Quantum Spin Hall Effect and Topological Phase Transition in HgTe Quantum Wells
B. Andrei Bernevig1,2,
Taylor L. Hughes1 and
Shou-Cheng Zhang1*
1 Department of Physics, Stanford University, Stanford, CA 94305, USA.
2 Kavli Institute for Theoretical Physics, University of California, Santa Barbara, CA 93106, USA.
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Fig. 1. (A) Bulk energy bands of HgTe and CdTe near the  point. (B) The CdTe-HgTe-CdTe quantum well in the normal regime E1 > H1 with d < dc and in the inverted regime H1 > E1 with d > dc. In this and other figures, 8/H1 symmetry is indicated in red and 6/E1 symmetry is indicated in blue.
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Fig. 2. (A) Energy of E1 (blue) and H1 (red) bands at k|| = 0 versus quantum well thickness d. (B) Energy dispersion relations E(kx,ky) of the E1 and H1 subbands at d = 40, 63.5, and 70 Å (from left to right). Colored shading indicates the symmetry type of the band at that k point. Places where the cones are more red indicate that the dominant state is H1 at that point; places where they are more blue indicate that the dominant state is E1. Purple shading is a region where the states are more evenly mixed. At 40 Å, the lower band is dominantly H1 and the upper band is dominantly E1. At 63.5 Å, the bands are evenly mixed near the band crossing and retain their d < dc behavior moving farther out in k-space. At 70 Å, the regions near k|| = 0 have flipped their character but eventually revert back to the d < dc farther out in k-space. Only this dispersion shows the meron structure (red and blue in the same band). (C) Schematic meron configurations representing the di(k) vector near the 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  . We measure all Hall conductances in electrical units. All of these plots are for Hg0.32Cd0.68Te-HgTe quantum wells.
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Fig. 3. (A) Experimental setup on a six-terminal Hall bar showing pairs of edge states, with spin-up states in green and spin-down states in purple. (B) A two-terminal measurement on a Hall bar would give GLR close to 2e2/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 µ2 – µ1 and µ4 – µ3 vanish on the QSH side with a power law as the zero temperature limit is approached. The spin Hall conductance  has a plateau with the value close to 2e2/h.
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