Stochastic Coulomb interactions in ion projection lithography systems with aberration-broadened crossover

Abstract

The allowable current in ion beam projection lithography is limited by stochastic Coulomb interactions. Theories for the dependence of these interactions on the parameters of the system, i.e., length, beam energy, print size, crossover diameter, etc. are thus important to guide the design of future machines. All existing theories on blur from stochastic Coulomb interaction assume a homogeneous current distribution in the crossover, where a large part of the blur is produced. However, in actual lithography machines, the crossover may be substantially broadened by the spherical aberration of the ion lenses, thus giving a very inhomogeneous current distribution in the crossover. In most analytical theories, the stochastic blur is independent of the radius of the homogeneously field crossover rc for very small crossovers, and relates to rc as (1/rc)1/2 or (1/rc)1/3 for larger ones. If aberration broadening could do the same, the effect of broadening the crossover from rc=1.5 to 50 μm would decrease the blur by a factor as much as 6, without the negative effect of increasing the curvature-of-field blur. We find physical arguments that such a large factor is impossible. Monte Carlo simulations first of all show that the blur reduction is larger near the edges of the image field than in the center, near the optical axis. For the simulated model system, the reduction on axis can be increased to a factor 2 at very large aberration coefficients.