General Expression for the Size-Dependent Optical Properties of Quantum Dots.

While initial theories on quantum confinement in colloidal quantum dots (QDs) led to analytical band gap/size relations or sizing functions, numerical methods describe size quantization more accurately. However, because of the lack of reliable sizing functions, researchers fit experimental band gap/size data sets using models with redundant, physically meaningless parameters that break down upon extrapolation. Here, we propose a new sizing function based on a proportional correction for nonparabolic bands. Using known bulk parameters, we predict size quantization for groups IV, III-V, II-VI, and IV-VI and metal-halide perovskite semiconductors, including straightforward adaptations for negative-gap semiconductors and nonspherical QDs. Refinement with respect to experimental data is possible using the Bohr diameter as a fitting parameter, by which we show a statistically relevant difference in the band gap/size relation for wurtzite and zinc blende CdSe. The general sizing function proposed here unifies the QD size calibration and enables researchers to assess bulk semiconductor parameters and predict the size quantization in unexplored materials.