Determination of the rate equations for erbium nanoparticles at arbitrary concentrations and drastically enhanced...

Erbium (Er)-based NaYF₄ nanoparticles (ENPs) are important for many applications including imaging, communications, and biosensing, particularly at high Er concentrations. The rate equations (REs) are fundamental for understanding various transitions in ENPs. However, the REs and their various coefficients are determined only at the classical composition of ∼2% Er and ∼20% ytterbium (Yb) while focusing only on up-conversion processes. This paper aims to determine the coefficients of the REs through systematic characterization across the full range of Er levels (5%, 50%, 75%, and 100%), where both up- and down-conversions are important and careful calibration of visible and near-infrared emission bands is required. The parameter values of the REs were then obtained through curve fitting for arbitrary Er concentrations. We found that non-radiative transitions and energy-transfer processes increase quadratically with Er concentration. We discovered that the non-radiative transition from ⁴I_11/2 to ⁴I_13/2 increases with Er concentration and is orders of magnitude faster than other decay processes, exhibiting the highest down-conversion at 100% Er. Our study explains why high Er concentration nanoparticles typically exhibit weak up-conversion emission. Our results establish the REs for arbitrary Er concentrations for the first time, which can be used more generally for designing ENPs and understanding complex nonlinear processes.