Theory of beam‐induced substrate heating

In all electron, ion, and photon beam systems, the beam unavoidably deposits heat in the target substrate. Resist sensitivity varies with temperature, causing unwanted linewidth variation. Substrate heating has been studied by finite element analysis solution of the heat equation. The four‐dimensional nature of the problem, three spatial dimensions together with time, leads to prohibitively long computation time. This limits our ability to obtain a physical understanding of the problem. Alternatively one can superimpose elementary analytic solutions, which have simple space‐time dependency, to construct cases of practical interest. The purpose of this article is to describe mathematically the basic building blocks of a general analytic theory of beam heating, with the aim of efficiently and quickly computing temperature rise due to beam‐induced substrate heating. Beam heating is most critical in writing of quartz reticles. Because of the poor thermal conductivity of quartz, the temperature rise at 10 μC/c...