Application of Fermi–Dirac Continuous Distribution to Carrier Excitation in Semiconductors

The Fermi–Dirac continuous distribution is one of the most fundamental formulations in quantum statistics, describing the probability of electron occupancy in energy states for fermionic particles obeying the Pauli exclusion principle. Unlike classical Maxwell–Boltzmann statistics, the Fermi–Dirac distribution accounts for the discrete yet overlapping behavior of quantum states at finite temperatures, making it particularly relevant to solid-state physics, semiconductors, and nanoscale systems. This proposal aims to examine the theoretical formulation of the Fermi–Dirac continuous distribution, explore its analytical significance in condensed matter physics, and demonstrate its applications through selected numerical examples. The theoretical background section introduces the derivation of the Fermi–Dirac probability function, focusing on the interplay of temperature, chemical potential, and density of states. The experimental and methodological section outlines numerical modeling approaches, emphasizing MATLAB/Python-based solutions for simulating carrier concentration and occupation functions in semiconductor materials. Two numerical examples are presented to validate the framework: (i) temperature-dependent electron occupation probability in a conduction band, and (ii) numerical evaluation of density of states-integrated Fermi functions for nanostructured materials. The results highlight how small temperature variations drastically alter electron distribution near the Fermi level, illustrating the importance of Fermi–Dirac statistics in explaining optical absorption edges and electronic conductivity in semiconductors. The proposal concludes by emphasizing the predictive capability of the Fermi–Dirac continuous distribution in modern materials science, nanotechnology, and electronic device optimization, offering a robust statistical model for bridging theory with real-world device performance.