SCIENCE

Space-charge - limited bipolar current flowbetween coaxial cylindrical electrodes

C.B. Wheeler, D.Sc.

Indexing terms: Ionisation, Breakdown and gas discharges

Abstract: Poisson's equation in cylindrical symmetry is solved numerically assuming that at each electrodethere is an unlimited supply of charges of zero energy; positive charges at the positive electrode and negativecharges at the negative electrode. A solution is obtained corresponding to zero electric field at each electrodesurface; this defines the maximum possible current flow of each charge polarity. The numerical techniqueis based on a perturbation of the solution appropriate to parallel-plane geometry and is carried out for aratio of electrode radii ranging from 1 to 400. In the case of the external emitter configuration, the increasein charged-particle current afforded by bipolar flow in comparison to unipolar flow can be significant. Ap-plication to the electron-beam pumping of vacuum ultraviolet excimer lasers is briefly considered.

1 Introduction

The flow of charged particles between one electrode andanother in an evacuated diode has an upper limit dictatedby the effects of space charge. This limiting current canbe increased if the collector electrode is also an emitter ofcharged particles of the opposite polarity. Langmuir [1]considered such bipolar flow in a parallel-plane geometryand showed that the limited current of charges was a factorof 1.860 times greater than the unipolar limit. Recently,Wheeler [2] considered bipolar flow between concentricspherical electrodes and found that, for the external-emitterconfiguration, the increase of limiting current affordedcould be several orders of magnitude. This paper considerssuch bipolar flow between coaxial cylindrical electrodesusing the same mathematical technique as in Reference 2.One obvious application of these calculations is to thepumping of vacuum ultraviolet excimer lasers by radiallyconverging beams of electrons, Hutchinson [3]. In suchdevices, the gaseous lasing medium is contained within a thin-walled metal cylinder that acts as the anode of a cylindricaldiode. The cathode cylinder is between six and nine timesthe radius of the anode and is either perforated or linedwith razor blades to promote field emission. High voltage,between 0.25 and 1.5 MV, is applied across the cylindersfor a timescale between 3 and 100 ns. The electrons emittedfrom the cathode have sufficient energy to penetrate thethin-walled anode and excite the contained gas, Bradleyet al. [4, 5] , Edwards et al. [6]. Electron currents of manykiloamperes are required and in practice this current is limitedby space-charge effects in the coaxial diode. This currentcan be increased by raising the applied potential, but theelectrons may then be too energetic for optimum pumpingof the gas. The analysis here shows that the limiting electroncurrent can be increased by about a factor of three if theanode acts as a source of positive ions.

2 Mathematical formulation

Consider a coaxial cylindrical geometry comprising an innerelectrode of radius rx at a potential Vx, surrounded by anouter electrode of radius r2 at a potential V2. Suppose thatthe inner electrode emits singly-charged particles of massmx at zero energy, and the outer electrode emits singly-charged particles of the opposite polarity of mass m2, alsoat zero energy. It is assumed that both these emissions are

Paper 2293 A, first received 20th May and in revised form 27th July1982The author is with the Plasma Physics Group, The Blackett Laboratory,Imperial College of Science & Technology, Prince Consort Road'London SW7 2BZ, England

IEEPROC, Vol. 130, Pt. A, No. 1, JANUARY 1983

limited only by the effects of space charge and that thepotential difference is such as to accelerate the chargesaway from their source electrode. Let V be the potentialat radius r, where rx < r < r2. The radial velocities of thesecharges, vx and v2, respectively, at the radius r are given by

\mxv\ = e{V-Vx) \m2v\ = e(V2-V) (1)

It is assumed here that the potentials are sufficiently lowto warrant a nonrelativistic treatment and that the particlecurrents are sufficiently low for the associated magneticfields not to influence the particle motion. The densitiesof the opposing charges, px and p2, respectively, at the radius rare expressible in terms of the currents Ix and I2 of the in-dividual species per unit cylinder length

px = Ix/2nrvx p2 = I2/2nrv2 (2)

In the steady-state the potential and total charge densityin the region rx < r < r2 are related through Poisson'sequation

(3)7TrVdr-l=-^-p>)/€o

Eliminating charges and velocities between eqns. 1, 2 and 3and introducing the dimensionless potential variable y =(V-VX)/(V2 -Vx) leads to

d_dr

r Qdr

,(2*): (4)/

2Xwhere V2X = \V2 — Vx\. Introduce the dimensionless spacevariable x = In (r/rx) and define the currents Ix and I2 interms of the currents /10 and 720, where

4ec9

2e \vl

mxj— I V2?*r?

I2o/2nrx =2e_

m2

1/2- ~ I T/3/2 ..-2

I v2X rx

Thus Ix0/2rrrx is the Child-Langmuir space -charge-limitedcurrent density for unipolar flov of charges, mass mx,between parallel-plane electrodes separated by distance rx

and at a potential difference V2X. I20 is correspondinglydefined. With these introductions eqn. 4 can be written

dx2 ~ 9 - - —

0143-702X/83/010043 + 03 $01.50/0

(5)

43

Multiply eqn. 5 by 2dy/dx and integrate, introducing theboundary condition (dy/dx)t = 0. This condition correspondsto zero field at the inner electrode, radius rx, and to space-charge-limited emission from that electrode. Eqn. 5 becomes

'-1/2

(6)

The requirement of space-charge-limited emission from theouter electrode, radius r2, introduces the boundary condition(dy/dx)2 = 0. From eqn. 6, this condition defines the ratioof the space-charge-limited particle currents

(A/Ao)/(A/Ao) =Ce*y-y2dy/fex(l -yj1/2dy (7)

Solution of eqn. 6 subject to eqn. 7 yields, simultaneously,the maximum current of charged particles that can flow fromeither electrode to the other, when there is an unlimitedsupply of charges of appropriate polarity at each electrode.

3 Numerical solution

Integration of eqn. 6 gives

( o\i/2ry .yI) Jo [ U

<*y (8)

This equation is solved by the same technique as used forthe solution of the spherical problem, Wheeler [2]; namelyby considering the cylindrical geometry as a successiveperturbation of parallel-plane geometry.

For small values of x, i.e. for x2l = l n f o / ^ ) < 1, theexponentials can be approximated to unity. Eqn. 7 thenreduces to A/Ao = A/Ao and eqn. 8 can be integrated togive

=j (Ao/A)1'2 J 0 V -y")V7 -y")

(9)

The integral here has been evaluated by Langmuir [1] in histreatment of the parallel-plane geometry. Setting y = 1 ineqn. 9, corresponding to x = x2\ (i.e. to r = r2), determinesthe current ratio A/Ao for that value ofx2 1 . Consequently,the interelectrode potential distribution y = f(x/x21) isdetermined. A larger value of x21 is then chosen and theintegrals of eqns. 7 and 8 performed numerically using thepotential distribution y = f(x/x21) appropriate to the previoussmaller value of x2l. Setting x = x2l at y = 1 in eqn. 8 then-gives a first estimate of the parameter A //10 and also yieldsan improved potential distribution y = f(x/x21). The cycleof integration is then carried out again with this improveddistribution and so on until successive cycles are consistent.The whole procedure is then repeated for larger values ofx21. In this manner, solutions were evaluated over the range0 < x2i < 6, i.e. 1 < r2fri < 400, yielding V A o and I2/I20 to an accuracy of ± 0.5%.

Fig. 1 shows the potential distribution obtained for selec-ted values of x2l. The curve for x21 -*• 0 corresponds toplane geometry (r2 ^/"i) and in this case the abscissax/x21 reduces to the distance from one plane electrodedivided by the electrode separation. The most concise wayof presenting the parameters A/Ao and I2/I20 is in terms

of a simple analytic function multiplied by a factor nearunity that is a weak function of the electrode geometry.Over the range of radii embraced by the calculations a goodfit was obtained with the expressions

= 1.860 axaf exp( -0 .41 x21

(10)

A /Ao = 1.860 7x 2{ exp ( - 0.60 x21)

Fig. 2 shows how the factors a and y vary with the ratioof electrode radii. (This parameter should not be confusedwith the ot used by Langmuir [1 ] in his parallel-plane analysis.)

Fig. 1 Interelectrode voltage distribution

0.8 1.0

l),x2l =ln(r2/r.)ax2l =6,6*2, = 4 . c x 2 , =2,dx2l -*• 0

1.0

0.9

0.80 1

x21 r

Fig. 2 Parameters a and y of eqn. 10 for the internal emitter andthe external emitter, respectivelyThe circled co-ordinates are taken from the results of Amemiya [7]for the internal emitter configuration.

4 Discussion

It is interesting to compare the potential distributions ofFig. 1 with the distribution relevant to spherical symmetry,Wheeler [2]. In spherical symmetry the distributions deviateless from the curve for parallel-plane geometry and are allsituated above this curve (x21 -*0), whereas in cylindricalsymmetry the distributions are all below this curve. Thiswas surprising because the results for the cylindrical systemswere expected to be intermediate between those for planeand spherical systems.

Amemiya [7] has considered the coaxial cylinder problemfor both internal and external emitter configurations. Hiscalculations, based on the Ruge-Kutta-Merson technique, arenot pursued far enough with the external emitter for thecurrent to be fully space-charge limited. For the internalemitter the values of a inferred from his calculations areshown as circled co-ordinates in Fig. 2. They are up to 12%

44 IEEPROC, Vol. 130, Pt. A, No. 1, JANUARY 1983

different from the present calculations and similar dis-crepancies were found in his results for the spherical problem,Wheeler [2].

For many applications, it is useful to determine whetherbipolar current flow results in a significant increase in currentover the unipolar case. The limited unipolar flow of currentper unit length of cylinder has been evaluated by Langmuirand Blodgett [8] and is expressible as

/,„ =1/2

3 / 2 Ir

\ 1/2

I V21'2 lr\ ( ~~ fi)

where 02 and ( — 0)2 are tabulated functions of r2 lrx foran internal and external emitter, respectively. Fig. 3 showsthat the ratio Iy\lu\ for the internal emitter is less thanthe factor of 1.860 appropriate to parallel-plane geometry,whereas for the external emitter the ratio / 2 / / u 2 can besignificantly greater than 1.860. A similar trend was foundfor the spherical geometry, Wheeler [2], but in that casemuch larger ratios were found for the external-emitterconfiguration.

1.9

1.8

1.7

1.6

1.5

Fig. 3 Ratio of bipolar current to unipolar current

Subscripts: 1 indicates the internal emitter2 indicates external emitter

5 Conclusions

For the coaxial diodes used to electron pump excimer lasersthe ratio of electrode radii is typically r2/rl = 8. From Fig. 3it is seen that at this aspect ratio, and for the external-emitterconfiguration, bipolar flow gives a current increase of theorder / 2 / / u 2 = 3 . The magnitude of the ion current Ix re-quired to realise this electron current I2 follows from eqn. 10as of the order Ix = {m2lmx)

V2/2 and therefore decreaseswith increase of the ion mass m2. For ions of hydrogen therequired ion current is about 2% of the electron current,

and for ions of copper the ratio is about 0.3%. The ex-periments of Humphries et al. [9, 10] with diodes operatedin the range lOOkV to 1.8MV show that sufficient ion pro-duction at the anode can readily be generated by electronbombardment. Protons were liberated when the anode wascoated with hydrogenous material such as varnish, and alu-minium or copper ions were produced when the anodeswere foils or meshes of these metals. Ion currents as greatas 20% of the electron currents were recorded.

Although the treatment here is nonrelativistic, thecorrection to be applied to the electron current is not large.For example, the calculations of Wheeler [11] show that,for r2 \rx = 8 and at a potential difference of 1 MV, thespace-charge-limited unipolar electron current is over-estimated by 11%, if relativistic effects are neglected. Powersin excess of 109 W are required to energise these diodes;consequently they are necessarily pulsed in operation. How-ever, the foregoing analysis is steady-state and, therefore,only valid after the elapse of at least one ion transit time.For an electrode separation of 2 cm and an applied potentialof 1 MV, the transit time of a hydrogen ion is 3 ns and 23 nsfor a copper ion. The theory is therefore applicable to thepumping of noble-gas excimer lasers using protons as positiveions, as these lasers have a maximum efficiency when pumpedfor about 15 ns. For the noble-gas-halide excimer lasers, theoptimum pumping time is considerably longer and muchheavier ions may be used.

References

1 LANGMUIR, I.: The interaction of electron and positive ionspace charges in cathode sheaths', Phys. Rev., 1929, 33, pp. 954 —989

2 WHEELER, C.B.: 'Space-chaige-limited bipolar current flow be-tween concentric spherical electrodes', IEE Proc. A, 1982, 129,(6), pp. 387-390

3 HUTCHINSON, MlH.R.: 'Vacuum ultraviolet excimer lasers',Appl. Opt., 1980,19, pp. 3883-3888

4 BRADLEY, D.J., HULL, D.R., HUTCHINSON, M.H.R., and Mc-GEOGH, M.W.: 'Megawatt XUV xenon laser employing coaxialelectron beam excitation', Opt. Commun., 1974,11, pp. 335-338

5 BRADLEY, D.J., HULL, D.R., HUTCHINSON, M.H.R., and Mc-GEOGH, M.W.: 'Coaxially pumped, narrow band, continuouslytunable highpower VUV xenon laser', ibid., 1975,14, pp. 1-3

6 EDWARDS, C.B., HUTCHINSON, M.H.R., BRADLEY, D.J., andHUTCHINSON, M.D.: 'Repetitive vacuum ultraviolet xenon exci-mer laser', Rev. Sci. Instrum., 1979, 50, pp. 1201-1207

7 AMEMIYA, H.: 'Bipolar space charge limited current betweencoaxial cylinders and concentric spheres', 5c/. Pap. Inst. Phys. &Chem. Res. (JPN.), 1969, 63, pp. 1-6

8 LANGMUIR, I., and BLODGETT, K.B.: 'Currents limited byspace charge between coaxial cylinders', Phys. Rev., 1923, 22,pp. 347-356

9 HUMPHRIES, S., LEE, J.J., and SUDAN, R.N.: 'Generation ofintense pulsed ion beams', Appl. Phys. Lett., 1974, 25, pp. 20—22

10 HUMPHRIES, S., SUDAN, R.N., and CONDIT, W.C.: 'Productionof intense megavolt ion beams with a vacuum reflex discharge',ibid., 1975, 26, pp. 667-670

11 WHEELER, C.B.: 'Space charge limited current flow betweencoaxial cylinders at potentials up to 15 MV, /. Phys. A., 1977,10, pp. 631-636

IEE PROC, Vol. 130, Pt. A, No. 1, JANUARY 1983 45