Dependence of surface coverage on pore geometry in dispenser cathodes

Applied

358

Surface Science 24 (1985) 358-371 North-Holland, Amsterdam

DEPENDENCE OF SURFACE COVERAGE ON PORE GEOMETRY IN DISPENSER CATHODES * B.A. FREE COMSAT

Received

and R.G. GIBSON

Laboratories,

**

Clarksburg, Maryland 20871, USA

28 May 1984; accepted

for publication

28 March

1985

This paper shows that the surface coverage of Ba/BaO on tungsten is more complete when the activator is supplied via slotted pores rather than circular pores. Both theoretical and experimental evidence is given to support this contention. The effect is primarily a geometrical one, since the surface diffusion in the case of circular pores is two-dimensional, whereas the surface diffusion for slotted pores is linear. The contrast becomes less pronounced as the circular pore size decreases. For dimensions of the order of those found on cathode surfaces (e.g., 10 gm diameter pores), a hexagonal array of circular pores can be optimized to produce an emitting area of 88% of the total, with a pore open area of about 11%. For slotted pores, the slot widths can be made arbitrarily narrow, consequently, the emitting area approaches 100% while pore evaporation losses are minimized. A slotted-pore cathode should, therefore, be capable of higher and more uniform current density with less barium dispensation. When the pore geometry is controlled, either for round holes or slots, the cathode should be less prone to space-charge-limited slump than those based on random sintered pores

1. Introduction

Over 25 years ago, Rittner, Ahlert, and Rutledge [l] performed one of the classical experiments in surface diffusion of barium on tungsten. They defined “effective diffusion length” as the distance from a pore opening where the surface coverage was reduced to the point where space-charge-limited (SCL) emission could no longer be supported. They assigned a surface coverage value of - 0.67 monolayers as the minimum required for SCL current, and measured the effective diffusion length, so defined, from a slotted opening in a tungsten ribbon. Since their measurements showed an effective diffusion length of 200 to 400 pm at normal cathode operating temperatures, while the pore separation distance in sintered tungsten cathodes is on the order of only 10 pm, they predicted full surface coverage over the entire life of impregnated * This paper is based on work performed at COMSAT Laboratories under the sponsorship of the Communications Satellite Corporation. ** Mr. Gibson is currently with North American Philips Lighting at the Bath, NY location.

0169-4332/85/$03.30 0 Elsevier Science Publishers (North-Holland Physics Publishing Division)

B.V.

B.A. Free, R. G. Gibson / Dependence of surface coverage on pore geometry

359

i

I

I

I

I

I

I ,

I

I

I

I

I

(a) CIRCULAR PORE Fig. 1. Idealized

cathode

(b) SLIT

pore geometry.

dispenser cathodes and, consequently, no reduction of SCL current with time. More recent studies, however, have firmly established the existence of slump in SCL current [2,3], and the fact of incomplete coverage at the beginning of life (BOL) which decreases with age [4-61. This apparent contradiction is reconciled by the central theme of this paper, namely, that effective diffusion lengths for slotted pores are greater than those for circular or irregular pores. Cathodes fabricated with the proper slotted-pore geometry are predicted to give better performance in terms of higher and more constant current density, more uniform emission over the area of the cathode, and lower dispensation rate of barium

2. Theory Consider the case of two sets of ideal cathode pores consisting of, respectively, thin slits and round holes, as illustrated in fig. 1. An activator is present in some form and concentration at the pore perimeters, and diffuses out over the hot surface, from which evaporation takes place. At equilibrium, the diffusion and evaporation terms are equal, so that v( Dve)

= e/7

(1)

holds, where 8 is the coverage, and both the diffusion coefficient D and the average residence time T are functions of 13. For slit geometry, this reduces to one-dimensional diffusion: D d28 I dD d8 dx -=-’ dx dx2 For round-hole force:

8 r geometry,

(2) however,

radial diffusion

in two dimensions

is in

360

B.A. Free, R.G. Gibson / Dependence

This can be written [7] as:

in polar coordinates

ofsurfacecooerage on pore geometry for the special case of axial symmetry

D$+(;+dJ$=$

(4)

The rigorous treatment cannot be pursued further with profit at this time, since both the residence time and the diffusion coefficient are strongly affected by both B and surface oxygen. Adequate values, especially for r, are simply not available. 2.1 Approximate

Treatment

It has been customary, following Moore and Allison [8], Rittner et al. [l], and more recently, Fote and Luey [9] to replace eqs. (2) and (4) with simpler forms:

d28

1 df? r dr2+--=r. dr

8P gc

(6)

in which constant values of both D and 7, determined at 8 = 1.0, are used, and p is a constant whose value is empirically chosen to compensate for real variations in D and 7. This technique works reasonably well for the case of metallic barium on tungsten [8,9]. For this case, as shown in table 1, the values of D and T do not vary too wildly as the coverage decreases, and the value of O/D7 (where both D and T are functions of (3) can be approximated by 8P/rcDc. Comparison of the last three columns of table 1 shows that p = 8 provides a much better fit to real behavior than does the more conventional value of P = 5. The fit is purely empirical, based on the referenced values.

Table 1 Coverage dependent values of D and 7 and determination of best value for p for barium on tungsten

1.0 0.8 0.6 0.4

5.7 2.2 7.7 4.1

x x x x

lo2 103 10’ 104

4.2~10-~ 5.6~10-~ 1.1 x10-s 3.OXlO~”

4.1 6.5 7.1 3.3

x x x x

105 lo4 103 102

p=5

p=8

4.1 x lo5 1.4x105 3.2 x i04 4.2.x lo3

4.1 x 7.0x 7.0 x 2.7 x

105 104 103 lo2

‘) Reciprocal of slopes of Moore and Allison evaporation curve at 1100 K, fig. 9 of ref. [8]. b, Values taken from Fote and Luey, fig. 8 of ref. [9], and corrected to 1100 K by eq (8) of ref. [l].

B.A. Free, R. G. Gibson / Dependence OJsurface cooerage on pore geometry Table 2 Values of r for Ba/BaO

e

on pure and oxidized tungsten near monolayer coverage

Values of r(s) Baon W (1100 K)

1.0 0.8

361

5.7x102 2.2 x 10s

Ba on oxidized tungsten ‘) (1348 K)

(1400 K)

9.4 x 10’ 3.3 x 103

2.3 x lo4 1.2x108

L-cathode effluent on oxidized W b,

a’ Secured graphically from fig. 8 and 13 of ref. [l].b) Estimated from rs,, = ra exp(e/k)(&/Ts E,/Ta), where E is the adatom binding energy, as given in table 3 of ref. [l], and B is related to the factor f of table 3 through fig. 8 of ref. [l].

These barium on tungsten values cannot be used to estimate diffusion lengths in real cathodes. In table 1,. the diffusion length for 8 = 1, estimated as 6, is only 15 pm, even at the relatively low temperature of 1100 K. Under conditions approaching those of real, oxygenated cathode surfaces, the diffusion length for barium desorption is many times this value. As shown in table 2, the presence of oxygen not only has a strong influence on T, but also increases the dependence of 7 on 8 dramatically, so that even near monolayer coverage, the value of r varies so wildly as to preclude the use of approximation treatments. From the above considerations, it is apparent that we cannot use closed-form approximations such as eqs. (5) and (6) to describe simultaneous diffusion and evaporation of Ba/BaO mixtures on refractory metal surfaces with any semblance of accuracy. Nor can we use numerical solutions to the more exact equations, since at present we have sparse information on the variation of T and D with 8, for the case of Ba/BaO mixtures. However, we can gain considerable insight into the effects of one- and two-dimensional diffusion by means of a worst-case approach. 2.2. A conservative approach Consider eq. (5) again, with the value of p set to unity, i.e., no compensation for real variation in D and 7. Since both D and 7 increase as the coverage goes down, solution of the equation in this form gives pessimistic (i.e., low) values of 8 as a function of distance from a slotted pore. Now reconsider eq. (6) modified in the same manner and applied to very small pores. Near the pore edge, where r is very small, the radial expansion effect is more im$ortant than evaporation in reducing the value of 8 with increasing r. In this region, neglecting the dependence of D and r on B will not greatly affect the results and the calculated values of 8 will be more realistic. Somewhat further out, the radial expansion term becomes less important, until at large r the equation

362

B.A. Free, R.G. Gibson/ Dependenceof surfacecoverageon pore geometry

- - - SLOTTED PORES O-0 CIRCULAR PORES, --. CIRCULAR PORES,

50

200 150 100 DISTANCE FROM PORE EDGE II

r = 172 r =5~

250

p

300

Fig. 2. Calculated surface coverage for slotted and circular pores.

approaches eq. (5). Thus, for circular pores also, but to a lesser extent, low values of 8 will be calculated for constant values of D and 7. 2.2.1. Theoretical estimates The solutions to eqs. (5) and (6) for constant D and r are: 19/e, = exp( -x/a),

(7)

where x is the distance from the slotted pore, r is the distance from the center of a circular pore, and Ho is the zero-order modified Bessel function (Hankel function). Approximate values for a, the true diffusion length, range from Rittner’s “effective diffusion lengths” of 360 pm at 1333 K, 200 pm at 1470 K to Thomas’ [ll] recent estimates of about 115 to 92 pm over the same range of temperature. In this subsection, we arbitrarily assume intermediate values of 250 and 150 pm. We also use a range of coverage, 0.67 and 0.50, to correspond to the effective diffusion length. Fig. 2 shows the coverage as a function of distance from slotted pores and two sizes of circular pores, as calculated with eqs. (7) and (8).

B.A. Free, R.G. Gibson / Dependence of surface coverage on pore geometry

363

Table 3 Effective diffusion lengths for several pore geometries Slots

6

250 150

Holes, r = 172 pm

Holes, r=5gm

t’= 0.67

B = 0.50

0 = 0.67

0 = 0.50

8 = 0.67

e = 0.50

100 60

173 104

69 45

118 68

14 11

34 25

Table 3 shows the larger effective diffusion lengths, corresponding to coverage of 0.67 and 0.50, for slots versus holes. For circular pores, at first glance it appears that the advantage rests with larger diameter holes. However, for equivalent geometries, the smaller diameter holes would be spaced correspondingly closer together and a more valid comparison is secured by considering the distance from the pore edge in terms of the pore radius, as shown in fig. 3. Here it is very clear that the coverage will be much better for small circular pores. This effect will be covered in more detail in a later section. We conclude from the foregoing treatment that slotted pores should theoretically provide higher coverage than circular pores, but that small circular pores spaced in a regular array should give at least adequate coverage. Experimental evidence to support this is given in the next section.

0.7 ;

0.6

s k

0.5

6 0 0.4 0.3

0.5

1 .o

1.5

2.0

2.5

DISTANCE FROM PORE EDGE (radii)

Fig. 3. Calculated surface coverage for circular pore of various radii.

3.0

364

B.A. Free, R.G. Gibson / Dependence of surface coverage on pore geometry

3. Experimental The general approach here is to verify the method demonstrated in ref. [l] and then to modify the design so as to (a) broaden the scope of the experiment to include circular holes, and (b) to eliminate any potential source of error in the original design. 3. I. Apparatus

and procedure

The apparatus is sketched in fig. 4, and pertinent dimensions are given in table 4. The changes from ref. [l] are (a) a somewhat wider ribbon to avoid any effects of diffusion around the outer edges, (b) a narrower slot to prevent excessive buildup of activator on the anode, and (c) the use of a circular anode to simplify maintenance of the anode/guard gap. The variation of the emitting areas for round versus square anodes is negligible for these dimensions. The procedure is as follows: with the cathode temperature lowered for negligible dispensation, the ribbon and anode are cleaned at about 1500 to 1550 K until the gap current becomes negligible at a gap voltage of - 25 V (about 4 to 6 h). This treatment is adequate for removal of barium but not

-n A

0

(a) SIDE VIEW OF L-CATHODE C, RISBON R. ANODE A, AND ANODE GUARD G

(b) AXIAL VIEW OF SLOTTED RIBBON, ANODE, AND GUARD

(c) AXIAL VIEW OF RIBBON WITH HOLES, ANODE, AND GUARD

Fig. 4. Experimental

setup for migration

length determination.

B.A. Free, R. G. Gibson / Dependence of surface coverage on pore geometry Table 4 component

dimensions

Ref. 111 This work

for diffusion

length experiments

365

(mm)

Ribbon width

Slot width

Hole diam./spacing

Anode diameter

gap

5.0 6.4

0.25 0.06

_ 0.344/1.5

3.0 3.5

0.5 0.5

Anode/ribbon

oxygen. Then, in quick succession, the anode heater is turned off, the ribbon temperature is reduced to the test value, and the cathode temperature is raised to a value that will dispense sufficient Ba/BaO to the back side of the ribbon to ensure complete surface coverage at equilibrium. After a suitable interval, the L-cathode effluent diffuses around the edges of the slot or holes and onto the front surface of the ribbon in a narrow band surrounding the openings. As the activator diffuses out over the surface, evaporation takes place until at some distance, d, the coverage is reduced to a level that will not support SCL electron emission. It has been customary to associate this distance with a coverage equal to about 2/3 of a monolayer, where saturated emission is reduced to about 20% of the full coverage value. 3.2. Results Measurement of the steady-state gap current and estimation of the XL current density is all that is required to determine the emitting area and associated diffusion length. Herrmann and Wagner [lo] give a suitable relationship for SCL current: J SCL=

2.33 x 1O-6 (v, - +a + G*)~” (I + A),

(9)

(d-z)’

where VP is the voltage across the gap d; & is the anode work function; +* is the effective work function, including the space charge hill; z is the electron sheath width; and A is a correction for initial electron velocity. The diffusion length for the slotted ribbon is given by I= &,,/2DJs,,

3

where D is the diameter of the anode, and Zobs is the recorded equilibrium. For round hole geometries, the diffusion length is

00) current

at

where r is the radius of the pore, and n = the number of pores. The experimental values for one-dimensional diffusion length (i.e., slit geometry) are given in fig. 5, upper curve. The results are nearly identical with

B.A. Free, R.G. Gibson / Dependence of surface coverage on pore geometry

366

4-

o

REF 1

7

+ PRESENT

WORK

0.344-mm

4

I 6.6

I

I 6.7

I

I 6.8

I

I 6.9

104/T Fig. 5. Diffusion

lengths

I

I

7.0

I

I 7.1

dia

I I 7.2

(K)

for slits and circular

pores.

those obtained by Rittner [l], despite the fact that our slit width is less than a fourth of his. This is certainly the expected result for one-dimensional diffusion, viz., the diffusion length does not depend on the slit width. Experimental values for two-dimensional diffusion (i.e., round hole geometry) are shown in the lower curve of fig. 5. Even with these relatively large holes, 0.344 mm diameter, the radial diffusion has a pronounced effect. Diffusion lengths are only about one-third as much as for slit geometry. Note that the diffusion lengths (60 to 80 pm) are considerably smaller than the hole radius (172 pm). This is expected from the theory, since, for holes of this size, the evaporation term of the diffusion equations is very significant. Note also that the experimental effective diffusion lengths for slots are about twice those of the conservative theoretical estimates of table 3, whereas the observed effective diffusion lengths for 0.344 diameter holes are only about 20 to 30% larger than calculated. The next logical step was to extend these experiments to the micron region, the size of cathode pores. We conducted a few unsuccessful experiments with a laser-drilled array of more or less circular holes, supplied through the courtesy of the Naval Research Laboratory and Hughes Aircraft Company. The holes were approximately 3 to 5 pm diameter on 15 nm centers. Although the

B.A. Free, R. G. Gibson / Dependence of surface coverage on pore geometry

367

Table 5 Percentage open area for three geometries tested Percent open area

Geometries

Slot

2.0

Three 0.344 mm diameter holes Array of 3 to 5 pm holes

2.8 5.6

experiment was repeated exactly as above, results were different in several respects: the gap current and calculated diffusion length were much too low, the gap current generally showed a second increase after an initial short plateau, and the gap current (and calculated diffusion length) appeared to be greater when the dispenser temperature was raised. We attribute these anomalies to the faulty topology and geometry of the array of micron-sized holes. Laser drilling is a violent process. The surface adjacent to the holes is greatly distorted on the (laser) entrance size, and even the exit side, which we used as the emitting surface, is considerably disturbed for one or two microns away from the hole. It may be more than coincidental that the diffusion length, calculated from the height of the initial short plateau, is about 1 pm. The gradual increase in gap current after the initial plateau may arise from two mechanisms: either the surface diffusion process proceeds with more difficulty outside the disturbed surface adjacent to the hole, or the emitting surface gradually accumulates activator by some other process. In the latter case, it is interesting to note that the open area for the array of micron-sized holes is considerably larger than for the other two geometries tested, as shown in table 5. This may lead to excessive evaporation of activator directly onto the anode through the holes, and possibly radiation heating of the anode through the same holes. Under these circumstances, back evaporation of activator from the anode onto the emitting side of the ribbon could occur, thereby causing a rise in gap current which is sensitive to the dispenser temperature. Aside from this speculation, we have no present explanation of the large discrepancy between our low measurements and the values of 11 to 34 pm estimated from theory (table 3). However, since Thomas’ direct measurements [ll] show a coverage reduction of - 50% at tens of microns away from a 25 pm diameter circular hole, we think that our results with the laser drilled array are invalid. 3.3. Application

of results to sintered cathodes

The tungsten ribbon used for these experiments is, except for surface morphology, very like a sintered cathode surface. That is, the oxygen composition, aging in the presence of Ba/BaO flow, activation process, and operating temperature are all close to real cathode values. Therefore, the true diffusion length, \/07, should be about the same. The resulting coverage, however, is an

B.A. Free, R. G. Gibson / Dependence of surface coverage on pore geometry

368

open question. On the one hand, the irregular shapes of sintered pores (some approaching circular shape and some slot-like) would argue that effective diffusion lengths for real cathodes should be somewhere between that of slots and round holes. Conversely, the random scattering of the pore size and position in sintered cathodes might lead to flooding in pore-rich areas, and very poor coverage in areas where pores are scarce. The latter argument, which the authors favor because of the pictorial evidence of Schaefer and White [12], would lead to poorer overall coverage on sintered cathodes, relative to both ordered round holes and slots.

4. Optimization of pore geometry From fig. 3, it is obvious that small circular pores are superior to large ones. In this section, we examine slotted and small hole array geometry to maximize the emitting area. The following conservative assumptions will ensure that we do not overestimate the emission capabilities: (a) Space charge limited emission for typical applications requires surface coverage > 0.67. (b) The pores of mature cathodes do not emit. (c) The coverage adjacent to one hole is not reinforced by diffusion from nearby holes. 4. I. Circular pore geometry To simplify the example we will assume a hexagonal array of 10 pm diameter holes, assign a value of \/07 = 150 pm, and extract from fig. 3 a coverage of 0.67 at a distance r = 3.2r,, when 8, = 1.0, and r = 2.47r,, when 8, = 0.9. Under these conditions, the following relationships hold for the simple geometry shown in fig. 6, with the ratio r/s = n: A,,, = 0.433s2,

(12)

A holes

03)

=

sn2s2/2,

A CO”= avrn2s2,

b =Gs/2,

04)

Acov=A,o,-A,,o,es,

bad,

(15)

where a is 4.62 and b is 3.2 when 0, = 1.0, and a is 2.55 l9, = 0.9. Fig. 7 shows the fractional coverage estimated as radius/pore spacing. On the right, the common portion sents the condition when the entire solid area is covered monolayer, and only the area of the holes themselves

and b is 2.47 when a function of pore of the curves repreby at least 2/3 of a detract from full

B.A. Free, R.G. Gibson / Dependence of surface coverage on pore geometry

Fig. 6. Geometry of a unit cell of hexagonal array of circular holes.

0.9

c

I

I

I

0.2

0.3

0.4

PORE RADIUS/SPACING

Fig. 7. Effect of pore size and $ on pore array geometry

0.5

369

370

B.A. Free, R.G. Gibson / Dependence of surface coverage on pore geometry

Table 6 Coverage for slit geometry Coverage (monolayers) x=0

X=39pm

1.0 0.9

0.77 0.69

Open area fraction

0.025 0.025

Emitting area fraction

0.975

* 0.975

emission. To the left of the maxima, the pore radius/spacing is low enough so that the radial surface diffusion is not sufficient to cover the entire solid surface area, and both the open area of the pores and the area covered by less than 2/3 of a monolayer detract from full emission. For pores with 5 ym radius, the maximum emitting area fraction is about 0.88 at r/s = 0.18, with a pore open area of nearly 0.12, provided that the coverage at the pore edge remains at 1.0. If the coverage at the pore edge decreases, say to 0.9 over the lifetime of the cathode, the emitting area (coverage > 0.67) would drop to emitting area fraction 0.60. At a value of r/s = 0.23, the initial maximum would be only 0.80 but would remain virtually unchanged as the coverage at the pore edge decreased to 0.9. For comparison, a similar set of curves is shown for pores with 12.5 pm radius. As can be seen, estimates for the larger pores show smaller maxima at larger values of T/S, but are less sensitive to drop in emitting area as the coverage at the pore edge decreases. 4.2. Slit geometry The characteristics of slit effective diffusion length is example, a slit width of 2 pm easily derived. In this example, the initial no reduction as the coverage

geometry are much less complicated, since the the same for all slit widths. If we assume, for and a spacing of 78 pm, the values in table 6 are emitting area fraction is very high, and there is at the pore edge is reduced from 1.0 to 0.9.

5. Discussion and conclusions Diffusion theory clearly shows that properly spaced slotted pores should provide higher and more uniform surface coverage than arrays of circular pores. The effect is fundamental, deriving from the difference between one-dimensional diffusion (slots) and two-dimensional diffusion (round holes). Although the diffusion equations cannot be used quantitatively with present information, the conservative values calculated from a worst case approxima-

B.A. Free, R. G. Gibson / Dependence of surface coverage on pore geometry

371

tion are fairly good estimates of the experimentally determined values. As predicted in the discussion on theory, the calculated values are more conservative for slotted pores than for circular pores. The purely geometrical calculations of section 4 show that circular pore array geometry can be optimized to produce coverage characteristics approaching that of slotted pores, provided that very small circular pores can be fabricated. Though the calculations are not exact, the approach is conservative and the trends unmistakable. It is also clear from this section that regular arrays of pores, either circular or slotted, can be overdesigned so as to minimize space-charge-limited slump. The same cannot be said for the random size, shape, and spacing of pores in the case of sintered cathodes.

References [l] E. Rittner, R. Ahlert and W. Rutledge, J. Appl. Phys. 28 (1957) 156. [2] A.M. Shroff, P. PaIlueI and J.C. Tonnerre, Appl. Surface Sci. 8 (1981) 36. [3] R. Longo, Life Test Studies on Dispenser Cathodes, Tri-Service Cathode Workshop, Naval Research Laboratory, Washington, DC, February 1978. [4] G. Haas, H. Grey and R. Thomas, J. AppI. Phys. 46 (1975) 3293. [5] R. Forman, J. Appl. Phys. 47 (1976) 5272. [6] M. Greene, Dispenser Cathode Physics, Final Technical Report, RADC-TR-81-211, Varian Associates, Inc., July 1981. [7] W. Jost, Diffusion (Academic Press, New York, 1960). [8] G. Moore and H. Allison, J. Chem. 23 (1955) 1609. [9] A. Fote and K. Luey, Barium Transport Processes in Impregnated Dispenser Cathodes, Aerospace Corporation Preprint, 1980 T&Services Cathode Workshop, Rome Air Development Center, Griffis Air Force Base, Rome, NY, April 1980. [lo] G. Herrmann and S. Wagner, The Oxide Coated Cathode, Vol. 2 (Chapman and Hall, London, 1959). [ll] R. Thomas, Appl. Surface Sci. 538. [12] D. Schaefer and J. White, J. Appl. Phys. 23 (1952) 669.