X-ray mask membrane motion in narrow gap lithography: Hydrodynamic model and experiment

Abstract

Articles you may be interested inFabrication of high resolution x-ray masks using diamond membrane for second generation x-ray lithographyThe viscosity of the gas in the proximity gap between x-ray mask and silicon wafer causes damping of any membrane motion. As x ray is extended to future integrated circuit generations, this effect becomes increasingly significant, because the gap must be rapidly scaled down to permit resolution of finer lithographic features. Damping is much greater at a reduced gap. Damping can be beneficial in reducing unwanted fluctuations in the gap due to vibration, convection, or ambient sound. The longer settling time following gap adjustment, however, can be detrimental to the throughput of x-ray steppers. This article introduces simple hydrodynamic models, formulas, and numerical algorithms to calculate membrane response to adjustments in gap setting. It applies the analysis to a variety of mask formats, including advanced "pedestal" design. Time constants of several seconds are shown to result from gap setting < 10 /-lm, which may be needed for 0.12 /-lm proximity x-ray lithography.

Deformation of a tensile membrane adjacent to a moving wall caused by viscous flow

“…Assumption (iii) is also reasonable, since the actual membrane had a size of several centimeters and the gap width was of the order 10 -100 m. Assumption (iv) is introduced because the Reynolds number Re= Ud= of the ow in the gap is estimated to be in an order smaller than 10 −2 , where is the density (order 1 kg=m 3 ), U is the velocity (order 10 −3 m/s), d is the gap width (order 10 −4 m) and is the viscosity (order 10 −5 Pa s). Under assumptions (iii) and (iv), we can apply the Hele-Shaw ow theory to the ow in the gap, which was also applied by Yanof et al (1993).…”

“…Particular exact solution of the homogeneous equation is the same as that obtained by Yanof et al (1993), which is expressed in terms of the 0th Bessel function J 0 (k r) and the modiÿed Bessel function I 0 (k r) as follows:…”

“…At present we have only a few studies on the problem of membrane deformation. Yanof et al (1993) obtained an exact solution for relaxation of membrane deformation in the case where an initial deformation is given and the wafer is at rest. Mitsui et al (1996) did an experiment to measure the mask membrane deformation, where the gap width was changed by ±2 m during the time of 40 ms to a ÿnal gap width, which was varied from 10 to 200 m. They observed that the maximum deformation (de ection at the membrane center) depended on the membrane tension and the ÿnal gap width, and it became as much as 6 m for the ÿnal gap of 10 m. When the gap was increased by 2 m, the ÿnal gap width to produce maximum deformation varied within 30 -50 m.…”

Fabrication of x-ray masks using evaporated electron sensitive layers for back patterning of membranes

“…Furthermore, irregularities of the gap can cause up to 20% disturbance and fluctuation in the relaxation time of a Si 3 N 4 membrane since the relaxation time is inversely proportional to the third power of the mask/ wafer gap value. 9 Sudden gap reduction can trap gas under the membrane, and actually delay the attainment of the equilibrium. A gap increase which is too rapid could possibly result in collision between the mask and the wafer.…”