Strongly anisotropic spin-orbit splitting in a two-dimensional electron gas

Near-surface two-dimensional electron gases on the topological insulator ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$ are induced by electron doping and studied by angle-resolved photoemission spectroscopy. A pronounced spin-orbit splitting is observed for these states. The $k$-dependent splitting is strongly anisotropic to a degree where a large splitting ($\approx 0.06{\AA }^{-1})$ can be found in the $\overline{\Gamma }\overline{M}$ direction while the states are hardly split along $\overline{\Gamma }\overline{K}$. The direction of the anisotropy is found to be qualitatively inconsistent with results expected for a third-order anisotropic Rashba Hamiltonian. However, a $\mathbf{k}·\mathbf{p}$ model that includes the possibility of band structure anisotropy as well as both isotropic and anisotropic third order Rashba splitting can explain the results. The isotropic third order contribution to the Rashba Hamiltonian is found to be negative, reducing the energy splitting at high $k$. The interplay of band structure, higher order Rashba effect, and tunable doping offers the opportunity to engineer not only the size of the spin-orbit splitting but also its direction.

Figure 1

Electronic structure of electron doped ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$ for the low doping case (a–d) and for the high doping case (e–h) as obtained by ARPES. (a,e) Constant energy surfaces at different binding energies. (b,f) On the left: Fermi contour of electron doped ${\mathrm{Bi}}_2{\mathrm{Te}}_2\mathrm{Se}$. On the right: schematic representation of the Fermi contours. In black the topological state, in red (orange) the first 2DEG, in blue (light blue) the second 2DEG, in gray the third 2DEG. (c),(g) and (d),(h) Dispersion along the $\overline{\Gamma }\overline{M}$ and $\overline{\Gamma }\overline{K}$ directions, respectively.

Figure 2

Photoemission intensity cuts of the 2DEG in the high doping (a,b) and low doping (c,d) cases along (a,c) $\overline{\Gamma }\overline{M}$ and (b,d) $\overline{\Gamma }\overline{K}$. The red lines show the fitting of the first 2DEG branches using the model described in Eq. (3).

Figure 3

Energy splitting for the first and second 2DEGs as a function of $\left|k\right|$ for the high doping case. Dispersion shown by symbols is extracted from the photoemission intensities. Solid lines represent the best fit to the splitting dispersion using Eq. (3). Dashed lines represent the splitting extrapolated to first order. Both isotropic and anisotropic third order contributions are necessary for a precise description of the energy splitting.

Figure 4

$k$ splitting of the first 2DEGs as a function of energy for the low doping (a) and high doping (b) case. The zero energy is set to be at the degeneracy point of the two branches. Symbols represent the splitting amplitude obtained by fitting of the photoemission spectra. Dashed lines represent the $k$ splitting extracted from our model (dispersion shown in Fig. 2). Insets: Section of the Fermi surface that shows the anisotropy of the spin at the Fermi level.