J. Phys. Soc. Jpn. 86, 094704 (2017) [5 Pages]
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Received May 8, 2017; Accepted July 10, 2017; Published August 17, 2017
We investigate the spin Hall effect (SHE) of electrons described by the Dirac equation, which is used as an effective model near the L-points in bismuth. By considering short-range nonmagnetic impurities, we calculate the extrinsic as well as intrinsic contributions on an equal footing. The vertex corrections are taken into account within the ladder type and the so-called skew-scattering type. The intrinsic SHE which we obtain is consistent with that of Fuseya et al. [J. Phys. Soc. Jpn. 81, 093704 (2012)]. It is found that the extrinsic contribution dominates the intrinsic one when the system is metallic. The extrinsic SHE due to the skew scattering is proportional to Δ/niu, where 2Δ is the band gap, ni is the impurity concentration, and u is the strength of the impurity potential.
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References
- 1 J. Sinova, S. O. Valenzuela, J. Wunderlich, C. H. Back, and T. Jungwirth, Rev. Mod. Phys. 87, 1213 (2015). 10.1103/RevModPhys.87.1213 Crossref, Google Scholar
- 2 N. Nagaosa, J. Sinova, S. Onoda, A. H. MacDonald, and N. P. Ong, Rev. Mod. Phys. 82, 1539 (2010). 10.1103/RevModPhys.82.1539 Crossref, Google Scholar
- 3 S. Murakami, N. Nagaosa, and S.-C. Zhang, Science 301, 1348 (2003). 10.1126/science.1087128 Crossref, Google Scholar
- 4 J. Sinova, D. Culcer, Q. Niu, N. A. Sinitsyn, T. Jungwirth, and A. H. MacDonald, Phys. Rev. Lett. 92, 126603 (2004). 10.1103/PhysRevLett.92.126603 Crossref, Google Scholar
- 5 L. Berger, Phys. Rev. B 2, 4559 (1970). 10.1103/PhysRevB.2.4559 Crossref, Google Scholar
- 6 J. Smit, Physica 21, 877 (1955). 10.1016/S0031-8914(55)92596-9 Crossref, Google Scholar
- 7 J. Smit, Physica 24, 39 (1958). 10.1016/S0031-8914(58)93541-9 Crossref, Google Scholar
- 8 L. Shubnikov and W. J. de Haas, Comm. Phys. Lab. Leiden 207d, 35 (1930). Google Scholar
- 9 W. J. de Haas and P. M. van Alphen, Comm. Phys. Lab. Leiden 212a, 3 (1930). Google Scholar
- 10 M. H. Cohen and E. I. Blount, Philos. Mag. 5, 115 (1960). 10.1080/14786436008243294 Crossref, Google Scholar
- 11 P. A. Wolff, J. Phys. Chem. Solids 25, 1057 (1964). 10.1016/0022-3697(64)90128-3 Crossref, Google Scholar
- 12 Y. Fuseya, M. Ogata, and H. Fukuyama, J. Phys. Soc. Jpn. 84, 012001 (2015). 10.7566/JPSJ.84.012001 Link, Google Scholar
- 13 L. Wehrli, Phys. Kondens. Mater. 8, 87 (1968). Google Scholar
- 14 H. Fukuyama and R. Kubo, J. Phys. Soc. Jpn. 28, 570 (1970). 10.1143/JPSJ.28.570 Link, Google Scholar
- 15 Y. Fuseya, M. Ogata, and H. Fukuyama, J. Phys. Soc. Jpn. 81, 093704 (2012). 10.1143/JPSJ.81.093704 Link, Google Scholar
- 16 J. Inoue, G. Bauer, and L. Molenkamp, Phys. Rev. B 70, 041303 (2004). 10.1103/PhysRevB.70.041303 Crossref, Google Scholar
- 17 J. Inoue, T. Kato, Y. Ishikawa, H. Itoh, G. E. W. Bauer, and L. W. Molenkamp, Phys. Rev. Lett. 97, 046604 (2006). 10.1103/PhysRevLett.97.046604 Crossref, Google Scholar
- 18 W. Kohn and J. M. Luttinger, Phys. Rev. 108, 590 (1957). 10.1103/PhysRev.108.590 Crossref, Google Scholar
- 19 G. D. Mahan, Many-Particle Physics (Kluwer Academic/Plenum, New York, 2000) 3rd ed. Crossref, Google Scholar
- (20) This implies that the Born approximation is no longer valid for \(\tilde{\mu } \gg 1\) since the longitudinal charge conductivity (∝ U (μ)) for T = 0 becomes constant for \(\tilde{\mu } \gg 1\) as seen in Eq. (3.6). Google Scholar
- 21 H. Emoto, Y. Ando, E. Shikoh, Y. Fuseya, T. Shinjo, and M. Shiraishi, J. Appl. Phys. 115, 17C507 (2014). 10.1063/1.4863377 Crossref, Google Scholar
- 22 H. Emoto, Y. Ando, G. Eguchi, R. Ohshima, E. Shikoh, Y. Fuseya, T. Shinjo, and M. Shiraishi, Phys. Rev. B 93, 174428 (2016). 10.1103/PhysRevB.93.174428 Crossref, Google Scholar
- (23) For pure semimetallic bismuth, the energy gap and the chemical potential are evaluated as \(\Delta \simeq 7.7\) meV and \(\mu \simeq 35\) meV.15,24,25) Google Scholar
- 24 G. E. Smith, G. A. Baraff, and J. M. Rowell, Phys. Rev. 135, A1118 (1964). 10.1103/PhysRev.135.A1118 Crossref, Google Scholar
- 25 Z. Zhu, B. Fauque, Y. Fuseya, and K. Behnia, Phys. Rev. B 84, 115137 (2011). 10.1103/PhysRevB.84.115137 Crossref, Google Scholar
- 26 K. Sakai, C. Ishii, and H. Fukuyama, J. Phys. Soc. Jpn. 50, 3590 (1981). 10.1143/JPSJ.50.3590 Link, Google Scholar
- 27 O. Chalaev and D. Loss, Phys. Rev. B 71, 245318 (2005). 10.1103/PhysRevB.71.245318 Crossref, Google Scholar
- 28 O. V. Dimitrova, Phys. Rev. B 71, 245327 (2005). 10.1103/PhysRevB.71.245327 Crossref, Google Scholar