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Corrected field enhancement factor for the floating sphere model of carbonnanotube emitter

Evgeny G. Pogorelov,1 Yia-Chung Chang,1 Alexander I. Zhbanov,2,a and Yong-Gu Lee21Research Center for Applied Sciences, Academia Sinica, 128, Section 2, Academia Road Nankang,

Taipei 115, Taiwan2Department of Mechatronics, Gwangju Institute of Science and Technology (GIST), 1 Oryong-dong, Buk-

gu, Gwangju 500-712, Republic of Korea

Received 18 March 2010; accepted 24 June 2010; published online 17 August 2010We have corrected the field enhancement factor for the floating sphere at emitter-plane potentialmodel with the finite anode-cathode distance. If is the radius of sphere, h is the distance fromcathode to the center of sphere, and l is the distance from the center to the anode, then the fieldenhancement factor is given as the following expression sph= 2 +7

2 2 2+ 2 / 2 1 2 , where =/ h, =/ l. This expression demonstrates reasonable behavior for threelimiting cases: if h, if l, and if l. We have compared our factor sph with the fieldenhancement factor tube for the hemisphere on a post model and the factor ell for thehemiellipsoid on plane model. We have shown realization of the approximate evaluation tube sph+ell /2. 2010 American Institute of Physics. doi:10.1063/1.3466992

I. INTRODUCTION

Since the first reports on remarkable field emission prop-erties of carbon nanotubes CNTs Refs. 13 in 1994 andthe first journal papers dedicated to this problem47 in 1995,significant efforts have been devoted to the application ofCNTs for electron sources.

Now area of its practical application includes a widerange of field-emission-based devices such as flat-paneldisplays,810 backlight unit for a liquid crystal displays,11,12

electron microscopes,13 vacuum microwave amplifiers,14,15

x-ray tube sources,16,17 cathode-ray lamps,18,19 nanolithogra-phy systems,20,21 etc.

The electric field Etop at the CNT tip increases comparedwith the average field E0 according to the expression Etop=E0, where is a field enhancement factor. The emissioncurrent is very sensitive to the field enhancement factor which rises up to 3000 or more depending on the tube aspectratio.

Four of the simplest models for calculation the field en-hancement factor are the hemisphere on a plane model, thefloating sphere at emitter-plane potential model, the he-miellipsoid on plane model, and the hemisphere on a postmodel. We follow to the classification suggested by Forbes etal.

22 The schemes of models in diode configuration are illus-trated in Fig. 1, where L is the anode-cathode distance; H is

the total height of emitters; is the radius of spheres orradius of curvature for hemiellipsoid; h is the distance fromcathode to sphere centers or distance to focus for hemiellip-soid; and l =L h.

Three first models allow analytical solutions. Simpler isthe case when the anode-cathode distances, L are muchlarger than the height of emitters, H.

In the case of infinite anode-cathode distance, L thehemisphere on a plane model can be analyzed exactly for

example Refs. 2224 , yielding hemi0 =3. This solution is

very useful as a limit case for all other models under consid-eration when h =0.

The analytical approximation for the floating sphere atemitter-plane potential model22,23 in the case L andH/1 is

sph0

H

+ 2.5. 1

Unfortunately this approach does not work well for tubes ofsmall aspect ratio.

The exact analytical solution for the hemisphere on aplane model22,24 is as follows:

ell0 =

23

1 2 ln1 +

1 2

, 2

where =1 /H is the eccentricity of ellipse.

a Electronic mail: [email protected].

FIG. 1. Color online Schemes of simplest models for field enhancementfactor estimation: a hemisphere on a plane; b floating sphere at emitter-plane potential, c hemiellipsoid on a plane, and d hemisphere on a post.

JOURNAL OF APPLIED PHYSICS 108, 044502 2010

0021-8979/2010/108 4 /044502/7/$30.00 2010 American Institute of Physics108, 044502-1

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The model of a hemisphere on a post allows only thenumerical solution even in the case of infinite anode-cathodedistance, L. There are many numerical results obtainedby various researchers which have been generalized bysimple algebraic formulas of field enhancement factor for anindividual nanotube and assembly of nanotubes. The mainproblem for such algebraic fitting formulas is the lack of adefinitive proof of their accuracy. The very accurate and

popular formula belongs to Edgcombe et al.22,25,26

tube0 1.2

H

+ 2.5

0.9

. 3

All these factors submit to common sense

sph0 tube

0 ell

0 hemi

0 . 4

In the case of large anode-cathode distance, L compari-son of field enhancement factors for the floating sphere atemitter-plane potential model, the hemiellipsoid on planemodel, and fitting formula for the hemisphere on a postmodel gives us the approximate relation

tube0

sph0 + ell

0

2. 5

Next case is the case of a diode configuration when theanode-cathode distances, L is comparable with the height ofemitters, H.

For the floating sphere at emitter-plane potentialmodel Wang et al.23 supposed the analytical approximation

sph sph0 + 1.202

h

L

3

. 6

The Eq. 6 is accurate enough for H0.5L. If the anode-cathode distance tends to infinity, L then sphsph

0 ,that is right. If the floating sphere touches with anode thenthe field enhancement factor must tend to infinity. Accordingto Eq. 6 , if HL then sph=H/+3.702 is the limitedvalue that is incorrect.

For the hemisphere on a post model in diode configu-ration Bonard et al.27 introduced numerically fitted formula

tube 1.2H

+ 2.5

0.9 1 + 0.013 L HL

1

0.033

L H

L .

7

The Eq. 7 demonstrates reasonable behavior for both limitcases: if L then tubetube

0 ; if HL then tube.The inequality 4 has to be true for the floating sphere

at emitter-plane potential model and the hemisphere on apost model in diode configuration, sphtube. For somesets of parameters, we can show the violation of this inequal-ity. Thus we assume that Eq. 7 is not perfectly correct.

In the present work we reexamine the field enhancementfactor for the floating sphere at emitter-plane potentialmodel with the infinite and finite anode-cathode distance tosolve the problem mentioned above.

II. CALCULATION OF THE ENHANCEMENT FACTORWITH THE INFINITE ANODE-CATHODEDISTANCE

A. Hemisphere on a plane

The metallic sphere in a uniform electric field E0 Fig.1 a was considered in many papers for example Refs.2224 . We can replace the sphere by point electric dipole. If

the electric dipole moment is p0 then the dipole potential is

dip = p0

40

z

z2 + r2 3/2. 8

Equation of circle is dip+zE0 =0. From this equation, wecan find the relation between the electric dipole moment andthe sphere radius: p0 = 40E0

3. The electric field on the topof hemisphere reaches Etop=p0 /20

3 +E0 = 3E0. The fieldenhancement factor is hemi

0 =Etop/E0 =3. The field distribu-tion over the hyperboloid surface have the form E=3E0 cos , where is a polar angle.

B. Floating sphere at emitter-plane potential

The floating sphere at emitter-plane potential modelhas no body of the field emitter and possesses only itshead. This model gives too high estimation of electric fieldon the apex of nanotube but plausibly reproduce tendenciesof change in the field enhancement factor. Approximate ana-lytical solution for the floating sphere at emitter-plane po-tential model is well known for example, Refs. 22 and 23 .To solve this problem the method of images28 is usuallyused.

The charge q0 = 40hE0 and the electric dipole p0=40E0

3 placed at point A Fig. 2 create a sphere of

radius and potential =0 in uniform external electric field.The charge q0 and dipole p0 cause a potential variationacross the emitter plane. To correct this we have to place animage-charge q0 and image-dipole p0 at point A behind theemitter plane. The image-charge and image-dipole will dis-tort the surface of sphere. To restore the shape we shouldplace additional charge q1 = q0/ 2h and dipole p1=p0

3/ 8h3 at point B on the distance s1 =

2/ 2h from the

center of sphere see Fig. 2 . Next, we have to put q1 and p1at point B, after to put q2 and p2 at C and so on.

Neglecting terms of higher smallness in this series ofapproximation we find the electric field on the top of floatingsphere

FIG. 2. Color online Two conducting spheres of radius at cathode po-tential in uniform electric E0.

044502-2 Pogorelov et al. J. Appl. Phys. 108, 044502 2010



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Etop =1

40

q0

2+

2p040

1

3+

1

40

q1

s1 + 2 + + E0

E0h

+ 3.5 . 9

In the expression 9 , we assume s1 0 and neglect thecharges q2, q3 ,..., and the dipoles p1, p2 ,.... So we acceptthe idea of charge centralization in the center of sphere anduse only the initial charge q0, the image-charge q1, and theinitial dipole p0. Thus, the field enhancement factor is

sph0 =

Etop

E0

h

+ 3.5 =

H

+ 2.5. 10

The initial charge q0 gives us h /, the image-charge q1brings 2, the initial dipole p0 adds 1/2, and the external elec-tric field E0 contributes 1 to the field enhancement factor.

We can provide more accurate calculations. Recurringformulas for the distance si+1, the charge qi+1, and the dipolemoment pi+1 through si, qi, and pi are the following si+1=2 / 2h si , pi+1 =pi

3/ 2h si

3, and qi+1 = qi/ 2h si

pi/ 2h si2

, where the initial distance is zero: s0 = 0. Letsnote here that the dipole pi causes not only an image-dipolepi+1 but also an additional charge pi/ 2h si

2.The exact analytical expression for the field enhance-

ment factor is as follows

sphex = 1 +

i=0

iq + i

p , 11

where iq = qi /40E0 1 / + si

2 1 / 2h + si2 and

ip = 2pi /40E0 1 / + s i

3 + 1 / 2h + si3 .

Series expansion of the exact field enhancement factor is

sphex

= 1

+

7

2

1

2+

1

82

+

7

163

25

324

+

25

325

+ O 6 , 12

where =/h.We can see that exact analytical value of the field en-

hancement factor, sphex is very close to approximate one, sph

0 .Larger difference takes place for a smaller aspect ratio. Themethod of images does not suppose penetration of sphereinto the cathode. We can consider only point contact betweensphere and emitter plane. In the limit case h=, we calcu-lated that sph

ex =4.207 and sph0 =4.5. Thus, the approximate

value of the field enhancement factor, sph0 gives us accurate

estimation from above.

III. INFLUENCE OF THE FINITE ANODE-CATHODEDISTANCE ON THE FIELD ENHANCEMENTFACTOR

The presence of a flat anode placed at a distance com-parable with the height of the nanotube has strong influenceon the field enhancement factor. To estimate the influence ofthe finite anode-cathode distance, we accept the idea ofcharge centralization in the center of sphere. The basic con-ception of calculation consists in replacement of the cathodeand the anode by an infinite set of image charges as shownin Fig. 3. The negative and positive charges have the same

spacing, 2L. It is clear that planes z =0 and z =L are planes ofsymmetry for point charges and will have potentials 0 oncathode and V=E0L on anode.

We consider the central sphere between the anode andthe cathode and also its two nearest neighbors separately.These three spheres are marked in Fig. 3 by red circles.

The electrostatic potential created by all right imagecharges at point T is

right =

q

40n=0

1

2 l + n + 1 L

1

2L n + 1=

q

80

1 + l/L +

L, 13

where is digamma function, 0.57722 is the EulerMascheroni constant.

The electrostatic potential created by all left charges is

left =q

40 n=0

12 h + n + 1 L

1

2L n + 1

= q

8

0

1 + h/L +

L

. 14

The external potential from left and right sets of imagecharges is

ext = right + left = q

80

1 + l/L + 1 + h/L 2

L.

15

We shall take into account only influence of the point dipoleplaced at the central sphere and neglect all others. The po-tential created by a dipole at point T is

dip = p0

4

0

2 . 16

As the floating sphere has cathode potential we shouldequate to a zero the total potential created by a uniform elec-tric field, all electrical charges, and a dipole in point T as thefollowing:

E0 h + + ext q

40

1

1

2l

1

2h +

p0

402

= 0. 17

Substituting in Eq. 17 , the dipole moment, p0 = 40E03

and the external potential ext from Eq. 15 and solvingequality

FIG. 3. Color online Infinite set of image charges for the simulation ofa planar anode.




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E0h q

80

1 + l/L + 1 + h/L + 2

L

q

40

1

1

2l

1

2h + = 0, 18

we find the electric charge at the center of floating sphere

q = 80E0h

1 + l/L +

1 + h

/L + 2

L+ 2

2

2l

2

2h +

1

. 19

The electric field at the top of sphere is

Etop = E0 ext

z

z=h+

+q

40

1

2+

1

2l 2

+1

2h + 2+

1

40

2p03

. 20

It is easy to show that in a small neighborhood of point T theexternal potential ext changes slowly, ext / z z=h+0therefore we neglect a partial derivative in Eq. 20 . Thus,the electric field at point T is

Etop =q

40

1

2+

1

2l 2+

1

2h + 2+ 3E0. 21

Introducing the dimensionless parameters =/ h and = h /L, we find the field enhancement factor

sph =Etop

E0=

2

2+

22

2 2 2+

2

2 + 2

2 + 1 + + 2 +2

2

2 22

2 +

+ 3. 22

In the limit 0, we have the case of infinite anode-cathodedistance. The series expansion in small parameter gives ussph

0 1 /+7 /2, what coincides with Eq. 1 and 10 .The expansion of our solution 22 in two small param-

eters and gives the approximation of Wang et al.23

sph

1

+

7

2 + 3 3

, 23

where is the zeta function, 3 = 1 + 1 /23 + 1 /33 +1 /43

+ . . . 1.20206.In Fig. 4, we see that Wangs approximation works very

well in the case h0.6L.

The divergence becomes very large when the sphereclosely approaches to the anode. We used to this illustration=0.005 but behavior is the same for other ratios / h.

IV. RESULTS AND DISCUSSION

A. Comparison of enhancement factors for theinfinite anode-cathode distance

Comparison of field enhancement factors for the float-ing sphere at emitter-plane-potential model green solidline , the hemiellipsoid on plane model red dashed-and-dotted line , and fitting formula for the hemisphere on apost model blue dashed line is shown in Fig. 5. The purpledotted line corresponds to sph

0 +ell0 /2.

We see realization of the inequality 4 . Also we canconclude that approximate equality 5 is satisfied with verygood accuracy.

B. The field enhancement factors for the finite anode-cathode distance

Lets consider the behavior of field enhancement factorsfor the floating sphere at emitter-plane potential model, thehemiellipsoid on plane model, and the hemisphere on apost model with finite anode-cathode distance.

We will use Eq. 22 for the floating sphere at emitter-plane potential model and Eq. 7 for the hemisphere on apost model. For the hemisphere on a plane model withfinite anode-cathode distance we will apply the analyticalapproximation supposed by Pogorelov et al.24

FIG. 4. Color online Comparison of our solution 24 with the approxi-mation of Wang et al. Ref. 8 for the field enhancement factor vs = h /L.




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ell = 23

1 2 ln1 +

1 2 P

, 24

where

P =n=1

2n 1 ln2n 1 + 2n 1

2n

+ 1 ln2n+ 1 +

2n+ 1

and =L /H.

For comparison of factors, we have two limiting cases.The first limit corresponds to the hemisphere on a planemodel with infinite anode-cathode distance when h= 0, L, and =3. The second limit arises if the gap betweenthe anode and the apex of the emitter is very small as shownin Fig. 6.

In this case, we can consider a gap between the anodeand the emitter as the parallel-plate capacitor. Local fieldnear the apex is Etop= V/ l , where V is applied voltage.The average field is E0 = V/L. Thus, the asymptote for fieldenhancement factor is

asym =L

l =

1

l . 25

The field enhancement factors have to approach to thisasymptotic curve from above.

Also all factors must submit to clear inequality

sph

tube

ell

asym, 26

which is similar to inequality 4 for infinite anode-cathodedistance.

Comparison of field enhancement factors is shown inFig. 7. The purple dotted line corresponds to the asymtoticfield enhancement factor asym. Color and style of other linesare same as in Fig. 5.

If we keep the height and the radius of emitters and wevary the anode-cathode distance, then the field enhancementfactors change how it is shown in Fig. 7 a . Change in thefield enhancement factors with change in the height of emit-ter is shown in Fig. 7 b . We see that all functions demon-

FIG. 5. Color online The field enhancement factors vs aspect ratio forthree models: floating sphere green solid line , hemi-ellipsoid red dashed-and-dotted line , and hemisphere on a post blue dashed line . The purpledotted line is average between the floating sphere and the hemiellipsoid.

FIG. 6. Color online Illustration of small distance between the anode andthe top of the emitter

FIG. 7.

Color online

Comparison of the field enhancement factors forthree models: a the anode-cathode distance, L is changing, =0.005; bthe height of emitter, H is changing, /L =0.005. The purple dotted line isthe asymptote.




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strate right behavior for two limiting cases: h =0 and H=L.The approximate equation tube sph+ell /2 is carried outin a wide range 00.8.

Despite of validity of inequality 26 for full diapason01, we see in Fig. 7 that if0.97, then tubesph.In other words, if the gap between the emitter apex and theanode is small, then the field enhancement factor for thehemisphere on a post model exceeds the field enhancement

factor for the floating sphere at emitter-plane potentialmodel.For example, if nanotube emitter have radius of 1 nm

and length of 1 m then according to our model in the diodeconfiguration with cathode-anode distance 1.5 m the fieldenhancement factor sph is 1004.5, the factor tube for theBonards model27 is 620.3, and the factor ell for the hemiel-lipsoid on plane24 model is 326.6. The approximate equalityis satisfied with quite good accuracy, tube 1004.5+326.6 /2 = 665.5. For smaller cathode-anode distance of1.02 we have sph=1030.6, tube =1024.0, and ell=478.6. We see sphtube, that contradicts common sense.Thus we believe arithmetical mean tube 1030.6

+478.6 /2 = 754.6 gives more reasonable value for shortcathode-anode distance.

We assume that our Eq. 22 is accurate enough but Eq. 7 overestimates the field enhancement factor for small gapbetween the emitter and the anode.

C. Simplification for the floating sphere at emitter-plane potential model

Our Eq. 22 for the field enhancement factor holds twodisadvantages. First, this expression is very complicated forfast estimations. For example, it is impossible to find sph by

using the usual scientific calculator. Second, the field en-hancement factor from Eq. 22 tends to infinity not if H=L but if H=L +/2. So we have the infinite field enhance-ment factor only if the floating sphere penetrates into theanode on half of its radius.

To solve the first problem we shall try to express thefield enhancement factor as product of the factor for the in-finite anode-cathode distance and the correcting multiplierwhich accounts finite distance between the anode and thecathode, sph=sph

0 sph. This representation is similar to whatBonard et al.27 used for the hemisphere on a post model inEq. 7 .

We will use the three first terms of series expansion

12for the exact field enhancement factor with infinite anode-

cathode distance

sph0 1 +

7

2

1

2=

2 + 7 2

2. 27

To obtain the correcting multiplier, we shall find the limit

sph = lim0

sph

sph0 . 28

Substituting sph from Eq. 22 and sph0 from Eq. 27 into

limit 28 and introducing new dimensionless parameter, =/ l, we can find

sph =

2 2 + 2

1 2 . 29

Thus, the field enhancement factor for the floating sphere atemitter-plane potential model with the infinite anode-cathode distance has the form

sph = 2 + 7 2 2 2 + 2

2 1 2 . 30

This field enhancement factor is very close to the factor fromEq. 22 , sphsph. Distinction between these two factors isinvisible in Fig. 7, they coalesce in one line.

It is remarkable, that Eq. 30 allows us to solve thesecond problem mentioned above. The field enhancementfactor sph tends to infinity if l or HL.

V. CONCLUSIONS

Using the method of images we have exactly calculatedthe field enhancement factor for floating sphere at emitter-plane potential model with the infinite anode-cathode dis-tance.

We have corrected this factor for floating sphere in diodeconfiguration between a flat anode and cathode. If we applythe dimensionless parameters =/h and =/ l, where isthe radius of sphere, h is the distance from cathode to thecenter of sphere, and l is the distance from the center to theanode, then the field enhancement factor is given as the fol-lowing expression sph= 2+ 7

2

2 2+ 2 / 2 1 2 . This expression demonstrates reasonable behaviorfor limiting cases. If l then sph= h /+7 /2/2h. If l and h then sph =4. Ifl then sph.

We have compared the field enhancement factor sph forthe floating sphere at emitter-plane potential model, the

factor tube for the hemisphere on a post model, and thefactor ell for the hemiellipsoid on plane model. We havevalidated correctness of the inequality sphtubeell. Wehave shown realization of the approximate evaluation tube sph+ell /2.

Thus, we can conclude that the floating sphere atemitter-plane potential model is reasonable for calculatingthe enhancement factor of CNT.

ACKNOWLEDGMENTS

We gratefully acknowledge support through the NationalScience Council of Taiwan, Republic of China, under grantNSC 98-2112-M-001-022-MY3, the Asian Office of Aero-space Research & Development AOARD under grant no.FA2386-09-1-4128, and support given by the Basic ScienceResearch Program through the National Research Founda-tion of Korea NRF funded by the Ministry of Education,Science and Technology grant no. 2009-0088557 .

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