First-principles explanation of the luminescent lineshape of SrLiAl$_3$N$_4$:Eu$^{2+}$ phosphor for light-emitting diode applications

White light emitting diodes are gaining popularity and are set to become the most common light source in the US by 2025. Their performance is still limited by the lack of an efficient red-emitting component with narrow band emission. The red phosphor SrLiAl$_3$N$_4$:Eu$^{2+}$ is among the first promising phosphors with small bandwidth for the next-generation lighting but the microscopic origin of this narrow emission remains elusive. In the present work, density functional theory, the $\Delta$SCF-constrained occupation method, and a generalized Huang-Rhys theory are used to provide an accurate description of the vibronic processes occurring at the two Sr$^{2+}$ sites that the Eu$^{2+}$ activator can occupy. The emission bandshape of Eu(Sr1), with a zero-phonon line at 1.906 eV and high luminescence intensity, is shown to be controlled by the coupling between the 5d$_{z^2}$-4f electronic transition and the low-frequency phonon modes associated to Sr and Eu displacements along the Sr channel. The good agreement between our computations and experimental results allows us to provide a structural assignment of the observed total spectrum. By computing explicitly the effect of the thermal expansion on zero-phonon line energies, the agreement is extended to the temperature-dependent spectrum. These results provide insights into the electron-phonon coupling accompanying the 5d-4f transition in similar UCr$_4$C$_4$-type phosphors: they highlight the importance of the Sr channel in shaping the narrow emission of SrLiAl$_3$N$_4$:Eu$^{2+}$, and shed new light into the structure-property relations of such phosphors.