The quantum anomalous Hall (QAH) effect has recently been realized in thin films of intrinsic magnetic topological insulators (IMTIs) like MnBi$_2$Te$_4$. Here we point out that that the QAH gaps of these IMTIs can be optimized, and that both axion insulator/semimetal and Chern insulator/semimetal transitions can be driven by electrical gate fields on the $sim 10$ meV/nm scale. This effect is described by combining a simplified coupled-Dirac-cone model of multilayer thin films with Schr{o}dinger-Poisson self-consistent-field equations.
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