This paper describes a model that simulates etching profiles and process latitudes in glow-discharge bombardment-induced reactive-etching processes. Numerical results are presented for the pattern-transfer step in trilayer lithography, but this analysis is applicable to many other pattern-transfer processes. The inputs to the interface-evolution model described here are a kinetic model for the yield per incident energetic particle and a statistical mechanical model that relates the incident-yield-weighted angular distribution to the pressure, sheath thickness, and sheath voltage drop. The kinetic model is based on experimental evidence and assumes that the yield per bombarding particle is proportional to its energy. The resulting interface-evolution equation is mathematically analogous to a free-surface evolution equation in hydrodynamics. This convective partial differential equation is reduced to a coupled set of ordinary differential equations via the method of characteristics and solved numerically. More general energy-dependent yields are easily incorporated in the present formulation, but angle-dependent yields are more difficult and are not treated here. This model describes how shadowing of the surface being etched results in proximity effects in line etching and aspect-ratio-dependent etching rates in trench etching. Simulated profiles are compared to experimental trilayer etching profiles and qualitatively describe their shape and the trends that are observed as pressure or other processing parameters are varied. Simulations showing the effect of angular distributions, line proximity, and trench aspect ratio on process latitudes in trilayer lithography are presented and discussed.
The effect of sheath collision processes on the energy and directionality of surface bombardment in reactive ion etching is modeled. Although the methods used are generally applicable, all the numerical examples are for a low-pressure high-frequency oxygen plasma. Charge transfer is shown to be the dominant process controlling bombardment energies. The effect of momentum-transfer collisions on ion bombardment energies is shown to be negligible. Equations are derived for the average energy of ions and neutrals, the average ion energy, the average neutral energy, and the ion energy distribution function. The ion drift velocity at a point in the sheath is related to the voltage distribution by an equation that provides a rigorous basis for a self-consistent theory of the sheath voltage distribution. These equations are generally applicable to high-frequency, low-pressure plasmas where charge transfer is the dominant collision process. The angular distribution of energetic species is modeled using elastic scattering theory. These angular distributions can be used as input to etching models that calculate profiles and process latitudes.
An interface evolution equation has been formulated to describe bombardment-induced etching by an axisymmetric angular distribution of energetic particles where the yield per incident particle is assumed to be a function of its energy and its angle relative to the surface normal. These assumptions result in a nonlinear integro differential equation, but this equation reduces to a partial differential equation in several important special cases. At points that are not shadowed by a remote part of the surface, the interface evolution equation reduces to a nonlinear hyperbolic conservation law. Such equations have been applied to bombardment-induced etching by a monodirectionaI beam with angle-dependent yields; however, this form of equation applies more generally to raised isolated convex regions (e.g., etching masks) regardless of the angular distribution of the incident particles or the angle dependence of the yield. The essential qualitative feature of the solution in these cases is the spontaneous evolution of facet edges (slope discontinuities) from smooth initial conditions. Shadowing by remote parts of the surface may occur in concave regions (e.g., trenches; where it results in proximity effects.
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