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

The Poisson-Boltzmann (PB) relation is an elliptic nonlinear partial differential equation that describes the thermodynamic equilibria of conducting fluids. It can be justified through a quasi-thermodynamic variational principle that considers the balances of particle number, entropy, and electromagnetic enthalpy. This principle also applies to a class of unmagnetized technological plasmas that exist far from thermodynamic equilibrium [1]. In this study, the variational principle and resulting Poisson-Boltzmann relation are expanded to magnetized discharges encountered in planar high-power pulsed magnetron sputtering (HiPIMS) [2]. The focus is on a planar high-power circular magnetron, assuming axisymmetric discharge chambers and magnetic fields. However, it should be noted that the dynamics may not share this symmetry. The domain is divided into two regions: the ionization region near the cathode, where electrons are confined and can only escape their magnetic field lines through slow processes such as drift and diffusion, and the outer region, where electrons are free and the plasma is cold. When considering the dynamics of electrons and the electric field, a distinction is made between a fast thermodynamic regime and a slow dissipative temporal regime. The variational principle derived for the thermodynamic regime resembles its counterpart for unmagnetized plasmas, but it accounts for magnetic confinement by treating the magnetic flux tubes as distinct thermodynamic units. The resulting solutions adhere to a generalized Poisson-Boltzmann relation. These solutions represent thermodynamic equilibria in the fast regime, but in the slow regime, they should be understood as dissipative structures. The theoretical characterization of the dynamics of high-power magnetrons is supported by experimental results published in the literature.