# Analytic model of the energy distribution function for highly energetic electrons in magnetron plasmas

## Abstract

This paper analyzes a situation which is common for magnetized technical plasmas such as dc magnetron discharges and high power impulse magnetron sputtering (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.

- Authors:

- Institute of Theoretical Electrical Engineering, Ruhr University Bochum, Bochum D-44780 (Germany)
- Department of Electrical and Computer Engineering, University of Wisconsin-Madison, Madison, Wisconsin 53706 (United States)

- Publication Date:

- OSTI Identifier:
- 22412815

- Resource Type:
- Journal Article

- Resource Relation:
- Journal Name: Journal of Applied Physics; Journal Volume: 117; Journal Issue: 2; Other Information: (c) 2015 AIP Publishing LLC; Country of input: International Atomic Energy Agency (IAEA)

- Country of Publication:
- United States

- Language:
- English

- Subject:
- 71 CLASSICAL AND QUANTUM MECHANICS, GENERAL PHYSICS; ANALYTICAL SOLUTION; BOLTZMANN STATISTICS; CATHODES; COMPARATIVE EVALUATIONS; DISTRIBUTION FUNCTIONS; ELECTRIC DISCHARGES; ELECTRON COLLISIONS; ENERGY SPECTRA; MAGNETRONS; PLASMA; SPUTTERING; STEADY-STATE CONDITIONS; TAIL ELECTRONS; TIME DEPENDENCE

### Citation Formats

```
Gallian, Sara, E-mail: gallian@tet.rub.de, Trieschmann, Jan, Mussenbrock, Thomas, Brinkmann, Ralf Peter, and Hitchon, William N. G.
```*Analytic model of the energy distribution function for highly energetic electrons in magnetron plasmas*. United States: N. p., 2015.
Web. doi:10.1063/1.4905943.

```
Gallian, Sara, E-mail: gallian@tet.rub.de, Trieschmann, Jan, Mussenbrock, Thomas, Brinkmann, Ralf Peter, & Hitchon, William N. G.
```*Analytic model of the energy distribution function for highly energetic electrons in magnetron plasmas*. United States. doi:10.1063/1.4905943.

```
Gallian, Sara, E-mail: gallian@tet.rub.de, Trieschmann, Jan, Mussenbrock, Thomas, Brinkmann, Ralf Peter, and Hitchon, William N. G. Wed .
"Analytic model of the energy distribution function for highly energetic electrons in magnetron plasmas". United States.
doi:10.1063/1.4905943.
```

```
@article{osti_22412815,
```

title = {Analytic model of the energy distribution function for highly energetic electrons in magnetron plasmas},

author = {Gallian, Sara, E-mail: gallian@tet.rub.de and Trieschmann, Jan and Mussenbrock, Thomas and Brinkmann, Ralf Peter and Hitchon, William N. G.},

abstractNote = {This paper analyzes a situation which is common for magnetized technical plasmas such as dc magnetron discharges and high power impulse magnetron sputtering (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.},

doi = {10.1063/1.4905943},

journal = {Journal of Applied Physics},

number = 2,

volume = 117,

place = {United States},

year = {Wed Jan 14 00:00:00 EST 2015},

month = {Wed Jan 14 00:00:00 EST 2015}

}