Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells

TL;DR

Problem: demonstrate a Quantum Spin Hall state in HgTe/CdTe quantum wells and identify a thickness-driven topological phase transition. Approach: derive a 4×4 effective Dirac-like Hamiltonian for E1/H1 subbands using k·p theory and envelope functions, and show the mass term changes sign at a critical thickness. Findings: the inverted regime hosts a QSH phase with a single pair of helical edge states and a topological Z2 distinction from the normal insulator, accompanied by a predicted jump in spin-Hall conductance. Significance: provides a concrete material platform and experimental transport signatures for observing the QSH effect and thickness-tuned topology.

Abstract

We show that the Quantum Spin Hall Effect, a state of matter with topological properties distinct from conventional insulators, can be realized in HgTe/CdTe semiconductor quantum wells. By varying the thickness of the quantum well, the electronic state changes from a normal to an "inverted" type at a critical thickness $d_c$. We show that this transition is a topological quantum phase transition between a conventional insulating phase and a phase exhibiting the QSH effect with a single pair of helical edge states. We also discuss the methods for experimental detection of the QSH effect.

Figures (4)

• Figure 1: (A) Bulk energy bands of HgTe and CdTe near the $\Gamma$ point. (B) The CdTe/HgTe/CdTe quantum well in the normal regime $E1> H1$ with $d< d_c$ and in the inverted regime $H1> E1$ with $d> d_c$. In this, and all subsequent figures $\Gamma_8$/H1 ($\Gamma_6$/E1) symmetry is correlated with the color red (blue).
• Figure 2: (A) Energy (eV) of $E1$ (blue) and $H1$ (red) bands at $k_{\parallel}=0$ vs. quantum-well thickness $d$ ($\AA$). (B) Energy dispersion relations $E(k_x,k_y)$ of the $E1,H1$ subbands at $40 \AA,$$63.5 \AA$ and $70 \AA$ from left to right. Colored shading indicates the symmetry type of band at that $k$-point. Places where the cones are more red (blue) indicates that the dominant states are H1 (E1) states at that point. Purple shading is a region where the states are more evenly mixed. For $40\AA$ the lower (upper) band is dominantly H1(E1). At $63.5\AA$ the bands are evenly mixed near the band crossing and retain their $d<d_c$ behavior moving further out in $k$-space. At $d=70\AA$ the regions near $k_{\parallel}=0$ have flipped their character but eventually revert back to the $d<d_c$ further out in $k$-space. Only this dispersion shows the meron structure (red and blue in the same band). (C) Schematic meron configurations representing the $d_i(k)$ vector near the $\Gamma$ point. The shading of the merons has the same meaning as the dispersion relations above. The change in meron number across the transition is exactly equal to $1$, leading to a quantum jump of the spin Hall conductance $\Delta\sigma_{xy}^{(s)}=2e^2/h.$ We measure all Hall conductances in electrical units. All of these plots are for Hg$_{0.32}$Cd$_{0.68}$Te/HgTe quantum wells.
• Figure 3: (A) Experimental setup on a six terminal Hall bar showing pairs of edge states with spin up (down) states green (purple). (B)A two-terminal measurement on a Hall bar would give $G_{LR}$ close to $2e^2/h$ contact conductance on the QSH side of the transition and zero on the insulating side. In a six-terminal measurement, the longitudinal voltage drops $\mu_2 -\mu_1$ and $\mu_4 - \mu_3$ vanish on the QSH side with a power law as the zero temperature limit is approached. The spin-Hall conductance $\sigma_{xy}^{(s)}$ has a plateau with the value close to $2 \frac{e^2}{h}.$
• Figure 4: Dispersion relations for the $E1$ and $H1$ subbands for(A) $d=40 \AA$ (B) $d=63.5\AA$ (C) $d=70\AA.$