Weak localization effects as evidence for bulk quantization in Bi${}_2$Se${}_3$ thin films

Strong spin-orbit coupling in topological insulators results in the ubiquitously observed weak antilocalization feature in their magnetoresistance. Here we present magnetoresistance measurements in ultrathin films of the topological insulator Bi${}_2$Se${}_3$ and show that in the two-dimensional quantum limit, in which the topological insulator bulk becomes quantized, an additional negative magnetoresistance feature appears. Detailed analysis associates this feature with weak localization of the quantized bulk channels, thus providing evidence for this quantization. Examination of the dephasing fields at different temperatures indicates different scattering mechanism in the bulk vs the surface states.

Figure 1

(a) X-ray diffraction data on all samples showed clear $c$-oriented Bi${}_2$Se${}_3$ peaks. The streaky RHEED pattern of S1 (inset on the bottom left) taken in situ indicates good crystallinity and smooth surface. (b) ARPES of S4, showing a relatively low Fermi energy above the Dirac point.

Figure 2

Magnetoresistance of samples S1, S2, and S3, measured at 2 K. Colored lines (yellow/orange/brown) are the measured data, while the black dashed lines are the fittings, which assume two surface states (both with infinite SO interaction) and two bulk bands (both with zero or small effective SO interaction). The inset shows the same data and fit curves at low magnetic fields. The fitting parameters are given in Table .

Figure 3

Magnetoresistance of sample S3, measured at temperatures ranging from 2 to 24 K. Data are presented together with our fitting curve (dashed black line), which assumes two surface states (both with infinite SO interaction) and two bulk bands (both with zero or small effective SO interaction). The fitting parameters are given in Table .

Figure 4

Dephasing fields as a function of temperature, calculated from fitting the MR of sample S3. The top plot shows the full $H_{\varphi }$ range, up to 26 T, while the bottom plot zooms in on the low $H_{\varphi }$ range. The dashed lines are a guide to the eye, showing the linear temperature dependence of the surface states’ $H_{\varphi }$ and quadratic temperature dependence of the bulk channels’ $H_{\varphi }$ on the temperature.