Analytical expressions for the three-dimensional effect on the breakdown voltages of planar junctions in nonpunchthrough and punchthrough cases

Analytical expressions are derived for the three-dimensional effect on the breakdown voltage of the planar junctions, which have the finite lateral radius of window curvature, by employing the suitable approximations for the electric field. The analytical results for nonpunchthrough and punchthrough cases are in excellent agreement with the results of the quasi-three-dimensional device simulation by MEDICI. For nonpunchthrough case, the critical electric fields at breakdown and the breakdown voltages are expressed successfully in a form which is normalized to the parallel plane case. Also, the breakdown voltages in punchthrough case are given in terms of the punchthrough voltage and the parallel plane breakdown voltage of p⁺-i-n⁺ diodes including the lateral radius of window curvature. The results may be applicable to the estimation of breakdown voltages in many practical power devices