An exposure model for electron-sensitive resists

We present a mathematical model for the exposure of electron-sensitive resists where an electron beam is incident normal to a substrate coated with a thin layer of resist. We include both the scattering of the incident electrons as they penetrate the resist and the electrons backscattered from within the resist and from the substrate. The calculations yield contours of equal absorbed energy density, and these are interpreted as the contours which bound the resist after development. The absorbed energy density is found as the sum, for all electrons, of the product of the energy absorbed per unit length of trajectory and the flux density of electrons at the point in question. We first calculate the absorbed energy density for an electron beam of vanishingly small cross section (an incident delta function) and then convolve that result with a beam of Gaussian current-density distribution to obtain the reSult for a single beam location. For poly(methyl methacrylate) resist, we study the achievable dot resolution, as a function of the incident charge, for various incident energies-and substrates. Since our main interest is in computer-controlled resist exposures in which patterns are generated as a succession of dots, we calculate the absorbed energy density contours for a line generated in that manner. Detailed comparison is made with the experimental results of Wolf et al., by fitting a single point on one contour at one beam energy to account for the unknown developer sensitivity. The resulting contours predict the undercutting effect experimentally observed for the 5-20-keV beam energies studied. The developed shape and linewidth are found to be nonlinear functions of the incident charge per unit length. Experimental data for the linewidth at 20 keV are presented and compared with theory.