The behavior of metal oxide semiconductor field effect transistors (MOSFETs) with small aspect ratio and large doping levels is analyzed using formal perturbation techniques. Formally, we will show that in the limit of small aspect ratio there is a region in the middle of the channel under the control of the gate where the potential is one-dimensional. The influence of interface and internal layers in the one-dimensional potential on the averaged channel conductivity is closely examined in the large doping limit. The interface and internal layers that occur in the one-dimensional potential are resolved in the limit of large doping using the method of matched asymptotic expansions. The asymptotic potential in the middle of the channel is constructed for various classes of variable doping models including a simple doping model for the built-in channel device. Using the asymptotic one-dimensional potential, the asymptotic mobile charge, needed for the derivation of the long-channel I-V curves, is found by usingstandard techniques in the asymptotic evaluation of integrals. The formal asymptotic approach adopted not only provides a pointwise description of the state variables, but by using the asymptotic mobile charge, the lumped long-channel current-voltage relations, which vary uniformly across the various bias regimes, can be found for various classes of variable doping models.
Using the explicit solutions of some free boundary problems solved by Howison and King (1988), the two-dimensional equilibrium potential near the source and drain is constructed asymptotically in strong inversion in the limit of large doping. From the asymptotic potential constructed near the source and drain, a uniform analytical expression for the mobile charge valid throughout the channel is obtained. From this uniform expression for the mobile charge, we will show how it is possible to find the I-V curve in a particular bias regime taking into account the edge effects of the source and drain. In addition, the asymptotic potential for a two-dimensional n⁺-p junction is constructed.
