The self-consistent analysis of the on-set of strong inversion in an MOS transistor with double-layer substrate impurity profile

A unique criterion of the surface potential at the onset strong inversion case is obtained from the extension of the single layer case. The criterion is clearly defined by setting of the increasing rate of the total minority carrier concentration Q_n equal to the increasing rate of the total depletion impurity charge Q_B. The same criterion then applies to analyze the on-set strong inversion case for the double layer profile. The expression for the surface potential ψ_s has the form of $\text{(}\text{kT}\text{q}\text{) {l}{}_{\mathrm{n}}\text{(}\text{N}{}_1\text{N}{}_2\text{n}{}_{\mathrm{i}}{}^2$3 ? l_n[1 ? (q(N₁ ? n₂)$\text{W}\text{Q}{}_{\mathrm{B}}\text{)]}}$. The first term is the same as 3 the conventional expression; however, the second term is new. The expression is a continuous function of the width W of first N₁ layer, and gives a consistent prediction in the limiting case for a single N₁ layer. Therefore, the inconsistent prediction of a discontinuity in $\text{ψ}{}_{\mathrm{s}}\text{[=(}\text{kT}\text{q}\text{)l}{}_{\mathrm{n}}\text{(}\text{N}{}_1\text{N}{}_2$)] from the 3 conventional expression in the single layer limit is then removed. The corrected magnitude to the conventional expression increases with the product of N₁N₂, the ratio of ($\text{N}{}_1\text{N}{}_2$) and ($\text{W}\text{x}{}_{\mathrm{<mtext>d</mtext><mtext>1</mtext><mtext>max</mtext>}}$), and the correction will give a positive M (negative) value as N₁ > N₂(N₂ > N₁).