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The modified Becke-Johnson exchange potential [ F. Tran and P. Blaha Phys. Rev. Lett. 102 226401 (2009)] (TB-mBJ) yields very accurate electronic band structures and gaps for various types of semiconductors and insulators (e.g., sp semiconductors, noble-gas solids, and transition-metal oxides). However, the TB-mBJ potential has, for a few groups of solids, the tendency to underestimate the band gap. This has led us to examine the possibility to further improve over the original TB-mBJ potential by either reparametrizing its coefficients using a larger test set of solids or defining a parametrization for small-/medium-size band-gap semiconductors only. We also checked alternatives to the average of |∇ρ|/ρ in the unit cell for the determination of parameter c, which determines the amount of the screening contribution. Among these different possibilities, the best one seems to be a reparametrization of the coefficients, which leads to a much more balanced description of the band gaps.
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