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 mu C/cm(2), 30 A/cm(2), and 50 kV is 45-85 degrees C, depending on pattern density. Heating is also significant for direct writing on silicon. At 10 mu C/cm(2) 50 A/cm(2), and 100 kV, the temperature rise is 0.4-48 degrees C, depending on pattern density. Beam heating depends directly on the incident energy per unit area, which is given by the dose times the beam voltage. The dose is, in turn, proportional to the beam voltage, due to the increased transparency of resist layers at high voltage. Beam heating therefore increases with the square of the beam voltage. These trends suggest fast resist and low voltage as the best means of containing the effect. The latter conflicts with the need for high resolution, which requires high beam voltage. This suggests an optimum beam voltage for a given set of conditions. (C) 1996 American Vacuum Society.