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Quantum anomalous Hall effect (QAH) is induced by substitution of a certainportion, x, of Bi atoms in a BiTe-based insulating parent compound by magneticions (Cr or V). We find the density of in-gap states, N(E), emerging as aresult of statistic fluctuations of the composition, x, in the vicinity of thetransition point, where the average gap, E_g, passes through zero. Local gapfollows the fluctuations of x. Using the instanton approach, we show that, nearthe gap edges, the tails are exponential, ln N(E) \propto -(E_g-|E|), and thetail states are due to small gap reduction. Our main finding is that, even whenthe smearing magnitude exceeds the gap-width, there exists are semi-hard gaparound zero energy, where ln N(E) \propto -E_g/|E| (ln E_g/|E|). The statesresponsible for N(E) originate from local gap reversals within narrow rings.The consequence of semi-hard gap is the Arrhenius, rather than variable-rangehopping, temperature dependence of the diagonal conductivity at lowtemperatures.

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