Theoretical analysis of focus wedge mark for best-focus determination in photolithography

Abstract

The focus wedge mark (FWM), a periodic array of wedge shaped marks, is a tool used to determine the best-focus position in photolithography. Here, we develop a one-dimensional physical model for the FWM and analyze the focus dependence of the wedge length under coherent imaging conditions. The amplification factor of the FWM, which describes the superiority of the FWM compared to measurements based on line and space marks, is determined by the geometrical magnification of the wedge. We show that around the exposure dose energy typically used in lithography the amplification factor is much higher than the geometrical magnification. Numerical lithography simulations confirm that the focus sensitivity of the FWM is much higher than that of a line and space pattern.

Accuracy of focus wedge mark measurement using fitting function for partially coherent illumination

“…5 of Ref. 18. With increasing defocus, the focus variation of the wedge length shows a monotonic decrease.…”

“…For the object spectrum, we adopt the one-dimensional effective model for the FWM proposed in a previous study. 18) As reviewed in Appendix A, the imaging properties of the FWM are well approximated by the periodic L&S pattern. The spectrum of the L&S pattern with the opaque line width D and pitch P is given by…”

“…19) is equivalent to the result for the coherent illumination case. 18) The left-hand side of Eq. ( 2.…”

“…In the previous studies, 17,18) we proposed a new concept with a specific focus sensitive mark named focus wedge mark (FWM). The FWM consists of an array of wedges, where the length of each wedge is greater than the width.…”

“…In what follows, we clarify the dependency of the FWM metrology accuracy Δz 0 on two fitting functions introduced below. The quadratic polynomial fitting function is likely to be utilized, 17,18) and it is expressed by…”