Determination of the rate equations for erbium nanoparticles at arbitrary concentrations and drastically enhanced non-radiative transitions.

Erbium (Er) based NaYF4 nanoparticles (ENPs) are important for many applications including imaging, communications, and biosensing, especially at high Er concentrations. The rate equations (REs) are foundational for understanding various transitions in ENPs. However, the REs and various coefficients are determined only at the classical Er-composition of ~2% and Ytterbium at ~20% where upconversion processes were included only. This paper aims to determine coefficients of the REs through systematic characterizations at the full range of Er levels (5%, 50%, 75%, and 100%), where both up- and down-conversions are important and a careful calibration of visible and near-infrared emission bands is required. The parameter values of the REs are then curve-fitted to obtain their values for arbitrary Er concentration. We found that the non-radiative transitions and energy-transfer processes increase quadratically with Er concentration. We discovered the non-radiative transition from 4I11/2 to 4I13/2 increases with Er concentration and is orders of magnitude faster than other decay processes, exhibiting the highest down-conversion at 100% of Er. Our study explains why high-Er nanoparticles typically show weak upconversion emissions. Our results establish the REs for arbitrary Er-concentration for the first time and can be used more generally for designing ENPs and understanding complex nonlinear processes.