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

Abstract

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 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. A two-dimensional band insulator can support two types of conducting edge states: counterpropagating (helical) edge states in zero magnetic field and unidirectional (chiral) edge states in a sufficiently strong perpendicular field. These two topologically distinct phases are referred to as a quantum spin Hall (QSH) and quantum Hall (QH) insulator, respectively [1,2]. The physics of the QSH-to-QH transition is governed by band inversion [3][4][5]: The electronlike and holelike subbands near the Fermi level are interchanged in a QSH insulator, so that the band gap in the bulk becomes smaller rather than larger with increasing perpendicular magnetic field [6,7]. The gap closing at a characteristic field B c signals the transition from an inverted QSH gap with helical edge states to a noninverted QH gap supporting chiral edge states.The early experiments on the QSH effect were performed in HgTe layers with CdTe barriers (type-I quantum wells) [8,9]. Recently the effect has also been observed in InAs/GaSb bilayers with AlSb barriers (type-II quantum wells) [10][11][12][13]. Both types of quantum wells can have electron-hole subbands in inverted order, but while these are strongly coupled in type-I quantum wells, they are spatially separated and weakly coupled in the broken-gap quantum wells of type II (see Fig. 1). Although the difference has no consequences in zero magnetic field, we will show here that the breakdown of helical edge conduction in a magnetic field becomes qualitatively different.In both type-I and type-II quantum wells we find an increase with disorder of the characteristic field B c for the QSH-to-QH transition, as a consequence of the same mechanism that is operative in topological Anderson insulators [14]: a disorderinduced renormalization of the band gap [15]. Basically, in a narrow-gap semiconductor the effect of disorder on the bulk band gap is opposite in the inverted and noninverted cases. While a noninverted band gap is reduced by disorder, the inverted band gap is increased. Since B c is proportional to the zero-field band gap, it is pushed to larger fields by impurity scattering.As a consequence, disorder increases the robustness of helical edge conduction in type-I quantum wells, such as HgTe. In contrast, we find that in broken-gap quantum...

Magnetotransport in disordered two-dimensional topological insulators: signatures of charge puddles

“…This changes only if one increases the disorder to a strength at which the material is already on the phase transition to a topologically trivial Anderson insulator, a scenario which we do not find very likely to occur in the available materials as it would imply that one should not be able to observe the measured quantized conductance and non-local transport phenomena. This is in line with other numerical studies which also obtained robust edge states under the combined influence of magnetic field and disorder [8,11] but disagrees with [7] in which a magnetoconductance signature similar to our results for a charge puddle is observed in a setup without puddles, i.e., only considering magnetic field and disorder. However, this publication uses a modified calculation setup including an artificial mass term which might modify the magnetoconductance signature in a way that they are probing the regime of very strong disorder.…”

“…This is surprising as this theory assumes a fully coherent transport picture, something that one would rather assume to be fulfilled for the short but not for the longer samples. In addition, there are later theory calculations which use a comparable calculation setup [8,11] but show a much weaker influence of the magnetic field.…”

Effects of disorder on electron tunneling through helical edge states

“…Space ordering T in Eq. (19) is thus crucial in determining the evolution operator U p,E and gives rise to the local energy shift E → E − v F k (j ) p . In contrast, for piecewise constant φ p (x) and τ · b p,E (x), space ordering can be dropped within each interval, and no shift arises.…”

“…This property has inspired various investigations about the effects of disorder on helical states. On the one hand, various studies have tested such robustness to disorder when inelastic scattering is included [1,[5][6][7][8], possibly in interplay with Rashba impurities [9][10][11][12][13], or when time-reversal symmetry is broken either by magnetic impurities [14][15][16][17] or by applied magnetic fields [18][19][20]. On the other hand, it also been realized that disorder itself can cause an ordinary insulator to undergo a phase transition to a topological insulator (topological Anderson insulator) [21][22][23][24].…”

Phase-Tunable Thermal Rectification in the Topological SQUIPT

“…The magnetic field required to reach Φ 0 in a typical junction with W = 2 µm 2 , and L = ξ 0 = v F /∆ 0 ≈ 600 nm [54,76] is ≈ 1 mT. This field is sufficiently small such that superconductivity is not suppressed and no backscattering is induced or gap is opened in the helical edge channels, thus preserving topological properties of edge channels [58]. Taking the Doppler effect into account, one can derive the density of states in the upper edge of the junction in units of the edge state density of…”