Hidden edge Dirac point and robust quantum edge transport in InAs/GaSb quantum wells

The robustness of quantum edge transport in InAs/GaSb quantum wells in the presence of magnetic fields raises an issue on the fate of topological phases of matter under time-reversal symmetry breaking. A peculiar band structure evolution in InAs/GaSb quantum wells is revealed: the electron subbands cross the heavy hole subbands but anticross the light hole subbands. The topologically protected band crossing point (Dirac point) of the helical edge states is pulled to be close to and even buried in the bulk valence bands when the system is in a deeply inverted regime, which is attributed to the existence of the light hole subbands. A sizable Zeeman energy gap verified by the effective $g$ factors of edge states opens at the Dirac point by an in-plane or perpendicular magnetic field; however, it can also be hidden in the bulk valance bands. This provides a plausible explanation for the recent observation on the robustness of quantum edge transport in InAs/GaSb quantum wells subjected to strong magnetic fields.

Figure 1

The energy spectrum of InAs/GaSb quantum wells in different phases. (a) The energies of the lowest-energy subbands at the $\mathrm{\Gamma }$ point as functions of the broken gap $V_0$. The band structures at $k_y=0$ with (b) $V_0=-70$ meV, (c) $V_0=-90$ meV, and (d) $V_0=-120$ meV.

Figure 2

Energy spectrum of bulk and edge states of the system with periodic and open boundaries in the $x$ and $y$ directions, respectively. (a) For $\mathrm{\Delta }V=30$ meV. (b) For $\mathrm{\Delta }V=10$ meV. (c) For $\mathrm{\Delta }V=-20$ meV. (d) The energy position of the Dirac point ($E_D$) and the maximum point of valence bands ($E_M$) as functions of $\mathrm{\Delta }V$. $V_0=-100$ meV is taken for all figures.

Figure 3

Effective $g$ factors of edge states and Zeeman energy gaps for edge state spectra. The effective $g$-factor tensor elements of the edge states as a function of $\mathrm{\Delta }V$ for the magnetic field along (a) $x$, (b) $y$, and (c) $z$ directions, respectively. (d) The energy gaps ${\mathrm{\Delta }}_Z^{x,y,z}$ of the edge states opened by three principal magnetic fields of 0.5 T as functions of $\mathrm{\Delta }V$. $V_0=-100$ meV is taken for all figures.

Figure 4

Spatial distribution of wave functions of edge states. (a) The wave functions of edge states for different energies corresponding to Fig. 2. (b) The wave functions of edge states for different broken gap values at fixed $k_x=0{\mathring{\mathrm{A}}}^{-1}$ with $V_0=-100$ meV. (c) Energy spectrum similar to Fig. 2 but for the magnetic field $B_y=4$ T. (d) The wave functions of edge states for different energies corresponding to panel (c). Here $\mathrm{\Delta }V=-20$ meV and $V_0=-100$ meV are taken in panels (a), (c), and (d). Periodic and open boundary conditions are taken for the $x$ and $y$ directions, respectively.

Figure 5

The two-terminal conductance $G$ of InAs/GaSb quantum wells in magnetic fields. (a) $G$ as a function of the Fermi energy $E_F$ for various magnetic fields $B_x$ along the $x$ direction. (b) The same as panel (a) but for magnetic fields $B_y$ applied along the $y$ direction. $\mathrm{\Delta }V=-20$ meV and $V_0=-100$ meV are taken for all curves.

Figure 6

The two-terminal conductance $G$ of InAs/GaSb quantum wells in the presence of a magnetic field and disorder. (a) $G$ as a function of the Fermi energy $E_F$ for various magnetic fields $B_x$. (b) The same as panel (a) but for magnetic fields $B_y$ applied along the $y$ direction. $\mathrm{\Delta }V=-20$ meV, $V_0=-100$ meV, $W=10$ meV, and 50 disorder configurations are considered for all curves.