Modeling and Control of Thin Film Morphology Using Unsteady Processing Parameters: Problem Formulation and Initial Results
Martha A. Gallivan, David G. Goodwin and Richard M. Murray
Submitted, 2001 Conference on Decision and Control
Thin film deposition is an industrially-important process to which control theory has not historically been applied. The need for control is growing as the size of integrated circuits shrinks, requiring increasingly tighter tolerances in the manufacture of thin films. Our contributions in this study are two-fold: we formulate a model of thin film growth as a control system and we examine the effects of fast periodic forcing.
We choose a lattice formulation of crystal growth as our physical model, which captures atomic length scale effects at a time scale compatible with film growth. We focus on the control of film morphology, or surface height profile. Although the system dimension is high, the structure is simple: the dynamics and the output are linear in the state. We consider the process conditions as inputs, which alter the transition rate functions. In the evolution equation, each of these nonlinear functions is multiplied by a linear vector field, yielding a system with a structure similar to a bilinear system.
The process conditions in some deposition methods are inherently unsteady, which produces films with altered morphology. We use the model developed in this study to analyze the effects of fast periodic forcing on thin film evolution. With the method of averaging we develop new effective transition rates which may produce film properties unattainable with constant inputs. We show that these effective rates are the convex hull of the set of rates associated with constant inputs. We present conditions on the convex hull for which the finite-time and infinite-time reachability sets cannot be expanded with fast periodic forcing. An example in which this forcing increases the reachability set and produces more desirable morphology is also presented.