Hybrid modelling of low temperature plasmas for fundamental investigations and equipment design

The modelling of low temperature plasmas for fundamental investigations and equipment design is challenged by conflicting goals—having detailed, specialized algorithms which address sometimes subtle physical phenomena while also being flexible enough to address a wide range of process conditions. Hybrid modelling (HM) is a technique which provides many opportunities to address both fundamental physics and practical matters of equipment design. HM is a hierarchical approach in which modules addressing different physical processes on vastly disparate timescales are iteratively combined using time-slicing techniques. By compartmentalizing the physics in each module to accept given inputs and produce required outputs, different algorithms can be used to represent the same physical processes. In this manner, the algorithms best suited for the conditions of interest can be used without affecting other modules. In this paper, the basis and implementation of HM are discussed using examples from simulations of inductively coupled plasmas. Abbreviations and symbols (ε, r, φ) dependence of energy, position and phase in harmonic period (or time) f (ε, r, φ) distribution function (eV−3/2) in the gas phase or incident onto surfaces ke( r, φ) electron impact rate coefficients (cm3 s−1) Se( r, φ) electron impact source functions (cm−3 s−1) kSe( r, φ) collectively, ke( r, φ) and Se( r, φ) k( r, φ) heavy particle collisional rate coefficients (cm3 s−1) S( r, φ) heavy particle collisional source functions (cm−3 s−1) kS( r, φ) collectively, k( r, φ) and S( r, φ) E( r, φ), B( r, φ) electromagnetic fields (V cm−1, G) E B( r, φ) collectively, E( r, φ) and B( r, φ) ES( r, φ), BS( r, φ) electroand magnetostatic fields (V cm−1, G) E BS( r, φ) collectively, ES( r, φ) and BS( r, φ) S( r, φ) electrostatic potential (V) b(t) electric potential across sheath as a jump boundary condition (V) ES( r, φ) collectively, S( r, φ) and ES( r, φ) MS( r, φ) magnetization (e.g. dipole moments) of materials external to plasma v( r, φ) electron collision frequency (s−1) σ ( r, φ) conductivity (1 −1 cm−1) ρ( r, φ) charge density (C cm−3) ρS( r, φ) charge density on surface (cm−3) j( r, φ) current density (A cm−2) jE( r, φ) external (or non-plasma) current density (A cm−2) N( r) density (cm−3) 0022-3727/09/194013+20$30.00 1 © 2009 IOP Publishing Ltd Printed in the UK J. Phys. D: Appl. Phys. 42 (2009) 194013 M J Kushner φ( r) flux (cm−2 s−1) φP(ν, r) photon flux versus frequency (cm−2 s−1) φ phase in harmonic cycle T ( r) temperature (K or eV) NφT ( r) collectively, N( r), φ( r) and T ( r) α( r) reaction probability on surfaces βij ( r) probability of species i incident on surface producing species j tM, time between calls to a module CCP capacitively coupled plasma H(r ) change in enthalpy DSMC direct-simulation Monte Carlo EEE electron energy equation EMCS electron Monte Carlo simulation FD frequency domain FDTD finite difference, time domain HM hybrid modelling HPEM Hybrid Plasma Equipment Model IEAD ion energy and angular distribution LFA local field approximation LTP low temperature plasma MERIE magnetically enhanced reactive ion etching SS, HSS steady state, harmonic steady state Td E/N in Townsend (1 Td = 10−17 V cm2)