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We study the transverse spin-Seebeck effect (SSE) on the surface of athree-dimensional topological insulator (TI) thin film, such as Bi$_2$Se$_3$,which is sandwiched between two normal metal leads. The temperature bias$\Delta T$ applied between the leads generates surface charge current whichbecomes spin-polarized due to strong spin-orbit coupling on the TI surface,with polarization vector acquiring a component $P_x \simeq 60%$ {\em parallelto the direction of transport}. When the third nonmagnetic voltage probe isattached to the portion of the TI surface across its width $L_y$, pure spincurrent will be injected into the probe where the inverse spin Hall effect(ISHE) converts it into a voltage signal\mbox{$|V_\mathrm{ISHE}|^\mathrm{max}/\Delta T \simeq 2.5$ $\mu$V/K} (assumingthe SH angle of Pt voltage probe and $L_y=1$ mm). The existence of predictednonequilibrium spin-polarization parallel to the direction of electronictransport and the corresponding electron-driven SSE crucially relies onorienting quintuple layers (QLs) of Bi$_2$Se$_3$ {\em orthogonal} to the TIsurface and {\em tilted} by $45^\circ$ with respect to the direction oftransport. Our analysis is based on the Landauer-B\"{u}ttiker-type formula forspin currents in the leads of a multi-terminal quantum-coherent junction, whichis constructed using nonequilibrium Green function formalism within which weshow how to take into account arbitrary orientation of QLs via the self-energydescribing coupling between semi-infinite normal metal leads and TI.

physical review. b, condensed matter and materials physics

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