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.

% basic plasma knowledge \subsection{Basics Plasma Physics} \label{bplasma} 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.

% sheat info Figure \ref{fig:pot_distr} shows a sketch of the potential distribution for a generic case. The voltage is not evenly distributed but concentrated in the sheaths near the cathode and anode. As indicated above, the potential drop in the cathodic sheath is the most significant of the mechanism that liberates electrons from the cathode. In contrast, the anode drop, can be positive or negative depending on arc current, anode area, and other factors affecting the electrons current arriving at the anode.

% conditions 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) \footnote{The escape of electrons outside the solid due to the temperature can be understood by considering the Fermi-Dirac distribution $\frac{dN(E,T)}{dE}=\frac{4\pi(2m_e)^{3/2}}{h^3}\frac{E^{1/2}}{1+\exp{\frac{E-E_f}{kT}}}$, which describes the number of electrons with a kinetic energy in the range ($E,E+dE$). The electrons of the 'electron gas' is confined to the solid volume by a potential barrier (also known as work function), $\phi$, above the Fermi level. If the 'electron gas' is heated, some electrons (the ones in the tail of the energy distribution) will have enough energy to overcome the barrier.} and/or a high electric field \footnote{At high electric field strength, the potential barrier at the surface due to the mirror image field (the free electron in the metal rearrange to compensate a external free charge) and the external applied electric field become a hill of sufficiently small width through which electron can tunnel quantum mechanically.} 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.

%This leads to an emission current density of $j_{thermionic}=en(E)v_z$, with $v_z$ the velocity in the direction perpendicular to the surface. $N(E)$ can be obtained by integrating the distribution function from -$\inf$ to $\inf$ in the x- and y- direction, and from the $\phi$ to $\inf$ in the z-direction.

%glow %In glow discharges, ions coming from the bulk plasma are accelerated toward the cathode when they enter the cathode sheath, and arrive at the cathode surface with approximately the energy gained in the cathode fall, assuming that they are typically singly charged, and don't suffer collisions. The yield of secondary electron emission is greatly affected by the potential energy of the impacting ion. The voltage drop in a glow discharge usually exceeds 300 V in order to give secondary electrons enough energy to heat plasma electrons and to cause ionization in the plasma bulk.

% short introduction non stationary emission 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 $\sim1$ $\upmu$m. The explosion leaves a micro crater with a diameter of about $\sim1$ $\upmu$m.

According to the ecton model, arc operation is self-sustained, and occurs in stages \cite{Bar2011} (Figure \ref{fig:arc_mech}). 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 \label{arc_energy} The transition from the cathode's solid phase to the plasma phase requires energy, which is supplied via the power dissipated by the arc,

\begin{equation} P_{arc} = V I_{arc} \end{equation}

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:

\begin{equation} I_{arc} V \tau = E_{phon} + E_{CE} + E_{ionization} + E_{kin,i} + E_{ee} + E_{th,e} + E_{MP} + E_{rad} \label{Energy_bal} \end{equation}

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.