| A variational principle, based on classical physics, is introduced, which completely describes the overall equilibrium behavior of an MOS micro-electronic device. This principle is corrected, using a prescription proposed in another context, to account for the quantum-mechanically required redistribution of charge at the oxide-substrate interface. The resulting principle constitutes a nearly complete static description for MOS systems, comparable to the coupled Poisson-Schrodinger differential equation model.; In order to demonstrate the power of this method, a "trial function" describing the spatial distribution of electrical potential is constructed which captures the essential three-dimensional features of a simple MOS capacitor (MOSCAP). The trial function is used to effect the variational minimization of the Helmholtz Free Energy of the MOSCAP. The resulting estimate of Free Energy is then used as a basis to model the threshold voltage and capacitance. These variational models, which occur as closed-form analytical expressions, give the threshold voltage and capacitance for devices down to extremely small size (principle dimension <20 nm). This modeling captures the effect of the fringe field at the edge of the device in a simple and direct manner, an effect not easily addressed by any other modeling approach.; A benefit of the approach described above, is that, in addition to modeling the threshold voltage and capacitance, the free energy estimate also supports analysis of several issues of strong current (and near-term future) interest in the industry. These issues include the effect of statistical fluctuations in device characteristics induced by: fluctuations in dopant atom numbers in the depletion region under the gate; fluctuations induced by Line Edge Roughness (LER), due to unavoidable process precision outcomes; and to pseudo-random fluctuations in characteristics due to temperature differences on the die. Comparisons of VQM modeling with empirical results and detailed numerical simulations in the literature are given, where available.; A further benefit of the approach is that it provides insight into the behavior of small devices to a degree not available from other approaches. For example, the analytical expression obtained for the Free Energy of the MOSCAP decomposes into four distinct capacitive energy terms plus a potential energy term. The four capacitive terms clearly indicate that the capacitance equivalent circuit of small MOSCAPs must contain four separate lumped capacitances. These four capacitances include two, which are proportional to the perimeter of the gate, in addition to the two which are known from the standard model, and which are proportional to the gate area. This is an entirely new discovery. |
