Spin-Seebeck effect on the surface of topological insulator due to nonequilibrium spin-polarization parallel to the direction of thermally driven electronic transport

We study the transverse spin-Seebeck effect (SSE) on the surface of a three-dimensional topological insulator (TI) thin film, such as Bi${}_2$Se${}_3$, which is sandwiched between two normal metal leads. The temperature bias $\mathrm{\Delta }T$ applied between the leads generates surface charge current which becomes spin-polarized due to strong spin-orbit coupling on the TI surface, with polarization vector acquiring a component $P_x\simeq 60$ {\em parallel to the direction of transport}. When the third nonmagnetic voltage probe is attached to the portion of the TI surface across its width $L_y$, pure spin current will be injected into the probe where the inverse spin Hall effect (ISHE) converts it into a voltage signal \mbox{$\mathrm{\mid }{V}_{\mathrm{I}\mathrm{S}\mathrm{H}\mathrm{E}}{\mathrm{\mid }}^{\mathrm{m}\mathrm{a}\mathrm{x}}\mathrm{/}\mathrm{\Delta }T\simeq 2.5$ $\mu$V/K} (assuming the SH angle of Pt voltage probe and $L_y=1$ mm). The existence of predicted nonequilibrium spin-polarization parallel to the direction of electronic transport and the corresponding electron-driven SSE crucially relies on orienting quintuple layers (QLs) of Bi${}_2$Se${}_3$ {\em orthogonal} to the TI surface and {\em tilted} by ${45}^{\circ }$ with respect to the direction of transport. Our analysis is based on the Landauer-B"{u}ttiker-type formula for spin currents in the leads of a multi-terminal quantum-coherent junction, which is constructed using nonequilibrium Green function formalism within which we show how to take into account arbitrary orientation of QLs via the self-energy describing coupling between semi-infinite normal metal leads and TI.