Anomalous Edge Transport in the Quantum Anomalous Hall State

Jing Wang, Biao Lian, Haijun Zhang, and Shou-Cheng Zhang

Phys. Rev. Lett. 111, 086803 – Published 20 August 2013

Abstract

We predict by first-principles calculations that thin films of a Cr-doped $(Bi, Sb)_{2} {Te}_{3}$ magnetic topological insulator have gapless nonchiral edge states coexisting with the chiral edge state. Such gapless nonchiral states are not immune to backscattering, which would explain dissipative transport in the quantum anomalous Hall (QAH) state observed in this system experimentally. Here, we study the edge transport with both chiral and nonchiral states by the Landauer-Büttiker formalism and find that the longitudinal resistance is nonzero, whereas Hall resistance is quantized to $h / e^{2}$ . In particular, the longitudinal resistance can be greatly reduced by adding an extra floating probe even if it is not used, while the Hall resistance remains at the quantized value. We propose several transport experiments to detect the dissipative nonchiral edge channels. These results will facilitate the realization of pure dissipationless transport of QAH states in magnetic topological insulators.

Received 7 June 2013

DOI:https://doi.org/10.1103/PhysRevLett.111.086803

Authors & Affiliations

Jing Wang, Biao Lian, Haijun Zhang, and Shou-Cheng Zhang

Department of Physics, McCullough Building, Stanford University, Stanford, California 94305-4045, USA

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Vol. 111, Iss. 8 — 23 August 2013

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Images

Figure 1
Band structures for five QLs and three QLs ${Cr}_{0.15} ({Bi}_{0.1} {Sb}_{0.9})_{1.85} {Te}_{3}$ without an exchange field are plotted in (a) and (c), respectively. The dashed line indicates the Fermi level. The inset of (a) shows the 2D Brillouin zone with high-symmetry $k$ points $Γ (0, 0)$ , $K (π, π)$ , and $M (π, 0)$ labeled, and that of (c) is the top view of a 2D thin film with two in-plane lattice vectors $a_{1}$ and $a_{2}$ . The 1D edges are indicated by the dashed lines, edge A and edge B. The energy dispersion of a thin film along edge A is plotted for (b) five QLs with exchange field 0.02 eV and (d) three QLs with exchange field 0.05 eV. Here, the warmer colors (lighter gray in gray scale) represent the higher local density of states, with red (light gray) and blue (dark gray) regions indicating 2D bulk energy bands and energy gaps, respectively. The gapless edge states can be clearly seen around the $Γ$ point as red (light gray) lines dispersing in the 2D bulk gap. One gapless chiral edge state $Σ_{1}$ and one pair of gapless quasihelical edge states $Λ_{1}$ coexist in (b), while only one gapless chiral edge state $Σ_{1}$ exists in (d).Reuse & Permissions
Figure 2
Hall bridge and transport properties. (a) Schematic drawing of a Hall bar device with both quasihelical edge channels (dashed lines) and a chiral edge channel (solid lines). The current is from terminals 1 to 4; voltage leads are on electrodes 2, 3, 5, and 6. (b) Voltage at terminals 1–6 for transport of the chiral, helical, and chiral and quasihelical edge states. The transport with both chiral and quasihelical edge channels (solid lines) show non-Ohmic behaviors of $ρ_{x x}$ . (c) $ρ_{x x}$ and $ρ_{x y}$ vs $r$ with different numbers of effective voltage leads on each side of the sample.Reuse & Permissions
Figure 3
Six-terminal Hall and nonlocal measurements. (a) Standard Hall measurement with six terminals and (b) corresponding voltages. The current is through 1 to 4, and the Hall voltage is measured between 2 and 6. Terminal 6 (denoted as $6^{'}$ ) is not be symmetric to terminal 2 due to misalignment, so the Hall signal may contain some longitudinal component. (c) Nonlocal measurement and (d) voltage. The current is through 1 to 2. In (b) and (d), the voltages with downward and upward magnetic orderings are denoted as solid red and dashed blue lines, respectively.Reuse & Permissions
Figure 4
Transport measurements with different numbers of terminals and device geometry. (a) Standard Hall measurement with extra floating terminals $2^{'}$ and $6^{'}$ inserted at the edges. $I$ , 1–4; $V_{x y}$ , 2–6; and $V_{x x}$ , 2–3. (b) The voltage at terminals 1–6 of (a) in the presence of extra floating probes. $n$ denotes the total numbers of voltage probes on one side. The dashed line denotes the chiral edge state transport, which is not affected by extra floating probes. (c) Nonlocal four-terminal resistance and two-terminal resistance measurements on the H-bar device.Reuse & Permissions

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