In this work, we investigated the response of two-dimensional topological insulators in high perpendicular magnetic fields. While band structure calculations suggest that the inversion of Landau levels, due to which the quantum spin Hall effect was observed in the first place, is destroyed at a critical magnetic field leading to a trivial insulator with a band gap. However, experimentally we observe the absence of a trivial insulating gap in finite-size 2D topological insulators based on HgTe quantum wells. Instead, we observe that the topological edge channel (from the quantum spin Hall effect) coexists with a quantum Hall edge channel at magnetic fields at which the transition from topological to trivial insulator is expected to occur. This happens due to a suitable potential landscape created by the charge puddles. Devices fabricated using a wet-etch process, where there are lesser charge puddles show the expected transition from topological to a trivial insulating state.

The conductance quantization due to the quantum spin Hall effect is robust even in the presence of magnetic impurities. Our experiments on (Hg,Mn)Te quantum wells with an inverted band structure reveal that the quantum spin Hall quantization is observed only at low temperatures (T < 400 mK), where Kondo screening of the magnetic impurities suppresses backscattering.

Alloying HgTe quantum wells with dilute concentrations of Mn atoms alter the band structure. We use k.p band structure calculations to map out the phase space of (Hg,Mn)Te quantum wells. The thickness at which the transition from trivial to topological insulator occurs now depends on the concentration of Mn atoms. The pink-shaded region in the left panel of the figure above shows the thickness and concentration of Mn atoms for which the (Hg,Mn) Te quantum wells have an inverted band structure and hosts the quantum spin Hall effect. The gray dashed line shows the transition from a direct band gap to an indirect band gap (examples of the two types of band structure is shown in the middle panel of the above figure). For a (Hg,Mn)Te quantum well with a direct band gap, but very close to the direct-to-indirect band gap transition, we observe quantum Hall effects at extremely low magnetic fields (below 50 mT). We name them emergent quantum Hall effects since their existence depends in the preexistence of the quantum spin Hall edge channels (and hence an inverted band structure). We observe two types of emergent quantum Hall effects:

 1. When the chemical potential is tuned to the bulk band gap, we observe a nu=-1 quantum Hall plateau at very low magnetic fields.

2. When the chemical potential is tuned to bulk valence band, we observe a series of quantum Hall plateaus from nu=-1 to nu=-5 starting from magnetic fields as low as 25 mT. 

The first effect to due to the removal of the degeneracy of the quantum spin Hall edge channels and the subsequent hybridization of one component of the quantum spin Hall edge channels with the bulk valence band states. The second effect is due to the formation of the extremely low-density two-dimensional hole gas (10^9 cm^-2), paired with extremely high mobility (1 million cm^2/Vs). This happens because of the pinning of the chemical potential by the band inversion-induced van Hove singularity in the density of states of the valence band (called as camel back structure) and subsequent localization of carriers in the camel back at the lowest temperatures (20 mK).                                                                              

The quantum spin Hall effect was first experimentally observed in 2007 in HgTe/HgCdTe quantum wells at the interface of two materials with mutually inverted band structures.  The conductance of the counterpropagating helical edge channels of the quantum spin Hall effect is quantized to 2e2/h in the absence of any magnetic field. The conductance quantization has been observed in devices whose dimensions are a few microns. Conventionally these devices were fabricated using dry-etching with high-energy Ar ions, which inevitably leads to a reduction in the mobility of the charge carriers, with the maximum reduction being in the devices of the smallest dimensions. Over the years, we have optimized a chemical wet-etching process, that causes minimal damage. The microstructures fabricated using wet-etching have mobility comparable to macrostructures and show clean SdH oscillations and quantum Hall plateaus. For more details see K. Bendias et al., Nanoletters 18, 4381 (2018).

We used universal conductance fluctuations (UCF) as a probe for the breakdown of time-reversal symmetry. The UCF magnitude (estimated by measuring the low-frequency noise) was measured as a function of a perpendicular magnetic field for devices of varying density. For low-doped devices, we find that the UCF magnitude is suppressed at zero magnetic fields which indicates a spontaneous breakdown of time-reversal symmetry. We attribute it to increased effective Coulomb interaction for low-doped devices.