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.