Martin P.M. Handbook of Deposition Technologies for Films and Coatings, Third Edition: Science, Applications and Technology
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Unfiltered and Filtered Cathodic Arc Deposition 471
Figure 10.2: Potential distribution between anode and cathode of a cathodic arc. In this
presentation, the common convention is adopted that the potential is plotted as negative
potential to account for the negative charge of the electron. For details see text.
electrons from the plasma arriving at the anode simply fall into the Fermi level of the anode
and the excess energy will become available (this explains why anodes tend to get hot without
intentional heating).
Figure 10.2 illustrates the situation using the common convention that a negative potential is
plotted to account for the negative charge of the electron. In doing so, a barrier for the electron
motion appears as a hill, and an electron-accelerating potential gradient goes downhill.
Consistently, the Fermi level of the anode is displayed lower than the Fermi level of the
cathode. Figure 10.2 shows the work functions of the two electrodes (a bit pushed out of the
metal for better visibility), the cathode fall (which in this negative potential presentation is
really a fall), a relatively flat potential in the interelectrode gap, and an anode fall. The anode
fall serves to balance the flow of plasma electrons to the anode in such a way that current is
equal to the current determined by the electrical circuit. The sign of the anode fall can be either
way. As shown, the anode ‘fall’ is actually a small barrier which will appear if the anode is
very large and there is an oversupply of electrons.
Let us return to the question of what mechanism allows electrons to overcome the cathode
work function. Electrons in a metal have a Fermi energy distribution function
f
F
=
(
E, T
)
=
1
1 + exp
E−E
F
kT
(10.1)
which describes the probability that a state of energy E will be occupied in thermal
equilibrium at a temperature T. The Fermi energy, E
F
, can be defined as the highest energy of
occupied states at temperature T = 0; it depends only on the density:
E
F
=
¯h
2
2m
e
(3π
2
n)
2/3
(10.2)
472 Chapter 10
where ¯h = h/2π, h is Planck’s constant, k is Boltzmann’s constant, m
e
is the electron mass,
and n is the electron density. The Fermi distribution implies that only a small fraction of
electrons can acquire energy of order kT above the Fermi energy. However, if the temperature
is sufficiently high, some electrons can overcome the barrier in a classical way, namely when
E
z
≥ φ (10.3)
where E
z
= m
e
v
2
z
/2 is the kinetic energy of an electron in the z-direction, the direction
perpendicular to the surface, and φ is the work function of the cathode material. The emission
current density is then
j
thermionic
= en
(
E
)
v
z
(10.4)
where n(E) is the density of electrons satisfying Condition (10.3). For T = 0 obviously no
electrons satisfy (10.3). One can calculate n(E) by integrating the distribution function from
−∞to + ∞ in the x and y directions and from φ to ∞ in the z-direction. At elevated
temperature, the high-energy tail of the Fermi distribution (10.1) is dominated by the
exponential term, where (E −E
F
)>kT and the quantum-statistical Fermi distribution can be
approximated by the classic Boltzmann distribution,
f
B
=
(
E, T
)
= exp
−
E − E
F
kT
(10.5)
The result of the integration, using the Boltzmann distribution, is
j
thermionic
= AT
2
exp
−
φ
kT
(10.6)
where
A =
4πem
e
k
2
h
3
= 1.202 ×10
6
A/m
2
K
2
(10.7)
Equation (10.6) is the well-known Richardson–Dushman equation for thermionic emission,
and A is called the universal Richardson constant. This is the mechanism for thermionic arcs,
which can be stationary: note that time does not appear in those equations.
For cathodic arcs we have to go further and consider the electric field that exists near the
cathode surface, E
c
= −dV/dz, which can be very strong even though the cathode fall is only
∼20 V because the cathode fall thickness can be extremely small. The electric potential
distribution associated with a strong field narrows the work function barrier, as already
indicated in Figure 10.2. Therefore, electrons have a chance not only to go classically over the
barrier but also to tunnel quantum-mechanically through the barrier. The latter is called field
Unfiltered and Filtered Cathodic Arc Deposition 473
emission. The derivation of the field emission current density and resulting Fowler–Nordheim
formula is rather complicated and the interested reader is referred to the specialized literature
[3, 8–10].
In a real cathodic arc situation, both high local temperature and high electric field strength are
present. This leads to a non-linear amplification of the emission process, which is called
thermo-field emission [11, 12]. The full equations are indeed lengthy so that researchers have
developed approximate expressions. For example, Hantzsche’s [9] formula for a typical
material with a work function φ = 4.5 eV is based on a number of additive and harmonic
combinations of thermionic and field emission,
j
TF
(
T, E
)
≈ k
AT
2
+ BE
9/8
exp
⎡
⎣
−
T
2
C
+
E
2
D
−1/2
⎤
⎦
(10.8)
where A = 120, B = 406, C = 2.727 ×10
9
, D = 4.252 ×10
17
; the units for Eq. (10.8) are j
TF
in
A/cm
2
, temperature T in K, electric field E in V/cm; the symbol E is used here to not confuse
it with the symbol for energy, E.
Up to now, time has not appeared in any of those emission mechanisms and related equations.
This changes completely when we consider the energy balance of an electron-emitting site.
Two main processes lead to temperature enhancement: ion bombardment from the plasma, and
Joule (resistive) heating of the cathode by the emission current (for a more detailed discussion,
including the Nottingham effect, see [3]). Cooling of a site is mainly by heat conduction,
electron emission, and radiation. Heating and cooling generally do not balance and therefore
the temperature becomes a time-dependent variable. In particular, a positive feedback can
evolve if the field is very strong and emission has already reached a high level: the Joule
heating of the emitting location leads to higher temperature, which further enhances the
emission, which in turn leads to greater Joule heating and yet higher temperature. Locations
where this occurs can evaporate in an explosive manner, leading to a new form of electron
emission called explosive electron emission [13].
Explosive electron emission is inherently non-stationary and always associated with the
formation of plasma of the emitting material. Therefore, it is not just electron emission but
emission of electrons and ions, and we simply refer to the process as explosive emission.
Mesyats and others [13–15] extensively studied explosive emission and concluded that a
minimum amount of energy needs to be invested to explode (ignite) an emission center. From
the theory of wire explosions [16] they adopted that the current density, j, and explosion delay
time, t
d
, satisfy
t
d
0
j
2
dt ≥
¯
h (10.9)
474 Chapter 10
Table 10.1: Specific action, h, for selected materials
(from [13, 17])
Material
¯
h
(A
2
s/m
4
)
C1.8
Al 18
Fe 14
Ni 19
Cu 41
Ag 28
Au 18
where
¯
h is called the specific action whose value depends on the cathode material but is
approximately independent on current density, wire cross-section, or other discharge quantities
(Table 10.1).
The minimum action can be considered the quantum of the explosive process, and therefore
they introduced the term ecton (derived from explosion center, with the ending -ton in analogy
with other particles or quasi-particles like photon, proton, exciton, etc.). To illustrate the
concept: under certain conditions, each ecton is associated with the emission of about 10
11
electrons within a time span of about 10 ns. The explosive or ecton phase is generally much
shorter than the overall cycle or lifetime of an emission center.
The existence of a minimum action for spot operation can also be put in relation to the
long-known fact that the arc current cannot be arbitrarily small. Depending on the cathode
material and surface conditions, and the kind of background gas, if any, the arc discharge tends
to chop and extinguish spontaneously. The minimum current where this occurs is called the
chopping current [18, 19]. One wants the chopping current to be small (e.g. a few amperes)
because the abrupt termination can generate high voltages by induction, which in fact may
exceed the safe operational specifications and can cause harm to the operator or equipment.
10.2.1.4 Stages of an Emission Center
At this point it is clear that electron emission in cathodic arcs is a non-stationary process. The
terms emission center and cathode spot were introduced, although their relationship requires
clarification, which will be given when discussing the fractal properties of cathode spot.
One may think of an emission center as a location on the cathode surface where one can
describe the evolution of electron and plasma generation. The evolution may be divided into
four stages: (1) the pre-explosion stage; (2) the explosive emission stage; (3) the immediate
postexplosion stage, where cool-down has started but electron emission and evaporation are
still large; and (4) the final cool-down stage. Each of the four stages is highly dynamic. The
Unfiltered and Filtered Cathodic Arc Deposition 475
ignition of an emission center is a very important element to understanding cathodic arcs. In
this context, ‘ignition’ is not the initial triggering of the arc discharge, but the arc’s perpetual
and repetitive mechanism to ‘stay alive.’
(1) The pre-explosion stage
Each location on the surface might be a candidate for the ignition of an emission center but in
reality each location has its own history and properties such as the local work function,
geometric features, surface coverage by oxide or water, etc. Some locations can be presumed
to be more favorable to experience the ignition of an emission center. The conditions will be
such that the local energy input will be higher than at neighboring locations owing to the
specific local properties and their relation to the plasma and sheath conditions produced by
predecessor emission sites. The conditions are favorable if the local work function is low and
the field is enhanced owing to the presence of dielectric contamination and/or the presence of
microprotrusions or nanoprotrusions. If such favorable conditions are coupled to a very high
electric field strength (i.e. thin sheath due to high plasma density), and a high intensity of ion
bombardment, the local energy input can lead to electron emission with thermal runaway,
bringing the location to stage (2).
(2) The explosive emission stage
This stage is at the heart of the ecton model developed by Mesyats and co-workers [13, 20]
mentioned above; the key feature is the thermal runaway in which electron emission and Joule
heating close an amplifying feedback loop. The microexplosion causes destruction (erosion)
of a microvolume, which is later evident as a microscopic crater on the cathode surface.
High-resolution, fast optical diagnostics indicate that the sequence of explosive events is rapid.
For example, very fast optical imaging of very low current arcs, with 3–12 A, shows bursts of
light every 50–70 ns, with the most intense phases having a duration of 10–20 ns [21].
(3) The immediate postexplosion stage
As long as the freshly formed crater is very hot, most likely still having a surface layer of
molten cathode material, and the nearby plasma is very dense, electron emission remains
strong by field-enhanced thermionic emission. The latter is thermionic emission in which the
work function barrier φ is reduced (Schottky reduction) but does not reach the level of
non-linear combination of thermionic emission and field emission (thermo-field emission).
Besides electron emission, evaporation occurs from the pool of liquid material [22]. The
neutral vapor represents a medium of low conductivity choking the flow of current. This
blockage or choking greatly contributes to the apparent spot motion, as explained later.
Yet another effect is the plasma pressure acting on the liquid, which yields and thereby gives
rise to the generation of microscopic droplets, generally known as macroparticles.
476 Chapter 10
(4) The final cool-down stage
In the final stage, thermal conduction increases the hot area but the temperature decreases, and
with it the electron emission ceases exponentially; see the temperature dependence of the
Richardson equation (10.6). The explosively formed plasma has expanded, its density is
lowered, therefore the cathode sheath thickness has increased, and the electric surface field is
also reduced. Yet, this stage can be important to the overall cathode erosion since the hot
surface may still deliver metal vapor, especially when the cathode material is of high vapor
pressure.
This qualitative description of the stages of an emission center does not really address why the
intense emission of this site cannot be maintained. We need to consider: What are the driving
forces for the inevitable dynamics of the emitting system?
There are several factors, as follows. First, and this is perhaps most easily to understand, heat
conduction leads to an increase in the area of the emission site, and this means that the
dissipated power is spread over a larger area, hence the areal power density decreases with
time.
Second, owing to the increase in resistance with temperature for all metals, dρ/dT > 0, the
region of the cathode bulk directly under the cathode spot is more resistive than all other areas
or parts of the cathode. Hence, if there was an alternative, less resistive way for the current to
flow, the current would switch to the new path.
Third, and most importantly, and somewhat ironically, the emission center itself builds a
highly resistive barrier against current flow. In the explosive stage, the cathode matter
transitions from the solid to the plasma phase, and it may initially bypass the gas phase by
circumnavigating the critical point in the phase diagram [23]. At this stage, the most resistive
zone for the current path is the non-ideal plasma phase. However, as time passes (and here we
consider some tens of nanoseconds!), the reduced areal power density leads to a change of the
path in the phase diagram: the material has now the time to transition through all conventional
phases: solid–liquid–gas–plasma. From those four phases, the gas phase is by far the most
resistive phase. Solid and liquid metal, and metal plasma are good conductors, but metal gas
(vapor) is not! The metal vapor chokes the flow of electricity. As the growth of the emission
area continues, the power density and related local surface temperature are reduced, electron
emission decreases rapidly, though there may be still significant evaporation from the hot
crater left by the explosion. At this point, the composition of the gas or plasma in front of the
site becomes increasingly influenced by neutral vapor, and indeed the current transfer
capability suffers greatly.
Fourth, in this situation, competition kicks in! The dense plasma near the emission site may
have started a microexplosion at a new site. Now, the new site and the older, much larger site
are electrically in parallel, and the path of lower resistance (i.e. where less metal vapor is
Unfiltered and Filtered Cathodic Arc Deposition 477
hindering) takes over the current, which accelerates the death of the original emission site
because less power is dissipated there. The low-conducting vapor mechanism took its toll.
Macroscopically, we see sequences of ignition, evolution, and death of emission sites, and we
interpret this as cathode spot motion: it is a virtual motion based on the life cycle of emission
sites.
10.2.1.5 Random Motion
In the absence of an external magnetic field, the probability of ignition of a new emission site
is isotropically distributed, i.e. a new emission site could ignite with equal probability in any
direction from the existing emission center. This makes the apparent motion of spots random,
and indeed a good approximation of the actual motion is given by the random walk model
[24, 25].
In this model, one assumes that each displacement is a small elementary step of step length s,
which takes an average elementary time τ. Considering a two-dimensional random walk (i.e.
a walk on a surface), the probability P(R) for a total displacement to be in the interval
(R, R+dR), with R = |R|, as measured from the starting point, is given by [26]
P
(
R
)
dR =
R
2Dt
exp
−
R
2
4Dt
dR for t>>τ (10.10)
The function P(R) is known as the Raleigh distribution. The diffusion constant, D, contains the
parameters of the elementary step:
D =
1
4
s
2
τ
(10.11)
The diffusion constant is material dependent; for example J
¨
uttner measured
D = (2.3 ±0.6) ×10
−3
m
2
/s for copper [27] and D ≈10
−3
m
2
/s for molybdenum [28], and
Beilis et al. [29] determined (1 ±0.3) ×10
−3
m
2
/s for copper and (4 ±1) ×10
−4
m
2
/s for
CuCr contact material.
The mean value for displacement is
R
=
∞
0
RP
(
R
)
dR =
(
πDt
)
1/2
=
s
2
π
t
τ
(10.12)
and the observable, apparent spot velocity follows as
v
spot
=
dR
dt
=
1
2
πD
t
=
s
4
π
tτ
(10.13)
478 Chapter 10
According to (10.13), and in contrast to steered motion which is dealt with in the next section,
the apparent velocity of random motion decreases as the observation time increases.
This approach can be refined by modifying the probability distribution. For example, one may
assume that the walker (i.e. the emission center) avoids locations for some time where ‘he’ has
been before, and/or previously visited places may become preferred places after some time.
Both features have been observed experimentally and can be explained by the time
dependence of surface conditions and emission properties [30].
The close relation of random walk and Brownian motion is one of the many arguments why
cathode spots have fractal properties [31]. The well-known Brownian motion is the scaling
limit of random walk. Random walk is a discrete fractal exhibiting stochastic self-similarity on
large scales, but self-similarity is cut off as scales approach the elementary step width s.
Brownian motion in two dimensions is a true fractal showing self-similarity on all scales [32].
Brownian motion has the fractal (Hausdorff) dimension 2.
10.2.1.6 Steered Motion
Anything that breaks the symmetry will lead to deviations from the random walk. For
example, the cathode could have a scratch, or an area of surface contamination, or an interface
to a different material, or an external magnetic field could be present. In all of these cases, spot
ignition may be preferred at certain locations, generally where the electric surface field is
enhanced. Of special interest is the use of an external magnetic field because plasma
production is not influence by any foreign material yet the most likely direction of the apparent
spot motion can be predicted and influenced. The mechanism of how the symmetry is broken
has stirred a lot of thinking and speculation, leading to a great proliferation of ideas and
models [3].
The most convincing approach to date is the J
¨
uttner–Kleberg model [30, 33] which is based on
high-resolution observations of the emission of the microscopic plasma jets from active
emission sites. The plasma density is higher in the direction of such jets, and therefore the
sheath thickness is thinner, which leads to a higher electric surface field and enhances the
probability of high electron emission and thermal runaway, i.e. the ignition of a new emission
site (Figure 10.3).
The ejection of macroscopic plasma jets is not isotropic when a transverse (parallel to the
cathode surface) magnetic component is present. This leads to a preferred direction in which
new emission sites are ignited. The spot appears to be magnetically steered. The plasma
column is bent in the Amperian j ×B direction, though the virtual motion is to the opposite
(the anti-Amperian or retrograde) direction.
As experimentally shown, the emission of the microscopic plasma jets is not exactly in the
retrograde direction but rather it is statistically distributed with a high likelihood to be ±45
◦
Unfiltered and Filtered Cathodic Arc Deposition 479
Figure 10.3: Ejection of two jets into approximate retrograde direction. The intensity of light
emission of the plasma jets is weaker than the emission from the spot itself, and therefore the
exposure was selected such as to make the jets visible, while the cathode spot is overexposed.
(Adapted from [33].)
with respect to the retrograde direction. Therefore, the retrograde motion is actually composed
of little microscopic zig-zag ignitions of new emission sites.
The average velocity of the steered motion increases with magnetic field in a roughly
proportional manner,
v
steered
= cB
t
(10.14)
where c is a material constant that also depends on the surface conditions, and B
t
is the field
parallel to the cathode surface. The value of c can vary greatly; for conditions where type 1
spots are observed (surfaces with non-metallic layers, non-conditioned surfaces), c can reach
1000 m/(s T) [34], whereas c-values of about 60 m/(s T) and 200 m/(s T) were determined for
type 2 (clean) copper and CuCr cathodes, respectively [35]. The value can be even smaller
when the cathode is very hot [36].
When the magnetic field vector has a transverse and normal component, i.e. it is tilted with
respect to the surface normal, the direction of the virtual motion is also tilted to the retrograde
direction (Figure 10.4). The angle between retrograde direction and actual direction of
apparent motion is often called the Robson angle [35, 37, 38].
Steered spot motion is often used to control arc cathode erosion in practical arc plasma
sources. It will be further discussed in Section 10.3.
480 Chapter 10
Figure 10.4: Illustration of the direction of the apparent motion when the magnetic field vector is
tilted relatively to the cathode normal.
10.2.1.7 Fractal Nature of Cathode Processes
Fractals are mathematical, or physical/chemical/biological objects of the real world that show
self-similarity [39]. That means they are invariant to scaling, i.e. to multiplicative changes of
scale. Self-similarity may be discrete or continuous, deterministic or probabilistic.
Power laws
f (x) = cx
α
(10.15)
are an abundant source of self-similarity [32], where c and α are constants, because
multiplication with a constant preserves that f (x) is proportional to x
α
. Therefore, one fruitful
approach to fractal modeling is to look for power laws describing the physical phenomena.
There are several phenomena in cathodic arcs showing fractal properties, among them the
random walk of emission centers [24, 25], the 1/f noise of ion current [40] and arc voltage
[41], the fractal dimensions of about 2 for arc traces left by the spot’s random walk [42], and
the self-similarity in the patterns of emitted light [43, 44].