Adaptive Grid Generation for Semiconductor Device Simulation

Abstract

In general the inim grid neecls modification at some biases in order to meet an allowable amount of discretization error. After each nonlinear solve. the grid is adapted - locally refin4umfined - and the solution is recomputed, using the final grid at each condition as an iiiiual estimate for the next. The key ID implementing an robust and efficient scheme is the selection of a proper error indlicator. We have found for instance that the local truncation error (LTE) is extremely poor as it frequently misses the proper regions to refine. Comparison of results on a linearly doped diode before and after refinement using LE and any estimate based on the solution elm (figure 4) show that LTE does not achieve the expected h2 (4X) reduction in error, due to the fact that it is non-’zero only at the ends of the space-charge region and where l for the standard box method, we solve 3x3 linear systems for the coefficients of quadratic basis functions. We note that, since the usual ScharfetterCummel discretization does not quite fit into a Galierkin hierarchy. special cam must be taken with the continuity equations; see 141. The calculations are completely vector/parallel and are a negligible expense. The advantage of BW is illustrated by the Poisson solution to a reverse biased p+n &ode (figure 6). The actual error extends throughout the depletion region, whereas the BW estimate is only large in the vicinity of the junction. However, both lead to a 4X reduction in error after refinement, although the BW estimate adds 25X less gnd points. l?re key observation is that the actual solution error can be influenced in regions not needing refinement by errors made some distance away. Furthmore, the actual error is often extremely expensive to estimate. The remaining CMOS grids (figures 6-8) show selected results ol adaptive refinmentslunrefinements at points in IV continuation simulation [5]. Figure 6 cormponds to the off-state wllae a large potential barrier exists at the tub-substrate junction: refined regions extend upward along the right sidewall and across the trench. Figure 7. a bias point near triggering, has moved the grid to follow the barrier to the epi-subrstmte interface (Kirk effect) and has refined the tubtrench interface due to sharper band bending. Figure 8, a point in the on-state, has completely unrefined dd junctions due to high-level injection, and the resulting carrier plasma butts against the substrate: the large difference in tub potentials sets up a barrier on the right,, including a point of near singularity at the n+ tub contact The average number of gnd points required to trace the IV curve to 5OmV accuracy in poten~tials is ~1700 using full adaption, and the overhead versus knowing the actual grid a priori is about 50%. If a static grid wen to be used, more than 4OOO @d points would have been required, and even then the error indicator would be nect!sq to pply piace them. If the initial grid were always used, the extracted triggeriholding points would have been in mor by as much as 3096.