<title>Optimum numerical aperture for optical projection microlithography</title>

Abstract

Given a resolution requirement and the imaging wavelength, there is an optimum NA. If the NA is too low, the resolution cannot be achieved, but if the NA is too high, the depth of focus, which is inversely proportional to NA2, becomes unacceptable. There is an optimum NA where the depth of focus is maximum. In this paper the optimum NA given by the aerial image is determined unambiguously by evaluating the normalized depth of focus k2 as a function of the normalized resolution k1, then identifying the k1 at which the function k2/k12 is maximum. The optimum NA is then simply the optimum k1 multiplied by (lambda) /NA. A substantial amount of work is required to evaluate k2 as a function of k1 by means of exposure-defocus trees and windows. In this paper, all k2 and k2/k12 as functions of k1 are given for line-space pairs, isolated line openings, isolated spaces, holes, islands, combination of the 3 long features, and of all the 5 features. A 10% exposure budget is used to simulate the situation of single layer resist systems and 30% exposure budget for multilayer resist systems. The results show optimum NA for individual feature shapes much lower than expectation, gaining insights to the problems occurring in manufacturing and in reducing the usable k1. They also lead to the following observations. The optimum k1 for single resist systems ranges from 0.57 to 0.87 depending on the feature shape. That for multilayer resist systems ranges from 0.42 to 0.7. Opaque spaces have the lowest optimum k1. Line openings have the largest depth of focus. Positive resists and negative masks are preferred to delineate contact holes. Negative resists and negative masks are preferred to delineate gates and metal lines. The opaque island is the limiting feature for the line-space pair, line, space, hole, and island combination at larger k1. That for the combination of line-space pair, line, and space is the line-space pair.