Semiclassical Boltzmann theory of spin Hall effects in giant Rashba systems

Abstract

For the spin Hall effect arising from strong band-structure spin–orbit coupling, a semiclassical Boltzmann theory reasonably addressing the intriguing disorder effect called side-jump has not yet been developed. This paper describes such a theory in which the key ingredient is the spin-current counterpart of the semiclassical side-jump velocity (introduced in the context of the anomalous Hall effect). Applying this theory to spin Hall effects in a two-dimensional electron gas with giant Rashba spin–orbit coupling, largely enhanced spin Hall angle is found in the presence of magnetic impurities when only the lower Rashba band is partially occupied.

Keywords

spin Hall effect / semiclassical Boltzmann theory / side jump / Rashba spin–obit coupling

References

[1] J.Sinova, S. O.Valenzuela,J.Wunderlich, C. H.Back, and T.Jungwirth, Spin Hall effects, Rev. Mod. Phys. 87(4), 1213 (2015)

[2] N.Nagaosa, J.Sinova,S.Onoda, A. H.MacDonald, and N. P.Ong, Anomalous Hall effect, Rev. Mod. Phys. 82(2), 1539(2010)

[3] A. A.Kovalev, J.Sinova, and Y.Tserkovnyak, Anomalous Hall effect in disordered multiband metals, Phys. Rev. Lett. 105(3), 036601(2010)

[4] D.Culcer, E. M.Hankiewicz, G.Vignale, and R.Winkler, Side jumps in the spin Hall effect: Construction of the Boltzmann collision integral, Phys. Rev. B81(12), 125332(2010)

[5] S. A.Yang, H.Pan, Y.Yao, and Q.Niu, Scattering universality classes of side jump in the anomalous Hall effect, Phys. Rev. B83(12), 125122(2011)

[6] N. A.Sinitsyn, Semiclassical theories of the anomalous Hall effect, J. Phys.: Condens. Matter20(2), 023201(2008)

[7] D.Hou, G.Su, Y.Tian, X.Jin, S. A.Yang, and Q.Niu, Multivariable scaling for the anomalous Hall effect, Phys. Rev. Lett. 114(21), 217203(2015)

[8] D.Culcer, A.Sekine, and A. H.MacDonald, Interband coherence response to electric fields in crystals: Berryphase contributions and disorder effects, Phys. Rev. B96(3), 035106(2017)

[9] C.Xiaoand Q.Niu, Semi-classical theory of spin-orbit torques in disordered multiband electron systems, Phys. Rev. B96(4), 045428(2017)

[10] J. M.Ziman, Electrons and Phonons, Oxford: Clarendon, 1960

[11] S.Zhangand Z.Yang, Intrinsic Spin and Orbital Angular Momentum Hall Effect, Phys. Rev. Lett.94(6), 066602(2005)

[12] For instance, one can verify that the semi-classical framework described in Ref. [11] cannot produce any extrinsic contribution to the SHE of the conventional spin current polarized in the zdirection in a disordered Rashba 2DEG [Eqs. (2)–(7) in that paper].

[13] S. V.Eremeev, I. A.Nechaev, Yu. M.Koroteev, P. M.Echenique, and E. V.Chulkov, Ideal two-dimensional electron systems with a giant Rashba-type spin splitting in real materials: Surfaces of Bismuth tellurohalides, Phys. Rev. Lett. 108(24), 246802(2012)

[14] M.Sakano, M. S.Bahramy, A.Katayama,T.Shimojima, H.Murakawa, Y.Kaneko, W.Malaeb, S.Shin, K.Ono, H.Kumigashira, R.Arita, N.Nagaosa, H. Y.Hwang, Y.Tokura, and K.Ishizaka, Strongly spin– orbit coupled two dimensional electron gas emerging near the surface of polar semiconductors, Phys. Rev. Lett. 110(10), 107204(2013)

[15] L.Wu, J.Yang, S.Wang, P.Wei, J.Yang, W.Zhang, and L.Chen, Thermopower enhancement in quantum wells with the Rashba effect, Appl. Phys. Lett. 105(20), 202115(2014)

[16] In the present paper we do not consider side-jump induced by spin–orbit scattering. A semi-classical treatment of this case can be found in: P. M. Levy, H. Yang, M. Chshiev, and A. Fert, Spin Hall effect induced by Bi impurities in Cu: Skew scattering and side-jump, Phys. Rev. B88, 214432(2013)

[17] C.Xiao, D.Li, and Z.Ma, Role of band-indexdependent transport relaxation times in anomalous Hall effect, Phys. Rev. B95(3), 035426(2017)

[18] The reason why the side-jump AHE and SHE induced by band-structure spin–orbit coupling is defined as the sum of these three semi-classical contributions was detailed in Ref. [2]. Simply, there are at least two motivations: one is the equivalence described in Ref. [26] and the other is that all these three contributions belong to the disorder-induced interband-coherence effect (see Refs. [2, 3, 7, 9]).

[19] C.Xiao, D.Li, and Z.Ma, Unconventional thermoelectric behaviors and enhancement of figure of merit in Rashba spintronic systems, Phys. Rev. B93(7), 075150(2016)

[20] H.-Z.Luand S.-Q.Shen, Extrinsic anomalous Hall conductivity of a topologically nontrivial conduction band, Phys. Rev. B88, 081304(R) (2013)

[21] J. I.Inoue, T.Kato, Y.Ishikawa, H.Itoh, G. E. W.Bauer, and L. W.Molenkamp, Vertex corrections to the anomalous Hall effect in spin-polarized two-dimensional electron gases with a Rashba spin-orbit interaction, Phys. Rev. Lett. 97(4), 046604(2006)

[22] C.Grimaldi, E.Cappelluti, and F.Marsiglio, Off-Fermi surface cancellation effects in spin-Hall conductivity of a two-dimensional Rashba electron gas, Phys. Rev. B73, 081303(R) (2006)

[23] P.Wang, Y. Q.Li, and X.Zhao, Nonvanishing spin Hall currents in the presence of magnetic impurities, Phys. Rev. B75(7), 075326(2007)

[24] K.Chadova, S.Wimmer, H.Ebert, and D.Kodderitzsch, Tailoring of the extrinsic spin Hall effect in disordered metal alloys, Phys. Rev. B92(23), 235142(2015)

[25] A.Fertand P. M.Levy, Spin Hall effect induced by resonant scattering on impurities in metals, Phys. Rev. Lett.106(15), 157208(2011)

[26] In the context of AHE induced by band-structure spinorbit coupling, it has been established that the disorderinduced interband-coherence contribution (side-jump) calculated in the semi-classical Boltzmann theory is equivalent to the ladder vertex correction in the weakdisorder limit to the bubble of the anomalous Hall conductivity in the nonchiral basis (s_zbasis for the Rashba model). The present calculations suggest that this equivalence is also valid for the spin Hall conductivity of the conventionally defined spin current. In fact, this equivalence has been employed in the statement of Ref. [1].

[27] W.Kohnand J. M.Luttinger, Quantum theory of electrical transport phenomena, Phys. Rev. 108(3), 590(1957)

[28] I. A.Ado, I. A.Dmitriev, P. M.Ostrovsky, and M.Titov, Anomalous Hall effect with massive Dirac fermions, Europhys. Lett. 111(3), 37004(2015)

[29] J. M.Luttinger, Theory of the Hall effect in ferromagnetic substances, Phys. Rev. 112(3), 739(1958)

[30] J.Shi, P.Zhang, D.Xiao, and Q.Niu, Proper definition of spin current in spin-orbit coupled systems, Phys. Rev. Lett. 96(7), 076604(2006)

[31] N.Sugimoto, S.Onoda, S.Murakami, and N.Nagaosa, Spin Hall effect of a conserved current: Conditions for a nonzero spin Hall current, Phys. Rev. B73(11), 113305(2006)

[32] K.Tsutsuiand S.Murakami, Spin-torque efficiency enhanced by Rashba spin splitting in three dimensions, Phys. Rev. B86(11), 115201(2012)

[33] C.Xiao, D.Li, and Z.Ma, Thermoelectric response of spin polarization in Rashba spintronic systems, Front. Phys. 11(3), 117201(2016)

[34] N.Zhang, Y.Wang, J.Berakdar, and C.Jia, Giant spinorbit torque and spin current generation in carriers at oxide interfaces, New J. Phys. 18(9), 093034(2016)

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