Analyzing Semiconductor Band Gaps in Ensemble Density Functional Theory using Thermodynamic Limits of Finite One-Dimensional Model Systems

Ensemble Density Functional Theory (EDFT) is one of the most promising extensions to Density Functional Theory (DFT) for analyzing excitation energies. So far, it has been shown to improve the accuracy of excitation energies in atoms, molecules and isolated model systems. However, it is uncertain whether EDFT can calculate band gaps of periodic systems. To investigate whether this is possible, we estimate the thermodynamic limit with increasingly wide finite one-dimensional systems. We use finite versions of the semiconducting Kronig-Penney (KP) model, which consists of repeating square wells, calculated in the Octopus real-space DFT code. This method has been used recently to study particle-in-a-box systems tending toward a metallic limit with no band gap [R. J. Leano et al., Electron. Struct. 6, 035003 (2024)]. Considering three different choices of centering of the square well within the repeating unit in the finite model, we find that the non-interacting excitation energies of the finite systems, for an appropriate number of electrons, approach the bandgap of the periodic KP model. These results serve as a basis for assessment of EDFT predictions of the bandgap of semiconducting systems.