Fundamentals of Modern VLSI Devices

The scale length model described in Appendix 9 is called the “one-region” model. It replaces the gate oxide with an equivalent region of the same dielectric constant as silicon, but with a thickness equal to (εsi /εox)tox or 3tox . As pointed out earlier, this treats the normal field (ℰ x ) correctly, but the tangential field (ℰ y ) incorrectly. The one-region approximation is valid only if the gate oxide is much thinner than the scale length λ, in which case the oxide field is dominated by its normal component. In 1998, a generalized scale length model was published (Frank et al., 1998) which extended the one-region model to two- and three-regions with arbitrary dielectric constants and thicknesses. It considers the different boundary conditions of the normal and tangential fields separately at the dielectric interfaces. These relations then lead to an eigenvalue equation that can be solved for the scale length λ for such general structures. The generalized scale length model is particularly important for high-κ gate dielectrics which can be physically thick (Section 3.2.1.5), as well as for SOI and double-gate MOSFETs (Section 10.3.2).

Two-Region Scale Length Equation

In this appendix, the derivation of a generalized MOSFET scale length is described. Consider the two-region MOSFET model depicted in Fig. A10.1. The gate insulator region is assumed to have a permittivity ε 1 and thickness t 1. The depletion region in the semiconductor has a permittivity ε 2 and thickness t 2. Note that the bottom boundary of the depletion region is simplified to a straight line in the same manner as in Fig. A9.1. In subthreshold, there are negligible mobile carriers in the channel. The electric potential is solved from the 2-D Poisson equation applied to the rectangular region (lightly shaded) in Fig. A10.1. This is a boundary value problem in which the potential on the four conductor sides of the rectangle, left (source), top (gate), right (drain), bottom (body), is specified. There are actually two small gaps not enclosed by conductors: on the top left between the gate and source and on the top right between the gate and drain. When these gaps are not excessively large (compared with, e.g., λ), it is a good approximation to assign potential values by linear interpolation between the gate and source potentials for the left gap and between the gate and drain potentials for the right gap.