ − iωτ v sja p × ∂ p (cid:15) p . (7)In these expressions ∂ p (cid:15) p is the usual group velocity ofelectrons in a given band, Q sk p we will refer to as skewacceleration, and v sja p is the side jump accumulation ve-locity familiar from the anomalous Hall effect context(see Ref. 6 for a review). The expressions for Q sk p and v sja p involve the symmetric and antisymmetric parts ofthe impurity scattering probability from p (cid:48) to p , w S,A pp (cid:48) ,see Eqs. (32) and (33), as well as the coordinate shiftof the electron in the same transition, δ r pp (cid:48) , Eq. (36).Specifically, Q sk p = (cid:90) ( d p (cid:48) ) w A pp (cid:48) δ ( (cid:15) p − (cid:15) p (cid:48) )( ∂ p (cid:15) p − ∂ p (cid:48) (cid:15) p (cid:48) ) , (8)and v sja p = (cid:90) ( d p (cid:48) ) w S pp (cid:48) δ r pp (cid:48) δ ( (cid:15) p − (cid:15) p (cid:48) ) . (9)The physical meaning of Q sk p and v sja p is clear from theirdefinitions: v sja p is a disorder-induced velocity of a wavepacket due to the accumulation of the side jump events;in turn, Q sk p is a disorder-induced acceleration of a wavepacket due to the velocity change accumulation uponskew scattering.These considerations make apparent the origin of ex-trinsic contributions to the magnetic moments, Eqs. (6)and (7): the product of Q sk p and min[1 /ω, τ ] , as well asthe product v sja p min[1 /ω, τ ] have the meaning of an aver-age displacement of a charge carrier due to accumulationof either skew scattering, or side jump events. The crossproducts of these quantities with the group velocity ∂ p (cid:15) p of the carrier determine the contribution of the extrinsiceffects to the effective “kinetic” magnetic moment of aquasiparticle.Finally, introducing the total magnetic moment m tot p = m int p + m sk p + m sj p , the gyrotropic tensor is writtenas g ab = e ( ω + iτ ) (cid:90) ( d p )( m tot p a ∂ b f p − δ ab m tot p · ∂ p f p ) . (10)Equations Eqs. (6), (7), and (10) are the main resultsof this work. Using these findings, we will describe thephenomena of natural optical activity and kinetic mag-netoelectric effect in metals in Sections V B and V C. FIG. 1: (Color online) Tellurium-type crystal structure.Atomic helices wind around the vertical edges of the hecago-nal prism representing the unit cell of the crystal.
A. Estimates for a typical helical metal
We conclude this Section with a brief qualitative dis-cussion of the origin of the skew scattering and side jumpeffects in a typical helical metal. To have a clear physicalpicture, as well as an opportunity to apply the obtainedresults to a concrete system, we consider a specific crys-tal structure of the hexagonal Tellurium, Fig. 1, which isone of the most important enantiomorphic crystals.We first consider skew scattering. It is well known that in the case of a free electron with spin s scattering offa spin-orbit coupled impurity, the scattering probabilitycontains an antisymmetric contribution ω A pp (cid:48) ∝ s · p × p (cid:48) .For an electron in a noncentrosymmetric crystal, in whichthe band structure is spin-split, one faces the questionof what plays the role of the spin s . Fig. 1 helps an-swer it: it is clear that an electron propagating alongthe c axis – that is, along the helices – obtains a mag-netic moment along the helix, which is proportional tothe momentum along the helix, at least for small mo-menta, m p ∝ p z z . It is then plausible to assume that m p plays the role of s in this situation, and the properlysymmetrized antisymmetric probability has the form of ω A pp (cid:48) ∝ ( p z + p (cid:48) z ) z · p × p (cid:48) . This form satisfies the time-reversal condition ω A p , p (cid:48) = ω A − p (cid:48) , − p , and is in fact dictatedby the symmetry of the lattice. It can also be confirmedwith a microscopic calculation .To calculate the skew acceleration, we assume anisotropic spectrum with a density of states ν p , and in-troduce the skew scattering time τ sk via ω A pp (cid:48) = 1 ν p τ sk ( z · e p + z · e p (cid:48) ) z · e p × e p (cid:48) , (11)where e p is the unit vector along p . Then we obtain forthe skew acceleration Q sk p = − ∂ p (cid:15) p τ sk ( z · e p ) z × e p , (12)and the skew scattering contribution to the magnetic mo-ment m sk p = 13 eτ (1 − iωτ ) ( ∂ p (cid:15) p ) τ sk ( z · e p ) e p × z × e p . (13) We will calculate the corresponding contribution to thegyrotropic tensor at zero temperature, and will use v F and ν F denote the Fermi velocity and the density of statesat the Fermi level, respectively. Then g sk ab = 145 e τ i (1 − iωτ ) ν F v F τ sk − . (14)Note that the tracelessness of the gyrotropic tensor is ageneral property of the extrinsic contributions describedabove (see the end of Section V D for further details).As far as the side jump mechanism is concerned, thereexist extensive discussions of the physics behind the sidejump effect itself , which we will not repeat here.To estimate the side jump contribution to the gyrotropictensor, we consider a weak spherically-symmetric impu-rity potential, which is smooth on the scale of the lat-tice spacing, producing mostly small-angle scattering. Inthis case the side jump length is independent of the spe-cific form of the potential, and is fully determined by theBerry curvature of the band, Ω p : Ω p = i (cid:126) (cid:104) ∂ p u p | × | ∂ p u p (cid:105) . (15)The coordinate shift δ r pp (cid:48) is given by δ r pp (cid:48) = Ω p + p (cid:48) × ( p − p (cid:48) ) . (16)We take Ω p ∝ p z z as appropriate for the upper valenceband of Tellurium to obtain δ r pp (cid:48) = a sj ( z · e p + z · e p (cid:48) ) z × ( e p − e p (cid:48) ) . (17)In the above expression a sj sets the overall scale of theside jump magnitude. Given the qualitative nature ofpresent considerations, we calculate the side jump accu-mulation velocity setting w S pp (cid:48) = 1 /ν p τ : v sja p = a sj τ ( z · e p ) z × e p . (18)The expression for the side jump magnetic moment is m sj p = − ea sj ∂ p (cid:15) p − iωτ ( z · e p ) e p × z × e p , (19)and the contribution to the gyrotropic tensor is g sj ab = − e τi (1 − iωτ ) a sj ν F v F − . (20)Finally, the intrinsic contribution comes from the in-trinsic magnetic moment, which for small p z can be writ-ten as m int p = µ int z ( z · e p ) . (21)The intrinsic part of the gyrotropic tensor is thus g int ab = − eµ int τi (1 − iωτ ) ν F v F . (22)Eqs. (14), (20), and (22) allow to draw some general con-clusions about the contributions of extrinsic effects tovarious physical effects. For instance, the polarization ro-tation and circular dichroism for light propagating alongthe optic axis of the crystal (cf. Section V B) are deter-mined by g zz . It is clear from Eq. (22) that at frequen-cies small compared to relevant band splittings, wherethe current theory is applicable, these effects are domi-nated by the extrinsic effects, regardless of the mobilityof a sample. This should be contrasted with the caseof optical frequencies , in which g zz is fully determinedby the band structure. On the other hand, both extrin-sic and intrinsic effects contribute to the current-inducedmagnetization (Section V C).It is also interesting to compare the typical values ofmatrix elements of the gyrotropic tensor that come fromvarious mechanisms. For ωτ (cid:28)
1, one has g sk ∼ e τ ν F v F τ sk ∼ e (cid:126) k F (cid:96) ττ sk , (23a) g sj ∼ e τ a sj ν F v F ∼ e (cid:126) k F (cid:96)a sj , (23b) g int ∼ eµ int τ ν F v F ∼ e (cid:126) k F (cid:96) µ int ev F . (23c)The first similarity sign (“ ∼ ”) pertains to the consid-ered simplified model of a helical metal; the expressionsfollowing the second one, expressed through the Fermiwave vector k F , and the elastic mean free path (cid:96) holdmore generally, and allow a comparison of extrinsic andintrinsic mechanism in various systems. Such compar-ison will be carried out in more detail in Sections V Band V C. Here we only note that for clean systems withlarge enough value of k F (cid:96) , skew scattering always domi-nates the physical phenomena related to the nonlocalityof the conductivity tensor.What is interesting in the present case is the factthat the different frequency dependence of the extrin-sic and intrinsic contribution allows the intrinsic onesto overpower skew scattering (the side jump contribu-tion appears to be small in all realistic examples ) atfrequencies still within the applicability range of thetheory. Consider, for instance, the case of a Weylsemimetal, in which µ int ∼ ev F /k F , and the intrinsiceffects are large. Comparing Eq. (14) and Eq. (22), andusing Eqs. (23a) and (23c), one can easily see that for ω (cid:29) ω ≡ (cid:112) k F (cid:96)/τ τ sk the intrinsic effects dominatethe skew scattering. The crossover frequency ω satisfies (cid:126) ω /(cid:15) F ∼ (cid:112) (cid:126) /(cid:15) F τ sk (cid:28)
1, hence the crossover regime lieswell within the applicability region of the present theory.
III. SEMICLASSICAL TRANSPORT THEORY
In this Section, we lay out the framework to describethe response of a crystal to a weak electromagnetic per-turbation varying slowly in space and time, which can be described with an effective single-band Boltzmann equa-tion. In this equation, the effective velocity of quasiparti-cles, the force that acts on them, as well as the impuritycollision integral are modified by the effects of interbandcoherence . The goal of this Section is to describe thistype of Boltzmann equation in the context of non-localconductivity calculation in metals.For the clean metal case, a detailed account of thekinetic equation formalism for a non-local conductivitycalculation was laid out in Ref. 24. Here we briefly re-peat the relevant equations, and in addition, introducethe appropriate form of the impurity collision integral totreat disorder.The main results of this Section are Eqs. (38), (39), and(40) for contributions to the impurity collision integral inthe Boltzmann equation, as well as Eqs. (42), (43), and(44) for the electric current in the presence of impurities.The semiclassical kinetic equation for the distributionfunction f n p of band n and momentum p has the stan-dard form: ∂ t f n p + ˙ r ∂ r f n p + ˙ p ∂ p f n p = I st , (24)where I st is the collision integral.The expressions for the velocity, ˙ r , of a quasiparticle,and its quasimomentum rate of change, ˙ p , are modifiedby the effects of interband coherence. These come fromseveral sources, and are listed below.Interband coherence induced by the acceleration dueto the Lorentz force manifests itself in the appearance ofthe Berry curvature Ω n p of band n . It can be expressedusing the periodic parts of the Bloch wave functions | u n p (cid:105) of band n , and is given by Ω n p = i (cid:126) (cid:104) ∂ p u n p | × | ∂ p u n p (cid:105) . (25)Spatial gradients of the distribution function of band n induce coherence that leads to the appearance of theorbital magnetic moment of quasiparticles , m n p = i (cid:126) e (cid:104) ∂ p u n p | × ( h p − (cid:15) n p ) | ∂ p u n p (cid:105) , (26)where h p is the Bloch Hamiltonian, and (cid:15) n p is the energydispersion of band n .The presence of this magnetic moment modifies theexpression for the band energy dispersion, E n p = (cid:15) n p − m n p B , (27)as well as the band velocity, v n p = ∂ p E n p . (28)Using these quantities, the semiclassical equations ofmotion for band n can be written as ˙ r = v n p − ˙ p × Ω n p , ˙ p = e E + e ˙ r × B . (29)The corresponding standard expressions for the velocityand the momentum rate of change are˙ r = 1 D B ( v n p − e E × Ω n p − e ( v n p · Ω n p ) B ) , ˙ p = 1 D B (cid:0) e E + e v n p × B − e ( E · B ) Ω n p (cid:1) ,D B = 1 − e BΩ n p . (30)The equation for ˙ r contains the group velocity v n p , aswell as the anomalous velocity due to the electric andmagnetic parts of the Lorentz force ; in particular, the last term in brackets is commonly associated with thestatic CME , but plays no role in the presentdiscussion. The equation for ˙ p , besides the usual Lorentzforce, contains an “ E · B ” term, which describes the chiralanomaly at the quasiclassical level. The factor D B is the phase space measure that ap-pears due to variables r and p not being a canonicalpair . It can be shown that the 1 /D B prefactor inthe expressions for the velocities in real and momentumspaces is compensated by D B appearing in the momen-tum integrals over the Brillouin zone, and does not affectthe linear response. Hence, we will omit it from now on.We now turn to the description of the disorder effects. We start with the discussion of the collision integral, whilethe question of velocity renormalization by the disorder is treated below, see Eq. (44).The central quantity describing a collision of an election with an impurity is the T-matrix of the latter, ˆ T imp . Below,the matrix elements of the T-matrix are defined as T n p ,n (cid:48) p (cid:48) = V (cid:104) n p | ˆ T | n (cid:48) p (cid:48) (cid:105) , (31)in which V is the volume of the system, | n p (cid:105) = √ N exp( i pr / (cid:126) ) | u n p (cid:105) are Bloch states normalized to unity, and indicesmust be read from right to left to infer the transition direction.For a randomly distributed ensemble of impurities with impurity concentration n i , the probability of the n (cid:48) p (cid:48) → n p transition is given by W n p ,n (cid:48) p (cid:48) = 2 π (cid:126) V n i | T n p ,n (cid:48) p (cid:48) | δ ( (cid:15) n p − (cid:15) n (cid:48) p (cid:48) ) ≡ V w n p ,n (cid:48) p (cid:48) δ ( (cid:15) n p − (cid:15) n (cid:48) p (cid:48) ) . (32)It is important for us that in a noncentrosymmetric crystal, the transition rate is not symmetric with respect to theinterchange of the initial and final states, W n p ,n (cid:48) p (cid:48) (cid:54) = W n (cid:48) p (cid:48) ,n p . Therefore, w n p ,n (cid:48) p (cid:48) in Eq. (32) can be written as asum of symmetric and antisymmetric components: w n p ,n (cid:48) p (cid:48) = w S n p ,n (cid:48) p (cid:48) + w A n p ,n (cid:48) p (cid:48) , w S,A n p ,n (cid:48) p (cid:48) = ± w S,A n (cid:48) p (cid:48) ,n p , (33)The antisymmetric part w A is responsible for the skew scattering.The conservation of the probability flux in impurity scattering (the optical theorem) imposes a restriction on theasymmetric part of the scattering rate. Indeed, probability conservation implies that the total transition rate out ofa given state is equal to the total transition rate into that state from all other states . In other words, the followingsum rule holds: (cid:88) n p W n p ,n (cid:48) p (cid:48) = (cid:88) n p W n (cid:48) p (cid:48) ,n p . (34)This immediately gives (cid:88) n (cid:90) ( d p ) w A n p ,n (cid:48) p (cid:48) δ ( (cid:15) n p − (cid:15) n (cid:48) p (cid:48) ) = 0 , (35)where ( d p ) = d p/ (2 π (cid:126) ) .Finally, we note that since the position of the center of a wave packet within the unit cell depends on its quasimo-mentum, the sudden change of the quasimonentum upon impurity scattering will lead to a shift in the wave packetlocation. This coordinate shift - commonly called “side jump” - is given for a weak impurity potential V imp by δ r n p ,n (cid:48) p (cid:48) = i (cid:126) (cid:104) u n p | ∂ p u n p (cid:105) − i (cid:126) (cid:104) u n (cid:48) p (cid:48) | ∂ p (cid:48) u n (cid:48) p (cid:48) (cid:105) − (cid:126) ( ∂ p + ∂ p (cid:48) )arg( V impn p ,n (cid:48) p (cid:48) ) . (36)In the presence of electromagnetic fields, one must take into account the work done by the electric field as an electrongets displaced within the unit cell, as well as the change in the Zeeman energy, Eq. (27), during the collision. Theseconsiderations lead to modifications in the energy-conserving δ -function in Eq. (32), and the collision integral canbe generally written as I st = − (cid:88) n (cid:48) (cid:90) ( d p (cid:48) ) D B ( w n (cid:48) p (cid:48) ,n p f n p − w n p ,n (cid:48) p (cid:48) f n (cid:48) p (cid:48) ) δ ( E n p − E n (cid:48) p (cid:48) − e E δ r n p ,n (cid:48) p (cid:48) ) , (37)The phase space measure D B will be omitted from now on, as it does not alter the linear response. We note in passingthat the accumulation of the subsequent side jumps also modifies the wave packet velocity, and this effect is discussedfurther below.To linear order in the electromagnetic fields, the impurity collision integral has four parts, I st = I S + I A + I m + I sj ,where I S,A are symmetric and antisymmetric parts of the usual collision integral, I S,A = − (cid:88) n (cid:48) (cid:90) ( d p (cid:48) ) D B ( w S,A n (cid:48) p (cid:48) ,n p δf n p − w S,A n p ,n (cid:48) p (cid:48) δf n (cid:48) p (cid:48) ) δ ( (cid:15) n p − (cid:15) n (cid:48) p (cid:48) ) , (38)in which δf n p ∼ O ( E , B ); I m is an effective generation term due to the Zeeman energy change during a collision: I m = − ∂ (cid:15) n p f n p (cid:88) n (cid:48) (cid:90) ( d p (cid:48) ) w S p , p (cid:48) ( m int p − m int p (cid:48) ) B δ ( (cid:15) n p − (cid:15) n (cid:48) p (cid:48) ) . (39)Finally, the side jump contribution I sj , which also plays the role of a generation term, is related to the work doneby the electric field as an electron completes a side jump : I sj = − ∂ (cid:15) n p f n p (cid:88) n (cid:48) (cid:90) ( d p (cid:48) ) w S n p ,n (cid:48) p (cid:48) e E δ r n p ,n (cid:48) p (cid:48) δ ( (cid:15) n p − (cid:15) n (cid:48) p (cid:48) ) . (40)For solenoidal fields, the standard interpretation suffers from the fact that the work done by the electric fielddepends on the integration path due to non-zero magnetic field. However, the difference of between the “straightpath” expression, e E δ r pp (cid:48) , and any other path is roughly determined by the time derivative of the magnetic field fluxthrough the scattering region, and is negligible.To complete the kinetic scheme, we have to write down the equation for the electric current. In the present case,there are three contributions, j = j qp + j m + j sj , (41)The first one, j qp , comes from the wave packet velocity of Eq. (30), and is given by j qp = e (cid:88) n (cid:90) ( d p ) (cid:0) ∂ p (cid:15) n p − ∂ p ( m int n p B ) − e E × Ω n p − e ( ∂ p (cid:15) n p · Ω n p ) B (cid:1) f n p . (42)The second term in Eq. (41) is the magnetization current due to the existence of the intrinsic orbital moment ofquasiparticles : j m = ∇ r × (cid:88) n (cid:90) ( d p ) m int n p f n p . (43)Finally, the last contribution to the total current, j sj , is the current due to the side jump accumulation, given by j sj ( ω, q ) = e (cid:88) nn (cid:48) (cid:90) ( d p )( d p (cid:48) ) w S n p ,n (cid:48) p (cid:48) δ r n p ,n (cid:48) p (cid:48) δ ( (cid:15) n (cid:48) p (cid:48) − (cid:15) n p ) f n (cid:48) p (cid:48) . (44)Qualitatively, it comes from the net displacement accumulated as an electron completes a series of side jumps.Formally, it stems from the interband coherence induced by collisions with impurities.The kinetic equation (24), together with the collision integrals (38) and (40), and the expression for the current (41)constitute the full kinetic scheme for the problem at hand.The intrinsic orbital moment of quasiparticles, m int n p , the antisymmetric part of the impurity scattering probabilitydetermined by w A n p .n (cid:48) p (cid:48) , and the side jump displacement, δ r n p ,n (cid:48) p (cid:48) , lead to the intrinsic, skew scattering, and sidejump mechanisms of nonlocal response, respectively. They all stem from the existence of Berry-Pancharatnam phasesin the band structure , and hence are geometric in nature. Since these effects are relatively small, the correspondingcontributions to λ abc , Eq. (1), are additive and can be considered independently, which is done in the subsequentSections. For the sake of brevity, we will suppress the band index n from now on. IV. NONLOCAL TRANSPORT IN METALS
In this Section, we use the formalism described in Section III to calculate tensor λ abc that determines linear-in- q effects of spatial dispersion in the conductivity tensor, Eq. (1). The main results of this Section are Eqs. (53), (61),and (69) for intrinsic, skew scattering, and side jump contributions to tensor λ abc . A. Intrinsic contribution in a disordered metal
We first consider the contribution to λ abc that comes solely from the p -dependence of the periodic parts of Blochfunctions, via the intrinsic orbital moment of quasiparticles, m p . To this end, we neglect the skew scattering andside jump effects by setting w A , δ r →
0, and call this contribution “intrinsic”, even though it does depend on themomentum relaxation time in a band, or w S .Since we are interested in the linear response to an electromagnetic field, it is sufficient to consider monochromaticfields, E , B ∝ exp ( i qr − iωt ). The distribution function response is then also monochromatic, and the kinetic equationhas the following form: − iωf p + i q ∂ p (cid:15) p f p + e E ∂ p f p = I S + I m . (45)The expression for the current contribution which is linear-in- q is j int g ( ω, q ) = e (cid:90) ( d p ) m int p B ∂ p f p + e (cid:90) ( d p ) ∂ p (cid:15) p f Bp + (cid:90) ( d p ) i q × m int p f Ep . (46)The subscript g stands for “gyrotropic”. The first term on the right hand side is the current due to renormalizationof the velocity expectation value for state p in the presence of magnetic field; the second term is the usual ballisticcurrent calculated using the nonequilibrium distribution function, f Bp , due to coupling to a time-dependent magneticfield, and the last term is the nonequilibrium magnetization current, appearing due to nonuniform acceleration ofelectrons by the electric field, described by f Ep . Because we work to linear order in the electric field gradients, theterms in j int g containing q can be dropped next to the magnetic field; the same is true for f Ep , since the magnetizationcurrent contains q in its definition.To proceed, we separate the non-equilibrium part of the distribution function δf p ≡ g p ∂ (cid:15) p f p , and define an integraloperator ˆ L that corresponds to the impurity collision integral via its action on a function of momentum φ p :ˆ L ◦ φ p = (cid:90) ( d p (cid:48) ) w S p , p (cid:48) ( φ p − φ p (cid:48) ) δ ( (cid:15) p − (cid:15) p (cid:48) ) . (47)Using this operator, and working to linear order in the external fields allows to write Eq. (45) as( iω − i q ∂ p (cid:15) p ) g p − ˆ L ◦ g p = e E ∂ p (cid:15) p + ˆ L ◦ m int p B . (48)It can be easily shown that e E ∂ p (cid:15) p and m int p B do not belong to the kernel of ˆ L , and that ˆ L is invertible on thesefunctions. Physically, this stems from the fact that both functions are odd in momenta, and the distribution functionperturbations that they induce can be relaxed by disorder.Then the solution of (48) is g Ep = ( iω − ˆ L ) − ◦ e E ∂ p (cid:15) p ,g Bp = ( iω − ˆ L ) − ◦ ˆ L ◦ m int p B . (49)Using f E , Bp = g E , Bp ∂ (cid:15) p f p , ∂ p (cid:15) p ∂ (cid:15) p f p = ∂ p f p , as well as Faraday’s law B = q × E /ω , the gyrotropic current becomes j int g ( ω, q ) = e (cid:90) ( d p ) ∂ p f p ω + i ˆ L ◦ m int p · q × E + e (cid:90) ( d p ) q × m int p ω + i ˆ L ◦ E ∂ p f p . (50)The corresponding tensor λ abc is given by λ int abc = e(cid:15) dcb (cid:90) ( d p ) ∂ a f p ω + i ˆ L ◦ m int p d + e(cid:15) acd (cid:90) ( d p ) m int p d ω + i ˆ L ◦ ∂ b f p . (51)This expression can be further modified using the obvious symmetry property of operator ˆ L : (cid:90) ( d p ) χ p ˆ L ◦ φ p = (cid:90) ( d p ) φ p ˆ L ◦ χ p , (52)which is inherited by ( ω + i ˆ L ) − . This brings λ int abc to its final form, λ int abc = e (cid:90) ( d p ) m int p d ω + i ˆ L ◦ ( (cid:15) bdc ∂ a (cid:15) p − (cid:15) adc ∂ b (cid:15) p ) ∂ (cid:15) p f p . (53)This form makes it apparent that regardless of the explicit form of w S p , p (cid:48) , the intrinsic contribution to λ abc in adisordered metal satisfies λ int abc = − λ int bac , as required by the Onsager relations. In the relaxation-time approximation,ˆ L ◦ φ p → φ p /τ , Eq. (53) reduces to the result of Ref. 24. B. Skew scattering mechanism
The skew scattering mechanism stems from the asymmetric part of the collision integral, and is determined usingthe following kinetic equation: − iωf p + i q ∂ p (cid:15) p f p + e E ∂ p f p = I S + I A . (54)The gyrotropic current due to this mechanism is given by the usual ballistic current calculated using the linear-in- q part of the solution of this equation, f sk p ( q ): j skg ( ω, q ) = e (cid:90) ( d p ) ∂ p (cid:15) p f sk p ( q ) . (55)Note that even for I A = 0 there exists a linear-in- q correction to the distribution function as determined by Eq. (54),but it is straightforward to show that this part of the distribution function does not make a contribution to thecurrent (55). Thus f sk p ( q ) is understood as the linear-in- q term in the distribution function that exists due to I A (cid:54) = 0.To solve the kinetic equation (54) to linear order in q , we treat I A as a perturbation: we solve the I A = 0 equation,and substitute the corresponding solution into I A , at which point the latter becomes an effective generation term. Toavoid extremely cumbersome expressions, we use the relaxation time approximation for the operator ˆ L that definesthe symmetric part of the collision integral, introduced in Eq. (47) of Section IV A.In the relaxation time approximation, the solution to Eq. (54) with I A = 0 is f p = 1 i (cid:0) ω + iτ − q ∂ p (cid:15) p (cid:1) e E ∂ p (cid:15) p ∂ (cid:15) p f p . (56)Substituting this expression back into I A , and using Eq. (35) to simplify Eq. (54), we obtain f sk p ( q ) = ∂ (cid:15) p f p (cid:90) ( d p (cid:48) ) w A pp (cid:48) δ ( (cid:15) p − (cid:15) p (cid:48) ) (cid:0) ω + iτ − q ∂ p (cid:15) p (cid:1) (cid:0) ω + iτ − q ∂ p (cid:48) (cid:15) p (cid:48) (cid:1) e E ∂ p (cid:48) (cid:15) p (cid:48) , (57)where we used the fact that δ ( (cid:15) p − (cid:15) p (cid:48) ) ∂ (cid:15) p (cid:48) f p (cid:48) = δ ( (cid:15) p − (cid:15) p (cid:48) ) ∂ (cid:15) p f p . Separating the linear-in- q part in f sk p ( q ), andsubstituting it into Eq. (55) for the current, we obtain j skg ( ω, q ) = e ( ω + iτ ) (cid:90) ( d p ) (cid:90) ( d p (cid:48) ) w A pp (cid:48) δ ( (cid:15) p − (cid:15) p (cid:48) ) ( q ∂ p (cid:15) p + q ∂ p (cid:48) (cid:15) p (cid:48) ) ( E ∂ p (cid:48) (cid:15) p (cid:48) ) ∂ p f p . (58)The corresponding contribution to tensor λ abc is λ sk abc = e ( ω + iτ ) (cid:90) ( d p ) (cid:90) ( d p (cid:48) ) w A pp (cid:48) δ ( (cid:15) p − (cid:15) p (cid:48) ) ∂ a (cid:15) p ∂ b (cid:15) p (cid:48) ( ∂ c (cid:15) p + ∂ c (cid:15) p (cid:48) ) ∂ (cid:15) p f p . (59)The antisymmetry of w A pp (cid:48) ensures that this tensor is antisymmetric with respect to the first pair of indices.The result for λ sk abc can be written more compactly if one introduces the “skew acceleration” Q p , which has themeaning of the rate of change of an electron’s velocity due to the skew scattering event accumulation: Q sk p = (cid:90) ( d p (cid:48) ) w A pp (cid:48) δ ( (cid:15) p − (cid:15) p (cid:48) )( ∂ p (cid:15) p − ∂ p (cid:48) (cid:15) p (cid:48) ) . (60)Note that the term with ∂ p (cid:15) p in the round brackets integrates to zero due to the probability flux conservationcondition (35), and has been added for clarity of the physical interpretation. This allows to write finally λ sk abc = e ( ω + iτ ) (cid:90) ( d p ) (cid:0) ∂ a (cid:15) p Q sk p b − Q sk p a ∂ b (cid:15) p (cid:1) ∂ c (cid:15) p ∂ (cid:15) p f p . (61) C. Side jump mechanism
The side jump contribution to the gyrotropic current comes from the following kinetic equation: − iωf p + i q ∂ p (cid:15) p f p + e E ∂ p f p = I S + I sj , (62)supplemented with the expression for the current that contains the contribution from the side jump accumulation: j sjg ( ω, q ) = e (cid:90) ( d p ) ∂ p (cid:15) p f sj p ( q ) + e (cid:90) ( d p ) (cid:90) ( d p (cid:48) ) w S pp (cid:48) δ r pp (cid:48) f Ep (cid:48) ( q ) δ ( (cid:15) p (cid:48) − (cid:15) p ) . (63)Here f sj p ( q ) is the linear-in- q correction to the distribution function due to the presence of side jumps in the externalelectric field, while f Ep ( q ) is the linear-in- q correction to the distribution function in the absence of side jumps.As before, we introduce f Ep ( q ) = g Ep ( q ) ∂ (cid:15) p f p , and f sj p ( q ) = g sj p ( q ) ∂ (cid:15) p f p , and treat the symmetric part of the collisionintegral in the relaxation time approximation. Then for the g ’s we obtain in the standard way g Ep ( q ) = 1 i ( ω + iτ ) ( q ∂ p (cid:15) p )( e E ∂ p (cid:15) p ) , (64)and g sj p ( q ) = 1 i ( ω + iτ ) ( q ∂ p (cid:15) p ) (cid:90) ( d p (cid:48) ) w S pp (cid:48) e E δ r pp (cid:48) δ ( (cid:15) p − (cid:15) p (cid:48) ) . (65)Now we can calculate the gyrotropic part of the current due to the side jump mechanism, Eq. (63). j sjg ( ω, q ) = ei ( ω + iτ ) (cid:90) ( d p ) (cid:90) ( d p (cid:48) ) w S pp (cid:48) ∂ p (cid:15) p ( q ∂ p (cid:15) p )( e E δ r pp (cid:48) ) δ ( (cid:15) p − (cid:15) p (cid:48) ) ∂ (cid:15) p f p + ei ( ω + iτ ) (cid:90) ( d p ) (cid:90) ( d p (cid:48) ) w S pp (cid:48) δ r pp (cid:48) ( q ∂ p (cid:48) (cid:15) p (cid:48) )( e E ∂ p (cid:48) (cid:15) p (cid:48) ) δ ( (cid:15) p − (cid:15) p (cid:48) ) ∂ (cid:15) p f p . (66)By interchanging p and p (cid:48) in the second term, and using δ r p (cid:48) p = − δ r pp (cid:48) , the above expression for the gyrotropiccurrent can be seen to be equivalent to the following contribution to tensor λ abc : λ sj abc = e i ( ω + iτ ) (cid:90) ( d p ) (cid:90) ( d p (cid:48) ) w S pp (cid:48) δ ( (cid:15) p − (cid:15) p (cid:48) ) ( ∂ a (cid:15) p δ r pp (cid:48) ,b − δ r pp (cid:48) ,a ∂ b (cid:15) p ) ∂ c f p , (67)which is obviously antisymmetric with respect to the first pair of indices.The expression for λ sj abc can be written in a more compact form if we introduce the side jump accumulation velocityaccording to v sja p = (cid:90) ( d p (cid:48) ) w S pp (cid:48) δ r pp (cid:48) δ ( (cid:15) p − (cid:15) p (cid:48) ) . (68)This brings λ sj abc to its final form: λ sj abc = e i ( ω + iτ ) (cid:90) ( d p )( ∂ a (cid:15) p v sja p ,b − v sja p ,a ∂ b (cid:15) p ) ∂ c (cid:15) p ∂ (cid:15) p f p . (69)Eqs. (53), (61), and (69) for intrinsic, skew scattering, and side jump contributions to tensor λ abc are one of the0main results of this paper. The physical origin and themeaning of the extrinsic mechanisms are elaborated uponin the next Section. V. PHYSICAL CONSEQUENCES OF THEOBTAINED RESULTS
In this section we discuss the physical content of thetensor λ abc . To this end, we focus on the kinetic magne-toelectric effect, and natural optical activity. The maingoal is to elucidate the relative role of the intrinsic andextrinsic mechanisms of nonlocality. We will show thatskew scattering dominates in clean noncentrosymmetricmetals – a familiar observation from the theory of theAHE. However, there are realistic examples of materi-als (e.g. Tellurium and TaAs-family Weyl semimetals) inwhich intrinsic effects dominate in experimentally avail-able samples.For continuity of the presentation, we defer the ques-tion of symmetry requirements for the physical phenom-ena we discuss till the end of this Section (see Sec-tion V D). A. Gyrotropic tensor
Being antisymmetric with respect to the first pair ofindices, λ abc contains nine independent components, andhence is dual to a second rank pseudotensor: λ abc = (cid:15) abd g dc . (70)In what follows, we call g , defined by Eq. (70), the gy-rotropic tensor. Explicitly, it is given by g ab = 12 (cid:15) cda λ cdb . (71)It is customary and simpler to use the gyrotropic tensorto describe the physical properties of a crystal. The in-trinsic, skew scattering, and side jump contributions to g ab = g int ab + g sk ab + g sj ab can be read off from Eqs. (53), (61)and (69). Using ∂ a (cid:15) p ∂ (cid:15) p f p = ∂ a f p and Eq. (71), weobtain g int ab = e ( ω + iτ ) (cid:90) ( d p )( m p a ∂ b f p − δ ab m p · ∂ p f p ) , (72a) g sk ab = − e ( ω + iτ ) (cid:90) ( d p )[ Q sk p × ∂ p (cid:15) p ] a ∂ b f p , (72b) g sj ab = − e i ( ω + iτ ) (cid:90) ( d p )[ v sja p × ∂ p (cid:15) p ] a ∂ b f p . (72c)The expressions for the intrinsic orbital moment ofquasiparticles, m p , the skew acceleration, Q sk p , andthe side jump accumulation velocity v sja p are given byEqs. (26), (60), and (68), respectively.Eqs. (72) make it obvious that both intrinsic and ex-trinsic contributions to the gyrotropic tensor have very similar structure. This similarity can be further empha-sized by introducing the effective kinetic orbital magneticmoment m kin p : m kin p = eτ (1 − iωτ ) Q sk p × ∂ p (cid:15) p + eτ − iωτ v sja p × ∂ p (cid:15) p , (73)the two terms in which correspond to the skew scatteringand side jump contributions to the orbital moment, seeEqs. (6) and (7) in Section II and the text around themfor further discussion. Using the total magnetic moment m tot p = m int p + m kin p , and noting that Q sk p × ∂ p (cid:15) p · ∂ p f p = 0and v sja p × ∂ p (cid:15) p · ∂ p f p = 0, the entire gyrotropic tensor –the sum of Eqs. (72)–can be written as g ab = e ( ω + iτ ) (cid:90) ( d p )( m tot p a ∂ b f p − δ ab m tot p · ∂ p f p ) . (74)in complete analogy with the intrinsic contribution,Eq. (72a).We note in passing that the kinetic part of the mag-netic moment, Eq. (73), does not make a contribution tothe trace of the gyrotropic tensor,Tr g = − e ( ω + iτ ) (cid:90) ( d p ) m p · ∂ p f p . (75)This can be shown to be a consequence of our restrict-ing treatment of the intrinsic effects to the linear orderin skew scattering probability, and the side jump length.This point is further elaborated upon at the end of Sec-tion V D. B. Natural optical activity and Faraday rotation
One of physical consequences of the linear-in- q spatialdispersion of the conductivity tensor is the phenomenonof natural optical activity, whereby a crystal respondsdifferently to right- and left- circularly polarized light.The difference in real parts of the refractive indices re-sults in polarization rotation upon transmission througha sample, and the difference in the imaginary parts, andhence absorption coefficients, leads to the circular dichro-ism. The goal of the present Section is to consider thecontribution of extrinsic effects to the natural optical ac-tivity in metals. The intrinsic ones have been previouslystudied in Refs. 24,25.Propagation of electromagnetic waves through a crys-tal is governed by the equivalent dielectric tensor ε ab ( ω, q ) that corresponds to the conductivity tensor (1): ε ab ( ω, q ) = δ ab + iωε σ ab ( ω, q ) . In general, in an anisotropic crystal the combined effectof optical activity and birefringence leads to tedious con-siderations of wave propagation. All the details can befound in textbooks . Here, we restrict ourselves to the1simple case of propagation along the optic axis of a uni-axial crystal with point group D . We did not choose amore symmetric group (like O ), in which the gyrotropictensor would be proportional to the unit tensor, sincethe extrinsic part of the gyrotropic tensor in Eq. (74) istraceless, and hence vanishes in isotropic crystals.In a crystal with D point group, the gyrotropic andlocal conductivity tensors are diagonal:[ g, σ ] = [ g, σ ] xx g, σ ] xx
00 0 [ g, σ ] zz (76)For propagation along the optic axis ( z -axis), the waveequation reads (cid:20) q (cid:18) δ ab − q a q b q (cid:19) − ω c ε ab ( ω, q ) (cid:21) E b = 0 , (77)and the electric field of the wave has only x- and y-components. The eigenvectors of the 2 × , ± i ) T ( T - transposition), and correspondto the left and right circular polarizations of light. For agiven ω , the solutions for the corresponding wave num-bers are given to linear order in g zz by q L,R = ωc (cid:115) iσ xx ( ω ) ωε ∓ µ ωg zz . Half of the difference between these wave numbers de-fines the complex rotatory power of the crystal, observedin Faraday rotation for transmitted light: ρ = 12 µ ωg zz . (78)To estimate the rotatory power due to the extrinsiceffects, we make several assumptions. First, we assumethat the skew scattering mechanism dominates over theside jump one, as is common under realistic circum-stances. Second, we consider a metal with Fermi en-ergy (cid:15) F , Fermi velocity v F , Fermi momentum p F , ef-fective mass m , and the density of states at the Fermilevel ν F ∼ mp F / (cid:126) . Note that the following estimatesalso work for Weyl metals, in which case one should set m ∼ p F /v F . Third, we introduce the skew scatteringtime τ sk via the typical value of the skew acceleration,Eq. (60), as Q sk ∼ v F /τ sk . Finally, in order to avoid thenecessity to discriminate between the reactive and dissi-pative effects, we consider ω ∼ /τ , where these effectsare of the same order. Then Eq. (72b) yields g zz ∼ e ( ω + iτ ) ν F v F τ sk , (79)and the corresponding contribution to the rotatory powerat ω ∼ /τ becomes ρ ∼ cτ e ε (cid:126) c ττ sk ( k F (cid:96) ) , (80) where c is the speed of light. For k F (cid:96) ∼ τ /τ sk ∼ . τ ∼ ρ ∼ . . C. Kinetic magnetoelectric effect and thecurrent-induced magnetization
Magnetoelectric phenomena are usually discussed inthe context of materials with broken time-reversal sym-metry, in which electric fields cause magnetic response,and vice versa. However, even in time-reversal invari-ant (yet noncentrosymmetric) crystals the response of themagnetization to an electric field is possible. In that case,different time-reversal parity of quantities in the left andright hand sides of a linear relationship M ∝ E impliesthat there must be a dissipative process underlying theresponse, that is, the current flow. Therefore, such ki-netic magnetoelectric response in time-reversal sys-tems can also be viewed as the phenomenon of current-induced magnetization. Below we will show that the dclimit of such a response is completely determined by thetensor λ abc , or equivalently, the gyrotropic tensor.Physically, such a conclusion is based on the expres-sion (74) for the total gyrotropic tensor: its similarity tothe expression for the intrinsic contribution, Eq. (72a),led us to identify the right hand side of Eq. (73) as an ef-fective kinetic magnetic moment of quasiparticles. If thisidentification is correct, one expects that the current-induced magnetization will be related to a finite densityof the total magnetic moment, given by the sum of theintrinsic, Eq. (26), and extrinsic, Eq. (73), contributions.Below, by arriving at this conclusion from the macro-scopic electrodynamics point of view, we will confirm theinterpretation of Eq. (73) once again.To determine the magnetization response to a dc trans-port current, we consider the “ q → ω → , while the quadrupolar and higherorder polarizations, of both electric and magnetic char-acter, can be neglected for the purpose of consideringlinear-in- q spatial dispersion effects.The general response of polarization and magnetiza-tion to electromagnetic fields is written as P a ( ω, q ) = χ e ab ( ω ) E b ( ω, q ) + iχ em ab ( ω ) B b ( ω, q ) , (81a) M a ( ω, q ) = − iχ me ab ( ω ) E b ( ω, q ) + χ m ab ( ω ) B b ( ω, q ) . (81b)We neglected the q dependence of the response tensors χ e , m , em , me : this approximation is sufficient to discuss theeffects of spatial dispersion to linear order in q . The mag-netoelectric susceptibility χ me ab describes the magnetiza-tion response to a transport electric field, the so-calledkinetic magnetoelectric effect . Our theory allows to2compare the intrinsic and extrinsic contributions to suchcurrent-induced magnetization.To relate χ me ab to the gyrotropic tensor g ab , we notethat the gyrotropic current is the linear-in- q part of themacroscopic electric current, j = − iω P ( ω, q ) + i q × M ( ω, q ), given by j g ,a = ( χ em ad (cid:15) dcb + (cid:15) acd χ me db ) q c E b . (82)Combined with the Onsager symmetry relation for themagnetoelectric response tensors, χ em ab ( ω ) = χ me ba ( ω ) , (83)Eq. (82) gives the following expression for λ abc in Eq. (1): λ abc = ( (cid:15) acd χ me db − (cid:15) bcd χ me da ) . (84)This allows to express the gyrotropic tensor via themangetoelectric one: g ab = 12 (cid:15) cda λ cdb = χ me ab − δ ab Tr χ me . (85)Conversely, using that Tr g = − χ me , one obtains χ me ab = g ab − δ ab Tr g. (86)This expression solves the problem of expressing themagnetoelectric susceptibility through the microscopi-cally calculated gyrotropic tensor. We note that foran isotropic system g ab = gδ ab , and Eq. (86) yields χ me ab = − gδ ab , in agreement with previously knownresults .One can make another step to bring out the essence ofthe phenomenon. To this end, we use Eq. (74) for thegyrotropic tensor to obtain χ me ab = e ( ω + iτ ) (cid:90) ( d p ) m tot p a ∂ b f p , which gives the following expression for the magnetiza-tion: M = (cid:90) ( d p )( m int p + m kin p ) e E · ∂ p f p ( iω − τ ) . (87)The meaning of this expression is obvious: the secondfactor inside the momentum space integral is the changein the electronic distribution function under the action ofan electric field, and the term in the brackets is the effec-tive magnetic moment of quasiparticles, which includesboth the intrinsic part, m int p of Eq. (26), and extrinsic- kinetic - contribution, m kin p of Eq. (73). This againconfirms our interpretation of Eq. (73).A detailed study of the current-induced magnetizationin three-dimensional metals will be presented in a sep-arate work , but it is obvious from Eq. (73) that in aclean enough sample the skew scattering contribution tothe kinetic momentum will always dominate the effect, since it is inversely proportional to the impurity concen-tration, unlike the intrinsic and side-jump contributions,which are independent of the impurity concentration.The phenomenon of current-induced magnetiza-tion has mostly been considered in two-dimensionalsystems , where it is of spin origin, simply becausereduced dimensionality does not allow electrons to “or-bit” around the direction of the current flow. For three-dimensional helical metals, there are theoretical stud-ies of the intrinsic mechanism (both of orbital and spincharacter) of the current-induced magnetization. On theexperimental side, the intrinsic contribution to current-induced magnetization was considered in Ref. 36 in Tel-lurium. Here, in order to have a rough estimate of the rel-ative magnitudes of extrinsic and intrinsic contributions,we present a comparison of intrinsic and skew scatter-ing mechanisms of current-induced magnetization in rel-atively low mobility (3300 V/cm s ) Tellurium samplesused in the experiments of Ref. 36.Near the top of the valence band of p-doped Tellurium,the intrinsic magnetic moments of quasiparticles pointalong the optical axis of the crystal, and can be estimatedas m int ∼ e (cid:126) m ∗ βk T ∆ , (88)where m ∗ ∼ . m , m being the bare electron’s mass, β = 2 . × − eV · m is a velocity related to the spin-orbit coupling, ∆ = 126meV is the gap between the twotop valence bands, and k T ∼ m − is the thermal wavenumber at T = 77K.In turn, the skew-scattering part of the kinetic mag-netic moment can be estimated by setting the skew-scattering acceleration to be Q sk ∼ v T /τ sk , where τ sk isthe skew scattering time, and v T is the thermal velocityof carriers. This yields m kin ∼ eτ τ sk v T . (89)Taking the ratio of the last two equations, we obtain m kin m int ∼ ∆ τ (cid:126) m ∗ v T βk T ττ sk . (90)Since the ratio τ /τ sk is independent of the impurity con-centration, and τ is inversely proportional to it, the ra-tio becomes large for clean samples. This, of course, isa manifestation of the general trend that extrinsic ef-fects dominate in clean materials . For samples usedin Ref. 36, τ ∼ × − s was reported, and onecan also estimate τ /τ sk ∼ . − .
01, which makes m kin /m int ∼ . −
1. That is, the role of the kineticmoments is substantial even relatively dirty samples.Note also that for electric fields perpendicular to theoptical axis of a Te crystal, the intrinsic contribution van-ishes identically, yet the extrinsic ones do not. Hence forsuch field directions the extrinsic effects completely de-termine the current-induced magnetization .3We can also present an estimate of the extrinsic contri-bution to the current-induced magnetization in a clean, k F (cid:96) ∼ τ /τ sk ∼ .
01, metal.Using the estimate for the skew contribution to thegyrotropic tensor, Eq. (14), and thetransport electricfield of 10 V/m, we obtain µ M ∼ v F ∼ m/s, then an extrinsic contribution of 5Gauss per 10 V/m of electric field would be equivalentto the Weyl point separation in energy of 0.1eV - a figuretypical for all known examples of time-reversal invariantWeyl metals.
D. Symmetry requirements
To complete the discussion of physical effects relatedto the linear-in- q spatial dispersion of the conductivitytensor, below we list the point groups of metals in whichone expects the phenomena of natural optical activityand current-induced magnetization to happen. Bothrequire broken inversion symmetry, which is necessaryto have a nonzero third rank tensor ( λ abc ). As far asthe current-induced magnetization is concerned, havingnonzero ( λ abc ), or equivalently, g ab tensors is in fact theonly requirement on the crystal symmetry, as followsfrom Eq. (86). Therefore, of the 21 noncentrosymmet-ric point groups (which can be found in Ref. 60, andwill not be listed here), 18 are gyrotropic, and allow atleast current-induced magnetization, but not necessarilynatural optical activity. The three noncentrosymmetricgroups that have zero λ abc are C h , D h , and T d .It is known that only the symmetric part of the gy-rotropic tensor contributes to the natural optical activity.Therefore, besides the nonsymmetricity, one has anothercondition that the gyrotropic tensor g ab has nonzero sym-metric part. Among the 18 gyrotropic groups only 15satisfy this condition, with C v , C v , and C v being theexceptions.In order to have robust dynamic Chiral Magnetic Ef-fect, which survives in polycrystalline samples , the gy-rotropic tensor of a crystal must have a nonzero trace.This further reduces the list of possible groups to 11 chi-ral ones, which describe enantiomorphic crystals. Suchcrystals do not possess any mirror symmetries. Thefour groups that do allow natural optical activity, butdo not have robust dynamic Chiral Magnetic Effect are C h , C v , S , and D d . S is excluded as having a roto-reflection axis, rather than a mirror symmetry.Finally, we would like to mention that the extrinsiccontributions to the gyrotropic tensor calculated in thispaper vanish in cubic crystals with point groups T and O . Indeed, in these groups g ab is proportional to theunit tensor, and since the traces of the extrinsic contri-butions, Eqs. (72b) and (72c), vanish, they in fact vanish identically. This can be traced back to our perturbativetreatment of the skew scattering and side jump processes,and does not depend on the relaxation time approxima-tion used to simplify the results.By going to the next order in w A and δ r pp (cid:48) , one canobtain nonzero extrinsic contributions. This was donefor an isotropic single-band crystal with chiral impuritiesin Ref. 52. In that case, the result for the skew scat-tering contribution, the larger of the two extrinsic ones,contains an additional small factor of τ /τ sk . In realisticenantiomorphic cubic crystals, that contribution is likelyto be dominated by the intrinsic one . By the sameisotropy token, the intrinsic effects will also dominate inpolycrystalline samples. VI. CONCLUSIONS
We have considered the leading nonlocal correctionsto the conductivity tensor in noncentrosymmetric disor-dered metals, which are linear in the wave vector of anelectromagnetic perturbation. Our discussion pertains togeneral multiband systems, but is limited by the frequen-cies which are small compared to all band splittings atthe Fermi surface. The relation between the frequencyand the elastic scattering rate is arbitrary.The main result of this work is the identification of twodisorder-induced contributions to the magnetic momentof charge carriers, which come from the skew scattering,and side jump effects. These contributions appear onlyin the nonequilibrium response to the electromagneticfields (“ q → . ωτ (cid:28)
1, the current-induced magnetization in such a metal is around 5 Gaussper 10 V/m of applied electric field. These figures showthat the extrinsic effects – mostly related to skew scat-tering – are comparable to the intrinsic ones in realisticsamples, and both types of effects are expected to bestraightforward to measure.
Acknowledgments
This work was supported by the National ScienceFoundation Grant No. DMR-1409089.4 L. D. Landau and E. M. Lifshitz,
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