Speaker
Description
\textbf{1. Potential mechanisms of vacuum breakdown. }The field electron
emission (FE), where electrons are released from cold negative electrodes due
to the applied electric field, is a necessary step in the development of
vacuum breakdown. It is well known that the field emission current from cold
electrodes in vacuum exceeds, by orders of magnitude, values given by the
quantum mechanical Fowler-Nordheim formula, evaluated in terms of the applied
electric field (which in the simplest case of a parallel-plate gap equals
$V/d$, where $V$ is the interelectrode voltage and $d$ is the gap width).
Various mechanisms for this enhancement of electron emission have been postulated.
The most popular mechanism is the amplification of the applied electric field
by microprotrusions, pre-existing on the surface of the negative electrode or
resulting from the application of the electric field. The conventional problem
with this hypothesis, as far as pre-existing microprotrusions are concerned,
is that in order to explain values of the field emission current observed in
the experiment, the microprotrusions have to be assumed to be quite slender
(needle-like), and such protrusions are not normally seen on electrode
surfaces; e.g., section 3.1 of \cite{Latham1991} and
\cite{Descoeudres2009b,Zadin2014}. Still, this point is far from having been
clarified; for example, analysis of deviations from the gas discharge
similarity law, observed at high and very high pressures in experiments on
discharge ignition and breakdown in corona-like configurations, appears to
confirm the existence of the microprotrusions on the surface of the electrodes
\cite{2021b}.
Another popular mechanism is a local reduction of the work function of the
cathode material, caused by, e.g., lattice defects or adsorbed atoms. However,
this effect seems to be insufficient to explain the observed values of the
field emission current; e.g., \cite{Sinelnikov2014,Wuensch2019}. Other
interesting hypotheses proposed in the literature include `nonmetallic'
electron emission mechanism \cite{Latham1991}, enhancement of field emission
by waves confined to the metal surface (plasmons) \cite{Wuensch2019}, and
mobile dislocations near the surface of electrodes \cite{Engelberg2020}. Thus,
there is still no widely accepted understanding of the mechanism of
enhancement of field electron emission from cold electrodes in vacuum, in
spite of several decades of active research.
What other mechanisms may play a role at the initial vacuum stage of
breakdown, apart from field emission, if any? The most popular mechanism is
fast heating of cathode protrusions due to shape-related runaway process;
works \cite{Kyritsakis2018, Veske2020, Barengolts2019a, Barengolts2019b,
Mofakhami2019,2021k} may be mentioned as recent examples. Other mechanisms
mentioned in the literature (e.g., \cite{Descoeudres2009b}) include gas
desorption at the anode caused by an intense FE current, melting of a spot at
the anode by a heavy bombardment of FE electrons, macroparticles that are
released from the electrodes by field induced stresses and subsequently partly
evaporated by the FE current, direct field evaporation of surface atoms. The
critical transition in the density of the mobile dislocations within a metal
was proposed as a vacuum breakdown mechanism in
\cite{Engelberg2018,Engelberg2019}.
Other mechanisms come into play as the breakdown develops, in particular, a
transition from the field to thermo-field to thermionic emission, vaporization
of the electrode material, production of plasma by ionization of the metal
vapor, melting of the electrode material, and formation of a crater on the
electrode surface with eventual droplet detachment.
\textbf{2. Difficulties in numerical modelling. }The initial stage of
breakdown in high-electric field has been simulated in a number of works
(e.g., \cite{Kyritsakis2018, Veske2020, Barengolts2019a, Barengolts2019b,
Mofakhami2019,2021k}), however it appears that the whole process, including
the formation of craters, has not been simulated in a self-consistent manner
up to now. This is in contrast with the cases of spot ignition in low-voltage
vacuum arcs and unipolar arcs in fusion devices, where several works dedicated
to the modelling of the whole life clycle of a spot have been published; e.g.,
\cite{2019d} and references therein. Apart from the uncertainty and diversity
of the dominating physical mechanisms, the main difficulty lies in the
presence of very different length scales. For example, microprotrusions
simulated in \cite{Kyritsakis2018} have the tip radius of $3\mathrm{nm}$ and a
total height of $93.1\mathrm{nm}$, while the crater radius is typically a few
micrometers or bigger$.$
\textbf{3. Phenomenological approach. }A phenomenological description of the
field electron emission is used in practice: experimental current-voltage
characteristics of field emission from cold electrodes in vacuum are fitted by
the Fowler-Nordheim formula with the applied electric field being multiplied
by the so-called field enhancement factor $\beta$, which has to be of the
order of $10^{2}$ or higher; e.g., reviews
\cite{Latham1991,Latham1995,Wuensch2019} and references therein.
A correlation of the vacuum breakdown field with the field enhancement factor
$\beta$, determined by means of analysis of the measured field emission
currents, was reported in \cite{Descoeudres2009b}. It was found that the
product $\beta E_{b}$, where $E_{b}$ designates the applied DC breakdown field
(and hence the product $\beta E_{b}$ may be interpreted as an "effective"
average microscopic local breakdown field inside the emission center), is a
constant value only dependent on the material and not on $\beta$ or the gap
spacing. The value of $\beta E_{b}$ around $1.1\times10^{10}$ \textrm{V/m
}$\mathrm{V/m}$ was found for copper electrodes.
The existence of this correlation makes it natural to explore the possibility
to describe the initial stage of vacuum breakdown within the framework of the
same phenomenological approach, i.e., in terms of the field enhancement factor
$\beta$ without invoking any special mechanism for the breakdown apart from
the mechanism responsible for the enhancement of field emission. Such an
attempt is described in this contribution. A half-space filled with a metal is
considered. A surface-directed electric field $E_{w}$ exists at the half-space
surface. The temperature distribution inside the half-space is uniform at
$300\mathrm{K}$ at the initial moment and evolves with time under the action
of electron emission heating (the Nottingham effect) or cooling at the surface
and the Joule heating inside the metal. The evolution of the temperature
distribution is simulated by solving the heat conduction and current
continuity differential equations in the half-space. The thermal and
electrical conductivities of the half-space material are given functions of
the local temperature; the dependencies for copper are employed. The electron
emission current density at each point of the surface, which serves as a
boundary condition for the current continuity equation, is evaluated by means
of the Murphy-Good formalism in terms of the local value of the surface
temperature and of the electric field equal to $\beta E_{w}$. The heat flux
from the surface inside the metal is evaluated with account of electron
emission heating or cooling; the boundary condition for the thermal conduction equation.
In the framework of this model, the parameter governing the temperature
evolution inside the metal is the product $\beta E_{w}$, rather than $\beta$
and/or $E_{w}$ separately, in agreement with the above-mentioned experimental
findings \cite{Descoeudres2009b}. Two cases have been considered: $\beta$ is
the same at each point of the surface, and $\beta$ varies along the surface.
In the first case, the solution is 1D: heat propagation inside the metal is
the same at each point of the surface. If $\beta E_{w}$ is high enough, there
is a very fast increase of the temperature and the current density at, or very
near, the surface: the maximum temperature reaches the critical temperature of
copper ($8390\mathrm{K}$) within approximately $2\mathrm{ns}$ for $\beta
E_{w}=10^{10}\mathrm{V/m}$ (and within $1$ microsecond for $\beta
E_{w}=0.7\times10^{10}\mathrm{V/m}$, while there is virtually no heating on
the microsecond time scale for $\beta E_{w}$ below $0.5\times10^{10}%
\mathrm{V/m}$). At such applied electric field values, the Joule heating
underneath the surface comes into play and overtakes the Nottingham effect.
This is a manifestation of the so-called thermal runaway. This instability has
been extensively studied in the context of low-voltage vacuum arcs (e.g.,
review \cite{Hantzsche2003}); its shape-related version was studied in the
context of high-voltage vacuum breakdown initiated by microprotrusions (e.g.,
\cite{Kyritsakis2018, Veske2020, Barengolts2019a, Barengolts2019b,
Mofakhami2019,2021k}).
Also considered in the modelling was the axially symmetric case where $\beta$
varied along the surface as a Gaussian function of the distance from the
origin with the height of the curve's peak of $150$ and the standard deviation
of $40\mathrm{nm}$. (Note that the latter value was chosen in accordance with
the measurements \cite{Descoeudres2009b}, which gave the diameter of the
emitting area typically between $20$ to $80\mathrm{nm}$). For $E_{w}%
=10^{8}\mathrm{V/m}$, the critical temperature was reached within
$8\mathrm{ns}$.
\textbf{4. Conclusions. }It appears to be possible to describe the initial
stage of high-voltage vacuum breakdown within the framework of the
phenomenological approach, i.e., in terms of the field enhancement factor
without invoking any special mechanism for the breakdown apart from the
mechanism responsible for the enhancement of field emission. It is of interest
to exploit this option in the modelling of advanced stages of vacuum
breakdown, which would take into account a transition from the field to
thermo-field to thermionic emission, vaporization of the cathode material,
production of plasma by ionization of the metal vapor, melting, and formation
of a crater with eventual droplet detachment, on the level comparable to that
reached in the modelling of the whole life-clycle of spots in low-voltage
vacuum arcs and unipolar arcs in fusion devices (\cite{2019d} and references therein).
\textbf{Acknowledgments }This work was supported by FCT of Portugal under
project UIDP/50010/2020 and by European Regional Development Fund through the
program Madeira 2014-2020 under project
PlasMa-M1420-01-0145-FEDER-000016.
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| Topic | Modeling and Simulations |
|---|
