Cross-Relaxation from Er3+(2H11/2,4S3/2) and Er3+(2H9/2) in β-NaYE4:Yb,Er and Implications for Modeling Upconversion Dynamics

We have implemented both the Dornauf- Heber (D-H) model and a brute force statistical (BFS) model to characterize the quenching by neighboring Er and Yb ions of the green and blue emitting states of Er in beta-NaYF4:Yb,Er@NaYF4 nanoparticles. Each of these models explicitly considers the radial distribution of Er and Yb acceptors around the two crystallographically distinct types of Er donor sites out to a user-specified donor-acceptor (D-A) cutoff distance, R-a, beyond which the acceptor distribution is approximated as a spatial continuum. An algorithm is proposed for the BFS model, which makes it tractable to deal with the enormous number of possible acceptor distributions around the donor sites. The two models are compared for the convergence of fit values of critical energy-transfer distances, R-0, as a function of the D-A cutoff distance, R-a, for switching to the continuum approximation. The D-H model requires a much higher R-a value (R-a >> R-0) to achieve convergence compared to BFS, primarily because of the specific mathematical construct of the D-H model. The BFS model achieves convergence at R-a approximate to R-0, indicating that, on a purely physical basis, the continuum approximation is acceptable for D-A separations larger than the critical energy transfer distance. Both models do, however, ultimately converge to similar R-0 values. Our analysis shows that the green emission from Er ions, which have other Er ions occupying nearest neighbor sites, is almost completely quenched. In modeling upconversion, this subpopulation of Er ions should be treated as dark quenching sites for Yb3+(F-2(5/2)) excitation. For an 18% Yb, 2% Er doping composition, approximately 10% of Er ions are dark.