Oscillatory angular dependence of the magnetoresistance in a topological insulator Bi$_{1}$$_{−}$ x Sb x A. A. Taskin, Kouji Segawa, and Yoichi Ando Institute of Scientific and Industrial Research, Osaka University, Ibaraki, Osaka 567-0047, Japan The angular-dependent magnetoresistance and the Shubnikov-de Haas oscillations are studied in a topological insulator Bi 0 . 91 Sb 0 . 09 , where the two-dimensional (2D) surface states coexist with a three-dimensional (3D) bulk Fermi surface (FS). Two distinct types of oscillatory phenomena are discovered in the angular-dependence: The one observed at lower fields is shown to originate from the surface state, which resides on the (2 ¯ 1 ¯ 1) plane, giving a new way to distinguish the 2D surface state from the 3D FS. The other one, which becomes prominent at higher fields, probably comes from the (111) plane and is obviously of unknown origin, pointing to new physics in transport properties of topological insulators. PACS numbers: 71.18.+y, 72.25.-b, 73.25.+i, 85.75.-d The three-dimensional (3D) topological insulator hosts a novel surface state whose existence is guaranteed by a topological principle characterized by the Z 2 invariant [1– 3], which makes it an interesting playground for exploring the role of topology in condensed matter [4–20]. The key feature of the topological insulator is that an odd num- ber of massless, spin-helical Dirac cones comprise the 2D surface state, whose gapless nature is protected by the time-reversal symmetry. Such an intrinsically conduct- ing surface is supported by an insulating bulk, where the energy gap is created by a strong spin-orbit cou- pling. To understand the macroscopic properties of this novel surface state and to investigate the possibility of its device applications, transport studies are obviously important. However, in actual samples of topological in- sulators, there is always some bulk conductivity due to residual carriers, and separating the contributions from 2D and 3D states turns out to be challenging [13–15]. Recently, we observed [13] strong de Haas-van Alphen (dHvA) oscillations in high-quality bulk single crystals of Bi 1 $_{$_{−}$x}$Sb$_{x}$ alloy in the “insulating” regime (0.07 ≤ x ≤ 0.22), which is the first material to be known as a 3D topological insulator [2, 7]. The dHvA oscillations sig- nified a previously-unknown Fermi surface (FS) with a clear 2D character that coexists with a 3D bulk FS [13]. Since Bi 1 $_{$_{−}$x}$Sb$_{x}$ is a 3D material, the observed 2D FS is naturally assigned to the surface. In the present work, to specifically probe low-dimensional properties of this material, we have extended our study to the angular de- pendence of the magnetoresistance. This method was successfully applied to the studies of quasi-2D organic conductors in the 1980s, resulting in the discovery of the celebrated angular-dependent magnetoresistance os- cillations (AMRO) [21–23], and later extended to quasi- one-dimensional conductors [24, 25]. In this Letter, we present a detailed study of the magnetotransport in a Bi $^{0}$.91 Sb $^{0}$.09 single crystal, where we have found oscilla- tory angular dependences in both the resistivity ρ$_{xx}$ and the Hall resistivity ρ$_{yx}$ . The oscillatory magnetoresis- tance (MR) is obviously different from known AMRO Figure 2 magnifies the “low-field” oscillations of ρ$_{xx}$ ( θ ), which become prominent above 1 T. These oscillations are symmetric with respect to the C 3 axis ( θ = 0 ◦ ) and show a clear tendency of shifting the peak positions closer to the center with increasing magnetic field. It is impor- tant to notice that this strong field dependence is very different from ordinary AMRO [23] where the angular positions of peaks are determined by commensurate mo- Let us start by presenting our main observations. Fig- ure 1(a) shows the angular dependences of the trans- verse MR measured in magnetic fields rotated within the trigonal-binary ( C 3 - C 2 ) plane perpendicular to the cur- rent. The magnetic-field strength, ranging from 0.01 T to 16 T, were kept constant during each rotation. As clearly seen in Figs. 1–3, pronounced oscillations of both ρ$_{xx}$ and ρ$_{yx}$ as a function of the rotation angle θ appear for B ≳ 1 T. Two different types of oscillations can be distinguished: The first type consists of oscillations ap- pearing at lower fields (Figs. 1 and 2), while the second one becomes prominent at higher fields (Figs. 1 and 3). These data were taken immediately after the sample sur- face was refreshed by a chemical etching [26]. High-quality Bi 1 $_{$_{−}$x}$Sb$_{x}$ crystals were grown from a sto- ichiometric mixture of 99.9999% purity Bi and Sb ele- ments by a zone melting method [13]. The resistivity was measured by a standard four-probe method on a rectan- gular sample. In this Letter, we focus on a x = 0.09 sam- ple in which the current was directed along the C 1 axis. Continuous rotations of the sample in constant magnetic fields was used to measure the angular dependence of the MR within two main crystallographic planes ( C 3 - C 2 and C 3 - C 1 ). To observe the SdH oscillations, ρ$_{xx}$ ( B ) and ρ$_{yx}$ ( B ) were also measured by sweeping B between +14 and -14 T along a set of magnetic-field directions. [25], and it is comprised of two components: one is a pe- culiar manifestation of the 2D FS, and the other appears to be a fundamentally new effect. The former provides a new way to distinguish the contributions of the 2D FS, while the latter points to new physics in topological in- sulators. arXiv:1001.1607v2 [cond-mat.mtrl-sci] 2 May 2010