A Unified Model for Size-, Shape-, and Composition-Dependent Bandgap in Semiconductor Nanocrystals: Beyond the Effective Mass Approximation

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Original abstract

Quantum confinement in semiconductor nanocrystals enables precise bandgap engineering for optoelectronic applications. The Brus equation provides a foundational description for spherical particles but cannot account for shape anisotropy, surface states, or compositional inhomogeneity in alloyed systems. This work presents a unified analytical model that extends the Brus equation by incorporating three physically motivated elements: a shape-dependent confinement energy derived from an infinite-barrier box model and expressed through an anisotropy factor α_conf; a negative surface-state correction proportional to the surface-to-volume ratio that defines a critical radius for surface-dominated behaviour; and three mixing rules for binary semiconductor nanoalloys. The most advanced mixing rule, Rule S, couples a modified Butler isotherm to the confinement model to capture surface segregation. Validation against 28 experimental data points encompassing CdSe spheres, cubes, and rods, PbS quantum dots, and alloyed Cd1-xZnxS nanocrystals yields an overall root-mean-square error of 0.07 eV, representing up to a factor-of-five improvement over the original Brus equation for anisotropic particles (3.2× for CdSe rods, 4.7× for PbS spheres). The fully analytical model evaluates a single composition–size–shape configuration in 2 - 5 ms, enabling the screening of thousands of candidates per minute on standard hardware. This framework provides a computationally efficient and physically transparent tool for predictive bandgap engineering in semiconductor nanocrystals.

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