An Analytical Model of the DC and Frequency-Dependent 2-D and 3-D Current Spreading in Forward-Biased Shallow p-n Junctions

We present an analytical model of the 2-D/3-D spreading of dc and small-signal minority carrier flow in forward-biased shallow finite-sized p-n junctions. The model achieves an analytical solution of the 2-D/3-D continuity equation by replacing a Dirichlet-Neumann mixed boundary condition by a Neumann condition. It expresses the current spreading in terms of the junction width, lateral (vertical) extent beyond the junction, diffusion length, lifetime, transit time, and frequency. It predicts that the small-signal spread of the minority carrier flow gets progressively restricted for frequencies greater than inverse lifetime in long diodes and inverse transit time in short diodes; at high frequencies, the minority carrier flow picture consists of a 1-D small-signal flow superposed over a 2-D/3-D dc flow. Under dc conditions, the flow is almost 1-D in short diodes, spreads with an increase in vertical extent, and saturates in long diodes. We give the critical lateral extent beyond which the spread saturates, in terms of the vertical extent. Our model includes a small-signal equivalent circuit, which considers both minority and majority carrier current spreading; the latter is frequency independent and becomes important at higher frequencies. We validate the model against numerical simulations, and show its application to a rectangular junction with rounded corners.