Rationalizing the structural changes and spectra of manganese and their temperature dependence in a series of garnets with first-principles calculations

Detailed first-principles calculations have been carried out to study the stabilization, excitation and luminescence mechanisms of the ion ${\mathrm{Mn}}^{3+}$ in a series of ${A}_{3}{B}_{2}{B}_{3}^{\ensuremath{'}}{\mathrm{O}}_{12}$ garnet hosts. The formation energy shows that ${\mathrm{Mn}}^{3+}$ is dominant and is situated at the octahedral $B$ site. The excited states, excitation, and emission energies of ${\mathrm{Mn}}^{3+}$ have then been calculated. The calculated energy levels of ${\mathrm{Mn}}^{3+}$ confirm that the red emission is due to the ${}^{5}{\mathrm{T}}_{2}\phantom{\rule{4pt}{0ex}}\ensuremath{\rightarrow}{\phantom{\rule{4pt}{0ex}}}^{5}{\mathrm{E}}^{\ensuremath{'}}$ transition and the near-infrared (NIR) emission arises from the ${}^{1}{\mathrm{T}}_{2}\phantom{\rule{4pt}{0ex}}\ensuremath{\rightarrow}{\phantom{\rule{4pt}{0ex}}}^{3}{\mathrm{T}}_{1}$ transition. The populations of the ${}^{5}{\mathrm{T}}_{2}$ and ${}^{1}{\mathrm{T}}_{2}$ excited states and the corresponding radiative rates lead to the temperature dependence of the red to NIR emission. Furthermore, the adiabatic potential energy surfaces along the ${A}_{1g}$ and ${E}_{g}$ moeity modes of [${\mathrm{MnO}}_{6}$] have been calculated and fitted well in the harmonic approximation. The high activation energy for ${\mathrm{Mn}}^{3+}$ indicates a low nonradiative multiphonon relaxation rate of ${}^{5}{\mathrm{T}}_{2}$ to ${}^{3}{\mathrm{T}}_{1}$. Hence, the ionization process was considered, and we show that it is responsible for the luminescence quenching of ${\mathrm{Mn}}^{3+}$, so that the luminescence has rarely been reported experimentally. This work illustrates a well-designed approach based on the density-functional theory framework to predict the optical transition properties of the transition metal ion ${\mathrm{Mn}}^{3+}$ by calculating the structural distortions due to the Jahn-Teller effect, the optical transitions, quenching processes and the influence of pressure.