Transport Theory of Half-Quantized Hall Conductance in a Semimagnetic Topological Insulator

Humian Zhou, Hailong Li, Dong-Hui Xu, Chui-Zhen Chen, Qing-Feng Sun, and X. C. Xie

Phys. Rev. Lett. 129, 096601 – Published 25 August 2022

Abstract

Recently, a half-quantized Hall conductance (HQHC) plateau was experimentally observed in a semimagnetic topological insulator heterostructure. However, the heterostructure was metallic with a nonzero longitudinal conductance, which contradicts the common belief that quantized Hall conductance is usually observed in insulators. In this work, we systematically study the surface transport of a semimagnetic topological insulator with both gapped and gapless Dirac surfaces in the presence of dephasing process. In particular, we reveal that the HQHC is directly related to the half-quantized chiral current along the edge of a strongly dephasing metal. The Hall conductance keeps a half-quantized value for large dephasing strengths, while the longitudinal conductance varies with Fermi energies and dephasing strengths. Furthermore, we evaluate both the conductance and resistance as a function of the temperature, which is consistent with the experimental results. Our results not only provide the microscopic transport mechanism of the HQHC, but also are instructive for the probe of the HQHC in future experiments.

Received 29 January 2022
Accepted 13 July 2022

DOI:https://doi.org/10.1103/PhysRevLett.129.096601

Physics Subject Headings (PhySH)

Electrical conductivity Quantum transport Topological insulators Topological materials Transport phenomena

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Humian Zhou¹, Hailong Li¹, Dong-Hui Xu ^2,3, Chui-Zhen Chen^4,5,*, Qing-Feng Sun^1,6,7, and X. C. Xie^1,6,7,†

¹International Center for Quantum Materials, School of Physics, Peking University, Beijing 100871, China
²Department of Physics, and Chongqing Key Laboratory for Strongly Coupled Physics, Chongqing University, Chongqing 400044, China
³Center of Quantum Materials and Devices, Chongqing University, Chongqing 400044, China
⁴School of Physical Science and Technology, Soochow University, Suzhou 215006, China
⁵Institute for Advanced Study, Soochow University, Suzhou 215006, China
⁶Collaborative Innovation Center of Quantum Matter, Beijing 100871, China
⁷CAS Center for Excellence in Topological Quantum Computation, University of Chinese Academy of Sciences, Beijing 100190, China

^*czchen@suda.edu.cn
^†xcxie@pku.edu.cn

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Issue

Vol. 129, Iss. 9 — 26 August 2022

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Images

Figure 1
(a) Schematic plot of a semimagnetic TI with a gapped Dirac cone on the top surface and a gapless Dirac cone on the bottom surface, where the red arrows indicate the magnetization on the top surface. (b) Schematic plot of a six-terminal Hall-bar device, where the red balls represent the sites attached to virtual leads. The distance of the two nearest balls is 4. (c) and (d) illustrate the Hall conductance $σ_{x y}$ and the longitudinal conductance $σ_{x x}$ against the Fermi energy $E_{F}$ for different dephasing strengths $Γ_{v}$ , respectively. (e) and (f) illustrate $σ_{x y}$ and $σ_{x x}$ against $Γ_{v}$ for different $E_{F}$ , respectively. Here, $N_{x} = 640$ , $N_{y} = 153$ , $N_{z} = 5$ , $n_{x} = 160$ , $n_{y} = 39$ , $n_{z} = 2$ , and the distance between lead 5 and 6 is $L = 40$ . The model parameters are fixed as $A = 1.0$ , $B = 0.6$ , $M_{0} = 1.0$ , and $M_{z} = 0.4$ .
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Figure 2
(a) The side and bottom surfaces of the sample in Fig. 1 laid on the $x y$ plane. The red balls represent the sites attached to virtual leads and the blue arrows indicate the chiral edge current. $a$ and $b$ are used to label the two adjacent black boxes, and the size of black box is $r_{c}$ . $(x, y)$ is the coordinate of the red ball in the lower right corner of the box $a$ . (b) The space distribution of the different of transmission coefficients $T_{d}$ with $Γ_{v} = 4.0$ and $E_{F} = 0.3$ . The inset show the value of $T_{d}$ in upper edge ( $y = {\tilde{n}}_{y} - r_{c}$ ), middle area [ $y = ({\tilde{n}}_{y} - r_{c}) / 2$ ], and the lower edge ( $y = 1$ ). (c) and (d) show $r_{c}$ and $t_{d}$ as a function of $Γ_{v}$ for different $E_{F}$ . Other parameters are the same as those in Fig. 1.
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Figure 3
(a) Red balls are divided in Fig. 2 by using blue boxes, and each blue box contains $r_{c}^{2}$ red balls. The numbers are used to label the boxes. (b)–(d) Comparison of the theoretical results and numerical results for a simple case with $r_{c} = 1$ . (b) $σ_{x y}$ as a function of ${\tilde{n}}_{y}$ with $t_{n} = 1$ . (c) and (d), $σ_{x y}$ and $σ_{x x}$ as a function of $t_{n}$ with ${\tilde{n}}_{y} = 200$ . Other parameters in (b)–(d) are $n_{x} = 7 {\tilde{n}}_{y}$ and $t_{d} = 0.5$ . (e) $σ_{x y}$ and $t_{d}$ as a function of ${\tilde{n}}_{y}$ . (f) $σ_{x x}$ and $t_{n}$ as a function of ${\tilde{n}}_{y}$ . The parameters of (e) and (f): $E_{F} = 0.2$ , $Γ_{v} = 3.0$ , $n_{z} = 2$ , $n_{y} = {\tilde{n}}_{y} - 2 (n_{z} - 1)$ , $n_{x} = 4 {\tilde{n}}_{y}$ , $N_{z} = 5$ , $N_{y} = 4 (n_{y} - 1) + 1$ , and $N_{x} = 4 n_{x}$ , and other parameters are the same as those in Fig. 1.
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Figure 4
Hall resistance $ρ_{x y}$ (a), longitudinal resistance $ρ_{x x}$ (b), Hall conductance $σ_{x y}$ (c), and longitudinal conductance $σ_{x x}$ (d) as a function of temperature $T$ . The curves correspond to different Fermi energy $E_{F}$ . The dephasing strength $Γ_{v} = 4.0$ , and other parameters are the same as those in Fig. 1.
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