Analytical solutions for the breakdown voltages of punched-through diodes having curved junction boundaries at the edges

Analytical expressions are derived for the breakdown voltages of punched-through diodes having a plane structure terminated with cylindrical and spherical curved boundaries at the edges, through the use of suitable approximations for the electric field in the depletion layer. The expressions derived include both p
⁺
-i-n
⁺
and p
⁺
-p-n
⁺
(or p
⁺
-n-n
⁺
) types and are given in terms of the middle-region (i-layer or p-layer) width, the radius of curvature of the junction edge, the punch-through voltage, and the plane parallel breakdown voltage of p
⁺
-i-n
⁺
diodes. The results obtained include a correlation between the middle-region (p-layer) width and the width of the depletion layer in the curved portions of the junction when the applied reverse bias across the diode is just sufficient so that punchthrough takes in the portions where the junction is plane parallel. These results are made use of in the breakdown voltage calculations.