Transverse averaging technique for the depletion capacitance of nonuniform PN-junctions

This paper 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). The solution of the quasi-1D Poisson's equation shows that, besides the usual depletion capacitance Cp and Cn due to the p- and n-layers, there is an additional capacitance Cs produced by the nonuniformity of the cross-section area S(z). The general expressions derived yield the particular formulae 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 formulae to the PN-junctions of exponentially varying cross section S(z) = S0exp(αz) as most universal and applicable to any polynomial approximation S(z) ≃ S0(1 + αz)n.