  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$_3$.
