Transverse Averaging Technique for Depletion Capacitance of Nonuniform PN-Junctions

This article evolves an analytical theory of nonuniform $PN$-junctions by employing the transverse averaging technique (TAT) to reduce the three-dimensional semiconductor equations to the quasi-one-dimensional (quasi-1D) form involving all physical quantities as averaged over the longitudinally-varying cross section $S(z)$. Solution of the quasi-1D Poisson's equation shows that, besides the usual depletion capacitance $C_p$ and $C_n$ due to the $p$- and $n$-layers, there is an additional capacitance $C_s$ produced by nonuniformity of the cross-section area $S(z)$. The general expressions derived yield the particular formulas obtained previously for the abrupt and linearly-graded junctions with uniform cross-section. The quasi-1D theory of nonuniform structures is demonstrated by applying the general formulas to the $PN$-junctions of exponentially-varying cross section $S(z)=S_0\exp(\alpha z)$ as most universal and applicable to any polynomial approximation $S(z)\simeq S_0(1+\alpha z)^n$.