The Temperature-Dependence of Multiphonon Relaxation of Rare-Earth Ions in Solid-State Hosts

Rare-earth ions are used in a wide range of emissive devices - ranging from lasers to displays - where high optical efficiency and narrow-linewidth are important. While their radiative properties are important, nonradiative properties are also critical since they can reduce optical efficiency and generate heat. In this paper, theories for multiphonon relaxation rate are reviewed for rare-earth excited states in solid-state dielectric hosts. A range of various approaches are used to simplify the mathematical form of the rate equations. The ²H_9/2 excited state of Er³⁺, responsible for a technologically significant green emission, is modeled to show how the various theories manifest an order-of-magnitude variation in the thermal dependence of the multiphonon relaxation rate, as well as anomalous local minima in phonon scattering for temperatures above 0 K. This work proposes a corrective term of two quanta (Δν= + 2) of the mediating phonon energy to energy gap, so the calculated and the experimentally determined relaxation rates are equal. Radiative quantum efficiencies of both the ¹G₄→³H₅ ~1.3 μm and ³F₃→³F₂ ~7 μm of Pr³⁺ are calculated to show the importance of both proper measurement of phonon energy and application of the multiphonon relaxation rate theory.