924 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 38, NO. 4, APRIL 1991

Optimum Neutralization of Negative Space Charge in the Parallel-Plane Diode

Clive B. Wheeler

Abstract-Poisson's equation, a third-order differential equation, is formulated for the steady-state Row of electron current in a parallel- plane diode with an arbitrary distributed source of positive ions throughout the interelectrode region. Solutions are obtained for one particular class of ion distribution and it transpires that there is no upper limit to the electron current density J - that can Row between the electrodes, provided there is sufficient ion production. The solution within this class that is most efficient in neutralization requires a total ion current density J , that is related to J - and the charge masses through J + / J - = 1.07 (m-/m+)'/*. For electron currents, a few mul- tiples of the Child-Langmuir limit, this ion source distribution has a single peak in the central region of the interelectrode space. As J - is further increased this peak moves progressively towards the cathode, increasing in height and decreasing in width. Methods of establishing the optimum ion distribution by ionization of a gas filling are consid- ered. Dependent on the diode anode voltage, this ionization can be achieved either by collisions with the cathode electrons or by interac- tion with a transversely injected electron or laser beam.

I. INTRODUCTION F the cathode of a parallel-plane vacuum diode emits an un- I limited supply of zero energy electrons, charge -e, mass m-,

then it was shown by Child [ 11 and Langmuir [2] that the effects of negative space charge limit the current density of electrons collected by the anode to

In this expression V, is the anode potential relative to the cath- ode and a is the electrode spacing. There are several applica- tions of the plane diode in which this limiting current is less than the emission currents obtainable from modern coated cath- odes and a means of raising this limit is highly desirable. For example, in the production of intense electron beams of pre- scribed energy eV,, it may not be possible to reduce the elec- trode spacing a in (1) in order to utilize the full cathode emis- sion since flashover might occur across the insulator separating the electrodes. In the case of the thermionic energy converter it is essential to utilize the full cathode emission, otherwise the overall efficiency is too low to be viable. For such devices the available anode potential is very low and (1) requires electrode spacings that are at the very limit of microengineering. An in- crease above the limit of (1) can be effected if the negative space charge is neutralized through the introduction of positive ions. Langmuir [3] showed that single positive ions would have a maximum effect in increasing the electron current if they were introduced at a point distant 0 . 4 4 4 ~ from the cathode.

Manuscript received July 20, 1990; revised October 22, 1990. The re- view of this paper was arranged by Associate Editor R. Temkin.

The author is with the Plasma Physics Group, The Blackett Laboratory, Imperial College of Science, Technology and Medicine, London SW7 2BZ, UK .

IEEE Log Number 9041420.

If ions, charge + e and mass m+, were continuously gener- ated in a plane parallel to the electrodes then Wheeler [4] showed that the electron current density J - could be raised to a maximum of 3- = 4.5865, if the ion plane was situated at a distance 0 . 4 4 1 ~ from the cathode. The required ion current den- sity in this case was J + = 1.000(m-/m+)1/2J- . Greaterelec- tron currents can be obtained by introducing further strategi- cally located planes of ion generation. With three such planes, Wheeler [5], a maximum electron current of J - = 18.823, can be obtained for a total ion current of J , = 1.306 (m- /m+ ) 1 / 2 J - . This present paper considers the logical ex- tension of the above analysis to an infinite number of ion planes, i.e., to a continuous distribution of ion generation throughout the interelectrode region. Particular attention is given to deter- mining the form of the ion distribution that maximizes the elec- tron current and also to the efficiency of the ions in neutralizing the negative space charge. The assumptions of the theory are very basic, namely, that the velocities of the charges at gener- ation are zero, both for the electrons liberated from the cathode surface and for the ions generated between the electrodes. Ap- plication of the results to certain experimental diodes may be somewhat limited by these assumptions, however the avoidance of two additional variables enables a clear physical understand- ing to be maintained throughout the analysis.

11. MATHEMATICAL FORMULATION Let the position variable z and the potential variable I/ be

defined with respect to the cathode as origin, the anode there- fore having the coordinates a, V,. If J - is the electron emission current density from the cathode then the associated negative space-charge density p- ( z ) at the plane z is obtained by divid- ing this current by the local electron velocity. This gives

p - ( z ) = -J-(m-/2e) ' / 2 "' I /2

Let S( z ) ion-electron pairs be generated per unit volume per second at the plane z . Consider ions generated between parallel planes positioned at z1 and z1 + d z l . These ions will be accel- erated towards the cathode and, at the plane z , where z < z I , will give a contribution d J , ( z ) to the ion current density J + ( z ) defined by

d J + ( z ) = eS(zl) d z l .

The contribution d p !+ ( 2 ) to the positive space-charge density p ( z ) at the plane z is obtained by dividing d J , ( z ) by the local ion velocity

d p : ( z ) = e(m+/2e)1/2(Vl - V)-1'2S(zl) d z l , zI > Z .

The total positive space-charge density at the plane z is obtained by integrating over all positions of the plane zI lying on the

0018-9383/91/0400-0924$01 .OO 0 1991 IEEE

WHEELER: OPTIMUM NEUTRALIZATION OF NEGATIVE SPACE CHARGE 925

anode side of the plane z (9/41- )(dy/dx)?; I / 2

= 4y’/2 - ( 2 e a / I - J O ) ( m + l m - ) p : ( z ) = e ( m + / 2 e ) 1 / ~ S a ( V I - ~ ) - l / * ~ ( z l ) dz l .

A similar treatment of the ionized electrons leads to an associ- ated negative space-charge density

* [ cs; (YI - Y)-1/2 F(Y1) dY dYl

+ s: 1: (YI - Y)-1/2F(Yl) dY dY1 - (m-/m+) 1 /2

* s y y (Y - Yl)-1/2F(Yl) dY dYl].

~

p ~ ( z ) = - e ( m - / 2 e ) ( V - v ~ ) - ~ ’ ~ s ( ~ ~ ) dz l .

(6) s:

The potentials and total space-charge density are related through 0 Y l

The function F ( yI ) can now be taken outside the inner integral. Integration of (6) with respect to y can then be performed, lead- ing to

Poisson’s equation

d 2 V / d z 2 = - [ p - ( z ) + P : ( z ) + P q z ) ] / E O .

Some simplification is obtained if the variables are made non- dimensional through putting x = z /a , y = V / V a , and express-

current density Jo of (1). Then Poisson’s equation becomes

(9 / 161- ) (dy / d x f I / 2 ing J _ in terms of the Child-Langmuir space-charge-limited = y1/2 - (ea/Z-Jo)(m+/m-)

* [ 1: Y:/2F(YI)dYl - s; (Yl - Y) I / 2 (9/41- )d2y/dx2

( ea /J - Jo 1 ( m + /m- ) - - y-1/2 - I / 2 I /2 - F(Y1) dYl - (m-/m+)

where I _ is a nondimensional cathode emission current density defined by I _ = J _ / Jo . In order to obtain effective space-charge neutralization it is evident that the first integral here, repre- senting the positive space charge, must have a dependence on y that can compete with the y-’/’ dependence of the principal negative space charge. This integral can be forced to be an ex- plicit function of y if the following particular form of source distribution is chosen:

where F( y) is some function of y. This does not necessarily mean that S(x) is zero when dy/dx is zero, as will be dem- onstrated in Section IV. However, such a choice does imply that the dependence of S ( x ) on x is not known until (3) has been solved to give the dependence of y and dy/dx on x. Sub- stitution of (4) into (3) gives

A first integration of ( 5 ) from the cathode to the point (x, y) can be carried out after multiplying through by 2 d y l d x . The order of the two successive integrations on the right-hand side can then be reversed, after making the appropriate changes to the integration limits. If the boundary condition (dyldx), = 0 is applied, corresponding to fully space-charge-limited emis- sion at the cathode surface, the result is

Each term on the right-hand side of (7) has dimensions that are proportional to a charge source function multiplied by the charge velocity. The complete equation expresses the invariance of momentum flux or pressure throughout the current flow. The first term on the right represents the momentum flux of the cath- ode emitted electrons at the coordinate y, while the fourth term is that due to the electrons generated between the electrodes. The third term represents the momentum flux of the ions at the coordinate y and the left-hand side of the equation is the stress due to the electric field. If these terms are added algebraically then (7) equates the result to the corresponding addition carried out at the cathode. This is given by the second term on the right which is the total momentum flux of the ions at the cathode. It must be remembered that the electric field and the electron emission velocity at the cathode surface have both been as- sumed zero, therefore there are no other momentum flux terms at that electrode.

The interelectrode spacing a can be removed from (7) by in- troducing the total ion current density J + reaching the cathode, defined by

J + = ea S(x) dx = ea F(yl) dy,. ( 8 ) Sd S: Equation (7) can then be expressed

(9/16Z-)(dy/dxf = y1 /2 - (1+/1-)WY) (9)

where

926 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 38, NO. 4. APRIL 1991

and I , is a nondimensional ion current density at the cathode, defined by

I , = ( m + / m - ) ' / 2 ( J , / J o ) . (11) Equation (9) is separable and can be arranged to express x as a function of y

(41'/2/3)x = K ( y , I + / Z - )

The space-charge-limited electron current I - corresponding to any particular value of ( I , / I - ) is determined by the boundary condition x = 1 at y = 1 , giving

I - = [ ( 3 / 4 ) K ( 1 , Z + / Z - ) I 2 . (13 )

It then follows that the distribution of potential between the electrodes is given by

x = K ( Y , I , / I - ) / K ( l , [ + / I - ) . (14 )

The form of (12) and (13) shows that I_, and therefore I , , in- crease as the parameter ( I , / I - ) increases.

For ease of analysis the function F ( y ) is now chosen such that all the integrals in (10) are analytic. The simplest form is a single power of y

F ( y ) = Ay". ( 1 5 )

If the index n is an integral multiple of one half then all the integrals in (10) are of standard form.

There are lower limits to the index n imposed by the condi- tion that the total ion current density J , should be finite and also that the ion source function S ( x ) should be finite for all values of x. Substitution of (15) into (8) shows that J , is finite providing n > - 1 . Equations (4), (8), and (15) give

( e a / J + ) S ( x ) = (n + l ) y " d y / d x (16 )

with d y / d x given by (9). Examination of the function H( y ) shows that, certainly for n > - 1 / 2 , it is finite for all values of y . In particular, in the vicinity of y = 0 it varies like the first power of y . Equations (9) and (16) therefore require that, pro- viding n > - 1 / 2 , S(x) is finite for all values of x if 2n + 1 / 2 1 0. This imposes the more stringent condition n 1 - 1 / 4 .

111. INTEGRATION OF THE EQUATIONS

In the case of very massive ions, m - / m , --* 0, the third integral in (10) is negligible. Substitution of (15) into (10) and integration for n = 0, 1 / 2 , 1 , and 3 / 2 gives

N Y , 0 ) = ( 2 / 3 ) [ 1 - (1 - Y P 2 1

H ( y , 1 / 2 ) = ( 3 / 8 ) 1 2 - ( 1 - y)'" - (1 - y f / 2

H ( y, 1 ) = (4 /15 ) [3 - 5 ( 1 - y?l2 + 2 ( 1 - y f 1 2 ]

H ( y , 3 / 2 ) = ( 5 / 4 8 ) / 8 - 3 ( 1 - y ) ' l 2 - 8(1 - y f l 2 1

+ 3 ( 1 - Y ) ~ " + 3y3 In

In general the nondimensional electric field d y l d x , given by (9), has a maximum in front of the cathode and a minimum in front of the anode. These maxima and minima become more pronounced as the parameter ( I , /I- ) is increased. Fig. 1 shows a plot of this field distribution in the case of n = 0 for selected values of ( I + / I - ). If follows from Poisson's equation that the total space charge is zero at each of these turning points and that the positive space charge exceeds the negative space charge throughout the region between the turning points. The abscissa presented in Fig. 1 is not the potential variable y but the space variable x as determined by the next, nonanalytic stage of in- tegration. The individual values of I - and I , also follow from this later calculation. For sufficiently large values of I - there is strict proportionality between electron and ion currents. In fact

I - = 0.68471, k O S % , for l - > 12.9 (withn = 0 ) .

In Fig. 1 the curve for ( [ + / I - ) = 0, ( I - = 1 , I , = 0 ) , cor- responds to the Child-Langmuir distribution, which is x = y3l4 or d y / d x = ( 4 / 3 ) x ' 1 3 . The final stage of integration, (12), must be carried out numerically, care being required in the two regions where the integrand can be infinite. The choice of boundary condition ( d y l d x ) , = 0 makes the integrand infinite at the cathode. However, the contribution to the integral is finite and difficulties in numerical integration can be circumvented by taking y314 as the variable.

In this manner K ( 1, I , /1-, n ) was evaluated by Simpson's rule as a function of ( I , / I - ) , thereby allowing both I - and I , to be determined from (1 3). For each value of n there is a crit- ical value of (1, / I - ) for which the field minimum in Fig. 1 is reduced to zero and the integrand of (12) again becomes infi- nite. However, unlike the former situation at y = 0, the integral in this case is also infinite, indicating that both I - and I , have infinite values. The rapid increase in I _ that takes place as the field minimum approaches zero is apparent in Fig. 1 . In this case the critical value of ( I , / I - ) is ( I , / I _ )- = 1.4679 which is the reciprocal of the proportionality constant in (17). As the value of the index n increases the value of ( I , / I - )- decreases. This is shown in Fig. 2 which has coordinates for selected val- ues of n that permit an analytic solution to (9), together with the result obtained by numerical integration for the minimum permitted value, n = - 1 / 4 . For these larger values of n the strict proportionality between electron and ion currents extends down to lower values of I -

I - = 0.92871, f O S % , for l - > 3.42 (with n = 5 ) .

The negative slope of Fig. 2 yields the important result that the efficiency of positive ions in reducing negative space charge increases as the index n increases, the ion current asymptoti- cally approaching the value I , = 1.07 I - .

Iv. OPTIMUM ION SOURCE DISTRIBUTIONS The positive ion source distributions corresponding to the

various values of n are given by (16) and (9), using values of ( l + / Z - ) appropriate to the required value of I - . Fig. 3 shows such ion distributions for seven values of n, all evaluated for the same value of electron current, I - = 10. It follows from (8) that the area under each curve is unity and all the curves, apart from n = - 1 / 4 , pass through the origin, the latter having its maximum at the cathode surface. The corresponding values of I , show a falloff of over 30% as the distribution changes from

WHEELER: OPTIMUM NEUTRALIZATION OF NEGATIVE SPACE CHARGE 921

0 02 O L 06 0 8 * 1.0

Fig. 1. The interelectrode electric field d y / d x as a function of distance x from the cathode, evaluated for index n = 0 and for selected values of the current ratio ( I + / I - ) .

A : ( I + / I - ) = 0 ( I - = 1, I + = 0 ) B: ( I + / I - ) = 1.3 ( I - = 4.026, I + = 5.234) C: ( I + / I - ) = 1.45 ( I - = 11.92, I + = 17.28) D: ( l + / I - ) = 1.4677 ( I - = 50.18, I , = 73.65)

1.6

( I&

1.L

1.2

1 .o

Fig. 2.

0 2 L 6 n

The critical current ratio ( I + / I - ) - required to produce an infinite electron current I - evaluated as a function of the index n .

L r 5

0 0.2 04 0.6 0.8 10

Fig. 3. The ion source distribution S ( x ) required to produce an electron current of I - = 10, evaluated for selected values of the index n .

928 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 38. NO. 4. APRIL 1991

a broad maximum at the cathode surface (n = - 1 /4, I + = 15.7), to a narrow peak near the center of the interelectrode region ( n = 5, I , = 10.8). The magnitude and position of this peak for n = 5 was studied as a function of I - , achieved by varying the parameter ( I , /Z-). It was found that the potential 9 corresponding to the maximum value of S ( x ) in (16) varied very little for sufficiently large values of !-. Equations (16) and (9) therefore imply that the maximum S of the distribution is proportional to J+Z!!2, i.e., S is proportional to Further- more, (14) and (13) imply that the spacial coordinate 2 , corre- sponding to this maximum, is inversely proportional to In numerical terms, for n = 5 and over the current range 3.9 5 I _ 5 03

(18a)

(18b)

(18c) Consequently, the peak in the distribution moves towards the cathode and increases in magnitude as the electron current in- creases. Moreover, the requirement of unit area under the curve implies that the distribution becomes more narrow as the elec- tron current increases. At the higher currents the n = 5 distri- butions become highly symmetiic about their maxima, as evi- denced in Fig. 3 for I _ = 10. This enables the profile to be well approximated by a simple analytic function. The shape of the profile strongly resembles that of the intensity distribution within the central maximum of the single-slit diffraction pattern, namely a sin2 x /x ’ dependence. Therefore, the ion source dis- tribution is well represented by

j = 0.878 f 0.002

(m+/m_)”2(ea/J,Z3/2)S^ = 1.317 f 0.013

Z!’2.f = 1.320 f 0.003.

sin2 b ( x - $) S ( x ) = S 2 ‘

[ b ( x T q j - The parameter b , determined by unit area under the curve, is proportional to the square root of the electron current, namely, b = 3.46Z!/2, and the distribution is to be cut off at its first zeroes, located at 2 k a / b .

V. CONCLUSIONS The foregoing calculations show that there is no upper limit

to the electron cuvent that can be attained in the parallel-plane diode with neutralizing ion source distributions of the form given by (4). The efficiency of the ions in neutralizing the neg- ative space charge increases as the index n of the power law in (15) increases, and approaches the limiting value I , = l.O7Z_. It is, of course, ’quite possible that more efficient neutralization can be obtained with ion distributions’differing from (4) and (15).

Probably the most convenient way of producing ions in the interelectrode region is by ionization of a neutral gas filling. This ionization can be achieved either by external means or by collisions between neutrals and the cathode-emitted electrons. At anode potentials too low to produce iovization of any gas filling there is a very ingenious technique due to Cook et al. [6] that utilizes the Ramsauer effect for the heavy inert gases, Mas- sey and Burhop [7, pp. 24-25]. Argon, krypton, and xenon have an anomalously high transparency for electrons of about I-V energy; the scattering cross section- being at least a factor of twenty less than the peak of the ionization cross section. Con- sequently, a diode can be gas-filled to such a density that the cathode electrons travel unimpeded to the anode, while the gas is sufficiently dense to be efficiently ionized by a transversely

injected electron beam of the appropriate energy. If the injected electron beam is profiled according to (19) and injected at the position given by (18c) then the current I _ can be realized with minimum ion production. This has the advantage of minimizing both the scattering of the cathode electrons and also the power consumption of the transverse beam. At higher anode poten- tials, say above the 4 V required to ionize the alkaline elements, the gas filling can be ionized by impact with the cathode emitted electrons. Equation (18a) indicates that a minimum number of ions is required when ionization takes place at the coordinate 9 = 0.878.

For a specified anode potential V, this can be achieved by choosing a filling gas that has the peak of its ionization cross section close to the potential 0.878 V,. Since this choice is based on I , / I - = 1.07, (1 1) therefore indicates that the probability P, of a cathode electron suffering an ionizing collision is

‘ / 2 P , = 1.07(m-/m,) . A rough estimate of the required filling density no can be ob-

tained by assuming that all the ions are generated within a vol- ume determined by the profile width of (19), where all the neu- trals have the peak ionization cross section U,. In terms of these parameters the probability P, can be expressed as

(20)

Pi = n,u ,aa/b = 0 . 9 0 8 n , ~ , a / Z ~ / ~ .

Equations (20) and (21) show that minimum gas density is ob- tained by selecting a heavy gas of large ionization cross section. However, in many applications, the quality of the electron beam at the anode is of great importance and it is essential to mini- mize scattering of the electrons by collisions with gas atoms. If uy is the maximum total scattering cross section of the gas atoms then an estimate of the probability P, that a cathode electron suffers a scattering collision is given by

P, = (u , /u i )Pi = 1 . 0 7 ( ~ , ~ / u ~ ) ( m - / m + ) I / 2 .

For the alkalie elements u,/u, is very high, typically 50, but for the other elements the ratio is typically 5 . Consequently, if low scattering of the cathode electrons is a requirement then it is preferable to select a suitable massive, nonalkalie, filling gas.

The range of ionization potentials exhibited by the elements places an upper limit on the anode potential for which this tech- nique of ion generation can be used. If only first stage ionization is contemplated then the tabulation of Massey and Burhop [7, pp. 128-1291 indicates that neon would permit the highest an- ode potential of about V, = 200 V . For operation at all higher potentials the gas filling must be of insufficient density to be significantly ionized by the cathode electrons and some external agency used to produce ionization in the region specified by (18c). Photoionization by an intense beam of UV radiation is an obvious choice, first reported in this context in 1930 by Whitman [8]. The present-day source of high-intensity UV ra- diation is the excimer laser. In particular the readily available krypton fluoride laser ( A = 249 nm), with its photon energy of 4.98 eV, can photoionize potassium, rubidum, and caesium. The minimum amount of photoionization required is that ap- propriate to ( I + /Z-) = 1.07 and (1 1) indicates that the total ion current density reaching the anode is then

J, = 1.07(m-/m+)‘ /2J_ .

Caesium has the largest photoionization cross section of the above three alkalies, Massey [9], and also the largest mass; therefore, on both counts a caesium filling would require the

WHEELER: OPTIMUM NEUTRALIZATION OF NEGATIVE SPACE CHARGE 929

smallest laser beam intensity. Laser beams usually have a Gaussian intensity distribution and for best correlation with the analysis the width should be matched to that of the profile of (19). If the diode is pulsed in operation it may be more eco- nomic to ionize the gas by spark breakdown using a &-switched solid-state laser rather than by photoionization.

Care must be exercised in applying these calculations to ex- perimental situations since the analysis has assumed that both polarities of the charged particle have zero energy at their re- spective points of production. In practice, the electrons emitted from the cathode have an energy spread appropriate to the emis- sion process and this must be several orders of magnitude less than the final electron energy eV, if the calculations are to be valid. Consequently, these results can only be regarded as a first approximation when applied to low potential diodes with ther- mally emitting cathodes, such as the thermionic energy con- verter.

REFERENCES

[ I ] C. D. Child, “Discharge from hot CaO,” Phys. Rev . , vol. 32,

[2] I. Langmuir, “The effect of space-charge and residual gases on thermionic currents in high vacuum,” Phys. Rev., vol. 2, pp.

[3] -, “The interaction of electron and positive ion space-charges in cathode sheaths,” Phys. Rev . , vol. 33, pp. 954-989, 1929.

[4] C. B. Wheeler “Space-charge limited electron current flow be- tween coaxial cylinders and concentric spheres with ion produc- tion at an intermediate radius,” Proc. Inst. Elec. Eng., pt. A,

[5] -, “Maximum electron currents in vacuum tubes with ion emitting electrodes,” J . Phys. D , vol. 19, pp. 1519-1527, 1986.

pp. 492-511, 1911.

450-486, 1913.

vol. 131, pp. 325-327, 1984.

[6] K. G. Cook, D. A. Fraser, and G. G. Isaacs, “A thermionic generator with ion injection,” Adv. En. Conv., vol. 3, pp. 323- 331, 1963.

[7] H. S . W. Massey and E. H. S . Burhop, Electronic and Ionic Impact Phenomena, vol. VI. Oxford, UK: Clarendon Press, 1969.

[8] V. E. Whitman, “The neutralisation of space-charge by positive ions in caesium vapour,” U.S. Bur. Stand. J . Res, vol. 4, pp.

[9] H. S . W. Massey, Electronic and Ionic Impact Phenomena, vol. 157-167, 1930.

VII. Oxford, UK: Clarendon Press, 1969, pp. 1129-1131.

*

Clive B. Wheeler received the first degree in physics from the Imperial College, London, UK, in 1954. In 1981, he received the D.Sc. degree in plasma physics, also from Imperial College.

He is Senior Lecturer at Imperial College. He was a founder member of the plasma phys- ics group and, with the advent of the laser, de- veloped several novel techniques for diagnos- ing plasma using the laser as a probe. His experience in constructing gas, solid-state, and

dye lasers enabled him to build several early forms of laser ophtamo- loscope for clinical use at Moorfield Eye Hospital. Since 1981, he has been particularly interested in techniques for reducing negative space charge in vacuum electron devices. Such a reduction enables the op- erating current to be raised to many multiples of the Child-Langmuir limit and has application to fusion pellet compression, pumping UV excimer lasers, thermionic energy converters, and high-power radio communication.