Charge density variation with fin width in FinFETs: an application of supersymmetric quantum mechanics

Abstract

FinFETs have become of great interest since it appears that they will be implemented within the next generation of integrated circuits. One point of interest is the transition from surface inversion on the faces of the fin to bulk inversion as the fin width is reduced. We have found that a good scaling approach can give estimates of this transition, with the scaling theory based upon supersymmetric quantum mechanics. For this, a double well potential is chosen whose supersymmetric partner is the harmonic oscillator potential. The depths and the separation between the wells were varied to simulate the change in fin widths. The fraction of charge residing inside the bulk, calculated as a function of fin width and electron density, was determined by calculating overlap area between the squared wave function magnitude of any state and that of a pure bulk state. More charges reside in the bulk for any electron density as the fin width decreases. On the other hand, for a fixed fin width, increasing electron density moves charge to the surface. It was found that in a Si FinFET, bulk inversion occurs at a fin width of about 8 nm for an inversion density of 3×10¹² cm⁻².