From coupled Rashba electron- and hole-gas layers to three-dimensional topological insulators

We introduce a system of stacked two-dimensional electron- and hole-gas layers with Rashba spin-orbit interaction and show that the tunnel coupling between the layers induces a strong three-dimensional (3D) topological insulator phase. At each of the two-dimensional bulk boundaries we find the spectrum consisting of a single anisotropic Dirac cone, which we show by analytical and numerical calculations. Our setup has a unit cell consisting of four tunnel coupled Rashba layers and presents a synthetic strong 3D topological insulator and is distinguished by its rather high experimental feasibility.

Figure 1

(a) Setup consisting of a stack of layers arranged in the $xy$ plane and tunnel coupled along the $z$ axis. The layers colored in green (orange) denote electron (hole) 2DEGs with Rashba SOI and at chemical potential ${\mu }_{\tau }$. The brightness of the color encodes two possible values of the Rashba SOI ${\alpha }_{\tau }$. (b) Dispersion of a 2DEG with Rashba SOI.

Figure 2

(a) Dispersion relation of each layer for fixed $\theta$. The chemical potentials ${\mu }_{\tau }$ are chosen such that inner and outer Fermi surfaces have the same radii across different layers. The arrows indicate where the tunneling between the layers opens up gaps (small green circles). Note that the bottom and top layers stay gapless and have a dispersion consisting of a single Dirac cone with spin locked to momentum due to time reversal invariance. (b) Interior and exterior Fermi surfaces of each layer with the cuts for $k_y=\mathrm{const}$. The fields for interior (exterior) left and right movers $S_{n{\theta }_i\eta \tau }^i,S_{n[\pi -{\theta }_i]\eta \tau }^i \left(S_{n{\theta }_e\eta \tau }^e,S_{n[\pi -{\theta }_e]\eta \tau }^e\right)$ have in general different spin orientations.

Figure 3

(a) Dispersion relation of the surface states (red) localized in the $yz$ plane as well as bulk states (blue) obtained numerically, see Eq. (15). At small momenta, the surface states form a single anisotropic Dirac cone, but merge with the bulk states at large momenta. b) Spin orientation (red arrows) of the first layer of the unit cell for a fixed energy $E/t_0=-0.03$ and at the position $x_0=3a_x$ away from the left edge. The parameter values assumed are ${\overline{\alpha }}_1=0.3t_0,{\overline{\alpha }}_{\overline{1}}=0.55t_0,|{\mu }_1|=|{\mu }_{\overline{1}}|=0.2125t_0$, and $t=0.1t_0$. The spin orientation is locked to the momentum direction, confirming the strong TI phase.

Figure 4

Dispersion relation of the surface states localized in the $yz$ plane for $k_{Fe}/k_{Fi}=3$, obtained analytically from Eq. (13) of the main text with $E/\mathrm{\Delta }=cos\mathrm{\Omega }$. We plot $k_y$ up to value of $0.9k_{Fi}$, the concrete range of validity of the dispersion depends on the value of $t$ and is given below Eq. (9).

Figure 5

Dispersion relation of the surface states localized in the $yz$ plane obtained numerically in the tight-binding model. The parameters of the system take the following values: ${\overline{\alpha }}_1=0.3t_0,{\overline{\alpha }}_{\overline{1}}=0.55t_0,|{\mu }_1|=|{\mu }_{\overline{1}}|=0.2125t_0$, and $t=0.05t_0$.

Figure 6

The (a) $k_z=0$ and (b) $k_y=0$ cuts of the dispersion relation of the surface states localized in the $yz$ plane obtained numerically in the tight-binding model. The parameters of the system take the following values: ${\overline{\alpha }}_1=0.3t_0,{\overline{\alpha }}_{\overline{1}}=0.55t_0,|{\mu }_1|=|{\mu }_{\overline{1}}|=0.2125t_0$, and $t=0.1t_0$.