Temperature-Induced Topological Phase Transition in HgTe Quantum Wells

We report a direct observation of temperature-induced topological phase transition between the trivial and topological insulator states in an HgTe quantum well. By using a gated Hall bar device, we measure and represent Landau levels in fan charts at different temperatures, and we follow the temperature evolution of a peculiar pair of “zero-mode” Landau levels, which split from the edge of electronlike and holelike subbands. Their crossing at a critical magnetic field $B_c$ is a characteristic of inverted band structure in the quantum well. By measuring the temperature dependence of $B_c$, we directly extract the critical temperature $T_c$ at which the bulk band gap vanishes and the topological phase transition occurs. Above this critical temperature, the opening of a trivial gap is clearly observed.

Figure 1

(a) Hall resistance $R_{xy}$ and (b) SdH oscillations, $R_{xx}=R_{28,34}$, at 1.7 K at zero gate voltage. The insert schematically shows a picture of the Hall bar. (c) Hall conductivity ${\sigma }_{xy}$ and its derivative $\partial {\sigma }_{xy}/\partial V_G$ as a function of gate voltage at 1.7 K and a magnetic field of 2.4 T.

Figure 2

(a) Evolution of $E1$ and $H1$ subbands (at $\mathbf{k}=0$) and band structure in the sample with temperature. Blue and red shading indicate contributions from electronlike and holelike states, respectively, at a given quasimomentum $\mathbf{k}$ point. (b)–(d) LL fan chart at different temperatures: (b) 1.7 K (TI phase), (c) 27 K (critical gapless state), and (d) 40 K (NI phase). A pair of zero-mode LLs is presented by red curves.

Figure 3

(a)–(d) Color map of $\partial {\sigma }_{xy}/\partial V_G$ as a function of both magnetic field and gate voltage, at (a) 1.7 K, (b) 20 K, (c) 27 K, and (d) 40 K. The white solid curves correspond to ${\sigma }_{xy}=(n+1/2)e^2/h$. In the NI state ($B>2.2\mathrm{T}$), the isolines of ${\sigma }_{xy}=\pm e^2/2h$ are linearly fitted by the red dotted lines giving the values of $B_c$. (e)–(h) The false-color maps show the longitudinal resistivity at the same temperatures as in the respective top panels. The white curves correspond to the isolines of ${\rho }_{xx}=h/e^2$. The red dotted lines are the linear fits of these curves in the field range of $B=2.2–6\mathrm{T}$. The crossing point of the two red lines gives another estimate for $B_c$.

Figure 4

Theoretical (solid curve) and experimental (open symbols) values of the critical magnetic field $B_c$ as a function of temperature. The values marked by the red symbols are obtained from the data set in the top panels of Fig. 3. The blue symbols mark the values of $B_c$ extracted from ${\rho }_{xx}$. The sparse and white regions correspond to the inverted and trivial band ordering, respectively. The inset shows $T_c$ as a function of $d$ in (013) $\mathrm{HgTe}/{\mathrm{Cd}}_{0.65}{\mathrm{Hg}}_{0.35}\mathrm{Te}$ QWs grown on a CdTe buffer. The open symbol represents the experimental value measured in our sample.