Numerical simulations of abrupt N-P-N structures with homogeneously doped layers show a systematic deviation from the analytical formulae, due to the application of the depletion approximation. The error in the depletion approximation, as a function of barrier doping, is found to reach a maximum in the limit of an intrinsic barrier region. The potential and charge distribution in the static N-i-N structure, including the effects of the charge distribution at the N-sides, is calculated analytically. For this structure a linear current-voltage relation is derived, and found in agreement with numerical results.
Bakker, J. G. C., Bisschop, J., & Schilders, W. H. A. (1992). Current-voltage relation for abrupt N+-P-N+ and N-i-N structures. Solid-State Electronics, 35(7), 897-904. https://doi.org/10.1016/0038-1101(92)90316-5