Size and shape effects on the measured peak temperatures of nanoscale hotspots

As device length scales trend downward, small feature sizes and steep temperature gradients require thermometers with increasingly fine spatial resolution in order to capture the true peak temperature. Here, we develop analytical expressions for the true and measured temperature rises as a function of thermometer size for Gaussian, disk-shaped, and rectangular surface heat sources. We find that even a thermometer the same size as the hotspot can underestimate the true peak temperature rise by more than 15%, and this error frequently exceeds 75% and can approach 90% for certain geometries when the thermometer is ten times larger than the measured hotspot. We show that a thermometer with resolution approximately two times smaller than the hotspot size is required to measure the peak temperature rise with less than 5% error for several common hotspot geometries. We also experimentally demonstrate that a 50x50x50nm(3) individual upconverting NaYF4:Yb3+,Er3+ nanoparticle thermometer captures the peak temperature rise due to laser heating more accurately than conventional diffraction limited optical techniques that our modeling results show would underestimate this value. In contrast to apparent self-heating effects that spuriously increase the nanoparticle thermometry signal at high excitation intensities, we measure true laser heating, as confirmed by comparing measurements on glass and diamond substrates.