A Phenomenological Mathematical Model on I–V Curve of p-n Junctions in Semiconductor Diodes

Abstract

A phenomenological mathematical (PM) model is proposed to describe the current–voltage (<inline-formula> <tex-math notation="LaTeX">$I$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">$V$ </tex-math></inline-formula>) characteristics of silicon (Si) p-n junction diodes, based on a self-feedback mechanism. The self-feedback principle assumes that the rate of change of a physical quantity depends only on the quantity itself, i.e., <inline-formula> <tex-math notation="LaTeX">$\text {d} {y}/\text {d} {x}={f}({y})$ </tex-math></inline-formula>. Experimental <inline-formula> <tex-math notation="LaTeX">$I$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">$V$ </tex-math></inline-formula> measurements are performed under both forward and reverse bias conditions. In the forward bias region, the carrier concentration growth rate <inline-formula> <tex-math notation="LaTeX">$\text {d} {n}/\text {d} {V}$ </tex-math></inline-formula> is modeled as proportional to <inline-formula> <tex-math notation="LaTeX">${n}+{n}_{{0}}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">${n}+{n}_{{0}}$ </tex-math></inline-formula> denotes the background carrier concentration, accounting for diffusion dominated current transport. In the reverse-bias regime, the <inline-formula> <tex-math notation="LaTeX">$I$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">$V$ </tex-math></inline-formula> behavior is divided into three distinct stages governed by different carrier dynamics: 1) Stage I: <inline-formula> <tex-math notation="LaTeX">$\text {d} {n}/\text {d} {V} \propto {n}_{{1}}-{n}$ </tex-math></inline-formula>, where <inline-formula> <tex-math notation="LaTeX">${n}_{{1}}$ </tex-math></inline-formula> represents the saturation carrier concentration, reflecting minority carrier diffusion; 2) Stage II: <inline-formula> <tex-math notation="LaTeX">$\text {d} {n}/\text {d} {V} \propto {n}$ </tex-math></inline-formula>, corresponding to avalanche multiplication; and 3) Stage III: <inline-formula> <tex-math notation="LaTeX">$\text {d} {n}/\text {d} {V} \propto {n}^{{(1+\delta)}} $ </tex-math></inline-formula>, with <inline-formula> <tex-math notation="LaTeX">$\delta \gt {0}$ </tex-math></inline-formula> as an empirical constant, capturing the effects of band-to-band tunneling and the steeper-than-avalanche rise in carrier concentration. Analytical expressions derived from the PM model exhibit excellent agreement with experimental data across the entire <italic>I–V</italic> curve. In contrast to conventional models that rely on idealized assumptions-such as constant carrier mobility or abrupt junction approximation-the proposed PM model captures the nonlinear evolution of carrier concentration through a self-feedback framework, enabling a unified description of both forward and reverse <inline-formula> <tex-math notation="LaTeX">$I$ </tex-math></inline-formula>–<inline-formula> <tex-math notation="LaTeX">$V$ </tex-math></inline-formula> characteristics. This work provides new insights into the mechanisms underlying charge transport in semiconductor diodes.