Spin Polarization and Transport of Surface States in the Topological Insulators ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ from First Principles

We investigate the band dispersion and the spin texture of topologically protected surface states in the bulk topological insulators ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ by first-principles methods. Strong spin-orbit entanglement in these materials reduces the spin polarization of the surface states to $\sim 50\%$ in both cases; this reduction is absent in simple models but of important implications to essentially any spintronic application. We propose a way of controlling the magnitude of spin polarization associated with a charge current in thin films of topological insulators by means of an external electric field. The proposed dual-gate device configuration provides new possibilities for electrical control of spin.

Figure 1

(a) Band structure of the 5 QL slabs of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and ${\mathrm{Bi}}_2{\mathrm{Te}}_3$ (lines) superimposed with the projected band structure of the corresponding bulk materials (shaded areas). (b) Evolution of the surface-state band dispersion in the vicinity of $\Gamma$ point as a function of slab thickness. Shaded areas show the bulk band gaps. (c) Band gaps at the $\Gamma$ point induced by the interaction between the surface states as a function of slab thickness.

Figure 2

(a)–(c) Expectation values of the spin operators $⟨S_{\alpha }(\overrightarrow{k})⟩$ ($\alpha =x$, $y$, $z$) for the topological surface states along the selected path ($A\mathrm{\text{−}}\Gamma \mathrm{\text{−}}B$) in momentum space (d) for 4 QL slabs of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ and ${\mathrm{Bi}}_2{\mathrm{Te}}_3$. Electron (solid lines) and hole (dashed lines) surface-state bands are distinguished. The surface normal corresponds to the $z$ direction. The coordinate system is right handed [(d), inset].

Figure 3

(a) Illustration of the effect of electric field ${\overrightarrow{E}}_{\mathrm{ext}}$ applied across a thin slab of bulk TI. The filling of the two surface-state bands and the corresponding spin textures for electrons and holes are shown. (b) Splitting of the surface-state bands by applied electric field ($\delta E=0.1\mathrm{eV}$, solid lines) in a 3QL slab of ${\mathrm{Bi}}_2{\mathrm{Se}}_3$ compared to the band dispersion at zero field (dotted lines). The inset shows a close-up view of band gaps at the charge-neutrality point. (c) Individual (surfaces 1 and 2) and total ballistic conductances in the $y$ direction per device width ${\overline{G}}_y$ and (d) spin polarization along the $x$ axis $P_x$ for $\delta E=0.1\mathrm{eV}$ and $0\mathrm{eV}$ as a function of chemical potential $\mu$ ($\mu =0\mathrm{eV}$ corresponds to the charge-neutrality point). The symbols have the same meaning in (c) and (d).