Generalised analytical model of the transition power densities of the upconversion luminescence and quantum yield

The quantum yield (QY) evaluation of upconverting nanoparticles (UCNPs) is an essential step in the characterisation of such materials. The QY of UCNPs is governed by competing mechanisms of populating and depopulating the electronic energy levels involved in the upconversion (UC), namely linear decay rates and energy transfer rates. As a consequence, at low excitation, the QY excitation power density (ρ) dependence obeys the power law ρ, where n represents the number of absorbed photons required for the emission of a single upconverted photon and determines the order of the energy transfer upconversion (ETU) process. At high power densities, the QY transits to a saturation level independent of the ETU process and the number of excitation photons, as a result of an anomalous power density dependence present in UCNPs. Despite the importance of this non-linear process for several applications (e.g., living tissue imaging and super-resolution-microscopy), little has been reported in the literature regarding theoretical studies to describe the UC QY, especially for ETUs with order higher than two. Therefore, this work presents a simple general analytical model, which introduces the concept of the transition power density points and QY saturation to characterise the QY of an arbitrary ETU process. The transition power density points determine where the power density dependence of the QY and the UC luminescence changes. The results provided in this paper from fitting the model to experimental QY data of a Yb-Tm codoped β-UCNP for 804 nm and 474 nm emissions (ETU2 and ETU3 processes, respectively) exemplify the application of the model. The common transition points found for both processes were compared to each other showing strong agreement with theory, as well as, compared to previous reports when possible.