Thermodynamic quasi-equilibria in high power magnetron discharges: a generalized Poisson–Boltzmann relation

The Poisson–Boltzmann (PB) equation is a nonlinear partial differential equation that describes the equilibria of conducting fluids. Using a thermodynamic variational principle based on the balances of particle number, entropy, and electromagnetic enthalpy, it can also be justified for a wide class of unmagnetized technological plasmas (Köhn et al 2021 Plasma Sources Sci. Technol. 30 105014). This study extends the variational principle and the resulting PB equation to high power magnetron discharges as used in planar high power pulsed magnetron sputtering. The example in focus is that of a circular high power magnetron. The discharge chamber and the magnetic field are assumed to be axisymmetric. The plasma dynamics need not share the symmetry. The domain is split into the ionization region close to the cathode where electrons are confined, i.e. can escape from their magnetic field lines only by slow processes such as drift and diffusion, and the outer region , where the electrons are largely free and the plasma is cold. With regard to the dynamics of the electrons and the electric field, a distinction is made between a fast thermodynamic and a slow dissipative temporal regime. The variational principle established for the thermodynamic regime is similar to its counterpart for unmagnetized plasmas but takes magnetic confinement explicitly into account by treating the infinitesimal flux tubes of as individual thermodynamic units. The obtained solutions satisfy a generalized PB relation and represent thermodynamic equilibria in the fast regime. However, in the slow regime, they must be interpreted as dissipative structures. The theoretical characterization of the dynamics is corroborated by experimental results on high power magnetrons published in the literature. These results are briefly discussed to provide additional support.