Fingerprints of the electron skew-scattering on paramagnetic impurities in semiconductor systems
FFingerprints of the electron skew-scattering on paramagneticimpurities in semiconductor systems
M.A. Rakitskii, ∗ K.S. Denisov, I.V. Rozhansky, and N.S. Averkiev
Ioffe Institute, 194021 St.Petersburg, Russia (Dated: November 18, 2020)
Abstract
In this paper we argue that the electron skew-scattering on paramagnetic impurities in non-magnetic systems, such as bulk semiconductors, possesses a remarkable fingerprint allowing todifferentiate it directly from other microscopic mechanisms of the emergent Hall response. Wedemonstrate theoretically that the exchange interaction between the impurity magnetic momentand mobile electrons leads to the emergence of an electric Hall current persisting even at zeroelectron spin polarization. We describe two microscopic mechanisms behind this effect, namely theexchange interaction assisted skew-scattering and the conversion of the SHE induced transverse spincurrent to the charge one owing to the difference between the spin-up and spin-down conductivities.We propose an essentially all-electric scheme based on a spin-injection ferromagnetic-semiconductordevice which allows one to reveal the effect of paramagnetic impurities on the Hall phenomena viathe detection of the spin polarization independent terms in the Hall voltage. ∗ [email protected] a r X i v : . [ c ond - m a t . o t h e r] N ov ntroduction. Nowadays, the spin phenomena physics in solids is particularly focused onthe effects arising from the combination of a spin-orbit interaction and magnetism. Thisinterplay lies in the basis of the modern spintronics and has proved to be a source of profoundphysics as well as device applications [1–4]. One example is the physics of the anomalousHall effect (AHE) which remains a vibrant topic since its discovery more than a centuryago. Not only it plays the key role for the electrical detection of the sample magnetization,but also it enriches our understanding of fundamental processes in solids [5]. The modernviewpoint is that the Hall effect measurements constitute a particularly effective tool allowingto probe rather subtle effects connected with both the exchange and spin-orbit interactions.In particular, the formation of magnetic skyrmions due to Dzyaloshinskii-Moriya interactionhas been observed be means of the topological Hall effects [6, 7]. The combined effect of themixed real-momentum space topology is predicted to influence the Hall resistivity [8–10] oftopological systems, such as the magnetic topological insulators [11–13].Various scenarios of AHE and the spin Hall effect (SHE) emergence have been widely dis-cussed, often classified into intrinsic and extrinsic mechanisms [1, 5]. One of the particularlyinteresting extrinsic mechanisms concerns the skew scattering of electrons on paramagneticimpurities embedded in a non-magnetic host. In the mid-70th it was argued to be respon-sible for AHE in dilute magnetic alloys [14–17], where the skew scattering turns out to bestrongly enhanced by the resonant coupling of itinerant electrons with magnetic impuritystates. Despite that AHE in typical ferromagnets is dominated by other mechanisms, theparamagnetic impurities preserve the key significance for AHE in non-magnetic materials.For instance, AHE due to paramagnetic centers has been recently confirmed in non-magneticZnO-based heterostructures [18]. Moreover, the skew scattering on paramagnetic impuritieshas lately experienced a new twist in view of the emergent spin Hall effect. This process isdebated [19] to be the key origin of the strong enhancement of SHE and the giant spin Hallangle observed in gold [20]Despite the high importance of the skew-scattering on paramagnetic centers in the for-mation of both SHE and AHE in non-magnetic systems, no clear physical concept allowingfor a direct experimental verification of this effect has been formulated so far. In this paperwe argue, that the electron scattering on paramagnetic centers has remarkable fingerprintsand can be differentiated from other mechanisms of SHE and AHE in an experiment.Indeed, the conventional mechanisms of AHE or SHE are based on the charge carrier2otion asymmetry due to spin-orbit coupling (SOC), which is sensitive to the carrier spin S . From the symmetry point of view, the skew-scattering leading to SHE can be describedas the contribution to the scattering rate of the form S · ( k × k (cid:48) ), where k , k (cid:48) are the incidentand scattered electron wavevectors, respectively. In fact, the product ( k × k (cid:48) ) responsiblefor the scattering asymmetry and Hall response must go along with some vector of magneticmoment to meet the rime-reversal symmetry requirement. Here S belongs to the incidentelectrons and the skew-scattering probability has opposite asymmetry for spin-up and spin-down electrons leading to SHE. The associated AHE results from conversion of the spincurrent to the charge current due to the spin polarization of the carriers in a magneticsystem. Following this mechanism AHE can emerge only provided that the charge carrierspossess a finite spin polarization; in particular, this conclusion is confirmed experimentallyfor semiconductor systems [21, 22]. However, this physical picture becomes different whena scatterer possesses its own magnetic moment J . Firstly, one can construct the similarcontribution to the skew-scattering rate by substituting the electron spin by that of theimpurity, i.e. J · ( k × k (cid:48) ). The important difference from the previously considered case isthat now the electron spin is decoupled from the quantities defining the asymmetry of thescattering. The fingerprint of this effect is that it leads to the same scattering asymmetryregardless the electron spin state, resulting in a finite AHE response even for completelynon-polarized electron gas. Secondly, the presence of an impurity magnetic moment J renormalizes the transport scattering times for the spin-up and spin-down electrons leadingto τ ↑ (cid:54) = τ ↓ and, consequently, to the spin dependence of the conductivity. As a result, theskew-scattering induced by the common SOC-term S · ( k × k (cid:48) ) leads not only to SHE, butalso induces the transverse charge current already for non-polarized electrons. Hence, if themagnetic impurities play a role in the observed Hall response, the latter should contain acontribution, which is independent of the electron spin polarization. Finding signatures ofthe AHE which remain unchanged upon varying carrier spins would be a direct confirmationof this effect.In this paper we theoretically demonstrate that paramagnetic impurities in typical zinc-blend semiconductors (ZBS) indeed give rise to AHE independent of the electron spin po-larization. We argue that the two microscopic mechanisms underlying this effect, namelythe J · ( k × k (cid:48) ) skew-scattering and the combination of SHE and the spin dependence of theconductivity have the same order of magnitude and should be treated on the equal footing.3oreover, these mechanisms appear to be differently coupled to the s − d and p − d types ofthe exchange interaction. We also discuss how the specific signatures of the skew-scatteringon paramagnetic centers can be revealed experimentally. We suggest to differentiate it fromother AHE mechanisms using a device scheme with the spin injection from a ferromagnetinto a semiconductor. Alternatively, the spin injection into the semiconductor can be doneby means of optical orientation. Microscopic consideration.
Let us consider the scattering of an electron from conductionband states described by Γ representation. The diagonal matrix elements for the core SOCterm in the conduction band are zero, hence, for the Γ band the effect of the SOC appearsdue to an admixture of the SOC split valence band states Γ and Γ via k · p term. Theconductance band wavefunction with account for the admixture of the valence band has theform [23, 24]: Ψ k s = e i kr (cid:16) S + i R ( A k − iB [ ˆ σ × k ]) (cid:17) | χ s (cid:105) ,A = P E g + 2∆3 E g ( E g + ∆) , B = − P ∆3 E g ( E g + ∆) , (1)where S and R = ( X, Y, Z ) are the conduction and the valence band Bloch amplitudes at theΓ–point, | χ s (cid:105) = |↑ , ↓(cid:105) denote the electron spin state, ˆ σ is the vector of Pauli matrices, E g isthe band gap between Γ and Γ bands, the real parameter corresponding to the momentummatrix element is P = ( i (cid:126) /m ) (cid:104) S | p x | X (cid:105) . The core SOC is taken into account via the energysplitting ∆ between Γ and Γ subbands.We model the paramagnetic impurity by a short-range potential acting on electrons bothby means of the electrostatic u ( r ) and the exchange interactions [17]:ˆ V ( r ) = u ( r ) ˆ I + ˆ u X ( r ) J · ˆ σ . (2)The second term describes the effective exchange interaction operator ˆ u X of an itinerantelectron and the impurity d-shell electrons. Since this term is essentially local one shoulddistinguish the matrix elements of ˆ u X for s − d and p − d exchange interaction types [25]. Inwhat follows we assume that some external magnetic field is applied to polarize paramagneticimpurities along z -direction. Given this condition we ignore the possible spin-flip processesfor the impurities assuming J = J e z to be a constant vector directed along z axis. Thescattering rate i.e. the number of transitions from ( k (cid:48) , s (cid:48) ) to ( k , s ) state per second isdetermined by the square modulus of the scattering T -matrix. To account for the skew4cattering in the leading order in the scattering potential it can be presented in the form [26]: | T ss (cid:48) kk (cid:48) | ≈ | V ss (cid:48) kk (cid:48) | + W ss (cid:48) kk (cid:48) , (3) W ss (cid:48) kk (cid:48) = 2 πg (cid:88) µ = ↑ , ↓ (cid:68) Im( V s (cid:48) sk (cid:48) k V sµkq V µs (cid:48) qk (cid:48) ) (cid:69) q , where V ss (cid:48) kk (cid:48) is the matrix element of the impurity potential, and W ss (cid:48) kk (cid:48) = − W s (cid:48) sk (cid:48) k is an asym-metric term responsible for the skew-scattering, here g is the electron density of states atthe Fermi level, (cid:104) . . . (cid:105) q means averaging over the intermediate momentum q directions.The matrix element V ss (cid:48) kk (cid:48) of the potential (2) between ( k (cid:48) , s (cid:48) ) and ( k , s ) states includesthe skew-scattering relevant contributions ( k × k (cid:48) ):ˆ V kk (cid:48) = u ˆ U kk (cid:48) + ˆ M kk (cid:48) , (4)ˆ U kk (cid:48) = 1 + i (2 AB + B )( k × k (cid:48) ) · ˆ σ , ˆ M kk (cid:48) = α ex J · ˆ σ + iβ ex (2 AB − B ) ([ k × k (cid:48) ] · J ) . Here we assumed that the matrix element of the electrostatic (spin-independent) part ofthe impurity potential u = (cid:104) S k | u | S k (cid:48) (cid:105) = (cid:104) X k | u | X k (cid:48) (cid:105) does not depend on either theelectron momentum ( u ( r ) is localized within the Fermi wavelength) or the Bloch amplitude( u ( r ) changes slowly within the unit cell) [24]. The exchange interaction, on the contrary, isdescribed by two parameters α ex = (cid:104) S | ˆ u X | S (cid:105) , β ex = (cid:104) X | ˆ u X | X (cid:105) corresponding to the s − d and p − d exchange coupling, respectively [25]. We assume that the electrostatic part significantlyexceeds the exchange one u (cid:29) u X . Keeping only the leading terms with respect to u X /u we find that the skew-scattering contribution to the scattering rates W ss (cid:48) kk (cid:48) includes two terms: W ss (cid:48) kk (cid:48) = − πg u [ k × k (cid:48) ] z (ˆ σ zss (cid:48) Z + δ ss (cid:48) Z X ) ,Z = u (cid:0) AB + B (cid:1) , Z X = J β ex (cid:0) AB − B (cid:1) + 2 J α ex (cid:0) AB + B (cid:1) . (5)The contribution Z is not aware of the impurity magnetic moment, it is therefore responsiblefor the spin-dependent skew-scattering, described as S · ( k × k (cid:48) ). The contribution Z X , onthe other hand, exists only if the impurity spin is taken into account and describes the spin-independent skew-scattering J · ( k × k (cid:48) ). Note, that the J -dependent skew-scattering appearsin the first order in u X /u . It is important to emphasize, that in this order there existsanother effect eventually leading to the similar spin-independent Hall response. The firstBorn approximation term in the T -matrix, which determines the transport scattering time5ppears to be spin-dependent (see Eqs. 3,4). Consequently, the scattering times (cid:126) / τ ↑ , ↓ = n i πg ( u ± J α ex ) ≈ ( (cid:126) / τ ) · (1 ± J α ex /u ) differ, here we defined the average scatteringtime as (cid:126) / τ = n i πg u , and n i is the impurity concentration. The difference between τ ↑ , ↓ is in the first order in u X /u appears to be∆ τ ≡ | τ ↑ − τ ↓ | ≈ τ (4 α ex J/u ) . (6)Due to this difference a spin Hall current due to the non-magnetic S · ( k × k (cid:48) ) skew-scattering( Z contribution) is converted into transverse electrical current. Anomalous Hall effect.
Let us now proceed with considering the Hall conductivity ofthe non-magnetic ZBS with degenerate electron gas doped by paramagnetic impurities de-scribed by Eq. 2. We assume that the impurity concentration n i is low enough so that theconductance band structure is not affected by the exchange part of the impurity and theelectrons remain non-polarized. Since the asymmetrical part of the scattering cross-sectionleading to the skew-scattering is typically small compared to the total cross-section, thelongitudinal conductivity is given by σ = σ ↑ xx + σ ↓ xx , where σ ↑ , ↓ xx = n ↑ , ↓ e τ ↑ , ↓ /m correspondto the spin-up and spin-down electron conductivities with the transport scattering times τ ↑ , ↓ ; the total electron concentration is n = n ↑ + n ↓ . The associated Hall conductivity foreach spin channel σ ↑ , ↓ H ≡ σ ↑ , ↓ yx can be expressed in terms of the Hall angle σ ↑ , ↓ H = θ ↑ , ↓ H σ ↑ , ↓ xx . Wecalculated the Hall angles on the basis of the Boltzmann kinetic equation and obtained thefollowing expressions: θ ↑ , ↓ H = (Ω X ± Ω ) τ ↑ , ↓ , Ω ,X = Z ,X · π τ g k F , (7)where k F is the Fermi wave-vector, and Ω ,X are the effective cyclotron frequencies, whichaccount for the S · ( k × k (cid:48) ) and J · ( k × k (cid:48) ) skew-scattering terms respectively. The totalanomalous Hall conductivity σ AH in the first order in u X /u can be expressed in the followingway: σ AH = σ ↑ H + σ ↓ H = σ (cid:18) P s · θ + θ X − θ · ∆ ττ (cid:19) , θ ,X ≡ Ω ,X τ, (8)where σ = ne τ /m ≈ σ xx is the total longitudinal conductivity, and P s = ( n ↑ − n ↓ ) /n isthe spin polarization of the electron gas, ∆ τ is defined by Eq. 6. Here the first contributionto σ AH is proportional to the spin polarization P s as it is associated with the spin-dependentskew-scattering ( Z ) and spin to charge transverse current conversion. In non-magnetic sys-tems the equilibrium concentrations of electrons in two spin subbands are equal n ↑ = n ↓ and6 a) (b) FIG. 1: Two mechanisms of the Hall effect due to skew scattering on a paramagneticcenter.this contribution vanishes. The other two contributions are sensitive to the J -dependentscattering. They contribute to the AHE provided the paramagnetic impurities are polarizedby an external magnetic field. The second term σ θ X arises from J · ( k × k (cid:48) ) skew-scattering(Fig. 1 (a)), which is unaware of the electron spin state thus leading to AHE even for un-polarized carriers. The third term stems from the combination of SHE scattering describedby the Hall angle θ and the J -dependent difference between the transport scattering times∆ τ . It can be interpreted as follows: as soon as the J -dependent scattering makes thelongitudinal currents of the spin-up and spin-down electrons different σ ↑ xx (cid:54) = σ ↓ xx , the cor-responding transverse currents arising from S · [ k × k (cid:48) ] skew-scattering and flowing in theopposite direction for spin-up and spin-down electrons become also different by their mag-nitude, thus restoring the transverse electric current, as schematically illustrated in Fig. 1(b). Essentially, these mechanisms contribute with the same order of magnitude to σ AH : σ AH = σ (cid:18) θ X − θ ∆ ττ (cid:19) = σ · J (cid:104) β ex (cid:0) AB − B (cid:1) − α ex (cid:0) AB + B (cid:1)(cid:105) · π g k F , (9)so they are equally important for producing AHE in a non-magnetic media. Note, how-ever, that the spin dependence of the conductivity is sensitive only to the s − d exchangeinteraction constant α ex , while the J -dependent skew-scattering is also affected by p − d cou-pling β ex . The theoretical considerations given above naturally suggest that contribution toAHE driven by paramagnetic impurities should exist even for zero electron spin polarizationand exhibit the magnetic-field dependence following the Brillouin function for the impuritymagnetic moments, similarly to the recently observed AHE in ZnO-systems [18]. Setting with nonequilibrium spins.
The skew-scattering on paramagnetic impurities can7 a) (b)
FIG. 2: Illustration of an experiment with a spin injection from FM to ZBS (a),polarization dependence of the magnetic dopant and injected electrons on an externalmagnetic field (b)be directly differentiated from other spin-dependent mechanisms of the Hall response whenoperating with nonequilibrium spins. Here we suggest an experimental setting shown inFig. 2 (a), which consists of a ferromagnetic injector in contact with ZBS doped with para-magnetic impurities. The Hall contacts are placed at the distance L away from the injector.We assume that a longitudinal current j x is injected from FM and flows across the semicon-ductor layer. This current carries non-equilibrium spin polarization P s , which is maintaineddue to the difference in the quasichemical potentials of the injected spin-up and spin-downelectrons µ s ≡ ( µ ↑ − µ ↓ ) /
2. The spin polarization is partially lost as the injected electronsreach the Hall contacts: P s ( L ) ∝ µ s (0) e − L/L s , here L s is the spin diffusion length, and µ s (0)is the spin accumulation at the interface between FM and ZBS. According to the spin injec-tion theory [27], µ s (0) is proportional to the total electric current flowing across the interface µ s (0) = − j x α s L s σ − where σ is the ZBS conductivity and α s is a material parameter de-scribing the efficiency of spin injection through the interface. The total longitudinal current j x = j ↑ x + j ↓ x = σ ↑ xx ∇ µ ↑ + σ ↓ xx ∇ µ ↓ , is accompanied by the appearance of the Hall currentscarried by the spin-up and spin-down electrons j ↑ , ↓ H = j ↑ , ↓ x θ ↑ , ↓ H , here θ ↑ , ↓ H are the correspondingHall angles, see Eq. 7. The condition of zero Hall electric current j H ↑ + j H ↓ + σ xx E H = 0 leadsto appearance of the transverse electric field E H ≡ ρ H j x , where for the Hall resistivity ρ H in the lowest order in u X /u we obtain: ρ H = ρ (cid:18) θ · α s e − L/L s + θ X − θ ∆ ττ (cid:19) , ρ = σ − . (10)Here the first term stems from the conversion of spin current to electrical current due to8he nonequilibrium spin polarization. The second and third terms reveal the presence ofmagnetic impurity induced skew-scattering and does not depend on spin polarization.Let us see how the two contributions to the Hall voltage can be separated using such aspin injection-based device. By applying a perpendicular magnetic field as shown in Fig. 2(a) the impurities get polarized. Also, the external magnetic field controls magnetization ofthe ferromagnet injector and, hence, the injected electrons. Note, that due to the hysteresisof the ferromagnet there is a range of the magnetic fields where the spins of the impuritiesremain oriented in the fixed direction while the magnetization of the ferromagnet gets flipped,see Fig. 2 (b). The flip of the magnetization reverses the sign of the injected electrons spinpolarization α s → − α s and thus inverts θ driven contribution to the Hall voltage. Takingthe difference between the Hall voltages at the two sides of the magnetization reversal point( ρ
Let us estimate the magnitude of the Hall angle θ H = σ AH /σ for typical ZBS due to zero spin polarization contributions to the AHE. We take the Fermienergy E F = 100 meV and get θ H ≈ · − for GaAs doped by Mn (∆ = 0 .
346 eV, N α ex = 0 . N β ex = − . θ H ≈ · − for CdTe also with Mn impurity(∆ = 0 .
95 eV, N α ex = 0 .
22 eV, N β ex = − .
88 eV [25]). The obtained value is aboutan order of magnitude smaller than the typical magnitude of the AHE angles observed invarious semiconductors [22, 29, 30]. The Hall angle magnitude can be further increased forsystems with smaller band gap, such as InAs or InSb, or due to a resonant state at theparamagnetic dopant. In II-VI dilute magnetic systems the developed theory can be appliedto the electron skew-scattering on bound magnetic polarons. In this case the effective polaronspin J can be up to three orders of magnitude larger leading to a strong enhancement ofthe spin-independent contributions to AHE.In summary, we have demonstrated that the combined action of SOC for the electronsin the conductance band and the exchange interaction with paramagnetic scatterers lead tothe emergence of the electrical Hall current even for zero electron spin polarization. We havedescribed two microscopic mechanisms behind this effect, namely the exchange interaction9ssisted skew-scattering and the conversion of the transverse spin current into the electricalone due to spin dependence of the scattering time. The proposed effect can be directlychecked in experiments by observing the corresponding spin-independent contribution tothe Hall voltage. Acknowledgements.
Authors acknowledge the financial support from from Russian Sci-ence Foundation project 17 − − − − [1] M. Dyakonov, in Spin Physics in Semiconductors (Springer, 2008).[2] J. Sinova and I. ˇZuti´c, Nature materials , 368 (2012).[3] I. ˇZuti´c, J. Fabian, and S. D. Sarma, Reviews of modern physics , 323 (2004).[4] S. D. Bader and S. S. P. Parkin, ARCMP , 71 (2010).[5] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. ,1539 (2010).[6] A. Neubauer, C. Pfleiderer, B. Binz, A. Rosch, R. Ritz, P. G. Niklowitz, and P. B¨oni, Phys.Rev. Lett. , 186602 (2009).[7] M. Raju, A. Yagil, A. Soumyanarayanan, A. Tan, A. Almoalem, F. Ma, O. Auslaender, andC. Panagopoulos, Nature Communications , 696 (2019).[8] F. R. Lux, F. Freimuth, S. Bl¨ugel, and Y. Mokrousov, Phys. Rev. Lett. , 096602 (2020).[9] C. Franz, F. Freimuth, A. Bauer, R. Ritz, C. Schnarr, C. Duvinage, T. Adams, S. Bl¨ugel,A. Rosch, Y. Mokrousov, and C. Pfleiderer, Phys. Rev. Lett. , 186601 (2014).[10] H. Ishizuka and N. Nagaosa, Science Advances , eaap9962 (2018).[11] K. Yasuda, R. Wakatsuki, T. Morimoto, R. Yoshimi, A. Tsukazaki, K. Takahashi, M. Ezawa,M. Kawasaki, N. Nagaosa, and Y. Tokura, Nature Physics , 555 (2016).[12] Y. Tokura, K. Yasuda, and A. Tsukazaki, Nature Reviews Physics , 126 (2019).[13] C.-Z. Chang, J. Zhang, X. Feng, J. Shen, Z. Zhang, M. Guo, K. Li, Y. Ou, P. Wei, L.-L.Wang, et al. , Science , 167 (2013).[14] A. Fert and O. Jaoul, Phys. Rev. Lett. , 303 (1972).[15] A. Fert and A. Friederich, Physical Review B , 397 (1976).
16] A. Fert, A. Friederich, and A. Hamzic, Journal of Magnetism and Magnetic Materials ,231 (1981).[17] B. Giovannini, Journal of Low Temperature Physics , 489 (1973).[18] D. Maryenko, A. Mishchenko, M. Bahramy, A. Ernst, J. Falson, Y. Kozuka, A. Tsukazaki,N. Nagaosa, and M. Kawasaki, Nature communications , 1 (2017).[19] G.-Y. Guo, S. Maekawa, and N. Nagaosa, Phys. Rev. Lett. , 036401 (2009).[20] T. Seki, Y. Hasegawa, S. Mitani, S. Takahashi, H. Imamura, S. Maekawa, J. Nitta, andK. Takanashi, Nature materials , 125 (2008).[21] J. N. Chazalviel and I. Solomon, Phys. Rev. Lett. , 1676 (1972).[22] J. Cumings, L. S. Moore, H. T. Chou, K. C. Ku, G. Xiang, S. A. Crooker, N. Samarth, andD. Goldhaber-Gordon, Phys. Rev. Lett. , 196404 (2006).[23] E. L. Ivchenko and G. Pikus, Superlattices and other heterostructures: symmetry and opticalphenomena , Vol. 110 (Springer Science & Business Media, 2012).[24] V. N. Abakumov and I. N. Yassievich, Soviet JETP , 1375 (1972).[25] J. A. Gaj and J. Kossut, Introduction to the Physics of Diluted Magnetic Semiconductors (Springer, 2010).[26] N. A. Sinitsyn, Journal of Physics: Condensed Matter , 023201 (2007).[27] J. Fabian, A. Matos-Abiague, C. Ertler, P. Stano, and I. ˇZuti´c, Acta Physica Slovaca. Reviewsand Tutorials , 565 (2007).[28] T. Dietl and H. Ohno, Reviews of Modern Physics , 187 (2014).[29] L. N. Oveshnikov, V. A. Kulbachinskii, A. B. Davydov, B. A. Aronzon, I. V. Rozhansky, N. S.Averkiev, K. I. Kugel, and V. Tripathi, Scientific Reports , 17158 (2015).[30] N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. ,1539 (2010).,1539 (2010).