A simple approximate model for potential distributions of a semiconductor sample having a surface inversion region

Abstract

Satisfactory approximate analytical solutions of potential distributions of an undegeneracy semiconductor sample having a surface inversion region and impurity density N≳102 ni were obtained by the following simple model d2u/dx2 = aN: un ≳u⩾0, bN: 2un ≳n⩾un, reu−u n: us ⩾u⩾2un, where un = arc Sinh(N/2) = ln N, us is the surface potential and a, b, and r are adjustable parameters. Particularly, the following formulas can provide quite good values of the parameters: a = 1−(1/un), b = 1+(g/un), and r = 1+(us −2un +1−g)(eus−2un −1)−1, where g is an adjustable parameter. For example, the error is not greater than about 2% when g = 1 and N≳103 ni and it decreases with increasing impurity density. The total amount of the space charges and the amount of carriers in the inversion region (us ⩾u⩾un) were accurately given. The model can be applied in a small current injection whose direction is perpendicular to the sample surface if the average quasi-Fermi energy level of the electrons and holes can be introduced instead of the Fermi energy level at equilibrium.