Dirac fermion approach and its application to design high Chern numbers in magnetic topological insulator multilayers

Dinghui Wang, Huaiqiang Wang, and Haijun Zhang

Phys. Rev. B 107, 155114 – Published 7 April 2023

Abstract

Quantum anomalous Hall (QAH) insulators host topologically protected dissipationless chiral edge states, the number of which is determined by the Chern number. Up to now, the QAH state has been realized in a few magnetic topological insulators, but usually with a low Chern number. Here, we develop a Dirac-fermion approach which is valuable to understand and design high Chern numbers in various multilayers of layered magnetic topological insulators. Based on the Dirac-fermion approach, we demonstrate how to understand and tune high Chern numbers in ferromagentic ${MnBi}_{2} {Te}_{4}$ films through the van der Waals (vdW) gap modulation. Further, we also employ the Dirac-fermion approach to understand the experimentally observed high Chern numbers and topological phase transition from the Chern number $C = 2$ to $C = 1$ in the $[3QL − (Bi, {Sb)}_{1.76} {Cr}_{0.24} {Te}_{3}]$ / $[4QL − (Bi, {Sb)}_{2} {Te}_{3}]$ multilayers. Our work provides a powerful tool to design the QAH states with a high Chern number in layered magnetic topological insulator multilayers.

Received 5 January 2023
Accepted 28 March 2023

DOI:https://doi.org/10.1103/PhysRevB.107.155114

Physics Subject Headings (PhySH)

Quantum anomalous Hall effect Topological insulators Topological phase transition

Condensed Matter, Materials & Applied Physics

Authors & Affiliations

Dinghui Wang ¹, Huaiqiang Wang¹, and Haijun Zhang ^1,2,*

¹National Laboratory of Solid State Microstructures, School of Physics, Nanjing University, Nanjing 210093, China
²Collaborative Innovation Center of Advanced Microstructures, Nanjing University, Nanjing 210093, China

^*zhanghj@nju.edu.cn

Article Text (Subscription Required)

Click to Expand

References (Subscription Required)

Click to Expand

Issue

Vol. 107, Iss. 15 — 15 April 2023

Reuse & Permissions

Access Options

Author publication services for translation and copyediting assistance advertisement

Images

Figure 1
(a) Schematic of a 3D topological insulator (TI) with two Dirac-cone surface states on the top and bottom surfaces. (b) When the 3D TI is reduced to a TI thin film, the two Dirac-cone surface states will gradually couple with each other, which is defined as a block layer (BL) with $Δ_{intra}$ denoting the intralayer coupling. (c) Illustration of a multilayer structure constructed through stacking BLs in panel (b). (d), (e) The wave-function profile from $k p$ analysis for (d) the normal insulator (NI) thin film reduced from a 3D NI and (e) the TI thin film. For the TI thin film, the wave function indicates that the low energy state is a surface state. (f)–(h) When the thickness of a TI thin film is gradually reduced, the Dirac-cone surface states get to couple with each other, seen in panels (f), (g), and (h), where the pink peaks schematically indicate the projected Dirac-cone surface states. The Dirac-cone states can be distinguished even at the ultrathin limit (h), which is essentially different from that in the NI thin film (i).
Reuse & Permissions
Figure 2
(a) Schematic of a BL with two local Dirac-cone states (D1 and D2). (b) Band structure from the Dirac model of a BL in the topological phase transitions from $C = 1$ to $C = 0$ . Here, $| 1 (2), ↑ (↓) 〉$ denote the Dirac-cone states D1 and D2, where $↑ (↓)$ indicate the spin state. $| \pm, ↑ (↓) 〉$ denote the bonding ( $+$ ) and antibonding ( $-$ ) states of $| 1 (2), ↑ (↓) 〉$ . For the condition of $Δ_{intra} = 0$ and $m \neq 0$ , both D1 and D2 open an energy gap and contribute the Chern number $C_{1, 2} = 1 / 2$ for a total $C = C_{1} + C_{2} = 1$ . A finite intralayer coupling $Δ_{intra}$ couples the D1 and D2 to form bonding and antibonding states $| \pm, ↑ (↓) 〉$ . If we gradually increase $Δ_{intra}$ from 0, then when $m > Δ_{intra}$ , the total Chern number preserves $C = 1$ ; when $m = Δ_{intra}$ , the energy gap is closed; and when $m < Δ_{intra}$ , the total Chern number becomes $C = 0$ . (c) An FM bi-BL multilayer with four Dirac-cone states (D1, D2, D3, D4). $Δ_{inter}$ denotes the interlayer coupling. (d) Schematic band structure of an FM bi-BL multilayer when $m = 0$ . V1,2 denote the valence bands, and C1,2 denote the conduction bands. If the magnetic exchange field is gradually induced, then the V1,2 and C1,2 bands will split and lead to two topological phase transitions from $C = 0$ to $C = 1$ at $m = m_{1}$ , then to $C = 2$ at $m = m_{2}$ . (e) The phase diagram on the parameters ( $m / Δ_{inter}$ , $Δ_{intra} / Δ_{inter}$ ) of the FM bi-BL multilayer, calculated by the Dirac-fermion model. (f) An FM 5-BL multilayer and its phase diagram by the Dirac-fermion model. (g) Schematic magnetic multilayers consist of magnetic TIs. Left: FM BLs seperated by TI/NI thin film (shaded in gray). Right: AFM-type BLs.
Reuse & Permissions
Figure 3
(a), (b) The phase diagrams of FM 5-BL and 13-BL multilayers. The black points denote the phase of ${MnBi}_{2} {Te}_{4}$ without the vdW gap modulation. The red lines denote the phase evolution due to the vdW gap modulation with fixed $Δ_{intra}$ and $m$ . $δ d > 0$ ( $δ d < 0$ ) indicates the vdW gap expansion (contraction). (c) The schematic of the vdW gap modulation ( $δ d$ ). (d) The phase diagram on the parameters (the number of SL $N$ , the vdW gap modulation $δ d$ ) of AFM ${MnBi}_{2} {Te}_{4}$ films is calculated by the Dirac-fermion model with $m = 40$ meV, $Δ_{intra} = 86.2$ meV, and $Δ_{inter} = 74.8$ meV from fitting first-principles calculations. The two red dashed lines in (d) correspond to the phase evolutions (black solid lines) in (a), (b).
Reuse & Permissions
Figure 4
(a) The schematic structure of [3QL-BCT]/[4QL-BT] multilayers. (b) The linearly increased Chern number with the number of [3QL-BCT/4QL-BT] layers, in the case of weak interlayer and intralayer couplings. (c) Chern number evolution from the 2-BL (red line) and 3-BL (blue line) Dirac model due to decreasing the Cr doping in the middle 3QL-BCT layer. We take the parameters (in arbitary unit) $m = 0.7$ , $Δ_{intra} = 0.2$ , and $Δ_{inter} = 0.1$ for the 2-BL model, and $Δ_{inter} = 1$ , and $m_{0} = 2 m^{'}$ for the 3-BL model. (d)–(f) Phase diagram for the 2-BL model with $m^{'} = m$ , and the interlayer coupling $Δ_{inter} = 0$ (d) and $Δ_{inter} = 1$ (e). A finite $Δ_{inter}$ induces a $C = 1$ phase (e). Topological phase transition from $C = 2$ to $C = 1$ arises when the interlayer coupling $Δ_{inter}$ is gradually increased from 0 to 1 (f).
Reuse & Permissions
Figure 5
The band gap for even-layer (a) and odd-layer (b) AFM structures, respectively.
Reuse & Permissions

Physical Review B

covering condensed matter and materials physics