Unified Mathematical Framework for Semiconductor Devices, Junction Dynamics, and Circuit Analysis

Abstract

Semiconductor devices form the fundamental building blocks of modern electronic and integrated circuit systems, where their electrical behavior is governed by coupled carrier transport, electrostatic, and circuit-level phenomena. This work presents a unified mathematical framework for modeling semiconductor devices, junction dynamics, and their interaction with electronic circuits. The formulation integrates fundamental semiconductor relations, carrier concentration, drift–diffusion transport, Poisson’s equation, continuity equations, PN-junction behavior, diode characteristics, bipolar junction transistor operation, and MOSFET charge-controlled dynamics. The framework further establishes mathematical relationships between device-level parameters and circuit-level voltage, current, power, switching, and frequency responses. Nonlinear device characteristics are incorporated to represent practical operating conditions, including recombination, generation, parasitic effects, temperature dependence, and transient behavior. The proposed formulation provides a systematic basis for deriving analytical and numerical models applicable to rectifiers, amplifiers, switching circuits, CMOS structures, and integrated semiconductor systems. The resulting mathematical framework enables consistent prediction of device characteristics and circuit performance while providing a foundation for advanced semiconductor-device optimization and reliability analysis.