Robust helical edge transport in quantum spin Hall quantum wells

We show that edge-state transport in semiconductor-based quantum spin Hall systems is unexpectedly robust to magnetic fields. The origin for this robustness lies in an intrinsic suppression of the edge-state $g$-factor and the fact that the edge-state Dirac point is typically hidden in the valence band. A detailed $\mathbf{k}·\mathbf{p}$ band-structure analysis reveals that both $\mathrm{InAs}/\mathrm{GaSb}$ and $\mathrm{HgTe}/\mathrm{CdTe}$ quantum wells exhibit such buried Dirac points for a wide range of well thicknesses. By simulating transport in a disordered system described within an effective model, we demonstrate that edge-state transport remains nearly quantized up to large magnetic fields, consistent with recent experiments.

Figure 1

Schematic depiction of edge-state dispersions: In the absence of a magnetic field, the edge-state crossing is topologically protected, but may (a) reside in the gap or (b) be hidden in a bulk band. In a finite magnetic field, a Zeeman gap opens and the edge-state spins become canted—permitting backscattering as in (c) However, when the edge-state crossing is hidden in a bulk band, spins within the gap are further away from the Zeeman gap and nearly antialign, greatly suppressing backscattering as in (d).

Figure 2

(a), (b) System geometries used for $\mathbf{k}·\mathbf{p}$ simulations. (c)–(f) Band structures for (c), (e) $\mathrm{InAs}/\mathrm{GaSb}$ and (d), (f) HgTe/CdTe. For both materials we observe Dirac points buried in a valence band, which obscures the opening of a Zeeman gap under applied in-plane magnetic fields as in (e) and (f).

Figure 3

(a) Topological gap of $\mathrm{InAs}/\mathrm{GaSb}$ as a function of InAs and GaSb well thicknesses. A red dot indicates a buried Dirac point. (b) Subband edges at the $\mathrm{\Gamma }$ point of $\mathrm{HgTe}$ as a function of $\mathrm{HgTe}$ well thickness. The Dirac point is buried for thickness $L_{\text{HgTe}}\ge 7.25\mathrm{nm}$.

Figure 4

(a) Band structure and (b)–(d) transport calculations for the BHZ model with (blue) and without (red) an additional edge potential $V_{\text{edge}}$. Transport calculations were performed for a disordered system at (b) zero field and (c) with an in-plane field $B_x=8\mathrm{T}$. (d) Transport calculation at fixed $\mu =3\mathrm{meV}$. All transport calculations are averaged over 50 different disorder realizations, with parameters $V_{\text{edge}}=-0.14\mathrm{eV}$, $U_0=0.05\mathrm{eV}$, $L=4000\mathrm{nm}$, $W=1000\mathrm{nm}$, and finite-difference grid spacing $a=4\mathrm{nm}$.