Smearing of the quantum anomalous Hall effect due to statistical fluctuations of magnetic dopants

Quantum anomalous Hall effect (QAH) is induced by substitution of a certain portion, $x$, of Bi atoms in a BiTe-based insulating parent compound by magnetic ions (Cr or V). We find the density of in-gap states, $N(E)$, emerging as a result of statistical fluctuations of the composition, $x$, in the vicinity of the transition point, where the {\em average} gap, ${\overline E}_g$, passes through zero. Local gap follows the fluctuations of $x$. Using the instanton approach, we show that, near the gap edges, the tails are exponential, $\ln{N(E)} \propto - \big({\overline E}_g-|E|\big)$, and the tail states are due to small local gap reduction. Our main finding is that, even when the smearing magnitude exceeds the gap-width, there exists a semi-hard gap around zero energy, where $\ln {N(E)} \propto -\frac{{\overline E}_g}{|E|}\ln \Big(\frac{{\overline E}_g}{|E|}\Big)$. The states responsible for $N(E)$ originate from local gap reversals within narrow rings. The consequence of semi-hard gap is the Arrhenius, rather than variable-range hopping, temperature dependence of the diagonal conductivity at low temperatures.