Advancing DFT predictions in Cu-chalcogenides with full-yet-shallow 3<i>d</i>-orbitals: Meta-GGA plus Hubbard-like <i>U</i> correction

The technologically important Cu-chalcogenides, such as Cu2Se and CuInSe2, usually have relatively small band gaps. Achieving a reliable yet efficient description of the electronic properties has proven to be quite challenging for the popular exchange-correlation functionals of density functional theory, primarily due to the involvement of full-yet-shallow Cu-3d orbitals. In this study, we evaluate the applicability of several meta-generalized gradient approximation (GGA) functionals that have been recently developed. We find that the r2SCAN (regularized-restored strongly constrained and appropriately normed) functional significantly improves upon conventional local density approximation and GGA in terms of geometry and electronic band structure; however, there is still a notable discrepancy with experimental results due to the remaining delocalization error. This error is mitigated by combining r2SCAN with a Hubbard-like U correction applied to the Cu-3d orbitals. For predicting band gaps, both the TASK functional and the mBJ potential, when combined with the U correction, demonstrate similar accuracies with a mean absolute error of 0.17-0.19 eV. This accuracy is lower than that achieved with the many-body Hedin's GW approximation method but more accurate than that of hybrid functionals. Moreover, the r2SCAN+U approach well reproduces the phonon dispersion in CuInSe2, revealing a neglected computational problem in previous reports. We conclude that the meta-GGA+U approach represents a significant advancement by striking a balance between reliability and computational effort, and further efforts are still required to describe the Cu-3d orbitals more accurately.