Unipolar arcing

Unipolar arcing is a phenomenon which may occur in plasma devices between the plasma and the cathode. This cathodic process features localized, bright, tiny spots on the cathode surface, which appear to move more or less randomly. At these spots, the cathode material makes a transition into dense plasma, which then expands rapidly into the vacuum or low-pressure ambient gas.

Thermo-ionic Emission

To understand what is going on, let us shortly recap some plasma physics. In a typical plasma device, the plasma is present between cathode and anode, which enables current to flow by motion of mobile charged particles. In the plasma, most of the electric current is carried by electrons because the electron mobility is much higher than that of the ions, due to the lower mass. The critical places of current continuity are the interfaces between plasma and metal. On the anode side, electrons fall into the conduction band, thereby liberating the potential energy known as the work function of the anode (about 4 eV per electron for most metals). On the cathode side, however, electrons are prevented from escaping by a potential barrier, the work function of the cathode.

The nature of the discharge may create conditions which enable a fraction of the electrons to overcome the potential barrier, leading to electron emission. Depending on the character of those conditions, we distinguish different electron emission mechanisms. Electrons can be emitted during individual events, such as ion impact, or by collective events, such as high cathode temperature (thermionic) and/or a high electric field on the cathode surface. The collective thermionic and field emission can non-linearly amplify each other known as thermo-field emission. The class of emission by individual events are called 'glow' discharges, and emission by collective events 'arc' discharges.

Collective thermionic, and/or field emission can be stationary. For arc discharges this is, however, not the case: the emission is related to energy dissipation and net heating of the cathode, which can enhance the temperature and associated electron emission, a so-called thermal run-away process. Locations where this occurs can explosively evaporate, leading to a new form of electron emission that is inherently non-stationary because the emission location is changed by the explosion, the plasma expansion and the increase of the hot spot area by thermal conduction. This non-stationary form of emission is called explosive electron emission or arcing.

Model of Arcing

The electron emission at the cathode spot occurs in the form of discrete explosive electron emission splashes, so-called 'ectons'. These quanta of the explosive process represent the minimum actions required for the explosive events. The duration of one ecton is about $\sim$10 ns, the current $\sim$1 A, and the size of the emission centers is about $\sim 1$ $\mu$m. The explosion leaves a micro crater with a diameter of about $\sim 1$ $\mu$m.

According to the ecton model, arc operation is self-sustained, and occurs in stages

. The first stage is the appearance of dense primary erosion plasma due to the external action, e.g. a laser pulse or ELM-plasma, onto the target. This dense plasma action results in a strong emission pulse ($10^{8}$ A/cm$^{2}$) that leads to a thermal explosion of the emitting local area, the start of stage two. The created dense plasma produces two important effects: 1) the sheath thickness reduces, leading to an increase in the electric field at the surface, and 2) (due to the electric field) the ion bombardment heating increases. Now, if the local electric field is additionally enhanced by the fine structure of the surface, e.g. tungsten fuzz, this can all together intensify the local energy input, leading to a thermal run-away process. If the energy input rate exceeds the energy removing rate, this can lead to a micro-explosion. The micro-explosion creates another dense erosion plasma, and hence creates another emission site, so this causes repeating ignition of micro-explosions. The dense plasma provides the conditions for the ignition while 'choking' the already operating emission center by its limited conductivity \footnote{During the explosive gas phase, material is evaporated which increases the gas density in front of the emission site. Since gas is a bad conductor, the current transfer capability suffers.}. Ignition in this sense is not just the triggering of the arc discharge but the arc's perpetual mechanism to 'stay alive.' The probabilistic distribution of ignition of emission centers can be associated with a fractal spot model\footnote{Fractals are mathematical or physical objects invariant to scaling, so called 'self-similar'. They occur in phenomena which are nonlinear, aperiodic, and chaotic, such as arcing \cite{And2008}.}

Finally, the electron emission, and evaporation ceases, because the thermal conduction has led to an increase of the spot area, lowered the power density, and hence lowered the surface temperature. The explosively formed plasma has expanded, its density is lowered, therefore the cathode sheath thickness has increased, and therefore the electric field at the surface is reduced.

Energy balance

The transition from the cathode's solid phase to the plasma phase requires energy, which is supplied via the power dissipated by the arc,

$P_{arc}=VI_{arc}$

where V is the voltage of the arc (measured between anode and cathode). The energy needed for the phase transition is only a fraction of the total energy balance. The total balance of the cathode region is given by:

$I_{arc}V\tau =E_{phon}+E_{CE}+E_{ionization}+E_{kin,i}+E_{ee}+E_{th,e}+E_{MP}+E_{rad}$

where $\tau$ is a time interval over which observation is averaged, $E_{phon}$ is the phonon energy (heat)transferred to the cathode material, $E_{CE}$ the cohesive energy needed to transfer the cathode material from the solid phase to the vapor phase, $E_{ionization}$ is the energy needed to ionize the vaporized cathode material, $E_{kin,i}$ is the kinetic energy given to the ions due tot the pressure gradient and other acceleration mechanisms, $E_{ee}$ is the energy needed to emit electrons from the solid to the plasma, $E_{th,e}$ the thermal energy (enthalpy) of electron in the plasma, $E_{MP}$ is the energy invested in melting, heating, and acceleration of marcoparticles, and $E_{rad}$ is the energy emitted by radiation. The input energy is mostly transferred to heat the cathode, to emit and heat electrons, and to produce and accelerate ions.