A phenomenological model for the description of rotating spokes in HiPIMS discharges

In the ionization region above circular planar magnetrons, well-defined regions of high emissivity are observed, when the discharge is driven in the HiPIMS regime. These regions are characterized by high plasma density and are often referred to as 'spokes'. Once their mode is stabilized, these structures rotate in the E x B direction with a constant rotation frequency in the hundreds of kHz range. A phenomenological model of the phenomenon is developed, in the form of a system of nonlinear coupled partial differential equations. The system is solved analytically in a frame co-moving with the structure, and its solution gives the neutral density and the plasma density once the electron density shape is imposed. From the balance of ionization, electron loss and constant neutral refilling, a steady-state configuration in the rotating frame is achieved for the electron and neutral densities. Therefore, the spoke experimentally observed can be sustained simply by the combination of this highly reduced number of phenomena. Finally, a study of the sensitivity of neutral and plasma densities to the physical parameters is also given.