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We have formulated a kinetic theory of bombardment induced interface evolution to describe etching by an axisymmetric angular distribution of energetic particles where the volume removed per particle is a function of energy and angle relative to the surface normal. Kinetic theory relates the angular distribution to the pressure, the voltage drop across the plasma sheath, the sheath thickness, and the cross sections describing the collision processes. The resulting interface evolution equation is a nonlinear partial differential equation that may be reduced to a coupled set of ordinary differential equations by the method of characteristics. Additional simplifications arise in two special cases that are relevant to the pattern transfer step in multi-layer lithography. One of these simplifications occurs when the yield is independent of angle as expected for the thermal spike mechanism of bombardment induced chemically assisted etching. This case applies to the planarizing layer in multi-layer lithography where the etching rate is proportional to the energy flux delivered by bombarding particles to a given point on the surface. Analysis of this case shows that shadowing of the angular distribution by adjacent features causes proximity effects in line etching and aspect ratio dependent etching rates in trench etching. For angle dependent yields, another simplification applies to regions that are not shadowed by a remote part of the surface. Analysis of this case shows that facet edges (slope discontinuities) will spontaneously develop from smooth initial conditions. Etching masks are not significantly shadowed by adjacent features, so this case describes mask erosion by physical sputtering mechanisms which are characterized by strongly angle dependent yields. Together these cases allow a complete description of the pattern transfer step in bi-layer lithography. We discuss the effect of mask selectivity, mask wall angle, trench aspect ratio, and etching conditions on etching profiles and process latitudes for the pattern transfer step in multi-layer lithography.
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