Electrostatic Coupling between Two Surfaces of a Topological Insulator Nanodevice

We report on electronic transport measurements of dual-gated nanodevices of the low-carrier density topological insulator (TI) ${\mathrm{Bi}}_{1.5}{\mathrm{Sb}}_{0.5}{\mathrm{Te}}_{1.7}{\mathrm{Se}}_{1.3}$. In all devices, the upper and lower surface states are independently tunable to the Dirac point by the top and bottom gate electrodes. In thin devices, electric fields are found to penetrate through the bulk, indicating finite capacitive coupling between the surface states. A charging model allows us to use the penetrating electric field as a measurement of the intersurface capacitance $C_{\text{TI}}$ and the surface state energy-density relationship $\mu (n)$, which is found to be consistent with independent angle-resolved photoemission spectroscopy measurements. At high magnetic fields, increased field penetration through the surface states is observed, strongly suggestive of the opening of a surface state band gap due to broken time-reversal symmetry.

Figure 1

(a) Colorized AFM image of device $A$, including schematic circuit elements describing the transport measurement, where $V_{\text{XX}}$ is the longitudinal voltage drop and $I_{\text{SD}}$ is the source-drain voltage. Red (dark horizontal bar) is BSTS, blue (wide vertical bar, variable brightness) is h-BN, and gold (bright) is $\mathrm{Ti}/\mathrm{Au}$ (contacts and gate electrode). The scale bar is 2 microns. (b) TRARPES measurement of a BSTS crystal. The white line indicates the chemical potential.

Figure 2

Gate dependence of the resistivity of devices $A$ and $B$. (a) Bottom gate dependence of resistivity at $V_T=0$ at low temperature (blue, green) and 270 K (dashed line) from cooldown 2. (c) Top gate dependence of resistivity at $V_B=0$ at low temperature (blue, green) and 270 K (dashed line) from cooldown 2. (b), (d) 2D map of resistivity while modulating both gate electrodes for devices $B$ and $A$, respectively, from cooldown 1. The black dots identify $V_{\text{peak}}(V_B)$, the top gate voltage at which $R_{\text{peak}}$ is found as a function of $V_B$.

Figure 3

(a) Schematic of the charging model used in this study with important parameters labeled. For comparison to the experiment, the upper surface state is kept at charge neutrality while charge is distributed between the lower surface state and the gate electrodes. Three voltage loops indicated by the blue dashed lines are used in deriving the charging model. (b) $V_{\text{peak}}(V_B)$, i.e., the trajectory in gate voltage space of the upper surface resistance peak $R_{\text{peak}}$ (left, black dots), and the value of $R_{\text{peak}}$ itself at those gate voltages (right, blue), as extracted from Fig. 2. (c) The fit of the energy density relationship to the ARPES data (red line) by the transformed $V_{\text{peak}}(V_B)$ data (black dots).

Figure 4

Effect of high magnetic fields on the transport data. (a) $V_{\text{peak}}(V_B)$ at 0 T and 8 T from cooldown 2. For comparison, the dashed pink line would be the gate-gate dependence if the lower surface has no electronic states, given by a ratio of geometric capacitances: $\mathrm{C}\text{ratio}=-(1/C_T)(C_B{C}_{\text{TI}}/(C_B+C_{\text{TI}}))$. The transport data approach this slope at 8 T. (b) The extracted energy-density relationship of the lower surface state at 8 T for the case of fixed intersurface capacitance $C_{\text{TI}}=C_{\text{TI}}^0$ (blue) and when using $C_{\text{TI}}$ as a fit parameter (green) to the zero-field density of states (ARPES model, red curve). Arrows indicate the increase in the total chemical potential change assuming fixed $C_{\text{TI}}$. (c) The difference in the total change of the chemical potential of the lower surface with magnetic field (blue, left axis, error bars are the standard deviation of possible values) and the best fit $C_{\text{TI}}$ as a function of magnetic field (green, right axis, error bars are 90% confidence intervals). (d) The temperature dependence of the resistivity at different magnetic fields and when both surfaces are at charge neutrality.