Quantum Spin Hall Effect in Inverted Type-II Semiconductors

The quantum spin Hall (QSH) state is a topologically nontrivial state of quantum matter which preserves time-reversal symmetry; it has an energy gap in the bulk, but topologically robust gapless states at the edge. Recently, this novel effect has been predicted and observed in HgTe quantum wells and in this Letter we predict a similar effect arising in Type-II semiconductor quantum wells made from $\mathrm{InAs}/\mathrm{GaSb}/\mathrm{AlSb}$. The quantum well exhibits an “inverted” phase similar to $\mathrm{HgTe}/\mathrm{CdTe}$ quantum wells, which is a QSH state when the Fermi level lies inside the gap. Due to the asymmetric structure of this quantum well, the effects of inversion symmetry breaking are essential. Remarkably, the topological quantum phase transition between the conventional insulating state and the quantum spin Hall state can be continuously tuned by the gate voltage, enabling quantitative investigation of this novel phase transition.

Figure 1

(a) Band gap and band offset diagram for asymmetric $\mathrm{AlSb}/\mathrm{InAs}/\mathrm{GaSb}$ quantum wells. The left AlSb barrier layer is connected to a front gate while the right barrier is connected to a back gate. The $E_1$ subband is localized in the InAs layer and $H_1$ is localized in the GaSb layer. Outer AlSb barriers provide an overall confining potential for electron and hole states. (b) Schematic band structure diagram. The dashed line shows the crossing of $E_1$ and $H_1$ in the inverted regime. Hybridization between $E_1$ and $H_1$, opens the gap $E_g$.

Figure 2

Energy spectrum of Eq. (1) on a cylinder with open boundary conditions in the $y$-direction and periodic boundary conditions in the $x$-direction. (a) The dispersion for a quantum well with GaSb layer thickness $d_1=10\mathrm{nm}$, InAs layer thickness $d_2=8.1\mathrm{nm}$, which is a NI with no edge states. (b) The dispersion for a $d_1=d_2=10\mathrm{nm}$ quantum well which is a QSH insulator with one pair of edge states. A tight-binding regularization with lattice constant $a=20\AA$ was used.

Figure 3

(a)–(c) The energy dispersions calculated from the 8-band Kane model for three well configurations, where $d_1$ and $d_2$ are the thickness of GaSb layer and InAs layer, respectively. (d) The energy gap variation in $d_1\mathrm{\text{−}}{d}_2$ plane, where brighter colors represent a smaller gap. A, B and C on the dashed blue line indicate, respectively, the place where (a),(b), and (c) are plotted. NI and QSH denote the phases in the corresponding region of parameter space.

Figure 4

Phase diagram for different front ($V_f$) and back ($V_b$) gate voltages. Regions I, II, III are in the inverted regime. The striped region II is the QSH phase with Fermi-level in the bulk gap, and I (III) is the $p(n)$-doped inverted system. Regions IV, V, VI are in the normal regime. The striped region V is the NI phase with Fermi level in the bulk gap, and IV (VI) is the $p(n)$-doped normal semiconductor. The black dotted line is the pase boundary between inverted and non-inverted band structures and the green circle shows the quantum critical point between the NI and QSH phases. The well configuration has $d_1=d_2=10\mathrm{nm}$, and AlSb barrier thickness is 30 nm on each side. $V_f$ and $V_b$ are defined with respect to the Fermi levels of the QW.