Equations of exposure time and X-ray mask absorber thickness in the LIGA process

Abstract

The LIGA X-ray exposure step was modeled into three inequalities from exposure requirements. From these inequalities, equations for the minimum and maximum exposure times required for a good quality microstructure were obtained. An equation for the thickness of an X-ray mask absorber was also obtained from the exposure requirement of threshold dose deposition. A power function of photon energy, approximating the attenuation length of the representative LIGA resist, PMMA, and the mean photon energy of the X-rays incident upon an X-ray mask absorber were applied to the above mentioned equations. Consequently, the trends of the minimum and maximum exposure times with respect to mean photon energy of X-rays and thickness of PMMA were examined and an equation for the maximum exposable thickness of PMMA was obtained. The trends of the necessary thickness of a gold X-ray mask absorber with respect to photon energy of the X-rays and PMMA thickness ratio were also examined. The simplicity of the derived equations has clarified the X-ray exposure phenomenon and the interplay of exposure times, the attenuation coefficient and the thickness of an X-ray mask absorber, the attenuation coefficient and the thickness of a resist, and synchrotron radiation power density.