Anderson localization induced by spin-flip disorder in large-Chern-number quantum anomalous Hall effect

We numerically investigate the localization mechanisms of the quantum anomalous Hall effect (QAHE) with a large Chern number $\mathcal{C}$ in bilayer graphene and magnetic topological insulator thin films doped with nonmagnetic or spin-flip (magnetic) disorder. By calculating the modified Berry curvature in real space, we demonstrate that QAHEs in both systems turn into Anderson insulators when the disorder strength is large enough in the presence of nonmagnetic disorder. However, in the presence of spin-flip disorder, the localization mechanisms in the two systems are completely distinct. For ferromagnetic bilayer graphene with Rashba spin-orbit coupling, the QAHE with $\mathcal{C}=4$ first enters a metallic phase and then turns into an Anderson insulator with the increase of disorder strength, whereas for magnetic topological insulator thin films, the QAHE with $\mathcal{C}=-\mathcal{N}$ first enters a metallic phase, then turns into another QAHE with $\mathcal{C}=-(\mathcal{N}-1)$ as disorder strength increases, and finally turns into an Anderson insulator after $\mathcal{N}-1$ cycles between QAHE and metallic phases. The phase transitions in the two systems originate from the exchange of Berry curvature between conduction and valence bands. In the end, we provide a phenomenological picture related to the topological charges to help understand the underlying physical origins of the two different phase-transition mechanisms.

Figure 1

The bulk band structures for bilayer graphene. (a) Pristine bilayer graphene. (b) When only $t_{\mathrm{R}}=0.4t$ is applied. (c) When only exchange field $M=0.4t$ is applied, the spin-up/spin-down bands are upwards/downwards lifted with the bands crossing near $K$ and $K^{\prime }$ points. (d) When Rashba SOC $t_{\mathrm{R}}=0.4t$ and exchange field $M=0.4t$ both exist, a bulk gap is opened and the system is a QAHE insulator with $\mathcal{C}=4$. (e)–(h) The bulk band spectra along the high-symmetry lines of the magnetic element doped topological insulator films in the presence of different exchange field strength (e) $m=2.465B$, (f) $m=2.2B, \mathcal{C}=-1$, (g) $m=1.963B$, and (h) $m=1.75B, \mathcal{C}=-2$. The corresponding system parameters are set as $C=0.3B, A=0.5B, N_z=2$ (layer number).

Figure 2

Averaged conductance $\langle G\rangle$ as a function of $W$ for different Fermi energies inside the bulk gap. The system is bilayer graphene with parameters set to be $N=60, t_{\mathrm{R}}=0.4t$, and $M=0.4t$. (a) and (b) For nonmagnetic and spin-flip disorders, respectively. 4000 samples are collected for each data point.

Figure 3

Averaged conductance $\langle G\rangle$ as a function of $W$ for different Fermi energies inside the bulk gap. The system is magnetic topological insulator thin films with width set to be $N=80$. (a) and (b) For nonmagnetic and spin-flip disorders, respectively. 4000 samples are collected for each data point. The inset of (b) is the conductance variation of the system with a higher Chern number in $E_{\mathrm{F}}/B=0.002$. 150 samples are collected for each data point. The parameters in the inset are $C=0.3B, A=0.5B, m=1.71B, N=60$ for $\mathcal{C}=-4$, and $C=0.3B, A=0.5B, m=1.7B, N=60$ for $\mathcal{C}=-3$, respectively.

Figure 4

$\left({\mathrm{a}}_1\right)$–$\left({\mathrm{l}}_1\right)$ Averaged Berry curvature density $\mathrm{\Omega }$ as a function of energy $E$ for different spin-flip disorder strengths $W$ in magnetic topological insulator thin films. $\left({\mathrm{a}}_2\right)$–$\left({\mathrm{l}}_2\right)$ Corresponding averaged Hall conductance ${\sigma }_{xy}$ as a function of energy $E$. $P_{D1}^{v,c}\left(P_{D2}^{v,c}\right)$ represent the Berry curvature contributed by the massive Dirac bands in the first (second) exchange process. $P_{R1}^{v,c}\left(P_{R2}^{v,c}\right)$ represent the Berry curvature contributed by the remaining bands in the first (second) exchange process. $v\left(c\right)$ denotes valence (conduction) bands. 30 samples are collected for each point.

Figure 5

$\left({\mathrm{a}}_1\right)$–$\left({\mathrm{h}}_1\right)$ Averaged Berry curvature density $\mathrm{\Omega }$ as a function of energy $E$ for different nonmagnetic disorder strengths $W$ in magnetic topological insulator thin films. $\left({\mathrm{a}}_2\right)$–$\left({\mathrm{h}}_2\right)$ Corresponding averaged Hall conductance ${\sigma }_{xy}$ as a function of energy $E$. 30 samples are collected for each point.

Figure 6

Schematic of topological charges carried by the spin textures of the valence and conduction bands in magnetic topological insulator thin films with spin-flip disorder. (a) In the absence of disorders, valence/conduction bands carry four skyrmions with topological charges of $+$1, $+$1, $-1$, and $-1$. (b) At weak disorders, four skyrmions are scattered to eight merons that carry half-integer topological charges and are labeled as “A1,”“A2,” “B1,” “B2,” “C1,” “C2,” “D1,” and “D2.” (c) By further increasing disorder strength, merons B2 and C1 move towards each other. (d) Merons B2 and C1 make an exchange, leading to Hall conductance falling on the next ladder. (e) and (f) Weak disorders hardly affect merons A2 and D1. When the disorder strength approaches the second critical point, merons A2 and D1 move towards each other and make an exchange, which results in the disappearance of integer conductance.

Figure 7

Schematic of topological charges carried by the spin textures of the valence and conduction bands in bilayer graphene with spin-flip disorder. (a) In the absence of disorders, valence/conduction bands carry four skyrmions with topological charges of $+$1, $+$1, $-1$, and $-1$. (b) At weak disorders, four skyrmions are scattered to eight merons that carry half-integer topological charges and are labeled as “A1,”“A2,” “B1,” “B2,” “C1,” “C2,” “D1,” and “D2.” (c) By further increasing disorder strength, merons, A2 and C1, B2 and D1 move towards each other. (d) At strong disorders, merons A2,C1 and B2,D1 and make an exchange, causing the disappearance of quantized Hall conductance.

Figure 8

Spin-flip disorder dependence of Hall conductance. (a),(b) Sketch of Hall conductance as a function of spin-flip disorder strength W, which can/cannot be realized in the magnetic topological insulator thin films and bilayer graphene.