Validity of double and triple Gaussian functions for proximity effect correction in X-ray mask writing

An improved function is proposed to express the deposited energy intensity distribution for fine patterns or that for heavy metals such as X-ray absorbers. An experimental method is also proposed for obtaining the function parameters, and the optimal parameters are obtained for Si and W-Ti substrates at acceleration voltage of 25 kV. Using the improved function with the best-fit parameters, the validity of the double and triple Gaussian functions for the proximity effect correction is evaluated. The dose modulation ratios for ninety line-and-space patterns on the two substrates are calculated using the double Gaussian, triple Gaussian, and improved functions. From the comparison among the ratios, the following are found: for the double Gaussian function the ranges of correction errors on W-Ti and Si substrates are 8 and 30%, respectively; for the triple Gaussian function the ranges of errors on W-Ti and Si substrates are 6 and 12%, respectively.