Impact of Size Distribution on Optical Properties of Quantum Dots: A Comparative Study of Statistical Models

This study investigates the effects of size fluctuations on the optical properties of PbS/ZnS quantum dots (QDs) by employing Gaussian, log-normal, Gamma, and Weibull probability density functions to model size distributions. Unlike most published studies, which apply averaging only to the gain coefficient, we perform statistical averaging across all terms in the rate equations—including stimulated emission, escape, and relaxation rates—to fully account for size fluctuations. This averaging is carried out using solutions of the Schrödinger wave equation as a function of quantum dot size. The resulting averaged rate equations are then consistently solved for various device parameters. The results reveal that the Weibull distribution exhibits the highest unsaturated optical gain, while the Gaussian distribution demonstrates lower gain with a peak shifted to higher energy. Log-normal and Gamma distributions produce similar optical spectra, with broader bandwidths but lower gain peaks. Larger size fluctuations result in reduced gain peaks and shifts toward lower energies. The Weibull distribution's pronounced contribution from larger particles explains its higher gain and shifted energy peak. Further analysis indicates that the log-normal distribution achieves the highest 3dB gain-saturation density product and the largest gain bandwidth, making it particularly advantageous for broadband applications. The Weibull distribution, while offering higher gain, exhibits narrower bandwidth and higher four-wave mixing efficiency. These findings underscore the critical influence of size distribution on the optical performance of QDs and provide valuable insights for designing QD-based devices, particularly at 1550 nm.