Super-quadratic upconversion luminescence among lanthanide ions

We explore the arguably most fundamental aspect of energy-transfer upconversion (ETU), namely the dependence of upconversion luminescence from a higher-energy level, following ETU excitation from a metastable lower-energy level, on direct luminescence from that metastable level. We investigate ETU among neighboring Nd³⁺ ions in single crystals of GdVO₄ and LaSc₃(BO₃)₄ with different doping concentrations by measuring, after short-pulse laser excitation with different pump energies, the infrared luminescence decay from the metastable ⁴F_3/2 level and the yellow upconversion luminescence decay from the ⁴G_7/2 level. We observe a highly super-quadratic dependence of upconversion on direct luminescence intensity. We conclude that the commonly assumed quadratic law of ETU, as proposed by Grant's model and frequently employed in rate-equation simulations, is inadequate to the description of ETU processes. Whereas Zubenko's model, which considers a finite migration rate, provides significantly better fits to the experimental luminescence-decay curves, also this model cannot accurately reproduce the measured decay curves, partly because it does not take the non-homogeneous distribution of active ions into account.