We predict that unpolarized charge current injected into a ballistic thin film of prototypical topological insulator (TI) Bi$_2$Se$_3$ will generate a {it noncollinear spin texture} $mathbf{S}(mathbf{r})$ on its surface. Furthermore, the nonequilibrium spin texture will extend into $simeq 2$ nm thick layer below the TI surfaces due to penetration of evanescent wavefunctions from the metallic surfaces into the bulk of TI. Averaging $mathbf{S}(mathbf{r})$ over few AA{} along the longitudinal direction defined by the current flow reveals large component pointing in the transverse direction. In addition, we find an order of magnitude smaller out-of-plane component when the direction of injected current with respect to Bi and Se atoms probes the largest hexagonal warping of the Dirac-cone dispersion on TI surface. Our analysis is based on an extension of the nonequilibrium Green functions combined with density functional theory (NEGF+DFT) to situations involving noncollinear spins and spin-orbit coupling. We also demonstrate how DFT calculations with properly optimized local orbital basis set can precisely match putatively more accurate calculations with plane-wave basis set for the supercell of Bi$_2$Se$_3$.
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