Nonlinear modeling and multivariable control of photolithography

This paper describes a novel approach for the control of the entire photolithography track using a combination of two methods: genetic programming (GP) and nonlinear model predictive control (NMPC). Here, the GP-NMPC approach is used to derive a multivariable control system to ensure the adequate regulation of the printed line width or critical dimension (CD) measured by metrology at the tail of the track. The genetic program is an optimization method motivated by natural evolution, which generates a model that best predicts the effect of process inputs on outputs. When applied to a simulated photolithography track, it identifies which of the process inputs have the greatest effect on CD and suggests the best empirical nonlinear model relating the inputs to the CD, which is then used in the development of the NMPC. Simulation runs using the multivariable controller demonstrate its superiority over that of a conventional feedback approach involving single-loop control. Since the multivariable control uses all available degrees-of-freedom and is designed to account for manipulated variable constraints, it enables the track to cope with unmeasured step- and drift-type disturbances of significantly greater magnitude.