Reliability of the Tran-Blaha functional in predicting band gaps and widths

For a set of oxides and semiconductors, we compute the electronic band structures (gaps and widths) within Density-Functional theory (DFT) using the Tran-Blaha (TB09) functional [Phys. Rev. Lett. {\bf 102}, 226401 (2009)]. We compare them with those obtained from (i) DFT using the local-density approximation (LDA), (ii) many-body perturbation theory (MBPT), and (iii) experiments. TB09 leads to band gaps in much better agreement with experiment than the LDA. However, the valence (and conduction) band widths are often underestimated (noticeably more than in LDA). MBPT corrections are obtained peforming one-shot $GW$ calculations using DFT eigenenergies and wavefunctions as starting point (both LDA and TB09 are considered). These corrections lead to a much better agreement with experimental data for the band widths. The MBPT band gaps obtained starting TB09 are close to those from quasi-particle self-consistent $GW$ calculations, at a much reduced cost. Finally, we explore the possibility to tune a semi-empirical parameter present in the TB09 functional aiming to obtain simultaneously better gaps and band widths. We find that these requirements are conflicting.