Anisotropic spin Hall and spin Nernst effects in bismuth semimetal

Abstract

Bismuth is an archetypal semimetal with gigantic spin–orbit coupling and it has been a major source material for the discovery of seminal phenomena in solid state physics for more than a century. In recent years, spin current transports in bismuth have also attracted considerable attention. In this paper, we theoretically study both spin Hall effect (SHE) and spin Nernst effect (SNE) in bismuth, based on relativistic band structure calculations. First, we find that there are three independent tensor elements of spin Hall conductivity (SHC) (σijs) and spin Nernst conductivity (SNC) (αijs), namely, Zyxz, Zxzy, and Zzyx, where Z=σ or α. We calculate all the elements as a function of the Fermi energy. Second, we find that all SHC elements are large, being ∼1000 (ħ/e)(S/cm). Furthermore, because of its low electrical conductivity, the spin Hall angles are gigantic, being ∼20 %. Third, all the calculated SNC elements are also pronounced, being comparable to that [∼0.13 (ħ/e)(A/m-K)] of gold. Finally, in contrast to Pt and Au where Zyxz=Zxzy=Zzyx, the SHE and SNE in bismuth are anisotropic, i.e., Zyxz≠Zxzy≠Zzyx. In particular, SNC is highly anisotropic, and αyxz, αxzy and αzyx differ even in sign. Also, such anisotropy in SHE can be significantly enhanced by either electron or hole doping. Consequently, the Hall voltages due to the inverse SHE and inverse SNE from the different conductivity elements could cancel each other and thus result in a small spin Hall angle if polycrystalline samples are used, which may explain why the measured spin Hall angles ranging from nearly 0 to 25 % have been reported. We hope that these interesting findings would stimulate further experiments on bismuth using highly oriented single crystal specimens.