Disorder and magnetic-field-induced breakdown of helical edge conduction in an inverted electron-hole bilayer

We calculate the conductance of a two-dimensional bilayer with inverted electron-hole bands to study the sensitivity of the quantum spin Hall insulator (with helical edge conduction) to the combination of electrostatic disorder and a perpendicular magnetic field. The characteristic breakdown field for helical edge conduction splits into two fields with increasing disorder, a field $B_c$ for the transition into a quantum Hall insulator (supporting chiral edge conduction) and a smaller field $B_c^{\prime }$ for the transition to bulk conduction in a quasimetallic regime. The spatial separation of the inverted bands, typical for broken-gap InAs/GaSb quantum wells, is essential for the magnetic-field-induced bulk conduction—there is no such regime in HgTe quantum wells.

Figure 1

Alignment of the conduction band (blue) and valence band (red) in a quantum well of (a) type I and (b) type II. Both quantum wells have electron and hole subbands in inverted order (dotted lines, red h-like above blue e-like). The band gap (gray) is broken in the InAs/GaSb quantum well of type II, providing for a spatially separated electron-hole bilayer. There is no such spatial separation in the HgTe quantum well of type I.

Figure 2

Landau-level spectrum in the two types of quantum wells, calculated from the Hamiltonian (1) for the parameters of Table 1 (nonzero $\delta \mu$ and $M_0=-0.01\mathrm{eV}$).

Figure 3

Same as Fig. 2, for a type-II quantum well with electron-hole symmetry ($\delta \mu =0$) and for a larger zero-field gap $M_0=-0.0325\mathrm{eV}$. The two characteristic fields $B_c^{\prime }$ and $B_c$ of the first and last Landau-level crossing are indicated.

Figure 4

Disorder-averaged conductance of a type-II quantum well with electron-hole symmetry ($\delta \mu =0$, other parameters as in Table 1), calculated numerically from the tight-binding Hamiltonian (1), with (a) hard-wall boundary conditions or (b) periodic boundary conditions. The Fermi level is set at $E_{\mathrm{F}}=8\times {10}^{-4}\mathrm{eV}$, slightly displaced from the center of the bulk gap to avoid the minigap in the spectrum of helical edge states. The disorder dependence of the characteristic fields $B_c$ and $B_c^{\prime }$ (white and green curves) is calculated from the renormalization of the band gap in the Born approximation, as described in the text.

Figure 5

Disorder-averaged conductance of a quantum well of (a) type I and (b) type II, with hard-wall boundary conditions. The parameters are those of Table 1, including the effects of broken electron-hole symmetry (nonzero $\delta \mu$). The conductance is calculated at the renormalized Fermi energy (5). The region of bulk conduction is present in the type-II quantum well, but not in type I, where instead a region of localized edge states appears. (This region turns black for periodic boundary conditions, so we know there is no bulk conduction there.)

Figure 6

Same as Fig. 4, but including the effects of the Zeeman energy with the effective $g$ factor $-15$.