Plasma Sources Science and Technology
TOPICAL REVIEW
Streamer breakdown: cathode spot formation,
Trichel pulses and cathode-sheath instabilities
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Topical Review
Streamer breakdown: cathode spot
formation, Trichel pulses and cathodesheath
instabilities
Mirko Černák , Tomáš Hoder and Zdeněk Bonaventura
Department of Physical Electronics, Masaryk University, Kotlářská 2, 61137 Brno, Czech Republic
E-mail: cernak@physics.muni.cz
Received 20 October 2017, revised 8 August 2019
Accepted for publication 22 October 2019
Published 20 December 2019
Abstract
The review provides an up-to-date overview and discussion of phenomena related to positive
streamer breakdowns in short uniform and non-uniform ﬁeld (corona) gaps. The terminology
used to specify different types of streamer phenomena is critically discussed in light of a uniﬁed
theory of high-pressure gas discharges describing the sequence of ionization events from initial
electron avalanches up to a partial or complete breakdown. The emphasis is given to the
understanding of the formation of an active cathode spot by the streamer arrival to the cathode,
which is the critical but still obscure phase of the breakdown development. Based on the analysis
including a computer simulation model a hypothesis is advanced that also such widely studied
and practically important gas discharge phenomena as negative corona Trichel pulses and fast
ionization instabilities in cathode regions of high-pressure gas discharges are due to the
formation of positive streamers in the immediate cathode vicinity. The proposed hypothesis
offers attractive feature of the uniﬁcation of a wide scale of high-pressure gas discharges within
the general class of positive streamer initiated breakdown phenomena. Moreover, it provides
indications for further study in the ﬁeld both by experiment and computer simulations.
Keywords: electron avalanche, Trichel pulse, streamer–cathode interaction, cathode spot,
cathode-sheath instability, breakdown, streamer and spark
1. Introduction
Already in 1857 Siemens [1] discovered that the ﬁlamentary
dielectric barrier discharge (DBDs) can be effectively used to
generate ‘ozonizing air’. In 1928 Rogowski [2] observed that
the pulsed voltage breakdown in wide ambient air gaps is two
orders faster than can be expected on the base of the gas
ionization by successive electron avalanches linked by secondary
electron emission from the cathode. Now, the gas
ionization phenomena related to these pioneering discoveries
in the ﬁeld of gas discharge physics are well interpreted and
understood in terms of the concept of ‘streamers’ ﬁrst proposed
by Raether [3], and by Loeb and Meek [4] in 1930s.
The streamer concept has proven very useful for fundamental
understanding of a multitude of gas discharge phenomena [5]
and has pushed the physics backing new applications of highpressure
gas discharges [6–8].
Nevetheless, in view of the maturity and proven usefulness
of the streamer concept, the scientiﬁc and engineering
literature on the subject is still surprisingly inconsistent,
which results in the persistent and even increasing gap
between the basic research and applications. For example, the
engineering community in the ﬁeld of applied gas discharge
physics is using the streamer concept often only in its obsolete
‘single-avalanche’ form [9, 10], or the term ‘streamer’ is
omitted even by respected authors in the ﬁeld of atmosphericpressure
plasma sources [11] and high voltage systems in air
[12]. In our opinion the problem is caused mainly by the lack
of a consistent and generally accepted streamer theory [13].
Plasma Sources Science and Technology
Plasma Sources Sci. Technol. 29 (2020) 013001 (31pp) https://doi.org/10.1088/1361-6595/ab5051
0963-0252/20/013001+31$33.00 © 2019 IOP Publishing Ltd1
In our topical review, an attempt is being made to clarify
the streamer terminology and to identify weakest links in the
understanding of ionization phenomena leading to the highpressure
gas breakdown starting from initiatory electrons, via
a streamer discharge phase, to the ﬁnal arc or spark.
Nowadays, the numerical simulations are crucial to our
understanding of complex streamer and breakdown phenomena
[6]. Therefore, the success of computer simulation
models in explaining the sequence of events leading to the
streamer breakdown can be regarded as a measure of the
theoretical understanding the phenomena leading to the
breakdown. Based on this, it appears that the positive streamer
arrival to the cathode, which is very difﬁcult to study
both by experiment and computer simulations, is the most
obscure, but very critical link in the chain of sequences
leading to streamer breakdown.
Therefore, in the review we will provide up-to-date
overview of phenomena related to the positive streamer
arrival to the cathode and attempt to clarify their role in highpressure
gas breakdowns. Also, contrary to commonly held
belief, we will advance hypothesis that such widely studied
and practically important gas discharge phenomena as negative
corona Trichel pulses (TPs) and instabilities in cathode
regions of high-pressure gas discharges resulting in arcing are
also due to a positive streamer generated in the immediate
cathode vicinity and its arrival to the cathode.
The paper is structured as follows. Section 2 presents the
historical progression and development of the relevant
experimental techniques and terminology. Section 3 reviews
the sequences of gas discharge phenomena more or less
already accepted as being associated with the generation of
positive streamers both for uniform (section 3.1) and nonuniform
(section 3.2) ﬁeld gaps. A computer simulation
model of the development of an active cathode region by the
streamer arrival to the cathode is presented in section 4. The
following section 5 discuss well-known gas discharge phenomena
as negative corona TPs and instabilities of the highpressure
cathode regions which, however, are not still
accepted by the gas discharge community as being associated
with the positive streamer formation. Section 6 summarizes
the discussions and conclusions presented in sections 2–5
attempting ﬁrst to remedy the existing terminological misunderstandings
and then to present an uniﬁed experimental
and theoretical picture of the sequence of ionization events
leading to the streamer breakdown.
2. Historical overview, experimental developments,
and basic terminology
The sudden transition of laboratory air to a highly conducting
state with a negative resistance phase termed the spark or
complete gas breakdown [14, 15], was studied already in the
middle of 18th century by Benjamin Franklin, who had
demonstrated that a laboratory spark and lightning were of
common nature [9].
The progress in vacuum technique enabled to ignite
steady gas discharges at reduced pressures and moderate
voltages, also often termed as ‘breakdowns’ [14]. In 1889,
Paschen measured the minimum voltage that was necessary to
ignite a discharge between two electrodes in a vacuum tube
[16]. He found that the minimum discharge ignition voltage
(the static breakdown voltage Vs) is a function of the product
between gas pressure p and gap distance d and measured
dependences =V f p ds ( · ) nowadays known as Paschen
curves.
In 1910, based on the extensive studies of low-pressure
gas dischages by measuring the discharge currents by an
electrometer and inductive balance, Townsend published his
‘Theory of Ionization of Gases by Collision’ [17] with the
equation describing the observed exponential spatial growth
of the ionization with the distance from the cathode attributing,
however, the ionization by collisions to the impact of
negative and positive ions. The following combination of
Townsend’s equation with the ﬁnding that the discharge
current is mainly carried by electrons generated predominantly
through the collision processes of electrons with
gas atoms and molecules [18, 19] resulted in what is currently
called the Townsend breakdown theory and in the well known
Townsend equation based on the exponential development of
current in a series of electron avalanche generations [20]
linked through secondary-emission feedback to the cathode.
The criterion for the self sustaining discharge resulting from
the Townsend equation is that each electron avalanche produces
at least one secondary electron at the cathode, so that
the discharge is self sustaining:
g -a e 1 1, 1d( ) ( )
where d is the gap spacing, α is Townsend’s ﬁrst ionization
coefﬁcient and γ is the number of secondary electrons produced
at the cathode per ionizing collision in the gap
(Townsend’s coefﬁcient of secondary emission).
The Townsend theory has been very successful in
explaining breakdown phenomena in uniform ﬁelds under
various discharge conditions as, for instance, explanation of
Paschen’s breakdown curves, and effects of electronegative
gases and cathode materials on the breakdown voltage. Following
the pioneering works by Rogowski and Schumann
[21, 22], in the 1930s and 1940s the ongoing development of
experimental equipment enabled the experimental study of
transient breakdown processes. By measuring cloud-chamber
tracks of electron avalanches and interrupted sparks it was
veriﬁed that at sufﬁciently high over-voltages the ﬁnal spark
breakdown occurs in 10−7
s or less [3]. Such fast spark
breakdown was found to be due an extremely rapid ionization
process requiring only a single large avalanche, rather than
the slow cumulative action of a sequence of avalanches with
the feedback to the cathode. This and others reasons to
question the validity of the Townsend theory at higher pressures
and larger electrode distances (p·d>103
Pa m) were
analyzed in details by Meek [23]. As a consequence, the socalled
streamer mechanism was proposed by Flegler and
Raether [24], Loeb [25] and Meek [23] particularly to explain
the fast single-avalanche breakdown observed in highly overvolted
gaps at near atmospheric pressures, where the pulsed
voltage magnitude is exceeding some 100% of Vs. In the
2
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
1950s the availability of high-speed oscilloscopes and photomultipliers
enabled Amin [26], Anderson [27], and Hudson
and Loeb [14] to make the ﬁrst observations of the full
sequence of events leading to the spark breakdown, where
during the ﬁrst phase of breakdown thin plasma channels the
so-called primary streamers were formed.
In the past four decades, the development of both streak
and gated intensiﬁed CCD cameras [28–31] has enabled such
phenomena to be observed with a temporal resolution on the
order of nanoseconds. The results obtained mainly in short
positive point-to-plane gaps in air indicate that a spark
breakdown at near atmospheric pressure is always preceded
by the formation of a cathode-directed primary streamer
propagating with the velocity in the range 107
–108
cm s−1
and
its arrival to the cathode. As studied in details [32] and
reviewed for example in [33] the primary streamer head
propagates as a luminous spot of the diameter in the range
10−5
–10−3
m. The streamer diameter, or parameters of its 3D
structure in general, is typically evaluated from CCD camera
images and, consequently, the results can be affected by
resolution and sensitivity of the optical apparatus used.
Moreover, different measures are used for its quantiﬁcation—
e.g. FWHM of the luminous ﬁngerprint, 10%-to-90% of
pixel-by-pixel summation, electrodynamic radius etc [34–37].
Other approaches investigated the quantiﬁcation of the relation
of the streamer diameter parameter to other qualities of
the propagating discharge [36, 38, 39]. In view of the accuracy
of both experiment and theoretical models used, the
sometimes claimed agreement between the measured and
computer simulated streamer diameters might be still not
sufﬁcient, as it was shown especially for lower pressures
in [35].
Other developments have taken place in the application
of modern sophisticated spectroscopic techniques and the
interpretation of their results for investigation of various
phenomena occurring during the streamer formation and
propagation [40–42]. For example, to study repetitive streamer
phenomena with high spatial and sub-nanosecond temporal
resolution, correlated single photon counting techniques
have been used, where the weak photomultiplier signals from
the space, time, and spectrally resolved light from millions of
streamer pulses were accumulated [43–47], also in twodimensions
(2D)48, 49]. Nevertheless, the transient feature
and the small dimensions still make some basic streamer
parameters difﬁcult to be accessible to the experiments.
Because of currently increasing availability of methods
and oscilloscopes allowing measurements of current pulses
with a sub-nanosecond rise time, large amount of information
on the streamer phenomena have been obtained by measurement
of the currents induced by the fast streamer phenomena
in a large variety of discharge gaps and probes [50].
The simplest way called by Raether ‘electric method’ [51] to
measure a streamer current is to use a low-inductance measuring
resistor in series with a homogenous ﬁeld gap, or
Rogowski coils for the time resolution not less than ∼1 ns.
However, using parallel plane electrodes the measured rise
time is critically limited by a high gap capacitance, even for
relatively small electrode areas. Moreover and unfortunately,
the difﬁculty to correctly measure and interpret the streamerinduced
currents with sub-nanosecond temporal resolution
has been often ignored by the workers in the ﬁeld of gas
discharges. At measurements of fast current signals induced
by the streamers, it must be taken into account that the nanoand
sub-nanosecond time resolution is provided only using
probes with small receiving areas, as sectional electrodes
[28, 46, 52–55], and using the input wave impedance equal to
the wave impedance of a coaxial cable through which signals
are transmitted to an oscilloscope. Moreover, as discussed in
more detail in section 3.2, particularly for non-uniform ﬁeld
gaps the interpretation of measured current waveforms is
complicated by a complex interrelation between conduction
and displacement currents which are produced by propagating
streamers in such probes [13, 28, 51, 56].
Along with the advances in experimental techniques,
starting in the 1970s the ﬂuid numerical modeling of the
primary streamers development and propagation has been
increasingly utilized [57]. Soon, also ﬁrst attempts [58, 59]
were made to use the more advanced kinetic approaches for
simulations of highly non-equilibrium, nonlinear, and nonlocal
conditions of a typical primary streamer formation and
propagation. Using faster and more accurate algorithms the
computer simulations become essential tools that can be used
in synergy with the experiment to increase our fundamental
understanding of the streamer-related phenomena.
At larger inter-electrode gaps in air, where secondary
electrons from the cathode are too delayed to play any role,
the positive streamer propagation is critically dependent on
the presence of seed electrons ahead of the streamer tip
[60–63]. However, the computer simulations by Kulikovsky
[64] and Pancheshnyi [38, 65] taken together with the current
experimental results in [65–68] quite persuasively indicate
just a weak sensitivity of the streamer characteristics to the
density of the seed electrons. The weak dependence of
streamer characteristics is due to the fact that the effective
parameter that appears in simple streamer theories is actually
a logarithm of the seed electron density [69, 70]. As stated by
Kulikovsky [64]: ‘In a weak ﬁeld the role of photoionization
is to provide at least one seed electron ahead of the tip.
Further multiplication of charges occurs due to collisional
ionization’. It appears that in many real situations the limiting
factor for the primary streamer formation and propagation is
rather too high ionization of the background gas [71, 72].
According to Wormeester et al [67] and Celestin et al [73] the
photoionization is the primary source of seed electrons in air
except for high background ionization above ∼1010
cm−3
.
Consequently, the rigorous deﬁnition of primary streamers as
‘Subset of ionizing potential waves propagating in initially
unionized gas’ [59, 74] is lacking practical relevance. This
fact can be documented, for example, by the results indicating
that the so-called plasma bullets can be successfully interpreted
as a positive streamer guided by a helium plasma jet,
whereby the streamer dynamic is not critically affected by the
preionization density [75, 76]. Also, it is generally accepted
that large-scale sprites propagating at a high background
electron density in lower ionosphere are physically similar to
3
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
small-scale streamer dischargers in air at atmospheric pressure
[77, 78].
Another controversial issue of a signiﬁcant practical
importance is the criterion for streamer-initiated breakdown.
In a historical context it should be noted that some of the
above mentioned discrepancies between the breakdown
experiments and the Townsend theory sumarized by Meek
[23] were later resolved [79]. Also, even when the Townsend
theory yields the criterion for establishing a low-current space
charge free Townsend discharge, it is often used as a working
breakdown criterion for the onset of streamer coronas and
complete breakdowns in non-uniform ﬁelds [13, 80]. This
quite surprising fact is clariﬁed by the excellent but often
overlooked work by Hodges et al [20], where the ‘pure’
Townsend and the ‘traditional’ single-avalanche streamer
mechanisms are just limiting cases accounted for the spark
breakdown. In fact, as it will be discussed in more details
later, the breakdown criterion, if considered correctly in terms
of the breakdown probability, does not depends on the
mechanism of streamer propagation, but rather on the conditions
under which the space-charge initiating the primary
streamers is generated: Generally, a streamer starts when the
charges that appear on the surface of a streamer-initiating
plasma become able ‘to shield the interior (that) unavoidably
enhance the electric ﬁeld over a limited region just outside of
the streamer’ [81]. Such streamer-initiating plasma region can
be generated not only by a single ‘critical’ avalanche as
considered by the traditional streamer theory, but particularly
in non-uniform ﬁelds more often by a sequence of avalaches
linked by a secondary electron emission from the cathode or
photoinization [82–85].
3. Streamer breakdown
Since the time when the streamer theory was ﬁrst postulated
for the single avalanche spark breakdowns in the over-volted
uniform ﬁeld gaps, being driven both by the advance in
fundamental discharge physics and applications, considerable
efforts have been devoted to examining how it can be generalized
and applied to a wide range of experimental and
practical conditions, such as different gases and pressures,
electrode geometries, over-voltages, etc. Nevertheless, despite
decades of intensive research efforts driven both by the
advance in fundamental discharge physics and applications it
appears that existing extensive scientiﬁc and technical literature
on the streamer breakdown of centimetric gaps suffers
from a lack of consensus on several central issues
[60, 80, 86]. As a consequence, a satisfactory understanding
of the sequence of events related to the streamer breakdown
from the initiation, by say a single electron, to the stage where
arc current of many amperes ﬂows in the gap, remains a
challenge for the future. To get an overall picture of the
streamer breakdown phenomena in the short gaps, where the
breakdown typically evolves without the formation of a leader
[87, 88], in this section we will attempt to summarize and
distill both older and contemporary ﬁndings and knowledges
in the ﬁeld.
For the sake of brevity and simplicity, we will concentrate
mainly on the DC, impulse, and low frequency prebreakdown
and breakdown phenomena in near-atmospheric
pressure air and nitrogen. Nevertheless, since the streamer
discharges are in general very robust against changes of gas
composition [66], we believe that many results and conclusions
presented will also be applicable, perhaps with some
modiﬁcations, to high-pressure gas discharges in other gases
and gas mixtures. In line with Kunhardt [89] we will assume
that the basic breakdown processes, including the streamer
phenomena, and their contribution to the development of
breakdown are in general common to both DC and pulsed
breakdown. Moreover, we will not consider the discharges
generated in atmospheric pressure air and other gases by
application of high voltage pulses with a nano- or subnanosecond
rise time, where the breakdown phenomena are
signiﬁcantly effected by beams of fast electrons [90–95].
An excellent starting point is the concept presented by
Marode in ﬁgure 1, [96], illustrating signiﬁcant similarities
between the sequences of events leading to the streamerinitiated
breakdown in the short parallel-plane and positive
point-to-plane gaps.
3.1. Streamer phenomena in uniform fields
3.1.1. Single avalanche streamer breakdown. The ﬁgure 1(b)
illustrates the already mentioned ‘classical’ single-avalanche
streamer mechanism proposed in the 1930s and 1940s to
explain the already mentioned fast breakdown observed in
cloud chamber experiments in the parallel plane gaps under
high impulse conditions. In the experiments often labeled as
‘single electron initiated’ [100] the single avalanche starts
from an electron emitted from the cathode. Till the net charge
in the avalanche head is not sufﬁcient to signiﬁcantly distort
the external electric ﬁeld, the avalanche grows exponentially,
and its head moves with the electron drift velocity in a
concentrated path due to the low lateral diffusion of electrons
at high pressures. Raether [101], still based mainly on the
early cloud chamber results and approximate calculations of
the space charge, concluded that the avalanche growth was
weakened after the avalanche reaching around 106
electrons
and stopped reaching a ‘critical size’ Necr at around 108
electrons, being transformed to a region of streamer-initiating
plasma at a distance a=x Nlnc ecr( ) from the cathode.
If the gap over-voltage is not too large, this can occur in
the anode vicinity resulting there in the initiation of a cathodedirected
streamer, or if the gap and voltage are large enough,
the avalanche-to-streamer transition can already take place
quite far from the anode. In the second case, as illustrated in
ﬁgure 1(b), both cathode and anode streamers directed are
initiated [102]. The single avalanche-to-streamer transition
occurs due to the generation of new avalanches initiated say
through photoioniziation, joining with the head of the primary
‘critical’ streamer-initiating avalanche to form a rapidly
developing streamer front propagating to the cathode, and
eventually also to the anode, which have come to be referred
to as the positive and negative primary streamers. Since the
streamer velocity is much higher than the electron velocity ve
4
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
in a Laplacian ﬁeld, the upper limit of the formative time of
the streamer breakdown is often considered [10] as:
a
t
N
v
ln
. 2d
ecr
e
( )
For an avalanche started from the cathode, this transition
corresponds to a value of a d· of about 20 in the undistorted
applied ﬁeld. This empirical criterion a >d 20· , nowadays
known usually as the Raether–Meek criterion for streamer
breakdown [102] is widely and successfully used for the
impulse breakdown in highly overvolted uniform ﬁeld gaps.
Note that starting from the work by Kunhardt and Tzeng [59]
many computer simulation works try to explain in details the
plasma processes related to the single avalanche-to-streamer
transition, which is more complex and less amenable to
experiment than the subsequent positive streamer propagation
already brieﬂy discussed in section 2. A detailed theoretical
discussion of the single avalanche to streamer transition
including the resulting breakdown criteria is presented, for
example, in a recent study by Rabie and Franck [103].
As illustrated in ﬁgure 1(b), in a single electron initiated
breakdown at the high over-voltage values, the Phase II is
starting by the single avalanche-to-streamer transition, and it
is terminated when a cathode spot of high electric ﬁeld and
high positive-ion densities is established by the positive
streamer arrival to the cathode. As seen from ﬁgure 1(b) the
fast cathode spot establishment is followed by a complex
sequence of secondary streamers. As suggested for example
by Llewellyn-Jones [104], the arrival of the streamer to
cathode can initiate the secondary streamers by ‘a burst of
electrons to give a high ﬁeld running back along the track’.
The subsequent bridging of the gap by secondary streamers
results in the establishment of a short-duration ﬁlamentary
‘quasi-glow discharge’ (see Phase III in ﬁgure 1(b)) followed
by a fast glow-to-arc transition (IV) that greatly increases the
conductivity of the gas if the power supply can deliver the arc
current above some amperes.
3.1.2. Multi-avalanche streamer breakdown. The above
discussed single avalanche streamer mechanism of the
highly-overvolted gaps illustrated by ﬁgure 1(b) constitutes
the fundamental characteristic of the ‘classical’ streamer
concept of breakdown generally accepted by the workers in
the ﬁeld of gas discharges, and taught in standard textbooks
and reviews on the gas discharge physics [9, 105]. However,
as reported by Hudson and Loeb already more than half of
century ago [14] for parallel-plane gaps: ‘The ﬁlamentary
spark, except at high, short impulse overvoltage, is now
recognized as being preceded by a low-order self-sustaining
discharge. These lower order discharges are at times nearly
imperceptible and at other times obvious as a Townsend glow
discharge in uniform ﬁelds ...’ (see also [13], discussed in
section 6).
Such low order Townsend type discharges are illustrated
in ﬁgure 1(a) that is based on the results of an excellent and
still very valuable study of the multi-electron initiated multiavalanche
breakdown by Doran [97]. As it is characteristics
for a Townsend discharge, the ionization during Phase I
initiated by ‘a pulse of some 400 electrons’ is due to the
superposition of several generations of electron avalanches
linked by the secondary electron emission. It is interesting to
note here that Marode in ﬁgure 1(b) assigned the single
electron initiated breakdown in ﬁgure 1(b) to the ‘streamer
mechanism’, while the multi-avalanche initiated breakdown
in ﬁgure 1(a), despite the evident formation of streamers (see,
also discussion to follow), as a ‘generation mechanism’. Such
a very common but misleading ‘streamerless’ breakdown
Figure 1. Comparison between typical streak photographs of the spark formation in different cases according to Marode [96]. (a) Uniform
ﬁeld gap: nitrogen, pulsed gap with small overvoltage (7.55%), generation mechanism, p=300Torr, d=2cm (after Doran in [97]), (b)
uniform ﬁeld gap: nitrogen, pulsed gap with high overvoltage (35%), streamer mechanism, p=300Torr, d=2cm, (after Koppitz [98] and
Chalmers and Duffy [99], (c) non-uniform ﬁeld gap: air, DC potential, p=760Torr, d=1cm, point radius 100 μm (after Marode [28]).
The picture is taken from [96].
5
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
terminology can be exempliﬁed by the following claim by
Cavenor [106]: ‘The ﬁlamentary spark channel, resulting
from the electrical breakdown of a gas, generally forms in
one of two ways. At voltages close to the minimum breakdown
potential the spark channel has been shown to develop from
the constriction of the diffuse glow discharge formed from the
superposition of many generations of electron avalanches. At
large over-voltages, however, single avalanches have been
observed to develop into luminous conducting ﬁlaments,
called streamers, which completely bridge the electrode gap
within the transit time of the initial avalanche’, and more
recently by the claim regarding the streamer formation in nonthermal
atmospheric pressure gas discharges in general ‘The
formation of a streamer requires the electric ﬁeld of the space
charge in the (single) avalanche to be of the order of the
external ﬁeld’ in [105]. Unfortunately these works, as well as
many textbooks and reviews on gas discharge physics ignores
the following facts and opportunities stated already by Loeb
in 1951 [107]: ‘Thus, spark discharges in uniform ﬁelds set in
at thresholds for low order Townsend discharges by creating
space charge ﬁeld distortions. These distortions increase to
the point where avalanches reach streamer-forming proportions
in middgap, thus yielding sparks as streamer-type
breakdown. ... The physicist must, therefore, reconcile himself
to the paradoxical situation of a streamer breakdown
mechanism for spark in uniform ﬁelds with a threshold
which is set by the conventional Townsend criterion and not
by that of Meek and the writer and of Raether. If this is done,
practically all of the difﬁculties of the past are reconciled in a
single consistent sparking theory.’ In this context the reader is
referred again to the paper by Hodges et al [20], where this
long-standing problem is treated correctly in terms of ‘a
uniﬁed (statistical) breakdown theory for which the Townsend
and streamer mechanisms are limiting cases’.
Turning back to Doran’s results [97] illustrated in
ﬁgure 1(a) who described the Phase II in terms of the
Townsend ionization mechanism: Following the Stage I
initiated by a pulse of some 400 electrons, at the beginning of
the Stage II ‘The results indicate a well deﬁned luminous
front which is ﬁrst observed in the midway between the anode
and cathode moving towards the cathode with a velocity
which increases from 5×107
cm s−1
to 108
cm s−1
. This
stage corresponds to the motion of the contracting leading
edge of the discharge shown in the intensiﬁer shutter
photographs. On its arrival at the cathode a ﬂash appears
at the surface of the electrode, coinciding with a marked kink
in the current curve (see ﬁgure 2). From this short duration
ﬂash at the cathode another front is launched, travels towards
the anode at a velocity of 2×108
cm s−1
and this in turn
initiates a further wave which returns towards the cathode
with a slightly higher velocity.’ Figure 2 shows the ‘marked
kink in the current curve’ measured by Doran together with
the results of early computer simulations by Davies et al
[108], who however termed the luminous front arrival
resulting in the short duration ﬂash at the cathode as the
‘cathode streamer arrival’.
As illustrated in ﬁgure 2, using several approximations,
as arbitrary changing the discharge radius according to
Doran’s data [97] and neglecting the space charge effects,
Davies et al [108] claimed to succeed (see ﬁgure 2) in
simulating the current kink due to the positive streamer arrival
and the development of subsequent secondary streamers.
Nevertheless, a later attempt to simulate ‘multi-electron
initiated’ breakdown by Doran in 2D and incorporating the
effects of space-charge distortion made also by Davies [109]
was not successful since ‘the calculations eventually break
down as the cathode streamer approaches the cathode
because of numerical instabilities.’ From the literature as,
for example, the recent review by Liu and Becerra [110], the
current calculations still tend to breakdown as the streamer
approaches the cathode because of numerical instabilities. As
a consequence, a satisfactory computer simulation ﬂuid
models describing in details all stages of the streamer
breakdown in uniform ﬁeld gaps, as illustrated in
ﬁgures 1(a) and (b), still remains a challenge for the future.
Moreover, as discussed in section 3.2.1 and in details shown
in section 4, the ﬂuid models assume more or less equilibrium
of electrons with the electric ﬁeld, which is not always true,
particularly for electrons in the strong ﬁeld in the streamer
head approaching the cathode, where the electrons can gain
more energy from the electric ﬁeld than they lose by
collisions.
The streamer mechanism is also responsible for abrupt
ionization in DBDs which are widely used as plasma sources
both in technological applications as well as fundamental
research ﬁelds. Although the detailed discussion of the
streamer phenomena connected to DBDs is beyond the scope
of this review, we address some important issues of general
interest later in the appendix of this manuscript.
In contrast to the above discussed low-overvoltage
multiavalanche-initiated (streamer) breakdown, the special
‘multiavalanche pulsed discharge’ (termed according to
[10, 100]) can be generated in pre-ionized highly-overvolted
Figure 2. Schematic representation of the experimentally measured
(—–, according to [97]) and theoretical (- - - -) discharge currents
and luminous fronts in conditions identical to ﬁgure 1(a). Taken
from [108].
6
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
gaps at the condition aN dln ecr  . Such situation of great
practical interest arises when preionization of laser gases is
used to generate pulsed volumetric glow discharges at gas
pressures greater than 1atm.: starting the avalanches in a preionized
highly over-volted gap by a large number of preionization
electrons, the heads of primary avalanches start
overlapping before the space charge ﬁeld becomes comparable
with the applied electric ﬁeld [111–113]. The preionization
will prevent the already mentioned situation [107]
‘where avalanches reach streamer-forming proportions in
middgap, thus yielding sparks as streamer-type breakdown’,
i.e. a sufﬁcient pre-ionization can prevent the formation of a
streamer initiating plasma by smoothing out the local
gradients of the space charge ﬁeld. In this way, the electron
density can grow uniformly in the discharge volume and a
homogeneous glow discharge with a high-ﬁeld electrondepleted
cathode region can be established without streamer
formation [111]. Nevertheless, the praxis show that even
under such optimized conditions in transversely excitedatmospheric
(TEA) lasers an undesired plasma ﬁlamentation
occurs, which is triggered by cathode spots quickly
(∼10–100 ns) developing in the initially laterally homogeneous
cathode sheath. It is interesting that even when the
cathode spot formation is caused by a stochastic electron
emission from randomly distributed cathode surface imperfections,
the spots are often regularly distributed at the
cathode surface [114].
According to computer simulations by Smy and
Clements [115] such instability occurs because the anodeside
of the established cathode sheath acts effectively as a
virtual anode, and ﬁnaly ‘the sheath can collapse catastrophically
with an accompanying decrease in its impedance of
many orders of magnitude’. Belasri et al [116] based on their
1D computer simulations suggested that a streamer collapse
of the cathode sheath was due to an anode-directed streamer
formation in the sheath. According to the 2D computer
simulations results by Černák et al [117], where the sheath
collapse was initiated by a 10 ns burst of electrons from a
typical cathode emission site, such electron emission forms a
space charge distortion on the cathode-faced surface of the
bulk plasma, i.e. on the virtual anode. In agreement with the
extensive and still unique experimental results by Makarov
et al [118, 119] and contrary to the model by Belasri et al
[116], the space charge distortion propagates not from the
cathode, but from the virtual anode taking the form of a
positive streamer-like ionizing wave. At the streamers
arrivals, similarly as in ﬁgures 1(a) and (b) cathode spots
like those seen in ﬁgure 3 are formed. In agreement with
experimental results by Makarov et al [118, 119] and
Drieskamper et al [120] the simulations in [117] indicate
that the cathode spot formation is an extremely rapid process
in the nanosecond range associated with a positive streamer
formation in the narrow cathode region, which can produce
local concentration of current in the cathode sheath leading to
the subsequent undesired glow-to-arc transition. Unfortunately,
the calculations by Černák et al [117], similarly as
those by Davies [109] and many others, broke down as the
cathode streamer approached the cathode so that the
subsequent formation of a cathode spot like those seen in
ﬁgure 3 was not treated in details.
Nevertheless, the authors suggested that their results (see
ﬁgure 4) taken together with the experimental results by
Makarov and Bychkov [119] provide a base for uniﬁcation of
the initiating plasma-to-arc transition that occurs in pulsed
high-pressure XeCl laser discharges into a general class of
streamer breakdown phenomena. Due to the practical interest,
the discussed works were done in conditions similar to those
in UV-preionized XeCl lasers. Thus it is worthwhile to note
that the results on breakdown formative time as a function of
electric ﬁeld in preionized pulsed air gaps can be found in
Figure 3. The formation of cathode spots in conditions simulating a TEA laser in HCl/Kr/Ne with partial pressures of (1/100/4899) mbar,
impulse voltage of approximately 25 kV and d=2 cm. (a) 100, (b) 200 (c) 300, and (d) 400 ns after the discharge onset. Taken from [114].
7
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
[121, 122]. Also, results by Baksht et al [123] on diffuse glow
discharges in air and N2 preionized at 1atm. by run-away
electrons generated by the discharge itself correspond well
with the work by Makarov and Bychkov [119] in Ne–Xe–
HCl excimer gas mixture at pressures approximately 2 atm.
3.2. Streamer phenomena in non-uniform fields
In non-uniform ﬁeld gaps the streamer initiating plasma is
generated in a close vicinity of the high ﬁeld electrode, and
the streamers then can propagate farther away into the gap
where the ﬁeld is typically too low for streamer initiation, but
high enough for streamer propagation due to its own space
charge ﬁeld. Figure 5 illustrates typical picture of such phenomena
for positive and negative point-to-plane gaps in
ambient air fed by voltage pulses with the rise times on the
order of ∼10 ns [32].
In electronegative gases (see ﬁgure 5) the streamers can
propagate and cross point-to-plane gaps for a signiﬁcantly
wide range of applied voltages generating a partial breakdown
without triggering a spark. Similarly to the breakdowns in
highly-overvolted uniform-ﬁeld gaps illustrated in ﬁgure 1(b),
when the streamer is initiated by a single critical avalanche,
the full breakdown in non-uniform ﬁeld gaps is also often
termed the ‘streamer breakdown’ [124, 125]. This, however,
contradicts to the fact that, as discussed below, the streamers
in ﬁgure 5 are not initiated by a single ‘critical’ avalanche
and, consequently, the ‘streamer breakdown criterion’ by
Raether and Meek is not fulﬁlled there.
In attempting to provide an uniﬁed model for the streamer
breakdown phenomena it is useful to use also the term
‘corona discharges’, even when the terminology is also
somewhat confused:
For example, according to Goldman et al [126] the corona
is a low-current discharge characterized by a faint visual
glow in the high-ﬁeld region: ‘A corona is self-sustained
electrical gas discharge, where the Laplacian electric ﬁeld
conﬁnes the primary ionization process to the regions close to
high ﬁeld electrodes or insulators’. Thus the authors included
in their deﬁnition also ‘DBD coronas’ formed close to insulator
surfaces, and termed the discharge with the streamers
propagating farther away into the gap like those in ﬁgure 5 as
the ‘streamer (bipolar) conduction coronas’. The deﬁnition by
Kogelschatz and Salge [127] is more general and more convenient
to be used in the following discussion: ‘Corona discharges
are self sustained gas discharges typically operated
in air at atmospheric pressure between metal electrodes. At
least one of the electrodes is of a geometric form which
causes locally high ﬁelds. Common conﬁgurations are pointed
electrodes facing a plane, ...etc. When the voltage is
raised in such a conﬁguration current starts to ﬂow at corona
onset and increases until the potential for spark is reached.
This intermediate range of corona activity is referred to as a
partial breakdown of the gap. The corona is characterized by
Figure 4. Simulation results of the positive streamer formation in the cathode region of a pre-ionized TEA discharge in neon gas at 3 bar
pressure. Electric ﬁeld (a)–(c) and electron density (d)–(e) at t=27 ns, t=28 ns and t=29 ns. The isocontours of density are taken from
1 × 1012
cm3
to 1 × 1013
cm3
with a space between two isocontours of 1 × 1012
cm3
and from 1 × 1013
cm3
to 1 × 1014
cm3
with a space
between two isocontours of 1 × 1013
cm3
. The lowest isocontour of electric ﬁeld is 10 kV cm−1
and the highest is 120 kV cm−1
, the space
between two isocontours is 10 kV cm−1
. Taken from [117].
8
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
a faint visual glow in the high-ﬁeld region occasionally
accompanied by luminous streamers propagating toward the
other electrode.’
3.2.1. Streamer phenomena in non-uniform fields with highly
stressed anodes. A detailed understanding of the positive
streamer initiation, propagation, and its arrival to the cathode
forming an active cathode spot, are principal for the general
understanding of the streamer breakdown. As a consequence,
the great majority of the study focused on these prebreakdown
events was done in the positive point-plane
gaps, where the whole sequence of events leading to the
breakdown, is better discernible than in the uniform ﬁeld
gaps, and thus more amenable to both experimental and
theoretical studies.
Loeb [60] was the ﬁrst who commented on ‘observed ﬁne
electric blue streamers darting outward from the point which
indicated a progressive projection of very high ﬁeld regions
from the positive point into the gap’ (see ﬁgure 5) and
‘pointed out that these streamers constituted a new type of
breakdown process.’ Even when this phenomenon was
considered by Loeb as a ‘new type of breakdown’ in praxis
it turned out that ambient air streamer coronas and breakdowns
in positive point-to-plane gaps are widely used for
basic studies of the streamer phenomena in general [28, 46].
A considerable amount of experimental, theoretical, and
numerical effort [128–131] has been devoted to the understanding
of the positive streamer initiation near the anode tip
in ambient air:
According to Gosho and Saeki [132], the discharge in
electronegative gases is triggered by electrons detached from
negative ions arriving to the high-ﬁeld region near the anode.
Subsequently, following a pre-corona sequence of avalanches
[82], a positive streamer is initiated close to the anode. Later
this initial discharge stage was studied in more details by
Laan and Paris who found that [84] ‘In a divergent ﬁeld the
streamer formation is preceded by the accumulation of space
charge, i.e. it has a multi-avalanche nature.’ and ‘A streamer
starts when the space charge density reaches a critical value
in a spatially localized region.’ This is in agreement, for
Figure 5. Time integrated photographs of positive (left column) and negative (right column) streamers in a 40 mm gap in air at 1 bar for
different pulse gap voltage values according to [32].
9
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
example, with the results of computer simulations [37, 133]
indicating moreover that the density of the initial seed
electrons hardly inﬂuences a positive streamer initiation and
propagation a positive poin-to-plane gap. The results in
[130, 134] also envisage the streamer initiation as a
detachment of the ionization layer from anode surface.
This is in general agreement with the description of the
streamer initiation by Nijdam et al [135]: ‘The discharge
starts with a small ball of light around the needle tip that was
called the initiation cloud. This ball expands and forms a
shell; this shell can be interpreted as a radially expanding
ionization front.’ Depending on the impulse voltage rise time
and magnitude, and gap length, the expanding shell breaks up
into a single or multiple streamer heads. This picture is in
general agreement also with the experimental results by
Lowke and D’Alessandro [136]. They found that the onset of
positive corona in ambient air occurs in a multiavalanche
process for avalanche sizes above 104
, that is signiﬁcantly
lower than the ‘critical size’ of 108
that would be expected
from the Raether–Meek streamer criterion. This conclusion is
in line, for example, with the theoretical consideration by
Warne et al [80] that ‘for increasing non-uniformity, the
results for the corona and spark branches of the breakdown
characteristics are shown inconsistent with a breakdown
criterion solely based on either the Townsend or streamer
mechanisms.’
Unfortunately, despite the existing experimental data
some of the current simulations still have attempted of to treat
such positive streamer initiation not in terms of a streamer
initiating plasma (‘initiating cloud’) generated by many
avalanches, but ‘traditionally’ in terms of a critical single
avalanche or of a chain of critical size avalanches [137–139].
The several streamers propagating from the ‘initiating
cloud’ towards the cathode with speeds of the order of
108
cm s−1
are well discernible in the streak camera record by
Bessieres et al [140] in ﬁgure 6(a) corresponding well with
ﬁgure 6(b) taken in similar experimental conditions [141] and
with the schematic picture in ﬁgure 1(c).
The formation of the secondary streamer starting from
the anode immediately after the primary streamer arrival,
ﬁrstly discussed in detail by Sigmond [142], can be properly
seen in the ﬁgures 5(c), (e) and 6(a).
From ﬁgure 6, it is apparent that several streamers
developed from the streamer initiating plasma and reached the
cathode at different times. The ‘ﬂashes of light’ [143] at the
streamer arrivals are well discernible in ﬁgures 5(a), (c), (e)
and 6. The streamer arrivals led to the ﬂash of light and the
formation ‘more or less spherical cathode spots’, as well as
the re-illumination of the primary streamer channel corresponding
to the secondary streamer launching are very well
documented also, for example, in [32, 144]. Nevertheless, the
secondary streamer formation due to the primary streamer
arrival to the cathode remains somewhat obscure, apparently
because of an incomplete understanding of properties of the
cathode spot formed by the primary streamer arrival. For
example, Suzuki [145] claimed that:‘After the primary
streamer crosses the gap, a return ionizing wave moves from
cathode to anode. This ionizing wave of potential gradient
comes from the photoelectrons liberated by the light from the
primary streamer tip and from the high ﬁeld between primary
streamer tip and cathode.’ However, in the fast streak-camera
experiments by Bessieres et al [140], such as that shown in
ﬁgure 6, the authors failed to observe the return ionizing wave
proposed by Suzuki. According to them (see [140]), ﬁgure 6
Figure 6. Streak camera record in a 2cm ambient air gap with an anode radius of 50 μm. The applied voltage and streak are 36.8kV and
20 ns, respectively [140]. ICCD camera records made in a 1.3 cm ambient air gap with an anode radius of 80 μm, applied voltage of 35 kV at
exposure times of 2 ns, part (b) taken from [141].
10
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
‘shows that the cathode spot formation seems to precede the
secondary streamer start but the time delay between the two
phenomena is too short to be measured on the streak. In fact,
this time delay is consistent with the propagation time of an
electromagnetic wave starting from the cathode toward the
anode. This wave should be radiated as the electrical ﬁeld in
the head of the streamer tends to zero at its arrival at the
cathode plane. Such a generated wave may be guided to the
point by the streamer channel. This radiation phenomenon is
considered to be close to the radiation of a monopole
antenna.’ The still not well understood formation of an active
cathode spot by the streamer arrival, however, marks a
turning point in the subsequent discharge development, since
as found out by Kondo and Ikuta that ‘almost of the electrons
making the secondary wave luminous are fed through the
cathode fall, and the ionization rate in the secondary wave
column must be negligibly low’ [52]. Unfortunately, as
discussed in details below, the current computer simulation
models describe the cathode spot formation starting the
secondary streamer only in the most superﬁcial terms, see
section 4.
The light reﬂection from the cathode surface, together
with the small size and short duration of the cathode spots
have made it too difﬁcult to obtain their time- and spaceresolved
optical measurements. Fortunately, one of the
advantages of short (∼1 cm) positive point-plane air gaps is
that, using proper pulsed voltage and optimized electrode
geometry, it is possible to minimize the generation of a multistreamer
bunch and the streamer branching [35, 146–148] like
those seen in ﬁgures 5 and 6. This, as reported by Larsson
[149], see ﬁgure 7, enables to measure the discharge current
corresponding to the single streamer propagation in a virgin
air and its arrival to the cathode with a nanosecond temporal
resolution:
The ﬁgure 7 illustrates the voltage and anode current
waveforms (in this case nearly identical with the cathode
current) corresponding to the streamer breakdown measured
by Larsson [149] in a rod-to-plane electrode conﬁguration
with a hemispherically tipped rod of 5mm radius and 1 cm
interelectrode gap, where ‘region 1 can be identiﬁed as the
streamer propagation phase of the discharge and region 2 as
the streamer’s arrival at the plane electrode (cathode)’, which
are corresponding to the stage labeled II in the schematic
streak camera record by Marode in ﬁgure 1(c). It is
noteworthy that the fast rising (∼2 ns) current pulse formed
by the streamer arrival to the cathode was in some 15 ns
followed by a second current peak. Such double-peak
structure was measured both in the pulsed voltage conditions
[150], and in the below discussed repetitive corona conditions([151]
see ﬁgure 4 there, and [152]) using several
different current measuring circuits. Thus it is apparent that
the observed complex structure of the current pulse generated
at the streamer arrival to the cathode hardly can be just an
experimental artefact. Note also that the constant gap voltage
during the stage 2 strongly indicates that the measured current
maximum is not due to a charging of stray capacitance in the
external circuit as suggested by Akishev et al [153].
The current signal in ﬁgure 7 is much better discernible
than a not well pronounced current hump induced by the
streamer arrival to the cathode in uniform ﬁeld gaps (see
ﬁgure 2) or in nearly uniform gaps [154], and reminds in the
shape the current peak in ﬁgure A2 induced by the streamer
arrival to cathode in DBDs. Since, as already mentioned
earlier, it is difﬁcult to obtain true temporal luminosity
corresponding to the streamer arrival, the current signal like
that in ﬁgure 7 can, as discussed below, provide important
information on the complex nanosecond-scale phenomena at
the primary streamer arrival to the cathode, subsequent
cathode spot formation, and secondary streamer onset:
The streamer arrival to the cathode forming an active
cathode region and signed in ﬁgure 7 by the sharp current
peak on the discharge current waveform, critically determines
the transition between the phases II and III illustrated in
ﬁgure 1. This is conﬁrmed, for example, by Marode [28] who
claimed that in a short point-to-plane gap in air, the spark
breakdown occurs only when the magnitude of the current
peak generated by the streamer arrival exceeds a critical
value, and that the current spike is due to an electron emission
from the cathode.
In this context it is noteworthy that Loeb [143] observing
the mentioned ‘ﬂashes of light’ suggested that ‘at a (streamer
head) distance of a millimeter or less from the cathode, the
intense ultraviolet radiation liberates a mass of electrons
from the cathode’. As a consequence a critical dependence of
the current spike shape on secondary photoemission coefﬁcient
of the cathode is to be expected. Nevertheless, the
experiments by Inoshima et al [147] comparing the current
peak measured by Cu and CuI coated cathodes, as well as
measurements by Johnson et al [155] revealed no signiﬁcant
dependence of the current peak like that in ﬁgure 7 on the
cathode material. Alternatively, Nasser [156] obtained an
experimental indication for a ﬁeld electron emission occurring
at the streamer arrival. In an attempt to test the effect of
cathode properties Černák et al [150] compared the discharge
current waveforms of a pulsed positive point-to-plane corona
measured using a small cathode probe for an freshly polished
unconditioned brass cathode with those obtained the wellconditioned
cathode. The results shown in ﬁgure 8 conﬁrm
the insensitivity of the current peak on the cathode surface
properties, but exhibit an unexpectedly strong effect that the
cathode surface properties have on the subsequent glow-toarc
transition. The observed effect on the glow-to-arc
transition is in some respect similar to that observed in
TEA laser discharges [118, 157], and contradict the current
physical description of the glow-to-arc transition in the
conditions considered [125, 144], where the effect of
processes in the cathode region created by the streamer
arrival in the glow-to-arc transition is neglected. An
interesting feature of the waveform 2 in ﬁgure 8 is the
double peak structure A–B resembling that apparent in
ﬁgure 7.
Černák et al [150] studied also the effect of a graphite
coating on the cathode probe, that is known to have
exceptionally low photoelectric yield and produces clusters
of ﬁeld emission sites. The graphite coating resulted in a
11
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
highly stochastic behavior of the current peaks induced by the
streamer arrival and strong acceleration of the glow-to-arc
transition like that in Curve1 in ﬁgure 8 measured using the
unconditioned cathode probe. Nevertheless, no relevant effect
of the graphite coating on the pulse magnitude was observed.
Note that the authors attributed the acceleration of the glowto-arc
transition observed using unconditioned and graphitecoated
cathode surfaces to the existence of positive-streamerlike
instabilities of the cathode sheath due to unstable and
inhomogeneous electron emission from the cathode, as
discussed in more details in section 5.2 (see also [158]).
The results by Černák et al [150, 158] exempliﬁed in
ﬁgure 8, call into question the 1D computer simulation
models like those by Marode et al [159] and Naidis [125], as
well as the more recent 2D simulations in [152, 160–162],
where using the Neumann boundary condition, the cathode
spot formed at the streamer arrival is considered to be just an
electron source with zero internal resistance feeding the
discharge during the glow-to-arc transition. For these works it
is typical that ‘The simulation of the streamer arrival to the
cathode obviously depends on the boundary conditions used.
This paper is not devoted to treat speciﬁcally the streamer hit
to the cathode; only the secondary emission by the ion
impacts is considered’ [161], and as noted also by Ducasse
et al [163], there is a need for ‘some speciﬁc analyses of the
boundary conditions at the cathode plane.’
To our best knowledge the only attempt to analyze in
details the streamer arrival at the cathode and its transformation
to the stationary cathode fall in a positive point-to-plane
gap was done by Odrobina and Černák [56] for the discharge
in nitrogen at 26.7 kPa. In the ‘1.5 dimensional’ model the
Figure 7. The gap voltage and the anode current waveforms corresponding to the streamer breakdown in a short (d=10 mm) positive rod to
plane gap in ambient air. The anode was a copper rod of 5 mm radius. Taken from [149].
Figure 8. Comparison of the cathode current waveforms measured in
a 1 cm positive point-to-plane gap in ambient air at a voltage of
20 kV. The current waveforms correspond to typical glow-to-arc
transitions observed using the freshly polished brass cathode (curve
1) and the well-conditioned cathode (curve 2) surface at the same.
(Scales: 400 mA and 50 ns per division.) Features of the waveform 2
denoted A and B are explained in the text. Taken from [150].
12
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
ions and electrons densities were limited to a cylindrical
channel with ﬁxed radius and the ﬁeld was computed using
the method of disks. The computed current induced by the
streamer arrival to a small cathode probe presented in ﬁgure 9
are in Ref. [56] compared with that measured experimentally.
As typical for the 1.5D models, the claimed good
matching of the magnitudes of the computed and measured
experimental current waveform was apparently achieved by
somewhat arbitrary choice of the constant discharge channel
radius. However, the overall physical picture and the
computed temporal development of the discharge current
can be more critically affected by the simplistic treatment of
the unrealistic high energy of electrons near the cathode
surface resulting from the modiﬁed local ﬁeld approximation
used. Without such treatment an excessive ionization is
generated immediately at the cathode, which in turn leads to
an enhancement of the ﬁeld etc, and the whole simulation
becomes unstable. Since the physical interpretation of the
current signal induced by the streamer arrival to the cathode is
central for our next considerations, the results in ﬁgure 9 and
conclusions by Odrobina and Černák [56] brieﬂy discussed
below are corroborated by a more advanced 1.5D model in
section 4, where the high-energy electrons are treated by a
more advanced and physically sound local mean energy
approximation.
The cathode probe current computed by Odrobina and
Černák [56], shown in ﬁgure 9, consists of an initial sharp
current spike due to the displacement current followed, some
20 ns later, by a lower current hump due to the ion arrival at
the cathode. The conclusion that the initial spike is due to the
displacement current is in contradiction to the generally
accepted assumption by Marode in [28], but has been
corroborated by very recent results by Belopotov et al
[164]. The computed current signal is relatively insensitive to
changes in the photon and positive ion secondary electron
emission coefﬁcients. The results obtained show that the
intense ionization indicated by the light ﬂash on the cathode
observed at the streamer arrival are not due to an intense
photoelectric ([143] on page 136, and [146]) or ﬁeld electron
emission [156], but rather due to a dramatic increase in the
multiplication factor and a release of electrostatic energy
accumulated in the streamer channel-cathode system. The
current hump delayed 20 ns after the current maximum is
claimed to be due the positive ion arrival to the cathode. The
very similar discharge current waveform with subsequent
damped oscillations was observed also by Ikuta and Kondo
[52] (see ﬁgure 5(a) there). Such observed current hump
seems to correspond well with the 2 ns delayed current hump
seen in ﬁgure 7 measured at atmospheric pressure in a highly
over-volted gap and a some 10 ns delayed current spike
marked B in ﬁgure 8. Note, however, that an alternative
explanation for such double peak structure presented by
Eichwald et al [152] is that a current hump delayed some
25 ns after the current maximum observed in the repetitive
discharge regime (see discussion below) is due to the
development of a secondary streamer starting from the anode
just after the arrival of the primary streamer.
An interesting feature of the results in ﬁgure 9 corresponding
well with the results shown in ﬁgure 8 is a 15 ns delay
between the maxima of the current and ﬁeld strength at the
cathode. Thus, if a signiﬁcant ﬁeld emission occurs as
indicated by the Curve 1 in ﬁgure 8, according to the
simulations, it is expected to take place some 10 ns after the
current maximum. The detailed discussion and theoretical
treatment of this issue is presented in section 4.
The discharges hitherto discussed in this section were fed
by single gap voltage pulses. In such highly transient,
probabilistic discharge phenomena the streamer radiation is
typicall studied by streak cameras and cameras in fast-framing
regimes (see ﬁgure 6) that can acquire up to 108
frames per
second with a high spatial resolution [31]. The discharge
currents are usually measured by fast digital oscilloscopes in
the single pulse acquisition mode with a temporal resolution
on the order of nanoseconds without the need to consider
shot-to-shot reproducibility (see ﬁgures 7 and 8).
Nevertheless, it should be noted that starting from the
pioneering work by Marode [28], the great majority of
positive streamer phenomena were studied in the high-stable
dc positive repetitive point-to-plane streamer coronas, where
using averaging acquisition modes the light and current
signals from millions of streamer pulses were accumulated,
improving the time resolution to 0.3–0.4 ns [46]. Unfortunately,
in such repetitive streamer experiments the earlier
discharges signiﬁcantly modify the discharge’s behavior. This
is because the ﬁeld and the background ionization, as well as
the gas composition, are affected by the previous discharges.
In this respect it is interesting to note also that the current
signal generated by the arrival of repetitive streamers
measured by small cathode probes in air [46] is very similar
in shape and just some 100% higher than the typical current
pulses of air DBDs exempliﬁed in appendix in ﬁgure A2.
3.2.2. Streamer phenomena in non-uniform fields with highly
stressed cathodes—complete breakdown. The streamer
phenomena in negative point-to-plane has attracted much
Figure 9. Time development of the total probe Ip, anode Ia, and
conductive probe Ii currents simulated for the streamer arrival to a
4 mm—diam. cathode probe in a short (S=20 mm) point to plane
gap in nitrogen at 26.7 kPa and the gap voltage of 4 kV. E is the
intensity of electrical ﬁeld at the axis of the probe. Taken from [56].
13
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
less attention than the above discussed ‘classical’ streamer
coronas and breakdowns in positive point-to-plane gaps.
Partly it is because the negative spark breakdown voltages are
signiﬁcantly higher, particularly for larger gaps [165, 166]. As
a consequence, comparing to the positive breakdown
phenomena, there is a lack of the time- and space-resolved
measurements of the initial ionization development, which is
necessary to shed light on the sequence of events leading to
the negative breakdown. Nevertheless, already in 1956, Fark
and Cones [167] obtained very insightful results using a 10cm
wide (gap spacing S = 100 mm) negative rod-to-plane gap in
laboratory air energized by the pulsed voltages chopped at an
accurately controllable time. Analyzing their still photos of
the discharge, the authors revealed the positive streamer
formation in a close vicinity of the 0.8 cm curvature radius
(r = 8 mm) spherical cathode: ‘With the sphere negative, the
electron avalanches formed in the region near the sphere
initiate positive streamers which quickly propagate the short
distance to the sphere. As the streamers approach the
(cathode) sphere, the high positive-charge densities at their
tips make already extremely high gradients near the sphere
even higher, and, largely because of these exceedingly high
gradients, electrons are immediately release from the sphere.’
This picture corresponds surprisingly well with the streamer
phenomena observed in a short rod-to-plane gap (S = 20 mm,
r = 10 mm) using an image converter camera together with an
image intensiﬁer and a photomultiplier by Akazaki and
Tsuneyasu [168].
Based on these results, we can hypothesize that the
negative breakdowns, as well as the breakdowns in uniform
ﬁelds gaps and positive point-to-plane gaps illustrated in
ﬁgures 1 are unavoidably associated with the formation of an
active cathode spot at the arrival of the primary positive
streamer. If this hypothesis is true, than it is reasonable to
expect also a similarity between the current signal induced by
the streamer arrival to a cathode probe in the positive pointto-plane
gaps, as those shown in ﬁgures 7 and 8, and the
initial cathode current preceding the negative breakdown.
Unfortunately the above described spatiotemporal measurements
of the discharge radiation were not published together
with the corresponding discharge currents, and in literature
there is only a limited information on such pre-breakdown
currents:
Both the discharge current and discharge light emission
during the negative breakdown in a short rod-plane gap
(S=10 mm, r=7.5 mm) were measured by Isa et al [170],
where the authors found that ‘Depending on the voltage pulse
magnitude a steady negative corona did or did nor exist prior
to the breakdown. However, in both cases the ﬁrst current
pulse was very similar to that of a negative corona TP with a
rise time less than 10 ns.’ Such resemblance of the prebreakdown
current pulses in a rod-to-plane gaps to the
negative corona TPs (discussed in detail in the following
section 5.1) was observed also by, for example, by Suzuki
[165]. The measured pre-breakdown current pulses had
magnitude of an order lower than the positive corona current
spikes in ﬁgures 7, 8, but similar to the repetitive negative
corona TPs. The transition from regular TPs to a glow
negative corona in a short point-to-plane gap (r=0.2 mm,
S=4 cm) in air is clearly seen in [171, 172]. Contrary to the
discharge development in air, the breakdown in short pointto-plane
breakdown in electron non-attaching nitrogen is not
preceded by a TP, but a fast-rising narrow current pulse
shown in ﬁgure 10.
4. Streamer-cathode interaction as a bottleneck in
computer simulations of the streamer breakdown
In the previous sections we summarized the sequence of
events leading to the streamer breakdown and identiﬁed the
streamer-to-cathode contact as a crucial bottleneck in understanding
the process. Due to the complexity of the ionization
processes at the streamer arrival to the cathode resulting in the
formation of a glow-discharge-type cathode spot numerical
simulations are very useful to provide additional insights into
the physical mechanisms that are not easily studied experimentally.
In this section, we present and quantify a theoretical
treatment of the streamer-cathode interaction including a
criticism of the simple local ﬁeld approximation often used to
simulate the streamer-cathode interaction and ionization
phenomena in the immediate cathode vicinity.
4.1. Model
The most widely used model for simulation of streamer discharges
at atmospheric pressure is based on truncated set of
moments of the Boltzmann equations for charged species
coupled with Poisson’s equation. The truncated set of
moments consists of continuity equation and momentum
equations. Continuity equations for density of charges species
nα are
 G
¶
¶
+ = -a
a a a
n
t
G L , 3· ( )
where Gα and Lα represent production and lost terms, and Ga
is a ﬂux of density nα due to macroscopic motion of the ﬂuid.
Usually the momentum equations at atmospheric pressure can
Figure 10. Oscillogram of the initial stages of a spark breakdown in
atmospheric-pressure N2 at a voltage of 6.5 kV, r=0.2 mm,
S=4 cm (scales 2.75 mA/div and 50 μs/div), taken from Černák
et al [169].
14
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
be efﬁciently approximated by a drift-diffusion approximation.
In this approximation, the momentum is considered
to be in steady state and inertial terms are neglected. This
approximation is valid when the mean free path of considered
species is signiﬁcantly smaller compared to characteristic
spatial scale of variation of other physical quantities, e.g. nα
and the electric ﬁeld E. Note that for electrons the mean free
path at atmospheric pressure is of order 10−7
m, while spatial
scale of variation of the electric ﬁeld in the head of the
ionization wave is in order of 10−5
m. Drift and diffusion
components of the momentum equations are then balanced by
collisions and the ﬂux is given by
mG = -a
a
a
a a a a
q
q
n D nE , 4
∣ ∣
( ) ( )
where nα, μα, aD and qα are density, mobility, diffusion
coefﬁcient, and charge of a specie α, respectively.
Moreover it is usually assumed that energy gain by
charged particles from the electric ﬁeld is locally balanced by
the losses due to collisions with neutrals. Note that collision
frequency of electrons with neutrals at atmospheric pressure is
in order of n = 10en
13 Hz and electron energy distribution
function then relaxes to equilibrium with local electric ﬁeld at
timescale n-
en
1
. The energy equation can be then replaced by
local ﬁeld approximation. Note that local ﬁeld approximation
is well justiﬁed for description of electrons in streamers that
are propagating in volume, as has been show by comparison
to particle in cell/Monte Carlo (PIC/MCC) simulation in
[173]. Transport parameters in (4) and reaction rates in gain
and lost terms in (3) then become a function of reduced
electric ﬁeld NE gas∣ ∣ .
In cases when local ﬁeld approximation is not sufﬁcient,
it is useful to add energy transport for electrons, leading to
local mean energy approximation. Energy equation can again
be obtained from electron Boltzmann equations [174] and can
be transformed to a form similar to (3):
 G G
¶
¶
+ + = -e
e e e
n
t
G LE , 5· · ( )
where nε is energy density, where Gε and Lε represent energy
production and lost terms due to collisions. The term G E·
represents heating of electrons by the electric ﬁeld, and on the
right-hand side is total energy transport by collisions. Energy
ﬂux Γε can be expressed as well in drift diffusion form [174]:
mG = -e e e e en D nE , 6( ) ( )
where με and Dε are energy mobility and energy diffusion
coefﬁcient. This formulation of energy equation is convenient
since it can be solved by the same numerical tools as the
continuity equation for charged species, e.g. [175]. Semiimplicit
time discretization is of great advantage because they
allow to avoid numerical instabilities and allows for much
larger timesteps compared to explicit methods [176]. Transport
parameters for electrons and reaction rates for electron
impact processes can be straightforwardly obtained from
corresponding cross sections sets using BOLSIG+ [174],
both for local ﬁeld and local mean energy approximation.
Transport equations (3)–(6) must be coupled to Poisson’s
equation. Note that Poisson’s equation is used on a basis of
electrostatic approximation: assuming the absence of external
magnetic ﬁelds and only small current densities so that selfconsistent
magnetic ﬁeld can be neglected. The Poisson’s
equation reads
åe j j  = - = -
a
a aq n E, , 70· ( ) ( )
where j is the electric potential and ε0 is the permittivity of
free space.
Electrode surfaces can also emit secondary electrons
under ion bombardment. Electron ﬂux emitted from the surface
Gse is assumed to be proportional the ﬂux of impinging
ions
ågG G= -
Î
n n , 8s
i
i sse se
ions
· · ( )
{ }
with secondary emission coefﬁcient γse. Note that value of γse
is poorly known, typical range of γse is Î 0.001, 0.1[ ]. Where
higher values of γse can be used to roughly account for other
electron emission processes such as the impact of excited
species and the ﬁeld emission. An alternative approach for
generation of secondary electrons is to consider Auger neutralization
process [90, 177].
Another important physical mechanism that is responsible
for generation of secondary electrons is photoemission.
The ﬂux of photo electrons is given by
jgG = -n n , 9s spe pe· · ( )
where γpe is a coefﬁcient for photoemission and j is photon
ﬂux at the dielectric surface. The photon ﬂux at point of
observation r due to the source point at ¢r is
òj
p
m=
¢ - ¢
- ¢
- - ¢ ¢
¢
I
Vr
r r r
r r
r r
1
4
exp d , 10
V 3
( )
( )( )
∣ ∣
( ∣ ∣) ( )
where ¢I r( ) is the production rate of photons and is often
assumed to be equal to the ionization production rate
[178, 179], and μ is photon absorption that can be usually
neglected [56], then the only coefﬁcient needed to model the
photoemission source term is γpe [180].
Higher moment ﬂuids models are also formulated
[181, 182]. Another alternative is an hybrid approach, coupling
a particle model for a streamer head region with a ﬂuid
model for a plasma channel [183], or beam-bulk model where
the bulk electrons are treated with a ﬂuid approximation,
while an electron Monte Carlo simulation is used to treat
energetic secondary electrons in a fully kinetic way [90,
184–186]. For comparison of outcomes of models with
varying complexity, see [187]. Alternative approaches to
account for non-local corrections in the ionization source
terms are also formulated and used, e.g. [56, 188–191].
Fluid models, even in simple local ﬁeld approximation,
can provide a fairly good description for streamer propagations
in volume [187]. While for description of streamer
contact with cathode, local ﬁeld approximation will necessarily
fail for simple reason: the closer the streamer comes to
the cathode, the higher is the electric ﬁeld resulting to higher
ionization rate and higher electron density. There is no
15
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
physical mechanism in the model with local ﬁeld approximation
to stop this growth. This leads to unphysical results
and collapse of the cathode sheath. The reason why the local
mean electron energy approximation can eventualy work is as
follows: as the streamer is coming closer to the cathode surface
a strong gradient of the electron density is created. The
strong gradient in density result in a strong diffusion ﬂux of
electrons against the electric force, thus electrons are loosing
their energy, being unable to produce strong ionization. Local
ﬁeld approximation lacks this effect of slowing down or
accelerating of electrons in electric ﬁelds, while this effect is
included in the local mean energy approximation, through the
G E· term in equation (5). This is because G contains both,
the drift in the electric ﬁeld and the diffusion of electrons, see
equation (4).
But even higher moment approximations, may be insufﬁcient,
and obtained results must be therefore treated with
highest caution, because very high electric ﬁeld can be produced
in the cathode sheath and transport parameters and
reaction rates calculated based on two term approximation of
the Boltzmann equation [192] may be incorrect since the
known validity limit up to 1500 Td for the two term
approximation of the Boltzmann equation solution [191, 192].
Secondly, the possibly very sharp variations of plasma parameters
within the cathode sheath, with characteristic space
scale down to 10−6
m, may become comparable with the
mean free path of electrons, and there ﬂuid approximation
becomes simply inappropriate. Moreover, the appearance of
runaway electrons at high values of the electric ﬁeld can be
expected [193].
4.2. Case study
We have implemented a local mean energy approximation
model in order to reveal the spatial scales that we need to deal
with during the streamer–cathode interaction. Conditions
considered in these simulations are similar to those in [56]
illustrated by the results in ﬁgure 9. Here, we use nitrogen at a
pressure of 26.7 kPa, point-to-plain electrode geometry with a
gap of 1 cm, and anode point radius of 0.2 mm, applied
voltage of 4 kV. The streamer is simulated using a 1.5D
model [194–199], the discharge is considered as a cylinder
with a ﬁnite radius of 0.6 mm, and uniform radial distribution
of charges and the evolution of the charged particle densities
is solved in one spatial dimension along the direction of the
streamer propagation. It is important to note that the 1.5D
model captures the main characteristics of streamer propagation.
With a 1.5D model, the electric ﬁeld is found by
slicing the plasma cylinder net charge density into discs.
Using a 3D analytical formulation for the axial electric ﬁeld
from a disc, the space charge electric ﬁeld in the electrode gap
is calculated by integrating the individual contributions of the
disks and their image charges in the electrodes. Similarly to
[56], the discharge propagation is initialized by low electron
ﬂux from the cathode, that produces background ionization at
level of 4.5×1012
m−3
. Transport parameters and ionization
rates used in this simulation has been calculated using
BOLSIG+ [174], with Biagi’s database of cross sections
[200, 201]. Note that resulting ionization coefﬁcient is substantially
higher than that used in [56]. In the following we
have used g = -10se
2 and γpe=0.
Considering that effective cross section of N2 molecule is
about 3×10−20
m2
, then for neutral gas density
6.1158×1024
m−3
, used in this simulation, the mean free
path is about λmfp=5.45 μm. We deﬁne a characteristic
spatial scale of electron density variation in the streamer head
as
L =-
n
n
x
1 d
d
, 11ch
1
e
e
( )
then obvious requirement for validity of the drift-diffusion
approximation is
lL . 12ch mfp ( )
Figure 11 shows electron density ne, positive ion density
ni, electric ﬁeld amplitude E and mean electron energy ε in the
vicinity of the cathode (placed at x = 0). Left panel of ﬁgure 11
shows simulation results at time 38.4 ns, when the streamer
head is 1 mm away from the cathode. Right panel of ﬁgure 11
shows simulation results at time 39.8 ns. At this moment the
maximum of the electric ﬁeld is no longer in the volume but at
the cathode. We also see a much steeper front of the electron
density at the rising edge of the ionization wave.
Figure 12 shows current and electric ﬁeld at the cathode
surface, Ec, during streamer–cathode interaction. Total current
Itot as to be measured on a cathode current probe consists of
displacement current Id and conductive current. High current
peak at around 39 ns is fully attributed to the displacement
current.
Figure 13 shows characteristic spatial scale of electron
density variation on the edge of the ionization front Λch and
corresponding electric ﬁeld in the streamer head. Condition
(12) is well satisﬁed when the streamer still propagates in the
volume, on the contrary, when the streamer approaches the
cathode (moment associated with the current peak at around
39 ns) this condition is clearly not fulﬁlled. At the same time,
the electric ﬁeld rises above critical limit Elim=1500 Td for
validity of transport coefﬁcients and source parameters used
in the model, calculated based on the two term approximation
for solution of Boltzmann equation.
This clearly shows that while the drift diffusion
approximation may be very well valid for studies of streamer
propagation in volumes, for streamer interaction with a
cathode the outcomes of the model must be regarded with
highest caution, because conditions for validity of the model
may be even strongly violated. Any interpretation of such
results should be accompanied with strong discussion
regarding validity and limits of the model.
5. Positive-streamer-like phenomena
In this section we will present a hypothesis congruent with
our experiences that some phenomena of signiﬁcant practical
importance as TPs in negative corona discharges and the
cathode-sheath instabilities affecting the high-pressure glow-
16
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
to-arc transitions can be interpreted in terms of the positive
streamer formation in the immediate (∼0.1 mm) cathode
vicinity and even quantitatively explained by the above presented
computer simulation model.
5.1. Negative corona TPs
It is well-documented that low-current initial stages of
breakdown in a short negative point-to-plane gap in nitrogen
[169, 202, 203] as shown in ﬁgure 10, and in other electron
non-attaching gases [204, 205] at pressures above the order of
10 kPa are discerned by a current peak marked as a stage I in
ﬁgure 10, where in several nanoseconds the discharge current
rises to a maximum of some 1–100 mA. Beyond the initial
peak (I), the discharge current is decreasing to a glow discharge
stage (II) followed by the rise to a spark (III). The
initial current peak formation is illustrated by ﬁgure 14 where
the initial discharge stage of the breakdown in nitrogen (1) is
compared with the ﬁrst corona TP in air (2) for r=0.2 mm,
S=4 cm. The current spike in air is some 30% higher than
that in nitrogen, in a correspondence with the higher values of
α in air. The peaked current is associated with the formation
of an abnormal glow-discharge-type cathode spot [206]. The
pulse amplitude increases with α, but is relatively insensitive
to the electron attachment coefﬁcient.
From ﬁgure 14 it is evident that the discharge in air is
within some 160 ns choked off by a negative space charge
formed due to electron attachment. A new discharge will start
again when the negative ions dissipate toward the anode in a
sufﬁcient extent [207]. In this way, regularly pulsed discharges
are generated resulting in the so-called negative
corona Trichel current pulses with amplitudes in the milliamps
range, such as shown in ﬁgure 15, and frequencies
extending to the megahertz range [166]. Note that the ﬁrst TP
of the pulse train such as that in ﬁgure 14, developing in a
Figure 11. Electron density ne, positive ion density ni, electric ﬁeld amplitude E and mean electron energy ε in the vicinity of the cathode at
times, (left) 38.4 ns, and (right) 39.8 ns. N2 at 26.7kPa, Gap: 10.0 mm, Ua=4 kV, streamer radius: 0.6mm.
Figure 12. Current and electric ﬁeld on the cathode surface during
streamer–cathode interaction. Itot: total current, Ic: conductive
current, Id: displacement current, and Ec electric ﬁeld on the cathode
surface.
Figure 13. Temporal evolution of the characteristic spatial scale for
electron density variation in the streamer head, Λch, and the streamer
head electric ﬁeld Esh, when the streamer approaches the cathode.
Elim=1500 Td is the limiting reduced electric ﬁeld for validity of
two term approximation of the Boltzmann equation solution. Critical
value of Λch is 5.5 μm, which corresponds to mean free path of an
electron.
17
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
negative ions free space, is always larger than the subsequent
regular pulses shown in ﬁgure 15.
It is an interesting and well-documented fact that the
regular TPs can be generated only using cathodes with some
surface imperfection as oxide spots or dust particles providing
a random ﬁeld (Malter) electron emission [143, 208, 209].
Since their ﬁrst observation in 1938 (see the original
article of Trichel, [211]) TPs have attracted a lot of attention
among the researchers and engineers working in the ﬁeld of
applied electrostatics [212]. Quite surprisingly, despite its
practical importance and the plethora of studies, the mechanism
of TPs in air is still one of the most controversial issues
in the physics of high pressure gas discharges.
A major source of controversy around the TP mechanism,
which has been well documented but often ignored, is
the very short rise time of TPs in ambient air (∼1.5 ns, see
ﬁgure 15) deﬁned as the time interval in which the pulse rises
from 10% to 90% of its amplitude. Apparently, it is because
the measurements of TP waveforms and the understanding of
their physical mechanism have always depended heavily on
the availability of fast oscilloscopes with a sub-nanosecond
time resolution and even more on experimental skills required
for such fast TP current measurements:
The ﬁrst systematic measurements of the regular TP
waveforms with a sub-nanosecond temporal resolution were
published in 1969 by Zentner in his excellent thesis [213] and
in the subsequent papers [171, 172]. Zentner used a sampling
oscilloscope with 0.35 ns rise time and found that the measured
TP rise time of 1.6 ns in atmospheric pressure air is
constant for the cathode curvature radii ranging from 15 μm
to 5 mm [171]. It should be noted that already forty years ago
Goldman and Goldman [214] stated that this quite surprising
but well documented fact can serve as a critical test for
models explaining the TP formation.
In papers by Zentner [171, 172], as well as in many other
works discussed below it was clearly demonstrated that to
follow the fast TPs current rise both the measuring resistance
in series with the discharge gap and the parasitic capacitance
have to be reduced to a values giving the nanosecond, or even
sub-nanosecond, time constant of the measuring circuit
required to follow the fast TPs current rise. In 1973, Torsethaugen
and Sigmond [215] used a sampling oscilloscope
with a rise time of 0.3 ns, and measured the rise times shorter
than 3 ns for sharp (r=0.074 mm) Mo and Au cathodes. In
1985 Gravendeel and Van der Laan published [216] the
measured rise time of 1.3 ns for r=0.0125 mm, lately
reduced to 1.15 ns [217] by considering the measuring system
inﬂuence. Their measurements were made by the measuring
system with the well veriﬁed time resolution of 0.7 ns. The
fact that TP rise time in ambient air is smaller than a few
nanoseconds at atmospheric pressures has been conﬁrmed by
many other authors [47, 203, 210, 218–221].
Nevertheless, it is necessary to note that in the last four
decades, starting by work Thanh [222] there is outburst of the
experimental and theoretical works devoted to TPs in atmospheric
pressure air with the measured and computed rise
times on the order of 10 ns, which neither refer to the above
presented results, nor consider their physical and practical
implications. We can exemplify it by several recent papers
[223, 224], where the measuring resistors used and the stray
capacitances were far to large to measure the 1–2 ns rise time
of TPs in atmospheric-pressure air. In fact we are not aware of
any published results, where the ‘slow-rising’ TPs in air at
atmospheric or near-atmospheric pressures were measured
using a measuring system with a properly tested for at least
nanosecond time resolution. Thus, we dare to claim that such
long TP rise times were not determined primarily by the
physical processes in the discharge itself, but by the improper
measuring circuits used. Therefore, also numerous computer
simulation models based on such disputable measurements
are not realistic in explaining the TP ionization mechanism.
Such models are in line with the obsolete physical picture
proposed a half of century ago by Loeb in his excellent and
still indispensable monograph [143], which is still widely
accepted by community of engineers and applied physics
researchers in the ﬁeld of high voltage and applied electrostatics.
For example, as supposed by Hogg et al ‘Negative
corona discharges in air initiate and sustain in accordance
with the Townsend primary electron avalanches emanating
from the tip of the cathode into the ionization zone. Much
slower positive ions traveling in the opposite direction
develop a cathode sheath which increases the ﬁeld in close
proximity to the sharp cathode.’ [12]. Unfortunately, as
mentioned in [220] and discussed in details at the end of this
section, such oversimpliﬁed physical picture can provide
misleading implications for important industrial applications
of TP coronas.
Beside their very short rise times, another surprising
property of the TPs is insensivity of the TP waveforms to the
cathode material [143, 220, 225, 226] observed particularly
for sharp cathodes with, say, r < 0.1 mm. As a consequence,
contrary to the above discussed models already in 1953 it was
Figure 14. Oscillograms of the initial stage of a spark breakdown in
N2 (1) and the ﬁrst Trichel pulse in dry air measured at atmospheric
pressure air at a voltage of 12 kV, r=0.2 mm, S=4 cm (scales
8 mA/div and 20 ns/div, taken from Černák et al [169].
18
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
speculated by Meek and Craggs [227] that ‘A Townsend γmechanism
may not be responsible by itself because of the
corona’s insensivity to cathode metal.’ In 1976 Ikuta and
Kondo [228] based on their fast (∼3 ns) spectroscopic study
in air at a pressure of 50 Torr advanced the hypothesis that
‘the development of (TP) cathode glow may be considered as
streamerlike, through the secondary effect from cathode may
also be active’. In context with the results discussed in
section 3.2.1 it is interesting to refer here to the results
[158, 229] illustrating a striking similarity between TP currents
and the current signal induced by the positive streamer
arrival in a positive point-to-plane gap in air. Many arguments
in favor of the hypothesis that, fast TP current rise is generated
by the arrival of a positive streamer formed in the
immediate vicinity of cathode to its surface can be found also
in extensive studies of current waveforms of the ﬁrst TPs by
Černák et al [220, 230–232].
The results of these studies are exempliﬁed by ﬁgure 16
illustrating the effect of the changing cathode secondary
emission on stepped waveform of ﬁrst TPs well discernible in
air at a reduced pressure of 40 kPa using a relatively blunt
cathodes [220]. The secondary emission of the brass cathode
was increased by a copper iodide coating that is known to
have exceptionally high photoelectric yield. The CuI coating
resulted in a signiﬁcant enhancement of a streamer-initiating
Townsend discharge. It, however, in agreement with the
computer simulation results in section 4 (see also ﬁgure 6 in
[56]) had no effect on the following TP current pulse maximum
generated at the streamer arrival to the cathode.
Comparing to the measurements of TPs current waveforms
in atmospheric pressure air, the related optical investigations
of TPs have been even more hindered by the speed
and sensitivity required of the diagnostic apparatus in order to
conﬁrm the hypothetical formation of the positive streamer in
the sharp cathode vicinity [47, 217, 233]. For sake of brevity,
we shall restrict ourselves to a brief discussion of recent
studies made using the time-correlated single photon counting
method [47, 210, 233] with tens of picoseconds temporal and
tens of micrometers spatial resolution used to study the regular
TPs in atmospheric pressure air:
Townsend discharge generating the streamer initiating
plasma, and the subsequent cathode and anode- directed
streamers are shown in ﬁgure 17 (see also similar results
presented by Akazaki et al in [168]). The measured radiation
of the ﬁrst negative system and the second positive system
intensities [47, 210] correspond well with typical positive
streamer radiation (see, for example [48, 234, 235]) in air.
Thus, the results by Hoder et al [47, 210] provide strong
experimental evidence that the ionization phenomena resulting
in the very short (∼1.3 ns) rise time and the subsequent
initial current decay (in some 10–20 ns) of the regular TPs in
air seen in ﬁgures 15 and 17 are due to the formation of a
positive streamer in the immediate (∼0.1 mm) vicinity of the
cathode and its arrival to the cathode such as simulated in
ﬁgure 9 and in section 4. In agreement with the results of
Sigmond in [206], Hoder et al [47, 210] observed that the
streamer ignition was preceded by the formation of streamer
initiating plasma during a Townsend discharge phase. This
sequence of events resembles closely the pre-breakdown
phenomena in positive point-to-plane gaps discussed in
section 3.2.1 and illustrated by ﬁgure 1(c). Note that the
spatial and temporal development of the discharge radiation
Figure 15. TPs current train (a) and the single current pulse (b) in atmospheric-pressure synthetic air using steel cathode with r=240 μm,
S=7 mm, and voltage 6.8 kV. Taken from Hoder et al [210].
Figure 16. First Trichel pulses measured in dry air at 40 kPa,
r=0.625 mm, S=10 mm and a gap voltage of 5.28 kV using the
brass (1) and the CuI-coated cathode (2). From Černák et al [220].
19
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
in ﬁgure 17 reminds closely the positive streamer development
in the close cathode vicinity (approx. 0.15mm) observed
very recently by Hoder et al [235] and Nemschokmichal et al
[71] in a pre-ionized DBD in atmospheric-pressure N2 + 0.1
vol% O2 gas mixture (see ﬁgure 3 in [71]). Also the
sequence of events in ﬁgure 17(a), namely a slowly developing
multi-avalanche Townsend discharge (approx. 1 μs, see
ﬁgure 17(a)) resulting in a critical space charge accumulation
and the subsequent positive streamer ignition resembles the
successive avalanches of ‘generation mechanism’ during the
pre-breakdown phenomena in low-over-voltage uniform ﬁeld
illustrated in ﬁgure 1(a) and the formation of a streamer
initiating plasma (‘initiating cloud’) by many ‘sub-critical’
avalanches in positive point-to-plane gaps discussed in
section 3.1.2.
The occurrence of a cathode-directed ionizing wave
during the TP rise, resembling that observed by Ikuta and
Kondo [234] at a pressure of 50 Torr and the results in
ﬁgure 17, has been indicated by several computer simulation
models [221, 236–239]. These simulation models are, however,
complicated by limitations of the local ﬁeld approximation
in the cathode vicinity moreover augmented by the
strongly non-uniform Laplacian ﬁeld. As a consequence (see
section 4), many very basic properties of TPs are inaccessible
in these simulations. For example, contrary to the experimental
facts, the computer simulation results are in general
[197] very sensitive to the assumed type and efﬁciency of
secondary electron emission from the cathode.
The results by Hoder et al [47, 210] taken together with
the observed [158, 229] similarities between the current signal
induced at the streamer arrival to the cathode in a short pointto-plane
gaps and TP waveforms (see also ﬁgure 20 in the
next section) invite a not unreasonable assumption that the
basic properties of typical TPs in air, particularly the fast
1.5 ns current rise and the subsequent initial (∼10 ns) decay,
can be understood in terms of the simulation model presented
in section 4. Such a hypothesis enables one to explain [220]
the fast ionizing phenomena resulting in the short rise time of
TPs in the terms very similar to the positive point-to-plane
streamer phenomena discussed in sections 3.2.1 and 4: after,
dEc/dt (and the positive streamer velocity) simultaneously
passes its maximum (see ﬁgure 12), the displacement current
stop to rise, and subsequently decreases rapidly as a result of a
reduction of electron multiplication path and saturation of the
α-coefﬁcient. As indicated by the experimental results of
Černák et al [220] in the case of TPs the streamer arrival is
preceded by a Townsend discharge feedback by cathode
photoemission. This provides also a plausible explanation for
stepped forms of TPs observed particularly at reduced pressures
and using blunt cathodes [172, 225, 230, 231, 240]
exempliﬁed by ﬁgure 16. For TPs from sharp cathodes with
small surface areas in atmospheric-pressure air such Townsend
ionization apparently manifests itself just as a short step
on the TP leading edge seen in ﬁgures 15(b) and 17(b).
As mentioned in the Introduction, our review is not
aimed to discuss the applications of streamer discharges.
However, at the end of this section it is useful to explain how
the recent theoretical models of ‘slow-rising’ TPs [224,
241–243] based on apparently incorrect experimental data
[223, 224] can mislead the workers in the ﬁeld of applied
electrostatics.
The simulation models in [243–246] (and similar others)
are based on the measurements of TPs current waveforms
made using measuring resistors 1 and 2 kΩ.
Even when the corresponding parasitic capacitances were
not mentioned, the measuring resistors used are evidently far
too large to measure the 1.5 ns rise time of a TP. As a consequence,
the measured long TPs rise times shown in the
ﬁgure 18, which disagree strongly with the results by Zentner
(see reference [171]) are not determined by the physical
processes in the discharge itself. In fact, the results ﬁgure 18
are due to the measuring circuit (integration) response to the
TP waveforms with the nearly constant ∼1.5 ns rise times
with magnitudes increasing with cathode radii [171, 172].
The all criticized computer simulation models of TPs
[242, 244, 245] based on the data like those in ﬁgure 18 (or in
[247]) assume that the TP current growth is in a short time,
which roughly equals to the TP rise time, choked off mainly
by a negative ion space charge formation, as explicitly stated
in [248]: ‘The impact of the ionization and attachment coefﬁcients
on the rise time of the pulses could be studied in the
Figure 17. The measured intensity of the second positive system of
molecular nitrogen using the time-correlated single photon counting
technique. In the panel (a), the phase prior the Trichel pulse is
shown. Slow increase of the emission intensity is visible. In the
panel (b), the light emission during the actual discharge is shown.
The corona discharge setup consisted of a grounded cathode with a
streamers tip curvature of 190 μm and a positive dc voltage
(+7.8 kV) connected plate, both made of stainless steel with a gap of
7 mm. Taken from Hoder et al [47].
20
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
light of the role the avalanche ionization plays in forming the
rising side of the pulse. The higher rate of the avalanche
ionization process, the sharper the rising side of the pulse.
Since increasing the attachment coefﬁcient enhances the rate
of dissipation of electrons, it will decrease the rate of the
avalanche ionization and hence, lengthen the rise time of the
pulse.’ However, the proposed role of the electron attachment
is in contradiction to the results shown in ﬁgures 10, 14.
Moreover, it assumes (particularly if not some 10–50 ns rise
times as in ﬁgure 18, but the real TP rise time of 1.5 ns is
considered), a very fast electron attachment in the cathode
ionization region, i.e. in less than some 0.4 mm according to
ﬁgure 17 and some 0.2–3 mm from the cathode surface
according to the simulation results in [241, 246].
A consequence of the fast electron attachment in the
cathode immediate vicinity assumed in the models
([242, 244, 245] and many others) is that the rest of interelectrode
space is supposed to be ﬁlled solely with negative
ions. On the other hand, there is a strong experimental evidence
for a signiﬁcant density of free electrons in electrostatic
precipitators and other practical devices based on the negative
corona charging process several cm away from the cathode
[249–251]. Such free electrons can be efﬁcient particularly in
charging nanoparticles [252, 253]. Thus the criticized models
[242, 244, 245] call away from considering the free electron
charging in electrostatic precipitators, which can results in
their poor construct and improper design.
5.2. Cathode-sheath instabilities in high-pressure discharges
Except for the so-called Penning gas mixtures characterized
by a high gas ionization at relatively low electric ﬁeld hindering
the formation of rapid ﬁeld gradients and large space
charges [8, 254], it is generally agreed to be difﬁcult to reliably
control diffuse discharges at near-atmospheric pressures.
It is because even small variations of the amplitude or repetition
frequency of the applied voltage, minor changes in the
electrode geometry and surface properties, in gas ﬂow and
pre-ionization, etc, can lead to discharge plasma instabilities
causing a transition from the relatively unstable diffuse mode
to that of a much more stable ﬁlamentary discharge.
As discussed in section 3.1.2 the gas discharge instabilities
resulting in the plasma ﬁlamentation have been extensively
studied to improve the performance of TEA lasers
[117, 118]. More recently, these phenomena are studied in the
context of large-area diffuse plasmas and micro-discharges
[5, 255, 256]. There is a large body of literature in this ﬁeld,
nevertheless it is still surprisingly contradictory:
For example, according to the current review by Bruggeman
et al [5] ‘The most common instability is the so-called
thermal instability. This instability is triggered by small
ﬂuctuations in the electron density that lead to the following
chain of events. An increase in electron density leads to
increased Joule heating and thereby a localized increase in
the gas temperature and decrease in gas background density.
... The above description of instabilities suggests that
instabilities occur in the bulk plasma and indeed contractions
of the positive column of atmospheric pressure glow discharge
have been observed.’ On the other hand, for example
according to the review by Kunhardt [255] ‘Since the largest
electric ﬁeld of a self-sustained discharge typically occurs in
the cathode boundary region, the glow-to-arc transition is
likely to be initiated by ﬂuctuations in the ﬁeld of this region.’
This assumption is supported, for example, by Akishev et al
[257] claiming that ‘Fast formation of the high-current density
current spots (which are not the hot arc spots) on the
electrodes strongly inﬂuences the homogeneity of the electrical
breakdown of the gas gap at higher pressures and
initiates the non-uniform constriction of the plasma column
farther if the initial breakdown was homogeneous.’ (see also
ﬁgure 4 and the related discussion in section 3.1.2). This is in
accord with the already a half of century old claim by Loeb
[258] that ‘The circumstance that in relatively high ﬁelds the
negative carriers are mobile free electrons, while the positive
carriers are the more sluggish gaseous ions, leads inevitably
to conditions which can make the cathode region a source of
instability in many gaseous discharges. Such instabilities now
appear to be much more common than heretofore suspected.
They manifest themselves in peculiar phenomena whose
nature and common origin have remained obscure.’, which is
supported by many more recent works as that of Akishev et al
[257].
In this section, we will attempt to discuss some of the
‘peculiar phenomena’ mentioned by Loeb, in terms of
instabilities related to the positive streamers formed in the
narrow cathode regions of high-pressure glow and Townsend
discharges. We suggest that such fast positive-streamer-like
phenomena associated with the cathode spots formation can
be very common but hardly observable initiators of ‘An
increase in electron density (that) leads to increased Joule
heating and thereby a localized increase in the gas temperature’
mentioned by Bruggeman et al in [5].
The following considerations are partly a continuation of
the discussion in section 3.1.2 related to the streamer breakdown
of the cathode region in conditions of TEA laser discharges
and TPs in section 5.1. In this context, an interesting
Figure 18. Rise times tr of the regular TPs in ambient air for different
cathode radii σ measured using a slow-response measuring circuit.
Taken from [245].
21
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
type of instabilities, that hitherto has attracted remarkably
little attention in the literature is an ionizing wave driven
cathode-sheath instability with a repetition period of 25 ns
theoretically proposed by Bityurin et al [259, 260] for a glow
discharge cathode region in N2 at atmospheric pressure. Such
proposed periodic ionizing waves can have much in common
with the current oscillations with the period of some 100 ns
‘reﬂecting the back and forth motion of the electric ﬁeld in the
cathode fall region’ computed by Morrow [197] at a reduced
pressure of 6.67 kPa for a negative point-to-plane gap.
Experimentally, similar current oscillations with a period of
20 ns were observed by Kondo and Ikuta [52] in a ﬁlamentary
glow discharge formed by the streamer arrival in air positive
point-to-plane gap at 100 kPa. Figure 19 illustrates the
peculiar discharge current oscillations with a period of some
100 ns observed by Zahoranova et al [261] on a glow-discharge
plateau of ﬁrst TP in hydrogen at a pressure of 35 kPa.
Still not well understood effects of the conditioning of
high-pressure discharge gaps by passing a number of sparks
are of a signiﬁcant practical interest, since the cathode conditioning
has been widely used to prevent or delay the glowto-arc
transition and to affect the breakdown voltage–time
characteristics [150, 262, 263] (see also ﬁgure 10). It is
generally accepted that the conditioning effect, such as illustrated
in ﬁgure 10, is due to the destruction of cathode
emission sites as protrusion sites, oxide spots, or free particles.
Nevertheless, as indicated by recent computer simulations
[264], the effect of cathode surface protrusions on highpressure
discharges, where the sheath properties are critical,
are still known only in the most superﬁcial terms, even for the
apparently simple steady state in argon. Unfortunately, the
related phenomena inside the narrow (∼10–100 μm) highpressure
cathode sheaths, which are typically random in time
and place, are not amenable to optical studies with the
necessary temporal and space resolution [119, 120]. However,
as already discussed in section 3.1.2, some insight into
the dynamics of high pressure cathode spots can be obtained
from the fast discharge current measurements.
As discussed in [169] (see in ﬁgure 3 there), the imperfections
of the cathode surface, at higher over-voltages in an
ambient air positive point-to-plane gap results in fast (∼50 ns)
glow-to-arc transition following the streamer arrival to the
cathode. However, at the lower over-voltages the cathode
surface imperfections result in a ‘peculiar current waveforms’
characterized by irregular narrow fast-rising current pulses of
few milliamperes magnitude riding on a glow discharge bias
current shown in ﬁgure 20. Note that such irregular current
pulses are discernible also in ﬁgure 7:
As illustrated by ﬁgure 21(a), at reduced pressures such
fast-rising pulses can be rather regular, and as shown in
ﬁgure 21(b) the similar regular pulses were observed also
using unconditioned cathodes in negative corona gaps (see
also ﬁgure 2 in [158], and ﬁgure 21 in [231], and here see
ﬁgure 20(b). A comparison of the smooth current oscillations
of the transient glow discharge currents computed for a
negative point-to-plane corona by Morrow (see ﬁgure 11 in
[197]) and measured in a positive point-to-plane corona in air
by Kondo and Ikuta [52] with the fast-rising regular pulses
from the unconditioned cathodes in ﬁgure 21 invites the
following speculation: it can be hypothetized that the smooth
electric ﬁeld and current oscillations due to ‘reﬂecting the
back and forth motion of the electric ﬁeld in the cathode fall
region’ [197] can be an intrinsic phenomena for high-pressure
ﬁlamentary glow discharges. However, if some cathode surface
imperfections are present, they can emit a burst of
electrons just at the maxima of the periodic ﬁeld oscillations.
The sudden electron ﬁeld (Malter) emission can, similarly as
Figure 19. Oscillations on the plateau of the ﬁrst Trichel pulse
current in hydrogen at a pressure of 35 kPa (brass cathode of
r=0.15 mm, S=10 mm). Taken from [261].
Figure 20. Peculiar current pulses observed during a glow discharge
phase using the unconditioned cathode surface at the same
conditions as in ﬁgure 12, but at relatively low gap voltage values.
(Scales 5 mA and 50 ns per division.) Taken from Černák
et al [150].
22
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
simulated by Černák et al [117] and illustrated by ﬁgure 4,
ignite a positive streamer-like ionizing wave in the cathode
region resulting in the fast-rising and narrow current pulse
characteristic (see ﬁgures 9 and 12) for the streamer arrival to
cathode, such as the regular pulses seen in ﬁgure 21. Note a
signiﬁcant similarity in shapes of the pulses (a) and (b) (of
ﬁgure 21) that corroborates the positive streamer mechanism
for negative corona TPs. As documented by ﬁgure 2 in [158],
at the static onset voltage such narrow ‘peculiar pulses’
observed using the unconditioned cathodes can be nearly
identical with the ﬁrst TP, and the subsequent regular TPs.
As a consequence, we can hypothesize that the regular
TPs, which are currently amenable to high-sensitivity
sub-nanosecond optical and current measurement (see
section 5.1), and the above discussed fast-developing
instabilities are of the same basic physical mechanism. If it is
so, then the more detailed study of the regular TPs in electronegative
gases can yield new insight into the hitherto
obscure instabilities in the high-pressure cathode regions, and
also into the cathode region dynamics in general. Note that
the techniques developed by Akishev et al [265] and Mizeraczik
et al [266] enable to generate and study repetitive
negative corona current pulses also in electron-non-attaching
gases and in space-charge free gaps.
In sections 3.2 and 4 a computer simulation model was
proposed for the cathode spot formation which can qualitatively
account also for the fast-rising current pulses of magnitudes
on the order of 1–100 mA corresponding to TPs and,
supposedly, also to many high-pressure discharges instabilities
and micro-breakdowns phenomena. An interesting
manifestation of such phenomena could be, for example, the
formation of a new glidarc cathode spot shown in ﬁgure 22
associated with irregular current pulses of some 100 mA
magnitudes, where ‘a kind of repeated breakdown is possible
between the cathode and the point on the primary positive
column’ [267]. Note, that the same team of authors observed
similar ‘characteristic oscillatory current waveforms’ also
during the glow-to-arc transition in a coaxial air plasmatron
[268].
6. Conclusions and summary
This review attempts to provide a uniﬁed physical picture of
ionization phenomena leading to electric gas breakdowns
associated with the formation of primary streamers in short
discharge gaps preferably with bare metal electrodes, the
emphasis is being laid on details of the formation of active
glow-discharge-type cathode spots.
Such awkward, but still not satisfactorily detailed speciﬁcation
is necessary to minimize misunderstandings related to
the basic terms ‘electric gas breakdown’, ‘streamer breakdown’
and ‘primary streamer’. As indicated by Hudson and
Loeb, and brieﬂy discussed in section 2, the need for more
correct terminology regarding the breakdown phenomena
emerged more than a half century ago [14]: ‘Some common
misunderstandings can be avoided if a careful distinction is
made between the terms breakdown and spark breakdown.’
Unfortunately, in the recent literature not only the spark
breakdown but many other not well deﬁned terms as low and
Figure 21. Comparison of (a) single positive-corona primary
streamer waveforms at 3.5 kV (r=0.02 mm, S=1 cm) with (b) the
ﬁrst Trichel pulses at 6 kV corresponding to a 6/3.35% over-voltage
(r=0.37 mm, S=1 cm) taken using the conditioned (traces 1) and
the freshly polished (2) Cu cathodes. The voltages were selected to
match magnitudes of pulses (a) and (b). All the waveforms were
taken in dry air at 26.6 kPa. Scales: 10 mA and 50 ns per division.
Taken from [158].
Figure 22. The formation of the new glidarc cathode spot
downstream. The air ﬂow velocity in the central part of the glidarc
gap is 20 ms−1
. The cathode is at the left. Taken from [267].
23
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
high pressure breakdowns [269], pre-Townsend [270] and
Townsend breakdowns [271], streamer and leader breakdowns
[272], partial [273] and corona breakdowns [274], and
streamer-like breakdowns [275] are used just within the
relatively narrow topic of our review. Nevertheless, since in
the technical literature the term ‘electric breakdown’ usually
means the process of plasma production in an external electric
ﬁeld, [276], as already suggested in the Introduction, the
simplest way to avoid of such complex and confusing terminology
is to apply the term ‘electric breakdown’ as a
synonym for all types self-sustained gas discharges, where the
gaseous insulation breaks down and an electric current ﬂows
the gap.
For the considered uniﬁcation of the ionization phenomena
discussed in our paper, crucial is the exact speciﬁcation
of the terms ‘streamer breakdown’ and ‘streamer
breakdown mechanism’. As discussed in section 3, these
terms were originally deﬁned to characterize exclusively the
fast ionization growth from a single electron avalanche to the
spark in short overvolted uniform ﬁeld gaps. Because of
experimental constrains the related single avalanche to
streamer transitions were directly observed only in cloud
chambers applied to uniform ﬁeld gaps at approximately p.
d>760 Torr cm (where d is the electrode gap, p the pressure)
[277]. Observations of the transition of a single avalanche
to streamer obtained exclusively in such rather special
conditions were used by Raether and Meek in their wellknown
empirical criterion for the breakdown process termed
as ‘streamer breakdown’. It was not realized at that time, that
the observed fast spark breakdowns can be obtained only with
over-voltages above some 100% of Vs [277]. Nevertheless,
later it was speciﬁed by Loeb [278] for the low-overvoltage
breakdowns that ‘Uniform ﬁelds at threshold and over-voltages
have been convincingly analyzed by Raether’s group at
Hamburg. At threshold, breakdown begins as a Townsend
glow discharge with secondary cathode action. Through
space charge accumulation this goes over to a streamer
spark’, which was subsequently afﬁrmed by other authors
[97] and theoretically explained by Hodges et al [20].
As indicated by ﬁgure 1 and the related discussion in
section 3.1.2, Marode used the term ‘streamer breakdown’
strictly in the traditional way only for the breakdowns initiated
by the critical single avalanche in a uniform ﬁeld gap.
However, as discussed in section 3.2.1 historically it has
developed that his pioneering work on the streamer phenomena
in short point-to-plane gaps [159] has motivated
researchers to use the terms streamer breakdown and streamer
breakdown mechanism for a large variety of ﬁlamentary gas
discharges, where the primary streamers are initiated from a
large variety of streamer-initiating plasmas. Such streamerinitiating
plasmas are typically created by multi-avalanche
ionization processed, such as those discussed for the uniform
ﬁeld gaps in sections 3.1 (see ﬁgures 1 and 4) and for the nonuniform
ﬁeld gaps in section 3.2 (ﬁgure 3), and section 5.1.
This more general terminology, where the term streamer
breakdown is used for all gas breakdowns associated with a
streamer formation, is increasingly common. Unfortunately,
such terminology is often used in an improper and misleading
combination with the single avalanche breakdown criterion.
The resulting misunderstanding is well apparent from considerations
presented in the following very recent review
papers:
In his ﬁrst topical review Brandenburg [8] ascribed the
streamer initiation to a multi-avalanche process ‘At medium,
normal and even higher pressures, gas discharges tend to
constrict due to the streamer breakdown mechanism. Electron
avalanches create a space charge and thus an additional
electric ﬁeld which enhances the growth of secondary electron
avalanches locally. Consequently, the ionized region and
the perturbation of the electric ﬁeld grows rapidly and forms
distinct plasma channels.’ In the second review by Bruggeman,
Iza, and Brandenburg [5] the authors also mentioned
such multi-avalanche streamer initiation (see ﬁgure 5 therein).
Nevertheless, subsequently the authors considered only the
single avalanche breakdown criterion ‘While the Townsendcriterion
provides the inception voltage for the Townsend
mechanism, the Meek criterion (sometimes also referred to as
the Raether criterion) describes the conditions for streamer
initiation. ’
As brieﬂy indicated in section 5.1 related to the TP
mechanism, the unclear deﬁnition of the term ‘streamer
breakdown’ traditionally associated with the single avalanche
breakdown (see ﬁgure 1(b) should not be regarded as merely
a terminology problem. This is a matter of great concern in
solving many other applied problems as, for example, the
long-standing problem of a general criterion for the corona
onset and for spark-breakdown in non-uniform ﬁelds. As
brieﬂy discussed in section 3.2.1, this problem is apparently
due to persistent unsuccessful attempts to develop new
empirical and semi-empirical breakdown criteria by oversimpliﬁed
modiﬁcations of the single avalanche Raether–
Meek criterion [80, 274] without accepting the multi-avalanche
streamer initiation.
It is hoped that the proposed speciﬁcation of the terms
‘streamer breakdown’ and ‘streamer breakdown mechanism’
can deﬁne away many difﬁculties and misunderstandings in
the theory of streamer breakdown phenomena and will make
it more understandable for the applied research scientists and
engineers. This paper is an attempt to step in this direction by
providing a uniﬁed model generalizing the sequence of events
leading to the spark breakdown for short uniform ﬁelds,
positive point-to-plane and negative rod-to-plane gaps, and
for streamer-like instabilities in the cathode region of highpressure
gas discharges. Based on the facts discussed in
sections 2 and 5 we claim that even when the streamer
breakdowns can be extremely varied and change a lot
according to the experimental conditions, in general any
streamer breakdown consists at least from the ﬁrst three of the
following most important stages:
(i) The avalanche stage, wherein the streamer initiating
charge in a localized region is formed by charges
generated in a single avalanche or more often
accumulated in a sequence of avalanches. This stage
which is often nearly imperceptible (see, for example
24
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
‘successive avalanches’ in ﬁgure 1(a) and transient
Townsend discharge in ﬁgure 20) and at other times
obvious (see, for example ‘initiating cloud’ in ﬁgure 6,
and ‘virtual plasma anodes’ in close proximity of the
cathode surface, see ﬁgures 6 and 22.
(ii) The positive streamer initiation: after an initial delay,
when the streamer initiating charge partially shields
itself from the external ﬁeld forming a ‘critical’ region
of relatively dense plasma (1013
–1015
cm−3
) resulting
in the primary positive streamer starts to propagate.
(iii) The positive streamer propagation, where the primary
streamer head propagates as a luminous spot of the
diameter typically less than 1 mm with the velocity
usually in the range 107
–108
cm s−1
followed by a less
luminous streamer trail. As discussed in section 1 the
streamer properties are rather insensitive to changes of
the gas composition or the source of the seed electrons.
As assumed in section 2, the success of computer
simulations in explaining the above sequences (i)–(iii)
can be regarded as a measure of their theoretical
understanding (see section 1). If it is true, then we can
claim that these initial stages are at present fairly well
understood, at least in general terms.
(iv) The streamer arrival to the cathode, forming an active
glow-discharge type cathode spot, which is effectively
producing the electrons by direct impact ionization in
the cathode fall and, consequently, marks a turning
point in the streamer-initiated formation of self
sustained discharges. At atmospheric pressure the
cathode spot develops in several nanoseconds and, as
discussed in sections 1–5, the fast-rising displacement
current peak (see ﬁgures 2, A2, 7, 8, 9, 10, 14, 15, 16,
17(b), 19–21) generated during this phase is rather
independent on the gas composition, cathode secondary
electron emission, and experimental conditions. As indepth
discussed in section 4, this stage has been a major
bottleneck in the computer simulations of the streamer
breakdowns resulting in an arc or spark formation since,
as considered in agreement with our opinion in [152]
(see section 4) ‘It is probably a kinetic description that
would be able to precisely simulate the cathode sheath
formation when the streamer arrives at the cathode.’
Note that although in homogenous or weakly inhomogeneous
ﬁelds the streamer arrival to the cathode is
sufﬁcient to induce the following Stage (v), in strongly
inhomogeneous ﬁelds it is not a sufﬁcient condition for
the arc formation.
(v) The ﬁlamentary glow to arc transition is often
characterized by the growth of secondary streamers
(see, for example, ﬁgures 1, 2 and 5). Because of its
signiﬁcant practical importance, it has attracted a great
deal of attention in the literature on both empirical and
theoretical planes, but the published results on the
subject are surprisingly inconsistent and, therefore are
not discussed in our review. Nevertheless, based on our
results discussed in sections 3.2.1 (see ﬁgure 8) and 5
we believe that a signiﬁcant source of the controversy is
that the existing models describing the glow-to-arc
transition rely on various processes in the residual
streamer channel, neglecting the processes taking place
in an abnormal glow-discharge-type cathode region
created by the streamer arrival including the positive
streamer-like instabilities just discussed in section 5.
We believe that the theoretical model using the above
speciﬁed terminology and based on the clearly deﬁned
streamer breakdown stages offers the attractive feature of the
uniﬁcation of a wide scale of the streamer breakdown phenomena
discussed in our topical review. Moreover, it can
serve as a necessary guide in the integration of our knowledge
and selection of cases for further study of streamer breakdown
phenomena both by experiment or computer simulation. For
example, as suggested in section 6 the periodic negative
corona pulses, which are good amenable to experimental
study, can be used to yield a new insight into the cathode spot
formation at the streamer arrival. Also, we believe that some
of the discussed experimental and theoretical approaches can
be applicable, for example to the streamer phenomena in RF
and microwave electric ﬁelds described in [279, 280].
The purpose of the review is not to compile a catalog of
references, but to give an overview of the ﬁeld and its trend in
line with several important topics in recent high-pressure gas
discharge physics. The current rapid proliferation of applications
of atmospheric pressure plasmas makes it difﬁcult to
cover both basic physics of the streamer discharges and their
application in a single review article. As a consequence, the
review has been preferably attempted to provide a more
consistent theoretical description of the streamer breakdown
phenomena. It is apparent that more experimental and
theoretical research is needed to establish a well elaborated
and veriﬁed theory in this ﬁeld. Nevertheless, we believe that
the recently developed experimental techniques and approaches
[141, 281–286] or advanced computer simulation
methods (e.g. to name a few [287–292]) when applied together
have the potential to meet this challenge.
Acknowledgments
This research was funded by the project LO1411 (NPU I) of
Ministry of Education Youth and Sports of Czech Republic
and by project LM2018097 funded by the Ministry of Education,
Youth and Sports of the Czech Republic.
Appendix. Implications for DBDs
Since the boundary conditions at the streamer arrival to the
cathode (dielectric and metal) and the subsequent discharge
development in the multitude [293, 294] of DBDs can be
rather different, a detailed discussion of the streamer phenomena
in DBDs is beyond the scope of this review.
Nevertheless, since the early phases of breakdown in the
volume DBDs burning between two parallel electrodes are
similar to those without dielectric [8, 295], it might be useful
to digress brieﬂy on the controversy in the streamer initiating
25
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
plasma formation in the volume DBDs in air, that is identical
to the above discussed ‘paradoxical situation of a streamer
breakdown mechanism for spark in uniform ﬁelds.’ [107]:
For a standard DBD ﬁlamentary discharge in air, the
streamer initiating plasma formation was for ﬁrst time discussed
in details by Kozlov [44] as the controversy between
‘Electrical breakdown via avalanche-to-streamer transition’
mechanism as used in, for example [295–299], and ‘Accumulation
of positive space charge followed by the cathodedirected
streamer’ mechanism that was for ﬁrst time considered
in 1986 by Yoshida and Tagashira [300].
Based on the experimentally observed weak but profound
pre-breakdown anode glow of a microsecond duration,
Kozlov suggested a more complex model based on the second
mechanism, i.e. the generation of the streamer initiating
plasma by a temporary Townsend discharge (i.e. a multiavalanche
charge accumulation). Kozlov’s model is in
agreement with the results of computer simulations by Yurgelenas
and Wagner [301] and accepted in reviews by
Kogelschatz and Salge [127] (see ﬁgure A1) and by Brandenburg
[8].
As illustrated by ﬁgure 2 (see, for example, also Williams
et al [29]), the current signal corresponding to the positive
streamer arrival to the cathode in an uniform ﬁeld gap with
bare electrodes is usually obscured by a superimposed
background discharge current. However, as illustrated by
ﬁgure A2, in DBDs it typically takes the form of a well
discernible current peak with a rise time of approximately 1ns
and magnitude of some 20–200 mA [55, 297, 302]. Generally,
the pulse rise times were rather insensitive to the gas
composition for 0.1–10 vol% O2 in N2 [303], but dependent
on the gas pre-ionization. Note also that the pulse magnitudes
increased strongly with increasing α coefﬁcient, and were not
signiﬁcantly affected by the electron attachment. The ﬁne
structure of the current pulse rising slope in DBDs is nevertheless
still an unclear subject which needs experimental
clariﬁcation. To resolve temporally this fast nanosecond
process in detail with sufﬁcient sensitivity, a properly
designed electrical system is needed. See e.g. the discussion
in [55]. It is worth noting that Höft et al [303–306] have
recently conducted systematic investigations to inspect the
inﬂuence of multiple parameters (gas composition and ﬂow,
pre-ionization, pulsed voltage steepness etc) onto the DBD
micro-discharge features: current signals, streamer velocities,
streamer diameter development or the spatiotemporal patterns
of the light emission.
Because of the practical importance to analyze partial
discharges in air-ﬁlled cavities of HV insulation systems, the
most detailed investigations, concerning DBD pulse shapes
have been reported for such discharges [307–309]. It is
notable that for this type of DBDs the terminology is well
established since the narrow current pulses with rapid rise
times as that in ﬁgure A2 are unambiguously associated to
streamer-like discharges and the streamer mechanism [307].
As observed by Morshuis and Kreuger (see ﬁgure 8 in [309]),
at certain conditions in thin dielectric voids (∼0.1 mm) the
streamer current pulse can be preceded by a well-discernible
slow rising Townsend discharge current.
In more complicated DBD geometries however, as for
instance in the case of the surface barrier discharge, the
agreement and/or the broad acceptance of the sequence of
fundamental ionization phenomena leading to the well
developed surface discharge is not apparent. This fact is
surprising as this kind of DBD is intensively studied for
multiple applied as well as fundamental research purposes [8,
310–313]. In our opinion, it is caused mainly by the asymmetry
of the arrangement where one electrode is exposed to
plasma while the other completely embedded into the dielectric
barrier. As a result, in each polarity the mechanism of
the discharge development is different (i.e. the measurable
parameters differ typically in orders of magnitudes). It is
complicated by effective charging of the dielectric surface by
species of different mobilities (electrons, positive and negative
ions). While the generation of the positive streamers in
positive surface electrode polarity is understood, the negative
polarity remains experimentally unveriﬁed. For negative
surface electrode the issue resembles that one as in the case of
metal electrodes discussed for TPs and cathode-sheath
instabilities earlier in the text. The behavior is similar (yet not
identical!) to the negative corona discharge in the Trichel
pulsed mode. The generation of a well localized microscopic
plasma on the dielectric surface in front of the cathode followed
by microscopic positive streamer propagation towards
the cathode (which is crucial for further discharge development)
was theoretically proposed in [311]. Nevertheless, this
signiﬁcant discharge phase is still not well validated by
experiment [314]. Typically the photoemission from surfaces
is assumed to play a major role.
Figure A1. Schematic representations of stages of an individual DBD
streamer discharge according to [127].
26
Plasma Sources Sci. Technol. 29 (2020) 013001 Topical Review
ORCID iDs
Mirko Černák https://orcid.org/0000-0003-0939-5013
Tomáš Hoder https://orcid.org/0000-0001-5346-6275
Zdeněk Bonaventura https://orcid.org/0000-0002-
9591-6040
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