Tunable surface conductivity in Bi${}_2$Se${}_3$ revealed in diffusive electron transport

We demonstrate that the weak antilocalization effect can serve as a convenient method for detecting decoupled surface transport in topological insulator thin films. In the regime where a bulk Fermi surface coexists with the surface states, the low-field magnetoconductivity is well described by the Hikami-Larkin-Nagaoka equation for single-component transport of noninteracting electrons. When the electron density is lowered, the magnetotransport behavior deviates from the single-component description and strong evidence is found for independent conducting channels at or near the bottom and top surfaces. The magnetic-field-dependent part of corrections to conductivity due to Zeeman energy is shown to be negligible for the fields relevant to the weak antilocalization despite considerable electron-electron interaction effects on the temperature dependence of the conductivity.

Figure 1

(a) Magnetoresistance [MR, defined as ${\rho }_{\mathrm{xx}}(B)/{\rho }_{\mathrm{xx}}(0)-1$] and Hall resistance of a typical Bi${}_2$Se${}_3$ thin film at $T=1.2$ K. The low-field MR (between $\pm 3$ T) is shown more clearly in the lower inset. The upper inset is an optical image of the Hall bar used for the transport measurements. (b) Magnetoconductivity data (symbols) from four samples fitted with Eq. (1) (lines). The dephasing field $B_{\varphi }$ varies by nearly a factor of 30. Extracted values of $\alpha$ for ten samples are plotted as a function of electron density $n_e$ and mobility $\mu$ in (c) and (d), respectively. They are evaluated with $n_e=1/({\mathit{eR}}_H)$ and $\mu =R_H/{\rho }_{\mathrm{xx}}(0)$ at low fields.

Figure 2

(a) Magnetoconductivity $\Delta \sigma (B)$ at $T=2$ K and $V_G=0\hspace{0.5em}\mathrm{V}$ plotted as a function of perpendicular magnetic field $B_{\perp }$ for several tilt angles. $\theta =0$ refers to $\mathbf{B}$ perpendicular to the thin-film plane. (b) Temperature dependencies of ${\sigma }_{\mathrm{xx}}$ (open symbols) recorded at $B=$ 0, 0.2, 1, and 5 T. The straight lines are linear fits of ${\sigma }_{\mathrm{xx}}$ to $\ln \hspace{0.5em}T$. In the upper inset, the slope, defined as $\kappa =(\pi h/e^2)d{\sigma }_{\mathrm{xx}}(B,T)/d(\ln \hspace{0.5em}T)$, is plotted as a function $B$. The electron density is about $2\times {10}^{13}$ cm${}^{-2}$ so that $E_F$ is located in the conduction band.

Figure 3

(a) Hall resistance curves for $V_G=-125,\hspace{0.5em}-90,\hspace{0.5em}0,\hspace{0.5em}-170,\hspace{0.5em}\text{and}\hspace{0.5em}-180$ V (from top to bottom). (b) Gate-voltage dependence of ${\rho }_{\mathrm{xx}}$ at $B=0$ and high-field Hall coefficient, defined as ${\mathit{dR}}_{\mathrm{xy}}/\mathit{dB}$ and fitted from the data in $B=16$–18 T. (c) $V_G$ dependence of $\alpha$ and $B_{\varphi }$ obtained from fits to Eq. (1). The left and right insets show the schematic sketches of the band diagrams for large and small negative gate voltages, respectively. The top (bottom) surface is depicted on the left (right). (d) $V_G$ dependence of $B_{\varphi 1}$ (hexagons) and $B_{\varphi 2}$ (triangles) obtained from fits to Eq. (2). $B_{\varphi 1}$ and $B_{\varphi 2}$ can be assigned to the bottom and top surfaces, respectively. This 10-nm-thick sample only has a back gate.