Model of the cathode fall region in magnetron discharges

Using a 1D, two-fluid description, the influence of the magnetic field on the plasma boundary region adjacent to the cathode target in a magnetron has been investigated.. Assuming the transport of magnetically confined electrons above the 'race-track' to be dominated by diffusion and mobility, with plasma ions falling directly to the cathode, the spatial variation in the electron and ion densities and velocities in the sheath and pre-sheath have been determined, together with the electric field structure. By assuming an electric field remnant in the plasma, the fluid equations have been integrated through the space-charge sheath up to the cathode, allowing smooth joining of the sheath and pre-sheath regions. Modelling with both constant and spatially varying magnetic fields has shown that realistic secondary electron fluxes are predicted for only a limited range of initial discharge parameters, indicating that Bohm diffusion of electrons is the most appropriate transport mechanism. Sheath widths are found to decrease with increasing magnetic field strength at the cathode and increase with cathode potential. In the constant-B-field case and for deep sheath potentials (>10 electron temperatures equivalent), the sheath behaviour is consistent with the Child-Langmuir law. However, the inclusion of a grad B force in the spatially varying magnetic field case gives rise to electron densities in the sheath, a significant fraction of the ion density (>10%).