Equilibrium crystal structures, electron band dispersions and band gap values of layered GaSe and InSe semiconductors, each being represented by four polytypes, are studied via first-principles calculations within the density functional theory (DFT). A number of practical algorithms to take into account dispersion interactions are tested, from empirical Grimme corrections to many-body dispersion schemes. Due to the utmost technical accuracy achieved in the calculations, nearly degenerate energy-volume curves of different polytypes are resolved, and the conclusions concerning the relative stability of competing polytypes drawn. The predictions are done as for how the equilibrium between different polytypes will be shifted under the effect of hydrostatic pressure. The band structures are inspected under the angle of identifying features specific for different polytypes, and with respect to modifications of the band dispersions brought about by the use of modified Becke-Johnson (mBJ) scheme for the exchange-correlation (XC) potential. As another way to improve the predictions of band gaps values, hybrid functional calculations according to the HSE06 scheme are performed for the band structures, and the relation with the mBJ results discussed. Both methods nicely agree with experimental results and with state-of-the-art GW calculations. Some discrepancies are identified in cases of close competition between the direct and indirect gap (e.g., in GaSe); moreover, the accurate placement of bands revealing relatively localized states is slightly different according to mBJ and HSE06 schemes.
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