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Analytical model of a longitudinal hollow cathode discharge
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2010 J. Phys. D: Appl. Phys. 43 465204
(http://iopscience.iop.org/0022-3727/43/46/465204)
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IOP PUBLISHING JOURNAL OF PHYSICS D: APPLIED PHYSICS
J. Phys. D: Appl. Phys. 43 (2010) 465204 (11pp) doi:10.1088/0022-3727/43/46/465204
Analytical model of a longitudinal hollowcathode dischargeG J M Hagelaar1,2, D B Mihailova3 and J van Dijk3
1 Universite de Toulouse; UPS, INP; LAPLACE (Laboratoire Plasma et Conversion d’Energie);118 route de Narbonne, F-31062 Toulouse Cedex 9, France2 CNRS, LAPLACE, F-31062 Toulouse, France3 Department of Applied Physics, Eindhoven Universite of Technology, PO Box 513, 5600 MBEindhoven, The Netherlands
E-mail: [email protected]
Received 22 June 2010, in final form 15 September 2010Published 4 November 2010Online at stacks.iop.org/JPhysD/43/465204
AbstractThis paper presents a simple analytical model of a longitudinal hollow cathode discharge usedin metal vapour lasers. The model describes the principle relations between the voltage,current, plasma density and axial structure of the discharge. Contrary to standard dcdischarges, this discharge does not require electron multiplication in the cathode fall (CF) toproduce ions, but rather to satisfy the electron energy balance. A self-sustainment condition isobtained from the energy balance per electron–ion pair. From this, it follows that there is amaximum voltage at which the CF thickness tends to zero and the current density tendsasymptotically to infinity. The discharge develops axial non-uniformity and an axial electricfield in order to evacuate the created electrons to the anode, such that the characteristic time fortransport losses is the same for electrons as for ions. The axial profiles of the current density,plasma density and potential are obtained from the electron continuity equation. It is shownthat additional energy absorption from the axial field, similar to electron heating in dc positivecolumns, modifies the self-sustainment condition and thus leads to a shift in thevoltage–current characteristic, depending on the cathode length.
(Some figures in this article are in colour only in the electronic version)
1. Introduction
Hollow cathode discharges (HCDs) are widely used inapplications in different fields: laser technology, atomicspectroscopy, UV generators, vacuum microelectronics,materials processing, etc. Although these discharges feature alarge variety of configurations, they are generally characterizedby the so-called hollow cathode effect: an exceptionally highdischarge current, compared with conventional dc dischargesat the same voltage, due to the cathode surface surrounding(a large part of) the plasma. The hollow cathode effect isaccompanied by exceptionally high plasma density, intensivelight emission and cathode sputtering, properties of greatinterest for the applications.
The general physical principles of HCDs have beenstudied for many decades and are discussed in many paperson the basis of experiments, analytical models and numericalsimulations [1–14]. Several phenomena are considered
responsible for the hollow cathode effect, in particular theelectrostatic trapping of fast electrons in an oscillating motioninside the cathode, known as the pendulum effect [4, 6], andthe enhanced secondary electron emission by UV photons [1]and ions [9] created in the negative glow plasma inside thecathode. A number of papers are devoted to the role of metalvapour atoms due to cathode sputtering [2, 10]. However, theimportance of each of these phenomena depends on the HCDgeometry [9].
In this paper we present a simple analytical model ofHCDs that are used in metal vapour lasers, both as a metalvapour source by cathode sputtering and as an active mediumto excite the laser transition of the metal atoms or ions [15].These HCDs have an elongated cylindrical cathode geometry,coinciding with the laser cavity. Different configurations areused for the anode. In the so-called transversal configuration,the anode is positioned beyond a narrow slit in the cathodecylinder all along its length. This paper, however, focuses on
0022-3727/10/465204+11$30.00 1 © 2010 IOP Publishing Ltd Printed in the UK & the USA
J. Phys. D: Appl. Phys. 43 (2010) 465204 G J M Hagelaar et al
Figure 1. Geometry of the longitudinal HCD: (a) schematicoverview of the discharge tube containing a cathode cylinder (C)and two anode rings (A); (b) definition of the coordinate axes usedin this paper.
the longitudinal configuration, where the anode is a ring at theend of the cathode cylinder and the discharge involves axialelectron transport.
The longitudinal configuration has a more stable dischargeoperation but leads to axial non-uniformity of the discharge.Experimentally, it was observed [11, 16–18] that (1) the currentdensity and optical emission decrease along the cathodecylinder axis as a function of distance from the anode;(2) the voltage–current characteristic is very flat but shiftedin voltage when the cathode length is changed; (3) thesefeatures are rather independent of pressure, gas compositionand metal vapour. In several previous publications [16–18],we reproduced these experimental findings by comprehensivetwo-dimensional numerical simulations. The analytical modelpresented here aims at interpreting our previous results. Werevisit and combine some elementary theories and extend themto account for the axial non-uniformity of the longitudinalHCD. In view of our earlier findings, we focus on themain discharge properties such as current density, plasmadensity and potential distribution, rather than plasma chemistryand metal vapour dynamics. The next section gives amore detailed introduction to the issues addressed in thispaper.
2. Discharge configuration and physical principlesconsidered in this paper
The geometrical configuration considered in this paper isshown schematically in figure 1. It consists of a coppercylindrical hollow cathode with an inner radius of a fewmillimetres and a length of a few centimetres, bounded oneither side by a thin dielectric ring and an anode ring of the sameinner radius. The geometry is symmetric around the centre ofthe cathode (both axially and azimuthally). A dc voltage of afew hundred volts is applied between the cathode and anodesto sustain the discharge. The discharge gas is typically heliumwith a small admixture of argon at an intermediate gas pressureof a few kilopascal, with argon ions dominating the discharge.
Table 1. Parameters of the HCD configuration considered in thispaper. The exact definition of the parameters is given further on inthe text. The values in the lower part of the table are estimates basedon our previous numerical simulations.
Parameter Symbol Value
Cathode radius R 2 mmCathode length 2L 2–8 cmApplied voltage V 350–500 VDischarge current I 0.2–2 AGas composition 95% He–5% ArGas density 1.666 × 1023 m−3
Gas temperature 1000 KElectron temperature (plasma) Te 3.67 eVElectron mobility µe 11.4 m2 s−1
Ion mobility (plasma) µi 0.29 m2 V−1 s−1
Ion mobility coefficient (CF) β 15 m3/2 V−1/2 s−1
Townsend coefficient α 1.3 × 103 m−1
Secondary emission coefficient γ 0.1Energy per electron–ion pair W 50 eV
Some typical parameter values are given in table 1; these valueswill be used to evaluate the analytical expressions derived inthis paper.
The discharge operation is illustrated in figure 2 by theresults of our previous two-dimensional numerical simulations[16–18]. These simulations are based on the self-consistentsolution of continuity and drift–diffusion equations fordifferent plasma particle species, an electron energy equationand Poisson’s equation. The particle species taken intoaccount are electrons and different ionic and excited neutralspecies of helium, argon and copper. Particle source termsdue to ionization, excitation and other plasma chemistryare calculated using rate coefficients as a function of theelectron mean energy. Wall recombination, secondary electronemission and copper sputtering are accounted for by wall-fluxboundary conditions.
Figure 2 shows that, owing to the presence of a plasma,the anode potential propagates along the cylinder axis insidethe cathode, such that most of the applied voltage falls in theradial direction across a thin space-charge sheath in front ofthe cathode, called cathode fall region or simply cathode fall(CF) in this paper. Outside the CF, the potential is close to theanode potential, with small radial variations of the order of theelectron temperature to ensure quasi-neutrality of the plasma,and a relatively small axial gradient to ensure continuity of theplasma current.
The electron and ion transport in the CF are mainly radialand control the current and plasma density. The plasma issustained by volume ionization (figure 2(e)) using the energyabsorbed by electron acceleration in the CF (figure 2(f )),similarly to the plasma in the negative glow region ofconventional dc discharges, and is therefore regarded entirelyas a negative glow plasma. What distinguishes this dischargefrom conventional dc discharges is that the ions created inthe plasma are all transported to the cathode and make animportant contribution to the secondary electron emission, sothat less electron multiplication in the CF is needed to sustainthe discharge. As a result, the CF becomes thinner, the ion
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J. Phys. D: Appl. Phys. 43 (2010) 465204 G J M Hagelaar et al
Figure 2. Spatial profiles of the main plasma parameters as obtained in our previous numerical simulations. The simulation domaincorresponds to figure 1(b). The cylinder axis is in the bottom of each plot, the cathode centre is on the right and the anode and cathodesurfaces are in the top as indicated in the first plot. The half cathode length L = 2.5 cm, the other discharge parameters are given in table 1.
density in the CF higher, and the discharge operates at highercurrent density. We will analyse this in the first part of thispaper, in sections 3 and 4.
The discharge develops non-uniformity along the axialdirection in order to generate an axial electron current thattransports all created electrons to the anode. This is analysedin the next part of this paper, in sections 5 and 6. We will showthat the non-uniformity involves an axial electric field fromwhich the plasma electrons absorb additional energy, leadingto a shift in the current–voltage characteristic as a function ofcathode length.
The potential profile inside the hollow cathode(figure 2(a)) traps the electrons in the radial direction, whichmakes it possible to sustain the discharge at low gas pressureswhere the electron mean free path exceeds the plasma size.The electrons accelerated in the CF then make multiple passesthrough the plasma at high velocity, each time reflected bythe CF on the other side. This phenomenon is known as thependulum effect and is generally considered essential for theHCD operation. The pendulum effect cannot be described byelectron fluid equations and was studied previously by electronparticle simulations [4, 7] and a nonlocal approximation of theelectron Boltzmann equation [6]. However, our previous simu-lations (figure 2) are based on electron fluid equations and thusneglect the pendulum effect. To get more insight in the role ofthe pendulum effect and the possible consequences of neglect-ing it (in our fluid simulations), we consider in this paper thefollowing two limit cases. First, we will derive the analyticalmodel (sections 3–6) by assuming that the electron mean freepath is much shorter than the plasma radius, so that the pendu-lum effect can be neglected, as in the fluid simulations. Then,in sections 7 and 8, we will revisit the equations derived in theprevious sections, assuming that the electron mean free pathis very much larger than the plasma radius. The reality can beexpected to be in between these two cases. We will show thatsome of the main principles of the HCD operation are similarfor the two cases.
3. Self-sustainment condition
It is well known that in dc discharges, the thickness of the CFadjusts to satisfy the self-sustainment condition: an electronemitted from the cathode by ion impact must cause, during itslife time, the creation of enough ions to ensure the emission ofa new electron. More precisely, if γ is the secondary emissioncoefficient characterizing the effective emission probability perion, then each emitted electron must create exactly 1/γ ions thatare transported to the cathode. In conventional dc discharges,most of the ions impacting the cathode are created in or near theCF. The emitted electrons must then multiply by a factor 1+1/γ
as they cross the CF region. Equating this factor to exp(αd),the Townsend expression for the electron multiplication factor,we obtain the well-known self-sustainment condition [19]
d ≈ 1
αln
(1 +
1
γ
), (1)
where d is the CF thickness and α is the Townsend coefficient,characterizing the mean ionization probability per unit traveledlength, which is relatively constant for the high electronenergies in the CF. In the steady state of conventional dcdischarges, the CF has developed such that its thicknessapproximately satisfies equation (1).
However, equation (1) does not hold for HCDs becausethe ions created in the plasma beyond the CF are nearly alltransported to the cathode and make an important contributionto the secondary electron emission, almost regardless of theirposition of creation. Let us analyse this for the HCD geometryof figure 1. The electron and ion transport across the CF isessentially radial. We assume that the CF thickness d �cathode inner radius R so that the curvature of the CF canbe neglected, i.e. the radial position coordinate r behaves asa Cartesian coordinate for R − d < r < R. For simplicity,we also assume that the coefficients α and γ are constant andthat the electrons have so many collisions that they quickly
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J. Phys. D: Appl. Phys. 43 (2010) 465204 G J M Hagelaar et al
lose their high energy after leaving the CF, so that they cannotpenetrate in the CF on the other side of the plasma (i.e. weneglect the pendulum effect).
According to the Townsend model [19], the electron flux�e in the CF grows exponentially as a function of distance fromthe cathode:
�e(r) = −γ�i(R) exp(α(R − r)); (2)
the ion flux �i follows from current conservation:
�i(r) = J
e
(1 − γ
1 + γexp(α(R − r))
), (3)
where J = e(�i − �e) is the current density which is constantacross the CF (for d � R). From equation (3) we can observethat if a significant ion flux enters the CF from the plasma,�i(R−d) > 0, then the CF must be thinner than in equation (1):
d <1
αln
(1 +
1
γ
). (4)
However, to find the value of the CF thickness d we need toknow how many ions are created in the plasma beyond theCF. This can be estimated from the electron energy balance,e.g. by balancing the average energy absorption and the energylosses associated with the creation and loss of a single electron–ion pair, as follows. We assume that the radial profile of thepotential � in the CF is parabolic so the electric field E islinear:
�(r) = −V (1 − (R − r)/d)2, (5)
E(r) = −∂�
∂r= −2V
d(1 − (r − R)/d), (6)
where V is the total voltage. Combining equations (2) and (6),the electric energy absorbed by electrons in the CF is then
HCF = − 1
�i(R)
∫ R
R–d
e�e(r)E(r) dr
= 2exp(αd) − 1 − αd
(αd)2γ eV, (7)
on average per ion impacting the cathode. Since eachelectron–ion pair created in the volume corresponds to an ionimpacting the cathode, HCF from equation (7) is directly theaverage absorbed energy per electron–ion pair. This absorbedenergy depends on the electron multiplication because alsothe new electrons created in the CF absorb energy, leadingto ‘multiplication’ of the absorbed energy. So the thicker theCF, the higher the absorbed energy per electron–ion pair. Thelower limit for the absorbed energy is γ eV, corresponding tothe case d = 0 when the emitted secondary electrons do notmultiply in the CF.
The electron energy losses can be convenientlyrepresented by the energy per electron–ion pair W , which is arather constant parameter, typically a few times the ionizationpotential, determined mainly by the discharge gas and weaklydependent on the electron energy (for the energy range ofinterest here). This parameter W is often used and given inthe literature for high-energy electron beams [20] but it canbe defined for gas discharges in general [21, 22]. For our
HCD, the total energy per electron–ion pair consists of differentcontributions due to collisions and transport [21]:
W = lost power
ionization rate�
∑j=collision εjχjkj∑i=ionization χiki
+ Wtr, (8)
where ε is the mean energy lost in a collision (constantfor inelastic collisions), k is the rate coefficient, χ is thefractional density of the target species and the sums run overall collision processes j and over the ionization processes i
only. The last term Wtr is the energy loss associated withelectron transport loss to the anode, which is effectively feltin the cathode region by thermal conduction. From thenumerical simulation results, the two terms of equation (8) canbe estimated to be approximately 37 and 13 eV, adding up toW ≈ 50 eV (see table 1). For simplicity we assume that W isconstant.
Equating the absorbed energy HCF and the lost energy W ,we find
2exp(αd) − 1 − αd
(αd)2γ eV = W. (9)
This energy balance can be seen as a self-sustainment conditionfor HCDs and controls the CF thickness d, meaning that d
adjusts such that the right amount of energy is absorbed tocreate the ions necessary to sustain the discharge by secondaryemission. The left-hand side of equation (9) is a monotonicallyrising function of αd that can be well approximated by Taylorexpansion (of its logarithm) so that d can be solved. This yields
d ≈ 6
α(√
1 − ln(V/Vmax) − 1) ≈ 3
α
(Vmax
V− 1
), (10)
where V cannot exceed the maximum voltage
Vmax = W
eγ. (11)
The first approximation in equation (10) is accurate to withina few per cent over the entire range of V . The second simplerapproximation is valid only in case the CF is so thin thatαd < 0.3 but in practice this is often the case; this expressionwas given previously in the context of HCDs [9]. The exactsolution of equation (9) and the approximate solutions (10) areshown in figure 3. The maximum voltage (11) corresponds tothe point where the energy balance is satisfied without electronmultiplication in the CF. Beyond this voltage the absorbedenergy always exceeds the energy loss, regardless of d, sothe energy balance cannot be satisfied. Substituting sometypical values γ = 0.1 and W = 50 eV from table 1, wefind a maximum voltage of 500 V.
4. Radial ion transport
Given the CF thickness, the ion density ni in the CF followsfrom Poisson’s equation:
eni = −ε0∂2�
∂r2= ε0
2V
d2, (12)
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J. Phys. D: Appl. Phys. 43 (2010) 465204 G J M Hagelaar et al
0.0 0.2 0.4 0.6 0.8 1.00
1
2
3
4
5
αd
V/Vmax
Figure 3. Relation between the CF voltage V and the CF thicknessd as given by the energy balance (9). Solid line: exact solution.Dotted line: first approximation in equation (10). Dashed line:second approximation in equation (10).
where we have used equation (5) and neglected the electroncharges and the curvature of the CF (as before). From this, thecurrent density can be estimated as
J = (1 + γ )e�i(R) � (1 + γ )eniui, (13)
where ui is the ion mean velocity in the centre of the CF fall.Note that the ion velocity can be expected to vary across theCF much more than the flux �i, which is not fully consistentwith our assumption of uniform ni and parabolic �(r), but theresulting errors are minimized if we take ui in the centre of theCF at position r = R − d/2. Assuming that ui is of the formβE1/2, which is a good approximation for collisional noble gasions in high field [19], the current density is then
J � (1 + γ )eniβ
√V
d= 2(1 + γ )ε0βV 3/2
d5/2. (14)
On the other hand, if the ion mean free path > d so that ui islimited by inertia rather than collisions, the current density is
J � (1 + γ )eni
√eV
2mi= 21/2(1 + γ )e1/2ε0V
3/2
m1/2i d2
, (15)
where mi is the ion mass. This latter situation can arise in gasmixtures where the dominant ions correspond to a minoritycomponent of the mixture so that they undergo a few chargetransfer collisions, e.g. Ar+ ions in He–Ar mixtures with asmall Ar fraction.
When substituting equation (10) in equation (14)/(15), asin figure 4, we see that the current density J increases rapidly asV approaches the maximum voltage Vmax and the CF thicknessapproaches zero. This is an important advantage of HCDs,essential for the hollow cathode effect: the current density canbecome much higher than in standard dc discharges becausethe self-sustainment condition allows a very thin CF. In fact,HCDs are usually operated at high current density close to themaximum voltage V ≈ Vmax to within 10% or so. This impliesthat αd < 0.3 and that most ionization happens in the plasma
0.75 0.80 0.85 0.90 0.95 1.00100
101
102
103
104
free fall
drift
J(A
/m2 )
V/Vmax
Figure 4. Current density at the cathode as a function of CF voltage,obtained by substitution of equation (10) in equation (14) (drift) andequation (15) (free fall), using the parameters in table 1 and the Ar+
ion mass. Only the lowest curve is physically valid, equation (14) inmost of the current density range.
rather than the CF, so that the ion flux is very nearly constantacross the CF:
�i(R − d) ≈ �i(R) = J
(1 + γ )e(16)
to within a few per cent as can be seen from equation (3). Inthe following we neglect that some ions are created in the CFand assume that the ion flux is totally created in the plasma;we will simply write �i for the ion flux anywhere in the CF.
Given the ion creation in the plasma, the plasma density n
is determined by the ion transport losses to the CF. Followingthe standard elementary theory for dc positive columns andmany other discharges [19, 22], we assume that the electronsin the plasma have a uniform temperature Te that controlsboth their pressure and ionization frequency νiz. The ions aretransported by the ambipolar electric field Eamb ≈ −Te∇n/n
resulting from the electron momentum balance. Neglectingaxial ion transport, the ion transport equation is
1
r
∂
∂r(rµiEambn(r)) = −µiTe
1
r
∂
∂r
(r∂n
∂r
)= νiz(Te)n(r),
(17)
where µi is the ion mobility at low electric field. The solutionis a Bessel profile:
n(r) = n0J0(krr) kr =√
νiz
µiTe. (18)
Neglecting the CF thickness (d � R) and the ion density inthe CF (ni � n0), we impose the boundary condition that theion density is zero at the cathode, which yields an implicitequation for the electron temperature:
νiz(Te)
Te= µi
(2.405
R
)2
. (19)
Since νiz(Te) is a very steep function, Te is determined quiteprecisely by this equation and not very sensitive to changes
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J. Phys. D: Appl. Phys. 43 (2010) 465204 G J M Hagelaar et al
in the gas density or radius R. The amplitude of the plasmadensity profile can be found by equating the total ionizationrate (integrated over the cross section of the plasma) to the ionflux in the CF (integrated over its circumference):
νiz2π
∫ R
0n(r)r dr = νizπR2n = 2πR�i, (20)
hence the radial-average plasma density is
n = 2
Rνiz�i = 2R
2.4052(1 + γ )eµiTeJ, (21)
i.e. the plasma density is proportional to the current density inthe CF. Equation (21) can be combined with equation (14)/(15)to find the plasma density from the CF voltage.
5. Axial electron transport
While the ions are lost by radial transport to the cathode, theelectrons are lost by transport in the axial direction towards theanode. This requires an axial electric field Ez in the plasma,corresponding to a decrease in the CF voltage V as a functionof the axial distance z from the anode, and leads to a decrease inthe plasma density n and the CF current density J as a functionof z. In this section we derive the axial profiles J (z), n(z),V (z), etc, from the equations for the axial electron transport.The axial coordinate z is defined in figure 1(b).
The axial electron flux results from both the electric driftand diffusion due to the density gradient. Integrated over thecross section of the plasma, the electron continuity equationwith drift–diffusion flux is
πR2 ∂
∂z
(−µeEzn − µeTe
∂n
∂z
)= 2πR(1 + γ )�i (22)
where µe is the electron mobility and the right-hand side isthe total rate of electron creation by secondary emission andvolume ionization. We assume that µe and Te are independentof z and substitute equations (21) and (16):
∂
∂z
(Ez
TeJ +
∂J
∂z
)= −(1 + γ )
(2.405
R
)2µi
µeJ, (23)
where we have chosen to work with the CF current density J ;however, this is directly proportional to the average plasmadensity n. To calculate the axial profiles, we could nowsubstitute Ez = −∂V/∂z and J (V ) from equation (14)/(15)and then try to solve for V (z), but this is cumbersome.Rather, we linearize equation (23) by assuming that Ez isuniform, in agreement with the numerical simulation resultsin figure 2. As a first boundary condition we assume thatJ = 0 at the cathode–anode interface at z = 0. This isconsistent with the fact that the plasma density drops to zerowithin a small distance ∼R inside the anode, as is shown bythe numerical results in figure 2. For the second boundarycondition, considering the axial symmetry of the geometry,we impose that the axial electron flux is zero at the positionz = L corresponding to the centre of the cathode:
Ez
TeJ (L) +
[∂J
∂z
]z=L
= 0, (24)
2 4 6 8 10
0.0
0.2
0.4
0.6
0.8
1.0
spac
eco
nsta
nt
k2/k0
k1/k0
k0L
Figure 5. Solution of equation (29). Solid line: exact solution.Dashed line: approximate solution (30).
where the two terms come from drift and diffusion. (Note that,taking Ez constant, we cannot impose the Neumann boundarycondition ∂J/∂z = 0 because this would lead to nonphysicalinflow of electrons at z = L.)
The appropriate solution of equation (23) is
J (z) = J0 exp(−k1z) sin(k2z), (25)
where J0 is a normalization constant and k1 and k2 are spaceconstants given by
k1 = Ez
2Te, (26)
k2 =√
k20 − k2
1, (27)
k0 =√
1 + γ2.405
R
õi
µe. (28)
In these equations, the axial electric field Ez is not a freeinput parameter, but an eigenvalue parameter (similar to Te inequations (18) and (19)) to be determined from the boundarycondition (24). Together with (27), this boundary conditionyields the following implicit equations for k1 and k2:
sin(k2L)
k2L= 1
k0Lk1 = −k0 cos(k2L), (29)
whose solution is shown in figure 5. In a good approximation(obtained by Taylor expansion around k2L = π) equation (29)yields
k1 ≈ k0
1 − π2
8
(√4
k0L+ 1 − 1
)2 . (30)
This determines the axial electric field through equation (25):
Ez = 2Tek1, (31)
i.e. the field adjusts to satisfy the boundary conditions. Notethat the value of Ez selected by our boundary conditions isalso the smallest Ez value allowed by the general solution
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J. Phys. D: Appl. Phys. 43 (2010) 465204 G J M Hagelaar et al
0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
L (mm) =
analyticalmodel
25
15
10
7.5
rela
tive
curr
ent
den
sity
0 5 10 15 20 250.0
0.2
0.4
0.6
0.8
1.0
L (mm) =
numericalsimulation
5
25
15
10
7.5
rela
tive
curr
ent
dens
ity
0 5 10 15 20 25-30
-20
-10
0
10
L (mm) =
analyticalmodel
5
25
15
10
7.5
pote
ntia
l(V
)
z (mm)0 5 10 15 20 25
-30
-20
-10
0
10
L (mm) =
numericalsimulation
5
15
10
25
7.5
pote
ntia
l(V
)
z (mm)
Figure 6. Axial profiles of the CF current density and the on-axis potential from the analytical model equations (25), (30), (31) and from thenumerical simulations, for different half cathode lengths L, indicated in mm with the curves.
of equation (23), corresponding to the most uniform currentdensity J (z).
Equations (30) and (31) show that the axial electric fielddepends on the cathode length. The field is weaker asthe cathode is shorter, decreasing from 2Tek0 for very longcathodes to zero for k0L = π/2. For still shorter cathodes theelectric field is reversed:
L <π
2k0⇒ Ez < 0. (32)
Substituting the values from table 1, we find that k0 = 201 m−1
so the field reversion occurs for L < 7.8 mm, which agreesquite well with our previous numerical simulation results[17, 18] (note that in these references, the cathode lengthl corresponds to the whole cathode cavity, i.e. l = 2L).Figure 6 shows a comparison, for different cathode lengths,of the analytical potential −Ezz and the on-axis potentialin the numerical simulations. This figure also compares theanalytical current density profile (25) with the current densityat the cathode in the numerical simulations. Note that forvery short cathodes with strong field reversion (L = 5 mm)the assumption of constant Ez is no longer reasonable and theanalytical solution (25) is no longer appropriate.
6. Influence of axial electric field on self-sustainment
To normalize the current density profile J (z) and calculatethe total discharge current, we use equation (14)/(15) obtainedfrom Poisson’s equation, in combination with the CF thickness
d from the energy balance. In order to do this rigorously, weshould establish the energy balance integrated (or averaged)over the axial position z, since the energy is not necessarily lostat the same position where it has been absorbed, but nearly thesame results can be obtained in a much simpler way by applyingthe energy balance locally at the position z = zm where J ismaximum. This maximum takes place some distance insidethe cathode (for long enough cathodes with k0L > π/2) andis given by
zm = π
k2− L, (33)
Jm ≡ J (zm) = J0k2
k0exp(−k1zm), (34)
so the current density profile (25) can be written as
J (z) = Jmk0
k2exp(−k1(z − zm)) sin(k2z). (35)
We now estimate Jm by substitution of equation (10) inequation (14)/(15), as shown in figure 4. The CF voltageV at z = zm differs from the applied discharge voltage bythe anode sheath voltage and by the axial voltage drop in theplasma Ezzm < 2Te but in total this voltage difference is sosmall that it can be reasonably neglected. We simply use theapplied voltage for V in equation (10).
However, the energy balance from section 3 requires animportant modification in order to account for the axial electricfield Ez. The electrons can absorb energy from this field, inaddition to the energy HCF absorbed in the CF. Although the
7
J. Phys. D: Appl. Phys. 43 (2010) 465204 G J M Hagelaar et al
axial field is relatively weak, it can have a significant influencebecause it acts upon all electrons everywhere in the plasma,whereas the CF field acts only upon part of the electrons ina small part of the volume. Per electron–ion pair, the energyabsorbed from the axial electric field is
Hz = − πR2
2πR�i
(−µeEzn − µeTe
∂n
∂z
)Ez,
= (1 + γ )4eTe
(k1
k0
)2 (1 +
1
2k1J
∂J
∂z
). (36)
The last factor in brackets, accounting for electron diffusion, isnegligible near the point z = zm as well as on average over thecathode length. Including the additional absorbed energy in theenergy balance, HCF + Hz = W , we see that Hz compensatesfor some of the energy losses, thereby reducing the maximumvoltage in equation (10) and further. For the point z = zm wefind
Vmax = W − Hz
eγ= 1
eγ
(W − (1 + γ )4
(k1
k0
)2
eTe
)(37)
rather than equation (11). Upon substitution of the ratiok1/k0 from equation (30) and figure 5, it turns out thatthe maximum voltage now depends on the cathode length,decreasing monotonically by approximately 4Te/γ as thecathode length is increased from very short to infinity.
The total discharge current I is directly related to Jm as
I = 2πR
∫ L
0J (z) dz = 2πRL
exp(k1zm)
k0LJm. (38)
Calculating this for different voltages V , we find the voltage–current characteristic V (I), which is shown in figure 7for different cathode lengths and compared with numericalsimulation results. The V –I curves for different cathodelengths are shifted in voltage due to different energy absorptionHz from the axial field, following the trend of the maximumvoltage (37). The shifts between the analytical curves aresomewhat larger than between the numerical curves, which isconsistent with the axial potential profiles in figure 6, varyingmore in the analytical model than in the numerical simulations.Similar voltage shifts are observed in experimental V –I curves[17, 18].
It is interesting to note that most of the results in this andthe previous section actually do not depend directly on thecathode length L, but rather on the parameter
k0L = 2.405√
1 + γ
õi
µe
L
R, (39)
i.e. on the aspect ratio length/radius, so decreasing the radiushas the same effect as increasing the length. This was foundearlier in [18]. The appearance of k0L as a scaling parameter isdue to the fact that the characteristic time for transport lossesis the same for the ions (lost by radial transport) as for theelectrons (lost by axial transport), which is necessary to obtaina quasi-neutral plasma.
300
350
400
450
500
0.0 0.2 0.4 0.6 0.8 1.0 1.2
analytical model
L (mm) = 7.5
10
15
25
40
volta
ge
(V)
300
350
400
450
500
0.0 0.2 0.4 0.6 0.8 1.0 1.2
numericalsimulation
4025
1510
L (mm) = 7.5
current (A)
volta
ge(V
)
Figure 7. Voltage–current characteristics for different half cathodelengths L from the analytical model, combining equations (14),(30), (37), (38), compared with numerical simulations [18]. Exceptfor the cathode length, the input parameters for the analytical curvesare constant as given in table 1.
7. Influence of pendulum effect on self-sustainment
In the previous sections we assumed that the electrons haveso many collisions that they rapidly lose their energy in theplasma and pass through the CF only once. This assumption isconsistent with our previous fluid simulations, but its validityis doubtful. In fact, the electrons accelerated in the CF canhave such a long mean free path that they penetrate in the CFon the other side of the plasma, and this is generally consideredessential for the HCD operation. Therefore, in this section, weinvestigate the possible consequences of a long electron meanfree path for the above analytical model. We consider theextreme limit case that the electron mean free path � cathoderadius. Then, the electrons coming from the CF oscillate inthe potential well formed by the CF surrounding the plasma,passing through the plasma at high velocity many times beforeloosing their energy, each time penetrating into the CF on theother side and reflected by it. We will call these electronsfast electrons; their oscillating motion is the pendulum effect.As in section 3, we describe the electron collisions by aconstant Townsend coefficient α (ionization probability perunit traveled length) and constant energy loss per ionization W .We assume that the fast electrons are only forward scatteredin the collisions, so that their velocities remain in the radialdirection, and that any new electrons are created with zeroinitial energy.
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J. Phys. D: Appl. Phys. 43 (2010) 465204 G J M Hagelaar et al
Let us first analyse how the pendulum effect modifies theself-sustainment condition (9) derived in section 3. Considera secondary electron emitted from the cathode, becoming afast electron. During a single pass through the plasma (fromthe CF to the CF on the other side) the electron is expectedto produce ∼2αR ionizations and lose an energy ∼2αRW ,where we neglect the CF thickness d � R. As the electronloses energy, it can penetrate less deeply into the CF. Afterm passes through the plasma the electron has lost an energy2mαRW and penetrates into the CF up to a position rm given by
e�(rm) = −eV (1 − (R − rm)/d)2 = −(eV − 2mαRW),
(40)
where we have used the potential profile from equation (5).Hence the number of times that the electron passes at a givenposition r in the CF is
2m(r) = eV
αRW(1 − (1 − (R − r)/d)2), (41)
where the factor 2 takes into account that the electron passestwice through the CF (forth and back) for each pass through theplasma. The expected number of ionizations that this electronproduces inside the CF is∫ R
R–d
2m(r)α dr = 2d
3R
eV
W. (42)
The resulting new electrons are accelerated through part of theCF voltage and also become fast electrons. The total energyabsorbed from the CF by the secondary electron plus the newfast electrons is
U = eV −∫ R
R–d
2m(r)e�(r)α dr = eV
(1 +
2d
15R
eV
W
).
(43)
Note that the new fast electrons create other fast electrons,which then also absorb energy, etc, but this additional energyis proportional to higher powers of d/R and can be neglectedfor the case d/R � 1 considered here. Dividing the absorbedenergy U by the energy per ionization W , we find the totalnumber of ionizations M in both the CF and the plasma, dueto a single secondary electron emitted from the cathode:
M = U
W= eV
W
(1 +
2d
15R
eV
W
). (44)
A similar expression was given by [9] as an upper limit ofthe multiplication factor (except that 2/15 is replaced by 1/4).This number is much larger than the number of fast electronscreated in the CF from equation (42), so most electrons areelectrons created outside the CF in the plasma, where theyremain at relatively low energy and form the bulk of theplasma. However, in our longitudinal HCD geometry thesebulk electrons can absorb energy from the axial electric fieldand cause additional ionization. Therefore, in order to find theself-sustainment condition and CF thickness, we write againthe electron energy balance per electron–ion pair, includingthe energy Hz absorbed from the axial field:
HCF + Hz = γU + Hz = W (45)
which yields
d = R15W
2eV
(Vmax
V− 1
)Vmax = W − Hz
eγ. (46)
This agrees with equation (10) in section 3, if we replace
α → 2eV
5WR. (47)
Substituting the values from table 1, we see that α = 1333 m−1
is to be replaced by a somewhat larger value ≈2000 m−1. Thissuggests that under these conditions, the pendulum effect isimportant but not completely dominant. It also suggests thatour previous fluid simulations somewhat underestimate theelectron multiplication.
Note that according to some previous papers [4, 6], theessence of the pendulum effect is that it allows the fast electronsto create new electrons inside the CF, which also become fastelectrons, and so on, leading to an exponential increase offast electrons and absorbed energy (per secondary electron).However, this mechanism is not specific for the pendulumeffect but is also present if the electrons pass through the CFonly once. This is the reason why equation (46) has sameform as equation (10) in section 3. The pendulum effect justallows the electrons to ionize the gas at arbitrarily low pressure.The discharge then becomes independent of the Townsendcoefficient α, according to equation (47).
8. Influence of pendulum effect on plasma properties
Finally, let us briefly discuss the consequences of the pendulumeffect for the plasma properties. We need to take into accountthat there are two groups of electrons with different kineticbehaviour. The fast electrons are responsible for most ofthe ionization but do almost not contribute to the plasmadensity and ambipolar field, which are determined mainly bythe bulk electrons. Assuming that the bulk electrons can becharacterized by a uniform temperature Te and neglecting thefast-electron density, the ion transport equation is
− µiTe1
r
∂
∂r
(r∂n
∂r
)= Sfast(r) + νiz(Te)n(r), (48)
where Sfast is the ionization rate due to fast electrons and n, Te
and νiz represent only the bulk electrons. The solution of thisequation is no Bessel profile as in section 4 because the fast-electron ionization Sfast(r) is not proportional to bulk-electrondensity n(r). In fact, in view of the essentially radial motion ofthe fast electrons, Sfast(r) can be expected to be peaked aroundthe axis due to focusing of the electron trajectories. We will notcalculate the radial profiles; we simply integrate equation (48)over the cross section of the plasma:
2πR�i = −2πRµiTe
[∂n
∂r
]r=R
= 2π
∫ R
0Sfast(r)r dr + πR2νizn. (49)
The density gradient at the boundary can be written as[∂n
∂r
]r=R
= Cn
R, (50)
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J. Phys. D: Appl. Phys. 43 (2010) 465204 G J M Hagelaar et al
where C is a numerical constant which is approximately 3–4for any kind of physically reasonable profile n(r); for a Besselprofile C = 2.4052/2 = 2.892. Hence, the first two membersof equation (49) yield
n = R�i
CµiTe= R
C(1 + γ )eµiTeJ, (51)
which remains close to equation (21) in section 4 even if n(r)
is not exactly a Bessel profile. Furthermore, the integral of Sfast
must be consistent with the number of fast-electron ionizationscalculated in equation (44):
2π
∫ R
0Sfast(r)r dr = 2πRγ�iM = 2πR�i
(1 − Hz
W
),
(52)
where we have also used equation (45). Substituting equations(51) and (52) in equation (49), we obtain an equation for thebulk-electron temperature:
νiz(Te)
Te= 2Cµi
R2
Hz
W. (53)
Comparing this with equation (19) in section 4, we see thatthe right-hand side is smaller by approximately a factor Hz/W ,leading to a lower electron temperature, and consequently to ahigher plasma density from equation (21)/(51). Expressingthe absorbed energy Hz in terms of the axial electricfield Ez, as in equation (36), equation (53) can also bewritten as
νiz(Te)W ≈ eµeE2z . (54)
This clearly shows that the bulk-electron temperature isdirectly controlled by the local axial field, exactly as in adc positive column, and contrary to the electron temperaturein section 4 without the pendulum effect. What is more, alower electron temperature can be expected to reduce the axialfield and the absorbed energy. According to section 5, bothEz and Hz are proportional to Te so that a decrease in Te
causes a proportional decrease in Hz, which reduces Te evenfurther, and so on. Substituting equation (36) in equation (53),we get
νiz(Te)
T 2e
� 8C(1 + γ )
(k1
k0
)2µi
WR2. (55)
The Te from this equation can indeed be much lower than thatfrom equation (19). However, this result depends strongly onour assumption that all bulk electrons are created with zeroinitial energy. A more quantitative estimation of the bulk-electron temperature would involve treating the energy transferfrom the fast electrons to the bulk electrons.
9. Conclusions
The analytical model presented in this paper gives thefollowing insights in the operation of longitudinal HCDs:
• Since the ions created in the plasma are all collectedby the cathode and cause secondary electron emission,electron multiplication in the CF is not necessary to sustain
the discharge, but rather to satisfy the energy balanceper electron–ion pair: the absorbed energy increases asa function of the electron multiplication in the CF. TheCF thickness adjusts such that the right amount of energyis absorbed to create the ions necessary to sustain thedischarge by secondary emission.
• There is a maximum CF voltage beyond which theabsorbed energy always exceeds the lost energy so thatthe energy balance cannot be satisfied. When increasingthe voltage up to this maximum voltage, the CF thicknessdecreases down to zero, in order to minimize the absorbedenergy. According to Poisson’s equation, the ion densityand current density in the CF then increase to infinity(or at least to such high values that the present modelis no longer reasonable). The typical operation of thelongitudinal HCD is close to this maximum voltage, witha thin CF where almost no electron multiplication takesplace.
• The longitudinal HCD develops axial non-uniformity andan axial electric field in order to evacuate the createdelectrons to the anode, such that the characteristic timefor transport losses is the same for electrons as for ions.The axial electric field necessary to achieve this is strongeras the cathode is longer, or more precisely, as the ratiocathode length/cathode radius is larger. For very shortcathodes the axial electric field is reversed to slow downelectron diffusion losses to the anode.
• Additional energy absorption from the axial electricfield in the plasma bulk, similar to electron heating indc positive columns, modifies the energy balance perelectron–ion pair and hence modifies the CF thickness.This leads to a shift in the maximum voltage dependingon cathode length, resulting in a voltage shift of thevoltage–current characteristic. This is also observed inexperiments and in our previous numerical simulations.
• In principle, the above mechanisms are not directlyaffected by or dependent on the pendulum effect. Thependulum effect just allows the ionization to take placefor arbitrarily long electron mean free paths, imposingan effective lower limit 2 eV/5WR on the Townsendcoefficient α. However, the pendulum effect can alsobe expected to reduce the temperature of the electrons inthe plasma bulk, thereby reducing both the axial and theradial electric field in the plasma, slowing down the radialion transport and consequently increasing the plasmadensity for a given current. Our previous numerical fluidsimulations could be unrealistic on these points. We notenevertheless that our simulation results were found in goodoverall agreement with experimental data. More work isneed to clarify this issue, e.g. using electron Monte Carlosimulation, as in [11, 12].
Acknowledgments
The authors thank L C Pitchford and J P Boeuf of theLAPLACE Toulouse, J J A M van der Mullen of the TUEEindhoven and M Grozeva of the ISSP Sofia for usefuldiscussions and remarks. This work was done as part of
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J. Phys. D: Appl. Phys. 43 (2010) 465204 G J M Hagelaar et al
‘Physical chemistry of plasma–surface interaction’ (PSI) in theframework of the Belgian federal programme ‘InteruniversityAttraction Poles’ (IAP). The authors acknowledge the supportof the Dutch National Programme BSIK, in the ICT projectBRICKS, theme MSV1 and the support of the National ScienceFund of Bulgaria, project DO 02-274/2008.
References
[1] Little P F and Engel A v 1954 The hollow cathode effect andthe theory of glow discharges Proc. Phys. Soc. Lond. A224 209
[2] Warner B E, Persson K B and Collins G J 1979 J. Appl. Phys.50 5694
[3] van Veldhuizen E M and de Hoog F J 1984 J. Phys. D: Appl.Phys. 17 953–68
[4] Boeuf J P 1990 Self-consistent models of DC and transientglow discharges Physics and Applications of Pseudosparks(NATO ASI Series B vol 219) ed M A Gundersen andG Schaefer (New York: Plenum) pp 255–75
[5] Schoenbach K H, Vahala L L, Gerdin G A, Homayoun N andLoke F 1990 The effect of pendel electrons on breakdownand sustainment of a hollow cathode discharge Physics andApplications of Pseudosparks (NATO ASI Series Bvol 219) ed M A Gundersen and G Schaefer (New York:Plenum) pp 293–301
[6] Kolobov V I and Tsendin L D 1995 Plasma Sources Sci.Technol. 4 551
[7] Donko Z, Rozsa K and Tobin R C 1996 Monte-Carlo analysisof the electrons’ motion in a segmented hollow cathodedischarge J. Phys. D: Appl. Phys. 29 105
[8] Arslanbekov R R, Tobin R C and Kudryavtsev A A1997 Self-consistent model of high currentdensity segmented hollow cathode discharges J. Appl. Phys.81 554
[9] Arslanbekov R R, Kudryatsev A A and Tobin R C 1998 On thehollow-cathode effect: conventional and modified geometryPlasma Sources Sci. Technol. 7 310
[10] Lichtenberg A J and Lieberman M A 2000 Modelling ametal-vapor buffer-gas hollow cathode discharge J. Appl.Phys. 87 7191
[11] Bogearts A and Grozeva M 2002 Appl. Phys. B: Lasers Opt.75 731–8
[12] Bogearts A and Gijbels R 2002 J. Appl. Phys. 92 6408[13] Sigeneger F and Winkler R 2005 Self-consistent analysis of a
helium plasma in a cylindrical hollow cathode PlasmaChem. and Plasma Proc. 25 147
[14] Sigeneger F, Donko Z and Loffhagen D 2007 Boltzmannequation and particle-fluid hybrid modelling of a hollowcathode discharge Eur. Phys. J. Appl. Phys. 38 161
[15] Ivanov I G, Latush E L and Sem M F 1996 Metal Vapour IonLasers (Chichister: Wiley)
[16] Mihailova D, Grozeva M, Hagelaar G J M, van Dijk J,Brok W J M and van der Mullen J J A M 2008 J. Phys. D:Appl. Phys. 41 245202
[17] Mihailova D, van Dijk J, Grozeva M, Hagelaar G J M andvan der Mullen J J A M 2010 J. Phys. D: Appl. Phys.43 145203
[18] Mihailova D 2010 Sputtering hollow cathode dischargesdesigned for laser applications; experiments and theoryPhD Thesis (Netherlands: Eindhoven University ofTechnology)
[19] Raizer Y P 1991 Gas Discharge Physics (Berlin: Springer)[20] Valentine J M 1952 Energy per ion pair for electrons in
gases and gas mixtures Proc. R. Soc. Lond. A211 75–85
[21] Lieberman M A and Ashida S 1996 Global models ofpulse-power-modulated high-density, low-pressuredischarges Plasma Sources Sci. Technol. 5 145–58
[22] Lieberman M A and Lichtenberg A J 2005 Principles ofPlasma Discharges and Materials Processing 2nd edn(New York: Wiley)
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