Tunable Coupling between Surface States of a Three-Dimensional Topological Insulator in the Quantum Hall Regime

The paired top and bottom Dirac surface states, each associated with a half-integer quantum Hall (QH) effect, and a resultant integer QH conductance ($\nu e^2/h$), are hallmarks of a three-dimensional topological insulator (TI). In a dual-gated system, chemical potentials of the paired surface states are controlled through separate gates. In this work, we establish tunable capacitive coupling between the surface states of a bulk-insulating TI ${\mathrm{BiSbTeSe}}_2$ and study the effect of this coupling on QH plateaus and Landau level (LL) fan diagram via dual-gate control. We observe nonlinear QH transitions at low charge density in strongly coupled surface states, which are related to the charge-density-dependent coupling strength. A splitting of the $N=0$ LL at the charge neutrality point for thin devices (but thicker than the 2D limit) indicates intersurface hybridization possibly beyond single-particle effects. By applying capacitor charging models to the surface states, we explore their chemical potential as a function of charge density and extract the fundamental electronic quantity of LL energy gaps from dual-gated transport measurements. These studies are essential for the realization of exotic quantum effects such as topological exciton condensation.

Figure 1

2D color maps of $R_{xx}$ as functions of $V_{\mathrm{tg}}$ and $V_{\mathrm{bg}}$ for BSTS devices with flake thickness of (a) 89, (b) 31, (c) 16, and (d) 10 nm. (e)–(h) Plots of $R_{xx}$ versus $V_{\mathrm{bg}}$ extracted from the corresponding maps in (a)–(d) at different $V_{\mathrm{tg}}$ indicated by the arrows. The inset in (e) is a schematic of cross-sectional illustration of the device.

Figure 2

2D color maps of ${\sigma }_{xx}$ and ${\sigma }_{xy}$ as a function of dual-gated voltages for the (a),(b) 89, (d),(e) 31, (g),(h) 16, and (j),(k) 10 nm BSTS devices measured at 18 T. The LL indices for top and bottom surfaces, $N_t$ and $N_b$ are labeled in the $y$ axis and $x$ axis, respectively. Line profiles of ${\sigma }_{xx}$ and ${\sigma }_{xy}$ as a function of $V_{\mathrm{bg}}$ extracted from the corresponding maps near the overall CNP (as indicated by the arrows) for the (c) 89, (f) 31, (i) 16, and (l) 10 nm BSTS devices. The vertical black dashed line and gray highlight in (c), (f), (i), and (l) denote the overall CNP and $\nu =0$ QH plateau, respectively. The LL filling factors for top and bottom surfaces (${\nu }_t$, ${\nu }_b$) and total, $\nu ={\nu }_t+{\nu }_b$ are indexed in the figure.

Figure 3

(a) Plot of $V_{tD}\text{−}{V}_D$ as a function of $n_b$ for different thickness BSTS at 18 T. Inset in (a) plots the splitting of $N_t$ and $N_b$ at overall CNP in total charge density ($\mathrm{\Delta }n$) as a function of flake thickness $d$. (b) Plot of $n_{tD}$ (as the $n_b$ is tuned from bottom Dirac point to $1\times {10}^{12}{\mathrm{cm}}^{-2}$) and ${ϵ}_{\mathrm{BSTS}}$ as a function of $d$. The black and blue dashed lines in (b) are the fittings of $n_{tD}$ and ${ϵ}_{\mathrm{BSTS}}$, respectively, with $d$. Inset in (b) is a schematic of the top and bottom Dirac surface states. (c) Plot of ${\mu }_b$ versus $n_b$ for the 16 nm BSTS device at magnetic field of 0 and 18 T. The color highlights in (c) display the LL indices of bottom surface $N_b$ from $-2$ (leftmost) to $+2$ (rightmost) formed at 18 T. Inset in (c) is the $\mathrm{\Delta }{\mu }_b={\mu }_b(18\mathrm{T})\text{−}{\mu }_b(0\mathrm{T})$ versus $n_b$. (d) Plot of $\mathrm{\Delta }{\mu }_b$ for $N_b=\pm 1$ versus magnetic field. Inset in (d) is the ${\mathrm{\Delta }}_{\pm 1}$ versus $B^{1/2}$.