  White light-emitting diodes are gaining popularity and are set to become the
most common light source in the U.S. by 2025. However, their performance is
still limited by the lack of an efficient red-emitting component with a narrow
band emission. The red phosphor SrLiAl$_3$N$_4$:Eu$^{2+}$ is among the first
promising phosphors with a small bandwidth for 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 band shape of Eu(Sr1), with a
zero-phonon line at 1.906 eV and a 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 with the 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 insight into the
electron-phonon coupling that accompanies the 5d-4f transition in similar
UCr$_4$C$_4$-type phosphors. Furthermore, these results highlight the
importance of the Sr channel in shaping the narrow emission of
SrLiAl$_3$N$_4$:Eu$^{2+}$, and they shed new light on the structure-property
relations of such phosphors.
