Analytic sheath resolution in high density plasma source simulations

Abstract

Summary form only given, as follows. Capacitive coupling involves the physics of external RF voltages coupling to the plasma through an oscillating sheath. For high density plasmas, the sheath dimension is much smaller than the scale lengths of density and field variation in the bulk of the plasma. To avoid costly numerical resolution of the sheath, we use analytic models to determine sheath voltage drops and power deposition due to Ohmic and stochastic heating at the sheaths. We use Metze's sheath model (1966) because it provides an ODE for the sheath voltage drop which can be directly incorporated into the numerical solution of Poisson equation. The model has been tested in the particle code PDPI by comparing simulations that resolve the sheath with simulations that use the analytic sheath. The analytic model reproduces the sheath drop fairly well. Alternatively, one can use Godyak and Sternberg's model (1990) for high frequency RF excitations. We use Lieberman's model (1988) to calculate Ohmic heating due to the substrate RF current and stochastic heating due to electrons bouncing of the oscillating substrate sheath. For a 13.56 MHz, 4 eV, 10 mTorr argon discharge 6 cm long with 8 inch substrate, P/sub stoc/=0.045(V/sub dc//T/sub c/) W and P/sub ohm/=0.078(V/sub dc//T/sub c/)/sup 1/2/ W, where V/sub dc/ is the time-average sheath voltage. These scalings indicate that stochastic and Ohmic heating at the substrate is typically only a few percent of the power coupled inductively.