  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) and spin Nernst conductivity (SNC), namely, $Z_{yx}^z$,
$Z_{xz}^y$, and $Z_{zy}^x$. We calculate all the elements as a function of the
Fermi energy. Second, we find that all SHC tensor elements are large, being
$\sim$1000 ($\hbar$/e)(S/cm) and comparable to that of platinum. Furthermore,
because of its low electrical conductivity, the corresponding spin Hall angles
are gigantic, being $\sim$20%. Third, all the calculated SNC tensor elements
are also pronounced, being comparable to that [$\sim$0.13 ($\hbar$/e)(A/m-K)]
of gold, although they are several times smaller than platinum and $\beta$-Ta.
Finally, in contrast to Pt and Au where $Z_{yx}^z = Z_{xz}^y = Z_{zy}^x$, the
SHE and SNE in bismuth are strongly anisotropic, i.e., $Z_{yx}^z$, $Z_{xz}^y$
and $Z_{zy}^x$ differ significantly. Consequently, the Hall voltages due to the
inverse SHE and 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 spin current experiments on bismuth using highly oriented
single crystal specimens.
