Gyrokinetic axisymmetric modeling of a HPPMS planar magnetron discharge - a valid approach?

Sara Gallian, Denis Eremin, Thomas Mussenbrock, Ralf Peter Brinkmann, William N. G. Hitchon

HIPIMS Conference 2012, Sheffield, UK, June 19-20


High Power Pulsed Magnetron Sputtering (HPPMS) is a novel Ionized Physical Vapor Deposition (IPVD) technique, able to achieve an ultra dense plasma with a high ionization degree among the sputtered atoms. This is accomplished by applying a large bias voltage to the target in short pulses with low duty cycle. \ In most HPPMS configurations the following ordering of spatial quantities holds: $lambda_{text{D}} ll s_text{t} ll r_text{L} ll L$, where $lambda_{text{D}}$ is the Debye length, $s_{text{t}}$ is the sheath thickness, $r_{text{L}}$ is the Larmor radius, $L$ is the system characteristic length. As concerns time scales, the ordering is: $omega_{text{pe}} gg omega_{text{ce}} gg nu_{text{ce}}$, where $omega_{text{pe}}$ is the electron plasma frequency, $omega_{text{ce}}$ is the electron cyclotron frequency and $ nu_{text{ce}}$ is the electron collision frequency. In the plasma bulk, the gyro-motion of the electrons around a magnetic field line provides the system smallest time and spatial scale. Therefore, it seems reasonable to describe the electron dynamics with gyro-averaged equations of motion. \ Moreover, given the axial symmetry of the set-up, one is induced to neglect the angular dependence, reducing the number of independent variables. \ Here, we individuate and discuss two reasons why this simplified approach may not apply or should be improved. \ Several authors have recently reported the presence of rotating structures during a HPPMS discharge. According to the experimental observations, these emission peaks rotate with constant angular velocity $Omega$, when the discharge parameters are held constant. This phenomenon clearly breaks the axial symmetry mentioned. We present here a simplified phenomenological model for these structures, based on the solution of a system of 1D Advection-Diffusion-Reaction equations to be solved analytically for the electron $n_text{e}(theta,t)$ and neutral $n_text{n}(theta,t)$ densities. We look for the existence and the stability of a set of solutions in a frame rotating with speed $Omega$. \ On the other hand, we focus on the electron trajectories. Most of the electrons are confined by the magnetic field, but a fraction enters the loss cone region and is reflected inside the thin sheath region or impacts on the target surface. The latter group is very small in number and can thus safely be neglected, since the target is negatively biased during the pulse duration. The electrons that are reflected inside the sheath cannot be described with the gyro-averaged equations of motion, since $r_{text{L}} gg s_{text{t}}$. If these unconfined electrons are statistically relevant and cannot be neglected, they represent a flux at the boundary of the gyro-averaged region, which should be described by appropriate boundary conditions. %Inside the sheath, however, $r_{text{L}}$ is no longer the smallest quantity and the gyro-average description is no longer applicable. The unconfined electrons represent a flux at the boundary of the gyro-averaged region, which should be described by appropriate boundary conditions. Moreover, we show that the electrons reflected in the sheath, which we represent as infinitely thin, have an average drift in the plane of the target surface, directed toward the external perimeter. If statistically relevant, this effect can potentially influence the magnetron performance. Therefore, we employ a single particle simulation to address the importance of this electron population in different magnetic field configurations.

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Tags: HIPIMS-Gyrokinetic