Parallel field magnetoresistance in topological insulator thin films

We report that the finite thickness of three-dimensional topological insulator (TI) thin films produces an observable magnetoresistance (MR) in phase coherent transport in parallel magnetic fields. The MR data of Bi${}_2$Se${}_3$ and (Bi${}_{1-x}$Sb${}_x$)${}_2$Te${}_3$ thin films are compared with existing theoretical models of parallel field magnetotransport. We conclude that the TI thin films bring parallel field transport into a unique regime in which the coupling of surface states to bulk and to opposite surfaces is indispensable for understanding the observed MR. The $\beta$ parameter extracted from parallel field MR can in principle provide a figure of merit for searching TI compounds with more insulating bulk than existing materials.

Figure 1

(a) A possible closed-loop electron trajectory in a topological insulator thin film (left) and its cross section projected perpendicular to the parallel magnetic field (right). The electron's motion may include transport on the surface and in the bulk, as well as surface-bulk and the intersurface scatterings. (b) Cross-sectional view of electron trajectories illustrating the Aharonov-Bohm (AB) phase induced by a parallel magnetic field in various transport regimes. Shown from top to bottom are AA (Ref. 54), DK (Ref. 63), BvH (Ref. 64), and RV (Ref. 65) regimes. See Eq. (1) for a definition of $\beta$.

Figure 2

Magnetotransport data of Bi${}_2$Se${}_3$ thin films in perpendicular and parallel magnetic fields. (a) Magnetoconductivity (MC) in perpendicular fields $\Delta {\sigma }_{\perp }\left(B\right)$, and MC in parallel fields $\Delta {\sigma }_{\parallel }\left(B\right)$ for films with thickness $d=7$ and 40 nm. (b) $\Delta {\sigma }_{\perp }\left(B\right)$ for various thicknesses (symbols: experiment; lines: fits to the HLN equation). (c) Extracted parameter $\alpha$ vs the film thickness $d$. (d) $\Delta {\sigma }_{\parallel }\left(B\right)$ for various thicknesses [symbols: experiment; lines: fits to Eq. (1)]. (e) Thickness dependence of $\beta$. The data in (a)–(e) were obtained at $T\approx 2$ K. (f), (g) $\Delta {\sigma }_{\parallel }\left(B\right)$ at various temperatures for films with (g) $d=7$ nm and (h) $d=30$ nm. (h) $T$ dependence of $\beta$ for $d=7$–30 nm. The inset shows a schematic band diagram of the Bi${}_2$Se${}_3$ films.

Figure 3

Magnetoconductivity of an 18 nm thick (Bi,Sb)${}_2$Te${}_3$ film at $T=1.5$ K. (a) Hall resistance $R_{\mathrm{xy}}$ at various gate voltages. (b) Gate voltage dependencies of the longitudinal resistance $R_{\mathrm{xx}}$ (squares) and the Hall coefficient $R_H$ (circles). In (a) and (b), the sign of $R_{\mathrm{xy}}$ and $R_H$ is set to $+1$ for electrons. (c) MC in perpendicular fields for various gate voltages. (d) MC in parallel fields. (e) Gate voltage dependence of $\alpha$. (f) Gate voltage dependence of $\beta$.

Figure 4

(a) Temperature dependence of $\alpha$ at $V_G=-150$ V (hexagons) and $+200$ V (squares). (b) Schematic band diagrams for these two gate voltages. (c) $T$ dependencies of $\beta$ at $V_G=-150$ V and $+200$ V. (d) $\beta$ plotted as a function of $1/T$ (squares, $V_G=+200$ V) and as a function of the coupling strength ${\tau }_{\varphi }/{\tau }_s$ [line, symmetric bilayer in RV theory (Ref. 65)].