Simulation of resist heating in electron beam lithography

A mathematical model of resist heating was developed to anticipate the resist temperature rise during electron exposure. This model that takes into consideration the heat propagation in both resist and substrate, each of which may consist of a number of layers. A tabulation of special functions used for numerical calculation allows the analytical solution to be simplified to a single integral for both a variably shaped beam and a Gaussian beam. As a result, a dynamic distribution of a temperature field in resist can be simulated for an exposure process in terms of thickness and characteristics of the resist and multilayer substrate, as well as size, placement, and sequence of flashes. Results of simulations are shown for both short and long exposure periods. It was demonstrated that heat from the substrate does not contribute noticeably at the beginning of the high throughput exposure at 50 kV, whereas heating of the resist itself is significant and highly nonuniform throughout the depth of the resist.