Potential Distribution and Capacitance of a Graded p-n Junction

1 1 his theory of p-n junctions in semiconductors, Shockley1 shows that 1 the potential distribution in a linearly graded p-n junction satisfies a oneparameter differential equation of the form = sinh U - Kz. (1) dzThe solution of this nonlinear equation cannot be obtained by analytical methods. However, Shockley gives two approximations, each applicable in an extreme case. The space-charge approximation (/v » 1) is valid in the case of a steep gradient at the junction, while the neutral approxi1573 1578 TIIE B E L L SYSTEM T E C H N I C A L J O U R N A L , NOVEMBER 1960 mation (K « 1) holds for a gentle gradient. Since in almost all cases of practical interest the gradients in p-n junctions are of such a magnitude that the space-charge approximation is valid, it has become the accepted form for the analysis of p-n junctions. Shockley derived the differential equation (1) for the case of equilibrium -- that is, for the case of zero bias across the junction. Recently Moll" has shown that the potential distribution in a p-n junction under bias satisfies the same one-parameter differential equation. The parameter K is now a function of the applied voltage, and it turns out that for most practical gradients in p-n junctions under forward bias the spacecharge approximation ceases to be valid. For a proper description of the potential distribution in a biased junction, a numerical solution of (1) is required. Such a solution is obtained in the present paper and applied to the computation of the small-signal ac capacitance of the junction as a function of the bias voltage.