All-electron topological insulator in InAs double wells

We show that electrons in ordinary III–V semiconductor double wells with an in-plane modulating periodic potential and interwell spin-orbit interaction are tunable topological insulators (TIs). Here the essential TI ingredients, namely, band inversion and the opening of an overall bulk gap in the spectrum arise, respectively, from (i) the combined effect of the double-well even-odd state splitting ${\Delta }_{\mathrm{SAS}}$ together with the superlattice potential and (ii) the interband Rashba spin-orbit coupling $\eta$. We corroborate our exact diagonalization results with an analytical nearly-free-electron description that allows us to derive an effective Bernevig-Hughes-Zhang model. Interestingly, the gate-tunable mass gap $M$ drives a topological phase transition featuring a discontinuous Chern number at ${\Delta }_{\mathrm{SAS}}\sim 5.4\text{meV}$. Finally, we explicitly verify the bulk-edge correspondence by considering a strip configuration and determining not only the bulk bands in the nontopological and topological phases but also the edge states and their Dirac-like spectrum in the topological phase. The edge electronic densities exhibit peculiar spatial oscillations as they decay away into the bulk. For concreteness, we present our results for InAs-based wells with realistic parameters.

Figure 1

(a) Proposed double-well structure and its potential profile showing the two lowest states split by ${\Delta }_{\mathrm{SAS}}$. The in-plane superlattice is denoted by the black dots on top of the structure. (b) Unit cell of the periodic “pits” in (a). The relative depth of the pits, indicated by different shadings, determine the values $V_1$ and $V_2$ that parametrize the periodic potential [15]. (c) Free- and nearly-free-electron bands (dashed and solid lines, respectively) in units of $\frac{{\hslash }^2{Q}^2}{2m^2}$ along the $k_x$ axis. The interband spin-orbit coupling $\eta$ will open up a gap at the crossing (circle) of the inverted bands; see Fig. 2.

Figure 2

In-plane superlattice band structure (red curves) for $V_1=3.5\text{meV},V_2=12\text{meV},{\Delta }_{\mathrm{SAS}}=4.5\text{meV}$, and $\eta =20\text{meV}\text{nm}$. The gray dashed curves are the energy bands for $\eta =0$. A finite interband spin-orbit coupling $\eta$ gives rise to the anticrossing of the inverted bands, thus generating an overall gap (shaded rectangular area). The shift ${\delta }_{\eta }$ is due to ISOC corrections to the band energies. The inset shows the band structure for different values of $M\left({\Delta }_{\mathrm{SAS}}\right)=-0.45\left(4.5\right),-0.1\left(5.2\right)$, and $0.1\left(5.6\right)\text{meV}$.

Figure 3

(a) The bandwidth ${\Delta }_V$ as a function of $V_2$ for three values of $V_1$. The dashed curves are the perturbative result obtained using Eqs. (11, 12, 13). The inset shows the mass gap $2M={\Delta }_{\mathrm{SAS}}-{\Delta }_V\left(V_2\right)$ as a function of $V_2$ around the value of $V_2=12\text{meV}$, showing that the bands can be inverted via the gate that defines the superlattice potential. (b) Numerical and perturbative results, Eqs. (14) and (15), for bands 2 and 3 and $\eta =0$. A nonzero $\eta$ opens up a gap at the crossing point; see Fig. 2.

Figure 4

(a) Vector field plot of $\mathit{d}$ and (b) the topological index $C_1$. In (a) the arrows denote the $xy$ components, and the color scale denotes the $z$ component of $\mathit{d}$. Note that $d_x \left(d_y\right)$ is linear in $k_x \left(k_y\right)$ only close to the $\Gamma$ point, in contrast to the BHZ model for which $d_x \left(d_y\right)$ is linear irrespective of the value of $\mathit{k}$. $C_1$ shows a jump as ${\Delta }_{\mathrm{SAS}}$ is varied (for fixed $V_1=3.5\text{meV}$ and $V_2=12\text{meV}$) and when $V_2$ is varied (for a fixed value of ${\Delta }_{\mathrm{SAS}}=5.4\text{meV}$).

Figure 5

Energy spectra resulting from the exact diagonalization of Eq. (4). (a) ${\Delta }_{\mathrm{SAS}}=4.5\text{meV}$ corresponds to the topological phase and edge states are visible. (b) ${\Delta }_{\mathrm{SAS}}=5.6\text{meV}$ gives rise to a normal insulator with no edge states.

Figure 6

The edge probability density $\rho \left(x\right)$ for ${\Delta }_{\mathrm{SAS}}=5.2\text{meV}$ for two strip widths, (a) ${\mathcal{L}}_x=80L$ and (b) $40L$. The curve for $80L$ is shifted up by 0.25 for clarity. The inset in (a) shows the edge states along with the BHZ dispersions obtained from the bulk spectrum. (c) Same as in (b), but for ${\Delta }_{\mathrm{SAS}}=4.5\text{meV}$. The blue curves in all panels denote the BHZ results.