TABLE OF CONTENTS In this appendix, we outline the mathematical approach that leads to the analytical expression, Eq. (3.66) in Section 3.2.1, for the short-channel threshold roll-off. The short-channel effect (SCE) is a very complex mathematical problem involving the solution of an irregular 2-D boundary-value problem. It is impractical to derive an exact analytical solution applicable to all general cases. Numerical simulations running on a finite-element program should be used to obtain accurate solutions for specific device geometries and doping conditions. Nevertheless, an analytical expression, even an approximate one, goes a long way in providing valuable insights into the fundamentals of the short-channel effect and its controlling parameters. The approach here essentially follows the Ph.D. thesis of Thao N.Nguyen published in 1984.
To simplify the 2-D boundary-value problem to a manageable level, we make a number of approximations so as to retain only the most basic aspects of the short-channel effect. A simplified short-channel MOSFET geometry is shown in Fig. A6.1 (Nguyen, 1984). The x-axis is along the vertical direction, the y-axis along the horizontal direction, and the origin at point A. As in Section 2.3.2, ?(x, y)= ? i (x, y) ? ? i( x= ?) is defined as the intrinsic potential at a point (x, y) with respect to the intrinsic potential of the p-type substrate. The substrate is assumed to be uniformly doped with a concentration N a. In the oxide region AFGH,
