Mathematical Modeling of Electrostatic Potential in Radial and Planar p–n Junctions: A Comparative Study
Abstract
This work presents a comprehensive mathematical and numerical study of electrostatic potential in planar and radial silicon p–n junctions, considering the combined effects of device geometry, temperature, and incomplete dopant ionization. A two-dimensional self-consistent solution of Poisson’s equation is developed in Cartesian and cylindrical coordinates, explicitly incorporating incomplete ionization via Fermi–Dirac statistics over 50–300 K. At 100 K, incomplete ionization reduces effective space-charge density by 38‑45%, increases depletion width by 55–70%, and modifies the built-in potential by up to 42% compared to room-temperature predictions. Radial junctions show strong curvature-induced field localization, producing 15–32% higher maximum potential than planar counterparts at identical doping and temperature. For N = 10²³ m⁻³, maximum potential rises from 1.95 → 2.85 V (planar) and 2.45 → 3.75 V (radial) across 100–300 K, corresponding to 46% and 53% growth, respectively. Peak electric fields reach 3.2×10⁶ V·m⁻¹, with radial junctions exceeding planar values by ~7–12%, consistently showing 25–32% stronger electrostatic confinement. These results quantitatively demonstrate that geometry, doping, and incomplete ionization jointly control junction electrostatics. Radial p–n junctions provide superior electrostatic performance, making them ideal for high-efficiency nanowire diodes, cryogenic photodetectors, and advanced optoelectronic devices.
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