Anomalous Conductance Oscillations in the Hybridization Gap of $\mathrm{InAs}/\mathrm{GaSb}$ Quantum Wells

We observe the magnetic oscillation of electric conductance in the two-dimensional $\mathrm{InAs}/\mathrm{GaSb}$ quantum spin Hall insulator. Its insulating bulk origin is unambiguously demonstrated by the antiphase oscillations of the conductance and the resistance. Characteristically, the in-gap oscillation frequency is higher than the Shubnikov–de Haas oscillation close to the conduction band edge in the metallic regime. The temperature dependence shows both thermal activation and smearing effects, which cannot be described by the Lifshitz-Kosevich theory. A two-band Bernevig-Hughes-Zhang model with a large quasiparticle self-energy in the insulating regime is proposed to capture the main properties of the in-gap oscillations.

Figure 1

(a) 2D conductance map as a function of the front gate bias $V_f$ and the perpendicular magnetic field $B_{\perp }$, which is measured with a Corbino device (optical microscope image in inset) at 300 mK. The conductance traces at three gate biases are plotted in (c) to represent the electron ($V_f=-0.2\mathrm{V}$, blue), the hole ($V_f=-0.9\mathrm{V}$, red), and the insulating ($V_f=-0.75\mathrm{V}$, black) regimes, respectively. The inverted band structure is schematically plotted in (b), where the Fermi energies of the three curves are labeled.

Figure 2

(a) Anomalous magnetic oscillations of $G_{xx}$ and $R_{xx}$ in the hybridization gap. (b) Filling factor index diagram. The fitting yields $n_0=(1.63\pm 0.05)\times {10}^{11}{\mathrm{cm}}^{-2}$. (Inset) Electron density $n$ in the conduction band (blue dots) and $n_0$ in the hybridization gap (red dot), and zero-field conductance $G_{xx}$ (black curve) vs $V_f$. The dashed line is a guide for the eye. (c) Energy dispersion (real part of the eigenvalues) of $H=H_0+\mathrm{\Sigma }$ with $n_0=1.6\times {10}^{11}{\mathrm{cm}}^{-2}$. Blue lines, ${\mathrm{\Gamma }}_e={\mathrm{\Gamma }}_h=0$; red lines, ${\mathrm{\Gamma }}_e=1\mathrm{meV}$, ${\mathrm{\Gamma }}_h=16\mathrm{meV}$. Also shown is the dispersion without hybridization (black dashed lines). (d) Electron density fitted from the density of states oscillations calculated from the two-band model. Blue dots, in the metallic regime, ${\mathrm{\Gamma }}_e=0.2\mathrm{meV}$, ${\mathrm{\Gamma }}_h=0.4\mathrm{meV}$. Red squares, in the insulating regime, ${\mathrm{\Gamma }}_e=1\mathrm{meV}$, ${\mathrm{\Gamma }}_h=16\mathrm{meV}$. $n_0$ is indicated by the gray line. The blue vertical line indicates the conduction band edge. The dashed line is the Fermi surface area in the conduction band when the self-energy is turned off.

Figure 3

Temperature dependence of the in-gap oscillations. (a) 2D conductance map in the insulator regime measured at different temperatures. In order to compare the oscillation amplitudes at different temperatures, we remove the background by taking the second-order derivative, $-{\partial }_{B^{-1}}^2{G}_{xx}$ in (b) at $V_f=-0.85\mathrm{V}$ and (c) at $V_f=-0.75\mathrm{V}$.

Figure 4

(a) The in-gap oscillations measured with different back gate biases $V_b$. The band inversion gradually decreases as $V_b$ is tuned from 3 to $-1\mathrm{V}$ and simultaneously $V_f$ is adjusted to hold the chemical potential inside the gap. (b) The in-gap oscillations measured in tilted magnetic fields.