Robust quantum anomalous Hall effect with spatially uncorrelated disorder

TL;DR

The paper addresses how the quantum anomalous Hall insulator phase in magnetic topological insulators withstands nonmagnetic disorder during a magnetization-rotation driven transition. It employs a two-dimensional four-orbital tight-binding MTI model with uniform magnetization at angle $\theta$, onsite disorder of strength $S_{\text{dis}}$ and correlation length $\Lambda$, plus transport calculations in a Hall-bar geometry mapped to a two-terminal picture and MacKinnon–Kramer finite-size scaling. The key findings show that spatially uncorrelated disorder yields strong QAHI resilience due to a confinement mechanism of low-energy bulk states in patches of size $\Lambda$, with a nonuniversally varying exponent $\nu$ that can exceed the quantum percolation value unless $S_{\text{dis}}$ is large; correlated disorder with $\Lambda \sim 10\ \mathrm{nm}$ exhibits scaling closer to the Chalker–Coddington universality class. These results provide a mechanism to realize robust QAHI in MTIs with atomic-scale defects and offer a practical experimental diagnostic via finite-size scaling to infer disorder correlations.

Abstract

In magnetic topological insulators, a phase transition between a quantum anomalous Hall (QAH) and an Anderson localization phase can be triggered by the rotation of an applied magnetic field. Without the scattering paths along magnetic domains, this phase transition is governed by scattering induced by nonmagnetic disorder. We show that the QAH phase is strikingly robust in the presence of spatially uncorrelated disorder. The robustness is attributed to the quantum confinement induced by the short correlation length of the disorder. The critical behavior near the phase transition suggests a picture distinct from quantum percolation. This provides new insights on the robustness of the QAH effect in magnetic topological insulators with atomic defects, impurities, and dopants.

Figures (8)

• Figure 1: QAH-to-trivial insulator transition driven by magnetization rotation in a MTI thin film.(a) The spectrum of a MTI ribbon along $k_x$ for different orientation angles $\theta$ of the magnetization in the $y\text{-}z$ plane. The spectral gap closes when the magnetization is directed along $y$ ($\theta = \pi/2$). (b) The size and position in reciprocal space of the spectral gap minimum as a function of $\cos\theta$. (c) Schematic of a MTI thin-film Hall bar in the single-channel regime with rotating magnetization angle $\theta$ and transmission probability $p$ for a chiral edge channel across the central section of the Hall bar. This central section is the scattering region of the effective two-terminal setup of our microscopic quantum transport simulation approach (see text for details). (d) The longitudinal and Hall conductivities of the MTI Hall bar as a function of the transmission probability $p$, satisfying a semicircle relation (see inset). (e) The longitudinal and Hall conductivities for disordered MTI thin-film Hall bars ($E_\mathrm{F} = 0.2 \, E_\mathrm{gap}^{\theta=0}$, spatially uncorrelated disorder with $S_\mathrm{dis} = 3 \, E_\mathrm{gap}^{\theta=0}$, and $y\text{-}z$ magnetization rotation plane) of different sizes (with identical aspect ratio $L/W = 80$) as a function of $\cos\theta$, based on microscopic quantum transport simulations. The transition value $\cos\theta^\ast$ is indicated for the different ribbon sizes with a vertical dashed line.
• Figure 2: Robustness of QAH effect with respect to correlated and uncorrelated disorder.(a) The transition values of $\cos\theta^\ast$ as function of disorder strength when rotating the magnetization in the $y\text{-}z$ plane ($W = 0.05\,\text{\textmu m}$, $L = 4\,\text{\textmu m}$). The shaded areas indicate the full width at half maximum of the $\sigma_{xx}$ curve (see examples in the lower panel of Fig. \ref{['fig:1']}e). The dashed lines indicate the expected phase boundaries based solely on conventional band-smearing picture. The data points for correlated disorder are obtained for $\Lambda = 10\,\text{nm}$. (b) Values of $\cos\theta^*$ as a function of the Fermi energy for different correlation lengths. Here, the disorder strength is fixed at $S_\mathrm{dis} = 0.2 \, E_\mathrm{gap}^{\theta=0}$. The dashed lines and the shaded areas are similar to those in (a). (c) An example of uncorrelated (left) and correlated (right) disorder completely smearing the topological band gap. The red–blue color scale indicates local fluctuations $\delta E$ with Fermi level remaining within the gap, whereas the gray color indicates local fluctuations exceeding the gap size. (d) The current density of a QAH edge state injected from the left in to the disorder profiles shown in (c). In both panels, we have $E_\mathrm{F} = 0.2 \, E_\mathrm{gap}^{\theta=0}$, $\cos\theta = 0.3$ ($y\text{-}z$ rotation) and $S_\mathrm{dis} = 2 \, E_\mathrm{gap}^{\theta=0}$.
• Figure 3: Robustness of spectral gap.(a) The bulk density of states for a MTI thin film at different magnetization orientations and disorder correlation lengths ($\Lambda$). The disorder strength is set at $S_\mathrm{dis} = 2 \, E_\mathrm{gap}^{\theta=0}$ for all three panels. (b) The effective mass along $x$ and $y$ at the bulk band edges (indicated by the arrows in Fig. \ref{['fig:1']}a). (c) The confinement quantization energy of a bulk state as a function of the disorder puddle size $d$ for different magnetization rotation angles. These confinement energies drop back below the band edge energy $E_\mathrm{gap}^{\theta=0}$ (horizontal gray dashed line) for large $d$.
• Figure 4: Finite-size scaling analysis of QAHI-ALI phase transition.(a) The localization length $\lambda$ as a function of $\delta\!\cos\theta \equiv \cos\theta - \cos\theta^\ast$ for different MTI ribbon lengths. (b) Collapsed values of $\lambda$ as a function of $L/L_\textrm{eff}$ using the extracted exponent for the effective size: $L_\mathrm{eff}(\cos\theta)=A |\delta\!\cos\theta|^{-\nu}$, with $A=1\,\text{\textmu m}$ here without loss of generality. We consider $E_\mathrm{F} = 0.2 \, E_\mathrm{gap}^{\theta=0}$, $S_\mathrm{dis} = 2\,E_\mathrm{gap}^{\theta=0}$ (uncorrelated), and $x\text{-}z$ rotation in both (a) and (b). (c) The scaling exponent $1/\nu$ as a function of disorder strength for spatially correlated (left, $\Lambda = 10\,\text{nm}$) and uncorrelated (right) disorder with different Fermi levels, considering $x\text{-}z$ rotation. The values of classical percolation $1/\nu_\textrm{perc}$ and the Chalker–Coddington value $1/\nu_\textrm{CC}$ are indicated by horizontal dashed lines for reference.
• Figure A1: Disorder-averaged transmission probability.(a) The average transmission probability $\langle p_\textrm{ext} \rangle$ as a function of the number of combinations of disorder configurations $N_\textrm{combi}$ (see text for details) up to a maximum of $N_\textrm{combi} = 5000$ for different values of $\cos\theta$. (b) The difference of the mean with respect to the overall mean ($\delta \langle p_\textrm{ext} \rangle \equiv |\langle p_\textrm{ext} \rangle_{N_\textrm{combi}} - \langle p_\textrm{ext} \rangle_{5000}|$) as a function of the number of combinations $N_\textrm{combi}$ up to a maximum of $N_\textrm{combi} = 5000$ for different values of $\cos\theta$ on a log-log scale. The slope of $1/N_\textrm{combi}^{1/2}$ and $1/N_\textrm{combi}$ is shown for comparison with purple and black dashed lines, respectively. The transmission probabilities here are obtained for an MTI ribbon with $W = 0.05\,\textnormal{\textmu m}$, $L = 0.2 \, (4) \, \textnormal{\textmu m}$ ($N_\textrm{ext} = 20$), $E_\mathrm{F} = 0.2 \, E_\mathrm{gap}^{\theta=0}$, spatially uncorrelated disorder with $S_\mathrm{dis} = 3 \, E_\mathrm{gap}^{\theta=0}$, and magnetization rotated in the $x\text{-}z$ plane.