Room-temperature spin thermoelectrics in metallic films

Considering metallic films at room temperature, we present the first theoretical study of the spin Nernst and thermal Edelstein effects that takes into account dynamical spin-orbit coupling, i.e., direct spin-orbit coupling with the vibrating lattice (phonons) and impurities. This gives rise to a novel process, namely, a dynamical side-jump mechanism, and to dynamical Elliott-Yafet spin relaxation, never before considered in this context. Both are the high-temperature counterparts of the well-known $T=0$ side-jump and Elliott-Yafet, central to the current understanding of the spin Hall, spin Nernst and Edelstein (current-induced spin polarization) effects at low $T$. We consider the experimentally relevant regime $T>T_D$, with $T_D$ the Debye temperature, as the latter is lower than room temperature in transition metals such as Pt, Au and Ta typically employed in spin injection/extraction experiments. We show that the interplay between intrinsic (Bychkov-Rashba type) and extrinsic (dynamical) spin-orbit coupling yields a nonlinear $T$ dependence of the spin Nernst and spin Hall conductivities.

Figure 1

Shown are the self-energies which determine the collision operators in the Boltzmann equation. The arrowed line represents the Green's function in Keldysh space, a cross (dot) the potential due to an impurity (a crystal displacement). The dashed line depicts the impurity correlation either for static (straight line) or for dynamical impurities (wavy line). The wavy solid line illustrates the phonon propagator and a box around a vertex the spin-orbit coupling due to the boxed potential.

Figure 2

${\mathcal{P}}^t$, compare Eq. (41), versus temperature, in units of $S_{0,r}(2m\alpha {\tau }_r/{\hslash }^2)(e/8\pi \hslash )$, split into its thermal and electrical contributions. The Elliottt-Yafet spin relaxation is chosen as $\tau /{\tau }_s=0.01$. For (a), we have ${\tau }_{s,r}/{\tau }_{\mathrm{DP},r}=1$, and for (b), ${\tau }_{s,r}/{\tau }_{\mathrm{DP},r}=20$. $T_r$ denotes the temperature scale (room temperature).

Figure 3

Spin Nernst conductivity in units of the “universal” value of the intrinsic spin Hall conductivity times the Seebeck coefficient at room temperature, $S_{0,r}e/8\pi \hslash$, against $T/T_r$, with $\tau /{\tau }_s=0.01$. We show the electrical and thermal contribution separately; the parameters are chosen as ${\tau }_{s,r}/{\tau }_{\mathrm{DP},r}=1$ for (a) and ${\tau }_{s,r}/{\tau }_{\mathrm{DP},r}=20$ for (b).

Figure 4

The spin Hall conductivity in 3D in units of $N_{0}e\hslash /4m$ for various ratios of ${\tau }_s/{\tau }_{\mathrm{DP}}$ vs $\omega \tau$, separated into its real part (solid lines) and its imaginary part (dashed lines). The extrinsic spin-orbit strength is chosen such that $\tau /{\tau }_s=0.01$.