Fundamental View of Electronic Structures of β-NaYF₄, β-NaGdF₄, and β-NaLuF₄

We discussed the electronic structures of β-NaLnF₄ (Ln = Y, Gd, and Lu). We found the band gap keeps nearly constant (8-9 eV). However, the difference of the experimentally observed band gap arises from the different positions of 4f orbital levels relative to the valence band maximum. The 4f empty state of Gd falls into the band gap, led to a decreased band gap for β-NaGdF₄, and is spin-polarized. In contrast, both filled and empty 4f levels of Lu widely separated below and above the valence and conduction band edges, respectively, which means they do not influence the optical transitions in the band gap of the host lattice. By projecting the components of self-energy and wave function relaxation in 4f orbitals, we indicated a hidden level of Gd and Lu ions in the β-lattices, giving three Gd/Lu ions in the lattice split into two different types of electronic levels. This analysis helped us understand the essential mechanism and modified the energy migration mediated upconversion (EMU) model. Different 4f levels of Gd ions have been updated. This reason may arise from different charge density overlaps of local F-Ln (Ln = Y, Gd, and Lu), given by 2p and 4d/4f orbitals, respectively. We thus discussed that the local disorder of fluoride modulates the electronic eigenvalues of the top of the valence band near the γ point within the first Brillouin zone (BZ). The conduction band minimum is always located at the γ point consisting of the d orbitals of Y/Gd/Lu, regardless of the site occupation disorder effect between Y/Gd/Lu and Na. One of the lattices possesses a direct band gap indicating a route to increase the efficiency of the vertical optical transition along the γ direction. This work proposed a convenient route for future investigation of the interface states that potentially quench the upconversion luminescence.