Microscopic origin of the relativistic splitting of surface states

Spin-orbit splitting of surface states is analyzed within and beyond the Rashba model using as examples the (111) surfaces of noble metals, ${\mathrm{Ag}}_2\mathrm{Bi}$ surface alloy, and topological insulator ${\mathrm{Bi}}_2{\mathrm{Se}}_3$. The ab initio analysis of relativistic velocity proves the Rashba model to be fundamentally inapplicable to real crystals. The splitting is found to be primarily due to a spin-orbit induced in-plane modification of the wave function, namely, to its effect on the nonrelativistic Hamiltonian. The usual Rashba splitting—given by charge distribution asymmetry—is an order of magnitude smaller.

Figure 1

Microscopic Rashba model. (a) Coordinate frame. (b) Model potential $V\left(y\right)=-V_{\mathrm{a}}{\sum }_{n}cos(2\pi ny/a)$ with lattice constant $a=\pi a_0$, $V_{\mathrm{a}}=0.1$ Ry, and vacuum level $V_0=1.5$ Ry. (c) Effective potentials ${\tilde{V}}_{\downarrow }\left(y\right)={\tilde{V}}_{\uparrow }(-y)$ in Eq. (4). (d) Surface-state wave function. (e) Complex band structure: inside the gap it is $\mathrm{Re}{k}_y=\pi /a$. (f) Energy dispersion $E\left(k_{\parallel }\right)$ of the surface states for the potential $V$ shown in graph (b). (g) The same for the potential $2V$. (h) $2{\alpha }_{\text{R}}=d(E_{\uparrow }-E_{\downarrow })/d{k}_{\parallel }$ for the split surface states for the potential $V_0$ (solid lines) and $2V_0$ (dashed lines). (i,j) Splitting parameter ${\alpha }_{\text{R}}^{\prime \prime }$ as a function (i) of the surface barrier $V_0$ and (j) of the eigenenergy at $\overline{\Gamma }$ for the potential with and without a “singularity” at the surface, graph (k).

Figure 2

Group velocity as a function of the Bloch vector in the direction $\overline{\Gamma }\overline{M}$ for the surface states on Au(111), left; Bi/Ag(111), middle; and ${\mathrm{Bi}}_2{\mathrm{Se}}_3$, right column. In the upper row, circles show a numerical derivative of the eigenvalues $dE\left(k_{\parallel }\right)/d{k}_{\parallel }$, and solid lines show the nonrelativistic part $v_{\text{c}}$. The middle row shows the relativistic part $v_{\text{r}}$, and the bottom row shows the dispersion $E\left(k_{\parallel }\right)$. Everywhere black color is used for the lower branch and red color is used for the upper one. For Bi/Ag(111), the broken line for the upper branch at $k_{\parallel }=0.07$ a.u.${}^{-1}$ indicates the interpolated values at the place of avoided crossing with a higher-lying surface state.

Figure 3

In-plane distribution of the ${\mathbf{k}}_{\parallel }$ projection of the classical current density $j\left({\mathbf{r}}_{\parallel }\right)$ (in units of Ry a.u.) for the surface state at Bi/Ag(111) with ${\mathbf{k}}_{\parallel }$ along the horizontal axis. (a) Relativistic $j\left({\mathbf{r}}_{\parallel }\right)$ map for $k_{\parallel }={10}^{-4}$Å${}^{-1}$ for the inner-circle surface state, i.e., for the one with the smaller $k_{\parallel }$ at a given $E$, hence the negative group velocity. (b) Scalar relativistic map for $k_{\parallel }={10}^{-3}$Å${}^{-1}$.