Effect of the expanding explosive-emission centers plasma on the impedance of a high-current diode

1572 IEEE TRANSACTIONS ON PLASMA SCIENCE, VOL. 27, NO. 6, DECEMBER 1999

Effect of the Expanding Explosive-Emission CentersPlasma on the Impedance of a High-Current Diode

Svyatoslav Ya. Belomyttsev, Sergei D. Korovin, and Igor V. Pegel

Abstract—Formulas for the space-charge-limited electron cur-rent in a planar diode with a small hemispherical or hemi-cylindrical plasma emitter have been obtained theoretically andby a numerical simulation. For a periodic cathode structure ofhemispherical emitters, an approximate expression for currentis derived. Based on these results, the increase in current withtime in high-current explosive-emission diodes is explained. Acondition has been found under which the expanding cathodeplasma has no effect on the diode impedance. This is related tothe electric field pressure on the cathode. A configuration of thecathode surface has been proposed which provides realization ofthis condition.

Index Terms— Blade-type cathode, Child–Langmuir’s law,cathode plasma expansion, diode impedance, diode perveance,discontinuity of emission surface, expanding hemisphericalplasma emitter, explosive-emission (cold) cathode, high-currentplanar electron diode, periodic cathode structure, space-chargelimited current, time dependence of current, time dependenceof diode impedance,3=2-power law, stability (time constancy)of current.

I. INTRODUCTION

PLANAR vacuum diodes with an explosive-emission cath-ode [1] produce large-cross-section high current electron

beams. Beams of this type are applied in high-power mi-crowave oscillators with a virtual cathode, for production ofintense bremsstrahlung, and in technology.

Many applications, especially the microwave ones, requirethat the electron beam current be stable during the pulse whichis typically of nanosecond duration. At the same time, theimpedance of a high current planar electron diode is knownto drop in time [2]. This effect leads to a mismatch of thediode with the pulse source, an increase in current, and drop inthe electron energy (Fig. 1). A similar problem also exists forcoaxial magnetically insulated diodes with edge-type cathodesin the case of a low surface density of emission centers [3].

The instability of the diode impedance is related to theevolution of the cathode emission surface. Early in theprocess, this surface exists as an aggregate of separateplasma blobs—emission centers. These centers expand in time,causing the current to increase. The typical plasma expansionvelocity for metals and graphite is about 2 10 cm/s[4]. It is this mechanism of current rise which is discussed inthis paper. Investigated was the dependence of the current in a

Manuscript received February 22, 1999; revised September 8, 1999.The authors are with the Institute of High Current Electronics, Siberian

Division, Russian Academy of Sciences, Tomsk 634055, Russia (e-mail:[email protected]).

Publisher Item Identifier S 0093-3813(99)09858-6.

Fig. 1. Typical time dependencies of the current and the voltage for aplanar vacuum diode with explosive-emission cathode (high current electronaccelerator SINUS-700).

planar diode with a discontinuous emission surface on the sizeof hemispherical emission centers (emitters), both in a single-emitter geometry (Section III) and for a periodic emittingstructure (Section IV). The situation is considered where theemitter size is much less than the diode gap:that means . The number of centers is assumedconstant in time.

The paper also formulates the condition for the impedanceconstancy for a high-current diode and proposes an exemplaryversion of an appropriate cathode design (Section V).

All numerical results presented were obtained using the-and - versions of the 2.5-dimensional self-consistent

stationary code SuperSAM [5]. The code employs the methodof electron trajectories to describe the electron flow. Theelectric field is calculated using the method of finite elementsover a nonuniform curved mesh with eight-point paraboliccells. This allows a simulation of diodes with extremely large

ratios 10 .

II. A SIMILARITY LAW FOR A DIODE

Consider a stationary electron flow in a diode of arbitrarygeometry. Let the cathode include an emitter with unlimitedelectron emission capacity. Assume zero start velocities foremitted electrons. Consider a nonrelativistic motion with

where , is the diode voltage, andand are the electron charge and mass. Let the external

magnetic field be absent; also neglect the effect of the selfmagnetic field of the electron flow. For simplicity, apply asingle-flow hydrodynamic model. Introduce the normalizedelectron velocity , the volumetric charge

0093–3813/99$10.00 1999 IEEE

BELOMYTTSEVet al.: EFFECT OF EXPANDING EXPLOSIVE-EMISSION CENTERS PLASMA 1573

(a) (b)

Fig. 2. Diodes with (a) point (hemispherical) and (b) linear (hemicylindrical)emitters.

density , and the electrostatic potentialwhere is a characteristic size of the diode. These functionsare related by the set of equations

where the differential operators andcorrespond to the dimensionless coordinate . Theboundary conditions are as follows: at cathode:

(nonemitting area), and(emitting area), and at anode: (subscript“ ” indicates a component tangential to the surface). Thegiven boundary conditions do not contain the diode voltage.Hence, the solution functions show no dependenceon as well. They depend only on the shape of the diodeelectrodes, but are independent of their absolute size. Thecurrent density is given by

where kA and . The total diodecurrent is

(1)

while the form factor depends only on theshape of the electrodes and remains constant if all dimensionsof the diode change proportionally. Here, is any surfacewhich encloses the full cross section of the electron flow, forexample, the emission surface.

Thus, along with the “3/2-power law” [6], [7], expression(1) demonstrates conservation of the current on proportionalscaling of the diode.

It is easy to verify that this result is also valid for amultiflow hydrodynamic model. (As follows from simulations,for a hemispherical emitter considered below, the electrontrajectories are crossing so that the number of flows is equalto two.)

III. SINGLE EMITTER CURRENT

Consider a planar diode with a point emitter of characteristicsize on the cathode surface [see, for example, Fig. 2(a)]. Letthe emitter have unlimited emission capacity. If the cross sizeof the diode is much greater than its gap spacingthen, since

Fig. 3. Dependence of current on the radius of a single hemisphericalemitter. Pluses—simulation, line—expression (3) with� = 0:47.

the emitter shape is fixed, the form factor in (1) depends onlyon . Therefore, for the nonrelativistic case we have

(2)

To determine the function we take into account thesmallness of the emitter size . Obviously, whenit tends to zero, so do the current and the thickness of electronflow, and the flow disturbs the electric field in the gap onlyslightly. The potential distribution is therefore almost linear:

. Obviously, in a diode with a linearly dis-tributed potential, the current remains constant if the diode gapspacing and voltage change mutually proportionally. Setting

in (2) we obtain const andtherefore . Thus, the single emitter current canbe expressed as

(3)

where the dimensionless factordepends only on the emittershape. Its value obtained in the simulation for a hemisphericalemitter [Fig. 2(a)] is , provided that is the emitterradius (see for example Fig. 3 for cm, kV).

Similarly, since the above reasoning about the smallness ofthe current and the quasi-linearity of the potential distributionat small is also valid for linear type emitters [forexample, for a hemicylindrical one; see Fig. 2(b)], the specificcurrent (per length unit of the emitter) for such emitters isexpressed as

For a hemicylindrical emitter, the numerical simulation gaveprovided that is the cylinder radius.

It should be specially noted that at small ratios, therelativistic deviation in the “3/2-power law” for a point-typeemitter occurs at much (roughly, times) higher voltagesthan for the classical planar diode. This is because the currentis limited by electron space charge in the immediate (size) proximity of the emitter. Although the diode voltage

is relativistic, the energy of electrons near the emitteris which may be far from relativism. Thus, thecondition of validity for expression (3) looks as

(4)


Fig. 4. Perveance of the hemispherical emitter: 1—expression (3), 2—byShubin and Yurike [9], and 3—by Djogo and Cross [10].

In the opposite case, when the diode voltage is so highthat electrons gain relativism already near the emitter, thedependence of current on voltage becomes linear, and thepower by in the expression for the current reduces by 1/2.So, in this case the current of a linear emitter does not dependon at all.

It is easy to check that condition (4) is satisfied in themajority of practically important situations with the plasmavelocity of 10 cm/s, the pulse duration of 10 10 s,the diode gap of about several centimeters, and the voltage ofup to several megavolts.

It should be stressed that the power of 3/2 in expression (3)is asymptotically exact in the limit of . Compare thisexpression with other known results. Fig. 4 gives the relevantdata for the diode perveance [8]. In the nonrelativistic case, theperveance is proportional to the form factor in (1). The methodof specific capacities [9] gives a quadratic dependence of cur-rent on . We should note that this method is insufficientlyjustified and for small it gives incorrect results. Thesimulation-based interpolation [10] gives a linear dependenceat small . Obviously, the corresponding simulation resultsfor are not precise enough. Note that theauthors of [10] gave an incorrect interpretation of data takenfrom [11] relating to the current of a spherical emitter situatedon the tip of a needle.

IV. CURRENT OF PERIODIC EMITTING STRUCTURE

Consider a periodic structure of emitters with a distancebetween them. Introduce . Thus, at

we have a planar diode with uniform emission. It isconvenient to approximately express the Child–Langmuir lawfor this diode as [12]

(5)

This expression has exact nonrelativistic and ultra-relativisticlimits and its accuracy in the intermediate range of voltages isabout 1% compared to the exact solution [13].

In the opposite case where the distance between the emittersis much less than the diode gap, at , the currents of the

Fig. 5. Schematic view of the simulated periodic emitting structure: 1—emit-ter, 2—implied neighboring emitters, 3—actual boundary of the hexagonalcell, and 4—simulated cylindrical boundary.

emitters tend to zero and their space charges do not interact.Thus, the currents are independently described by (3).

An expression describing the average current density in thediode in the entire range from up to musttransform in these limits into (3) and (5). This is inherent inthe expression

(6)

The weight function should satisfy the conditions, and its concrete form depends on the shape of

the emitter cell.Consider a six-border prismatic cell. To avoid three-

dimensional simulations, approximate it by a cylinder with aradius of , with Neumann’s condition for the electrostaticpotential and the mirror reflection for the electron trajectorieson its lateral surface (Fig. 5). Thus, the problem becomesaxisymmetric. The simulation has demonstrated that in thiscase the weight function could be chosen aswith the error below 5% (see Fig. 6 for cm and

kV).Fig. 7 presents per-unit voltage dependencies of the per-

veance for a diode with hemispherical emitters, calculated bothnumerically and using (6). The smaller is the relative sizeofthe emitter, the higher is the upper voltage limit for a constant


Fig. 6. Dependencies of emitter current on its relative size in a periodicstructure. Asterisks—simulation, curves—expression (6).

Fig. 7. Perveance versus voltage for a planar diode with hemisphericalemitters. The structure period isp = 0:1 cm. Emitter radii: 1–5�m, 2–25�m, 3–100�m (points—simulation, curves—expression (6); and 4—exactdependence for the classical planar diode.

perveance, which means that the voltage dependence of thecurrent is nonrelativistic. The difference between analyticaland numerical results observed at extremely high voltages re-sults from the violation of the nonrelativistic motion conditionfor the electrons in the emitter vicinity.

The simulations performed for a periodic system show that,at small , the dependence of current on diode gap spacingtends to which is peculiar to a single emitter.

Fig. 8 represents the waveforms predicted using (6) forthe average current density in a diode with hemisphericalexpanding plasma emitters. The plasma expansion velocity is

cm/s, the diode gap spacing is cm, andthe diode voltage is kV. Comparison of curves ofthis sort with experimental ones could allow an estimation ofactual state of cathode emission surface.

V. CONDITION OF DIODE IMPEDANCE CONSTANCY

Let us formulate the general condition of impedance con-stancy for a planar diode with an explosive-emission cathode.Remember that we consider a situation where the expansion ofthe cathode plasma during the pulse is negligible: .

Fig. 8. Calculated time dependencies for the average current density in adiode with hemispherical emitters.

For the classical planar diode, assume the electron trajec-tories are perpendicular to the electrodes. The electrostaticpotential satisfies the Poisson equation

Since , we have

Setting and taking into account that

we have

Multiplication by and integration yields

where is the integration constant. Substituting ,we obtain

const

At the electrodes, the left part of this expression has themeaning of a full pressure resulting from the electric fieldpressure and the pressure of electrons. Thus, the anode and thecathode pressures are equal to each other. The field pressureproduces a force directed inward the diode, and the beampressure gives a force of the opposite sign. Thus, the totalforce effecting the diode is zero which is a consequence of themomentum conservation in a stationary system (assuming theelectron flow is absorbed by the anode not leaving the diode).

If electrons leave the cathode with zero (in a certainapproximation) velocities, the cathode pressure is given by


where is the electric field strength at the cathode. If thecathode emission capacity is unlimited we have andthe full pressure on the cathode is zero as well. The anodepressure is

where is the anode electric field strength. Ifand, consequently, , the field pressure on the anodeis compensated by the electron pressure.

Consider a cathode with a nonuniform but periodic structure.Neglecting the magnetic field, we obviously have periodicityin the electric field and in the particle flow. Pick out a periodicprismatic cell in the diode with end faces on cathode andanode. It can readily be seen that the fluxes of thecomponentof the momentum, related to the field and to the particles,through the lateral (parallel to axis) surfaces of the cell arezero. Therefore, the components of the force acting on theanode and on the cathode faces of the cell are equal to eachother in magnitude but opposite in sign. The same is true forthe diode in average.

If the emission surface of the cathode is continuous, thediode current does not depend on time. However, it is hardto rapidly cover the entire cathode surface with plasma. Topromote the development of explosive emission, cathodesare usually made to have a nonuniform periodic emittingsurface. Let the cathode structure period be much less than

. Neglecting magnetic field, we can consider the electronflow normal to the electrodes (except near the emitters).

The number of emission centers increases rapidly onlywithin the first nanoseconds of the pulse. However, the emis-sion surface continues to increase due to the plasma expansionuntil the neighboring plasma clouds merge. So the current canrise even at a constant voltage (see Section IV).

It is generally believed that the increase in current isdue to the increase in cathode emission surface area. Definemore exactly the mechanism of this phenomenon, taking intoaccount the above said about the balance of forces. Theincrease in current occurs only if the expansion of the emittingsurface leads to a reduction of thecomponent of the fieldforce acting on the cathode. On the contrary, it is well knownthat in a coaxial diode with an edge-type cathode, the currentis invariable in time if the entire edge of the cathode is coveredwith plasma. Despite the plasma expansion, the longitudinalcomponent of the field force on the cathode is zero, and thisis just the reason for the current constancy [14], [15].

Thus, to obtain a constant diode impedance, is it necessaryto provide an invariable electric field pressure on the cathode.Below we give an example of how to realize this requirement.

Make a cathode as a plane with thin parallel blades (Fig. 9)with the distance between them and . Due tothe latter condition, the cathode base is well screened. Thus, ifthe blade edges are covered with plasma, the field pressure onthe cathode vanishes and it remains unchanged in the courseof plasma expansion.

However, a cause for the change in current still exists. Thematter is that the anode pressure is the sum of the field andthe particle pressures, and these two are mutually related. The

Fig. 9. Diode with blade-type cathode.

Fig. 10. Simulated value of specific current in a planar diode. 1—uni-form emission, 2—blade-type cathode, and 3—cathode with hemicylindricalemitters.

expanding plasma can distort the shape of the electron trajec-tories and change the ratio between the pressure componentsalthough the full pressure remains constant. However, at smallpulse durations, the front of the plasma covers a distance muchsmaller than the typical dimensions of the diode; so we canexpect that the change in current will be insignificant.

Computer simulations confirm this supposition. The lowercurve in Fig. 10 presents the current per length unit of theemitter for a planar diode with a hemicylindrical emitter[Fig. 2(b)] of radius . The distance between theemitters is 1 cm and the diode gap spacing is 3 cm. The uppercurve is for the blade-type cathode (Fig. 9) with cm,

cm, and cm. In both cases, the diode voltage iskV. As can be seen, the dependence of the current

on emitter size is much weaker for blade-type cathode.Also note that for the value of the current

for a blade cathode diode tends to that of the classical planardiode with the gap .

VI. CONCLUSION

Thus, the theoretical study has demonstrated a strong effectof the emission surface micro structure on the current in ahigh-current planar diode. If the distance between the emissioncenters far exceeds their size, the current depends on diodeparameters as , while the dependence of thecurrent on voltage remains nonrelativistic up to much highervoltages than for a diode with uniform emission. Probably,such a behavior is to a greater or lesser extent inherent in any


diodes with explosive emission at nanosecond pulse durationswhen the emission surface is substantially discontinuous.

It stands to reason that correct application of the presentedresults to real diodes requires additional information on thestate of the emission surface. The above model dependencieswere obtained assuming constant number of emission centersand their uniform positioning. At the same time, there is nodoubt that the emission centers emerge nonsimultaneously.The micro-points vary in size and their distribution over thecathode surface is not uniform. The self space charge of ajust born emission center suppresses the electric field in itsvicinity [16] increasing the time delay to the explosion ofneighboring nonuniformities. Taking into account these effectswould provide a subject for further studies.

As for producing a planar cathode with a minimum timedependence of the impedance, although the requirements onthe cathode surface are clear, the attainment of this goalcan meet several problems. Simple metal blades are hard touniformly explode, especially at high pulse repetition rates.Besides that, heat deformation of blades occurs at a highaverage beam power. Probably, application of metal-dielectriccontacts [17] could make the emission more uniform due toeasier development of explosive emission and faster prop-agation of the plasma along the dielectric and to mergingof neighboring emission centers. Employment of needle-typecathode surfaces with an optimum distance between the nee-dles is also considered a way to compromise between therequirements of electric field screening and uniform excitationof the explosive emission.

ACKNOWLEDGMENT

The authors are grateful to Prof. D. I. Proskurovsky forvaluable discussions.

REFERENCES

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[3] I. V. Pegel, “Study of transients in the formation of high-currentnanosecond electron beams,”Russian Phys. J., vol. 39, no. 12, pp.1186–1199, 1997.

[4] G. A. Mesyats and D. I. Proskurovsky,Pulsed Electrical Discharge inVacuum. Berlin, Germany: Springer-Verlag, 1989.

[5] D. G. Myakishev, M. A. Tiunov, and V. P. Yakovlev, “Code SuperSAMfor calculation of electron guns with high beam area convergence,”Int.J. Mod. Phys. A (Proc. Suppl.) 2B, vol. II, pp. 915–917, 1993.

[6] C. D. Child, Phys. Rev., vol. 32, p. 492, 1911.[7] I. Langmuir, “The effect of space charge and residual gases on

thermionic currents in high vacuum,”Phys. Rev., vol. 2, p. 450, 1913.

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[13] H. R. Iory and A. W. Trivelpiece, “Exact relativistic solution for theone-dimensional diode,”J. Appl. Phys., vol. 40, no. 10, pp. 3924–3926,1969.

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[16] S. Y. Belomyttsev, S. D. Korovin, and G. A. Mesyats, “Effect ofscreening in high-current diodes,”Pis’ma J. Tech. Phys. (Sov.), vol. 6,no. 18, pp. 1089–1092, 1980.

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Svyatoslav Ya. Belomyttsevwas born in Novokamenka, Altai region, Russia,in 1940. He received the degree in physics from Tomsk State University,Russia, in 1962 and the Kandidat degree in physical electronics from theInstitute of High Current Electronics, Russian Academy of Sciences, Tomsk,in 1984.

He has been with the Institute of High Current Electronics since 1977and is currently a Senior Scientist at the Theoretical Physics Laboratory. Hisresearch interests include the classical electrodynamics and the high currentelectronics.

Sergei D. Korovin received the degree in physics from Novosibirsk StateUniversity, Russia, in 1975, the Kandidat degree in 1981 and the Doctorof Sciences degree in 1991 from the Institute of High Current Electronics,Russian Academy of Sciences, Tomsk, Russia.

He has been with the Institute of High Current Electronics since 1975 and iscurrently Deputy Director and Head of the Physical Electronics Department.His research interests include microwave physics, relativistic high-currentelectronics, and physics of high-current Electron accelerators.

Igor V. Pegel was born in Tomsk, Russia, in 1964. He received the degreein physics from Tomsk State University, Russia, in 1986 and the Kandidatdegree in physical electronics from the Institute of High Current Electronics,Russian Academy of Sciences, Tomsk, in 1995.

He has been with the Institute of High Current Electronics since 1989 andis currently a Senior Scientist at the Department of Physical Electronics. Hisresearch interests are in high-power microwave electronics and the physics ofhigh-current electron beams, in particular, numerical simulations.

Documents

Effect of the expanding explosive-emission centers plasma on the impedance of a high-current diode