A time resolved analysis of the emission of HiPIMS plasmas reveals inhomogeneities in the form of rotating spokes. The shape of these spokes is very characteristic depending on the target material. The localized enhanced light emission has been correlated with the ion production. Based on these data, the peculiar shape of the emission profiles can be explained by the localized generation of secondary electrons, resulting in an energetic electron pressure exceeding the magnetic pressure. This general picture is able to explain the observed emission profile for different target materials including gas rarefaction and second ionization potential of the sputtered elements.

The effect of a magnetic field on the characteristics of capacitively coupled radio frequency discharges is investigated and found to be substantial. A one-dimensional particle-in-cell simulation shows that geometrically symmetric discharges can be asymmetrized by applying a spatially inhomogeneous magnetic field. This effect is similar to the recently discovered electrical asymmetry effect. Both effects act independently, they can work in the same direction or compensate each other. Also the ion energy distribution functions at the electrodes are strongly affected by the magnetic field, although only indirectly. The field influences not the dynamics of the sheath itself but rather its operating conditions, i.e., the ion flux through it and voltage drop across it. To support this interpretation, the particle-in-cell results are compared with the outcome of the recently proposed ensemble-in-spacetime algorithm. Although that scheme resolves only the sheath and neglects magnetization, it is able to reproduce the ion energy distribution functions with very good accuracy, regardless of whether the discharge is magnetized or not.

The simulation of magnetron discharges requires a quantitatively correct mathematical model of the magnetic field structure. This study presents a method to construct such a model on the basis of a spatially restricted set of experimental data and a plausible a priori assumption on the magnetic field configuration. The example in focus is that of a planar circular magnetron. The experimental data are Hall probe measurements of the magnetic flux density in an accessible region above the magnetron plane [P. D. Machura et al., Plasma Sources Sci. Technol. 23, 065043 (2014)]. The a priori assumption reflects the actual design of the device, and it takes the magnetic field emerging from a center magnet of strength mC and vertical position dC and a ring magnet of strength mR, vertical position dR, and radius R. An analytical representation of the assumed field configuration can be formulated in terms of generalized hypergeometric functions. Fitting the ansatz to the experimental data with a least square method results in a fully specified analytical field model that agrees well with the data inside the accessible region and, moreover, is physically plausible in the regions outside of it. The outcome proves superior to the result of an alternative approach which starts from a multimode solution of the vacuum field problem formulated in terms of polar Bessel functions and vertical exponentials. As a first application of the obtained field model, typical electron and ion Larmor radii and the gradient and curvature drift velocities of the electron guiding center are calculated.

This paper analyzes a situation which is common for magnetized technical plasmas such as dc magnetron discharges and HiPIMS systems, where secondary electrons enter the plasma after being accelerated in the cathode fall and encounter a nearly uniform bulk. An analytic calculation of the distribution function of hot electrons is presented; these are described as an initially monoenergetic beam that slows down by Coulomb collisions with a Maxwellian distribution of bulk (cold) electrons, and by inelastic collisions with neutrals. Although this analytical solution is based on a steady-state assumption, a comparison of the characteristic time-scales suggests that it may be applicable to a variety of practical time-dependent discharges, and it may be used to introduce kinetic effects into models based on the hypothesis of Maxwellian electrons. The results are verified for parameters appropriate to HiPIMS discharges, by means of time-dependent and fully-kinetic numerical calculations.

Abstract. 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 × 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.