Shielding effects in random large area field emitters, the field enhancement factor distribution and current calculation
SShielding effects in random large area field emitters, the field enhancementfactor distribution and current calculation
Debabrata Biswas
1, 2 and Rashbihari Rudra Bhabha Atomic Research Centre, Mumbai 400 085, INDIA Homi Bhabha National Institute, Mumbai 400 094, INDIA
A finite-size uniform random distribution of vertically aligned field emitters on a planar surface is studiedunder the assumption that the asymptotic field is uniform and parallel to the emitter axis. A formula for fieldenhancement factor is first derived for a 2-emitter system and this is then generalized for N -emitters placedarbitrarily (line, array or random). It is found that geometric effects dominate the shielding of field lines.The distribution of field enhancement factor for a uniform random distribution of emitter locations is foundto be closely approximated by an extreme value (Gumbel-minimum) distribution when the mean separationis greater than the emitter height but is better approximated by a Gaussian for mean separations close tothe emitter height. It is shown that these distributions can be used to accurately predict the current emittedfrom a large area field emitter. I. INTRODUCTION
The in-principle advantages of using field emissioncathodes over thermionic ones are manifold. Issues as-sociated with high temperature operation and tempo-ral response in thermionic cathodes clearly indicate thatnext-generation high performance electron emission sys-tems for use in vacuum devices must be based on fieldemission . The strides achieved in the past decadesin our ability to pattern arrays of pointed emitters and the discovery of carbon nanotubes (CNT) as a suit-able material for stable operation, have led to vig-orous research activity in this direction. These effortsare supported by theoretical studies on large area fieldemitters (LAFE) and corrections for nano-tippedemitters to the planar Fowler-Nordheim (FN) for-mula for current density . Despite this progress,there are several challenges in our ability to predict the-oretically, the current emitted by a single emitter or acluster of emitters placed randomly or in an array.The Fowler-Nordheim formalism continues to remainrelevant despite the vast change in experimental field-emission setups. This is surprising considering that itis based on a planar model for metallic emitters. Fieldemitters of today are highly curved leading to large lo-cal electric fields at the apex due to the phenomenonknown as field-enhancement. Thus, moderate asymp-totic fields ( ∼ V/ µ m) can lead to local fields as largeas 5-10 V/nm, at which appreciable field emission canoccur. The transition from planar to curved emittersin field-emission theory is generally made using a localapex field enhancement factor (AFEF) γ a . It doesnot change the shape of traditional FN-plots and allowsone to extract the parameter γ a . However, theoretical es-timates of the emission current require knowledge aboutthe AFEF and efforts in this direction depend either onanalytically tractable models such as the hemisphere orhemiellipsoid on a plane in the presence of an asymptoticfield E that is uniform and parallel to the emitter axis orrely on finite-element codes for particular emitter shapesand diode configuration. Using a different approach, a recent study using the line charge model (LCM), general-izes the known result for the hemiellipsoid and expressesthe apex field enhancement factor as γ a = 2 h/R a α ln(4 h/R a ) − α (1)where h is the height of the emitter, R a the apex radiusof curvature and α , α depend on the details of the linecharge and hence the emitter shape. It was also foundnumerically that the field enhancement factor is equallywell described by the simpler form γ a = 2 h/R a ln(4 h/R a ) − α (2)where α was found to depend on the emitter base (e.g.cone, ellipsoid, cylinder). Single emitter predictions foremitter current can thus be made under the conditionthat the image charges at the anode can be neglected(large anode-cathode separation) and the work functionand band-structure variations on the active emission sur-face is negligible .While single emitter setups are important in their ownright, an efficient and bright electron source requires alarge area field emitter comprising of numerous emis-sion tips (such as CNTs) placed in an array or evenrandomly. From a theoretical perspective, despite allsimplifying assumptions, there is the added complica-tion of each emitting site in a finite-sized patch, hav-ing a different enhancement factor due to the process ofshielding. Emitters in close proximity “shield” an emit-ter apex thereby lowering the enhancement factor fromits un-shielded value. While attempts have been madeto understand the shielding process using models such asthe floating-sphere on emitter plane potential as well asnumerically, a quick and accurate prediction of the apexfield enhancement factors in a LAFE based on the prox-imity of the other emitters, or even an estimate of theaverage enhancement factor for a given mean separation,is not available. a r X i v : . [ phy s i c s . a pp - ph ] J u l We shall deal here with a cluster of emitters, all havingthe same height and apex radius of curvature but placedrandomly following a uniform distribution on a rectangu-lar patch. Our methods allow us to deal with arrays aswell. Our interest is twofold. First, we shall try to under-stand the process of shielding and try to arrive at a sim-ple unifying picture. Next, we shall probe the existenceof a universal field enhancement factor distribution when the emitters are placed randomly. This is then usedto find the net emission current from a random LAFE.Our approach here is a generalization of the methodadopted recently to arrive at Eq. 1. We shall first in-troduce the line charge model and consider the case oftwo emitters. The result can then easily be generalizedto N -emitters. II. THE 2-EMITTER CASE - LINE CHARGE MODEL
Consider two emitters of height h , apex radius of cur-vature R a , separated by a distance ρ , placed on agrounded metallic plane and aligned along an asymptotic(away from the emitter tips) electrostatic field − E ˆ z . Ifwe assume the first emitter to be centred at the originsuch that its apex has co-ordinates ( ρ, z ) = (0 , h ) whilethe apex of the second emitter is located at ( ρ, z ) =( ρ , h ). This setup can be modelled by 2 vertical linecharge distributions and their image. Since the emittersare identical in every other respect, they possess iden-tical line charge density Λ( z ) of extent L . We shall as-sume that the line charge density is linear: Λ( z ) = λz .This puts a restriction on the shape of each emitter-basebut otherwise does not pose any limitation on the mainconclusions regarding shielding. Thus, in view of the lin-earity assumption, we are considering two ellipsoid-likeemitters placed a distance ρ apart. The potential atany point ( ρ, z ) can be expressed as V ( ρ, z ) = 14 π(cid:15) (cid:104) (cid:90) L − L λs (cid:2) ρ + ( z − s ) (cid:3) / ds + (cid:90) L − L λs (cid:2) ( ρ − ρ ) + ( z − s ) (cid:3) / ds (cid:105) + E z (3)where the integration is along the z − axis from [ − L, L ], L being the extent of the line charge distribution alongthe z − axis and E is the magnitude of the asymptoticfield or the external field in the absence of the ellipsoidalprotrusion. The parameter λ can be fixed by demandingthat potential vanishes at the apex. Thus, at the apex ofeither emitter, λ π(cid:15) (cid:104) (cid:90) L − L s (cid:2) ( ρ ) + ( h − s ) (cid:3) / ds + (cid:90) L − L s ( h − s ) ds (cid:105) + E h = 0 . (4) When the two emitters are well separated, the zero-potential contour of the above potential defines 2 ellip-soidal emitters, each with base radius b = ( h − L ) / and separated by ρ , mounted on a flat planar surface.As the emitters are brought closer, the zero potentialcontour keeps the apex invariant due to the impositionof Eq. 4, but its shape deviates slightly from an ellip-soid as it approaches the base. The effect gets especiallymarked when the separation is small ( ρ /h < .
2) andthe linear line charge density can no longer be used tomodel the 2-emitter ellipsoidal system. We shall there-fore steer clear of this regime. Furthermore, we shallassume that the deviation in the zero-potential contourof individual emitters away from the apex, introduces achange in the apex field enhancement factor that is smallcompared to the direct effect of neighbouring emitters atthe apex. Note that for an isolated emitter, the param-eter L = (cid:112) h ( h − R a ) . Since the imposition of Eq. 4preserves the height and apex radius of curvature of thezero-potential surface, this quantity remains invariant for ρ /h > . γ a . For axially symmetric emitters aligned along ˆ z , thisis defined as γ a = − E ∂V∂z | ρ =0 ,z = h . Our starting pointfor the AFEF is Eq. 3. At the apex, ( ρ, z ) = (0 , h ) ofemitter 1, ∂V∂z | ( ρ =0 ,z = h ) = − λ π(cid:15) (cid:104) (cid:90) L − L s ( h − s ) ds + (cid:90) L − L s ( h − s ) (cid:2) ρ + ( h − s ) (cid:3) / ds (cid:105) + E (5)which, on integrating, leads to ∂V∂z | ( ρ =0 ,z = h ) = − λ π(cid:15) (cid:104) hLh − L + ln (cid:16) h + Lh − L (cid:17) + (cid:90) L − L s ( h − s ) (cid:2) ρ + ( h − s ) (cid:3) / ds (cid:105) + E . (6)For ρ large compared to the base radius of the emitters,the integral in Eq. 6 is negligible compared to the firsttwo terms in the square bracket. Furthermore, for sharpemitters ( h/R a >> .Thus ∂V∂z | ( ρ =0 ,z = h ) = − λ π(cid:15) (cid:104) hLh − L (cid:105) . (7)It now remains to determine λ using Eq. 4. The integralsin Eq. 4 yield λ π(cid:15) (cid:104) − (cid:90) L − L h − s (cid:2) ( ρ ) + ( h − s ) (cid:3) / ds + h (cid:90) L − L ds (cid:2) ( ρ ) + ( h − s ) (cid:3) / + h ln (cid:16) h + Lh − L (cid:17) − L (cid:105) + E h = 0 (8)which further simplifies as λ π(cid:15) (cid:34)(cid:104)(cid:113) ρ + ( h − s ) + h ln (cid:12)(cid:12)(cid:12)(cid:115) h + s ) ρ + ( h + s ) ρ (cid:12)(cid:12)(cid:12) (cid:105) L − L + h ln (cid:16) h + Lh − L (cid:17) − L (cid:35) + E h = 0 . (9)Note that (cid:113) ρ + ( h − s ) (cid:12)(cid:12)(cid:12) L − L (cid:39) ρ (cid:104) − (cid:0) δ (cid:1) / (cid:105) (10)where δ = h/ρ . Also,ln (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:115) h + s ) ρ + ( h + s ) ρ (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) L − L (cid:39) ln (cid:12)(cid:12)(cid:12)(cid:113) δ + 2 δ (cid:12)(cid:12)(cid:12) (11)if ρ >> b . Finally, using h (cid:39) L for nano-tipped emit-ters, we have λ = − π(cid:15) E ln h h − L − α S (12)where the shielding term α S = 1 δ (cid:104) − (cid:113) δ (cid:105) + ln (cid:12)(cid:12)(cid:12)(cid:113) δ + 2 δ (cid:12)(cid:12)(cid:12) (13)Thus γ a = 2 h/R a ln (cid:0) h/R a (cid:1) − α S (14)is the enhancement factor of the 2-emitter system.A topic of recent interest has been the change in 2-emitter AFEF as compared to the single or un-shieldedcase ( γ (1) a ). Note that for the isolated or single case, α S = 0 so that the relative change γ a − γ (1) a γ (1) a = − α S ∆ (15)where ∆ = ln (cid:0) h/R a (cid:1) −
2. For large separations ( ρ large), δ is small and it is easy to verify that the α S = 23 δ + O ( δ ) . (16)Thus, at large separations, γ a − γ (1) a γ (1) a ∼ ρ (17)as deduced by other methods . A. The N-emitter case
The 2-emitter case was simple in that everything elsebeing equal, both emitters have the same line charge den-sity. As a consequence, the shielding term α S does notdepend on λ , . For a N-emitter system, N ≥
3, the linecharge densities are unequal since different emitters havedifferent degrees of shielding. The mutual shielding termthus generalizes as˜ α S ij = λ j λ i α S ij , where (18) α S ij = 1 δ (cid:104) − (cid:113) δ ij (cid:105) + ln (cid:12)(cid:12)(cid:12)(cid:113) δ ij + 2 δ ij (cid:12)(cid:12)(cid:12) and λ j /λ i (cid:54) = 1 in general. The net shielding due to N-emitters can be expressed as˜ α S i = (cid:88) j (cid:54) = i ˜ α S ij . (19)This follows on noting that Eq. 7 can be expressed forthe i th emitter apex located at ( ρ i , h ) with respect to anarbitrary origin, as ∂V∂z | ( ρ = ρ i ,z = h ) (cid:39) − λ i π(cid:15) (cid:104) hLh − L (cid:105) (20)since the other terms are small and can be neglected asbefore. Further, Eq. 4 can be expressed as λ i π(cid:15) (cid:104) (cid:88) j (cid:54) = i λ j λ i (cid:90) L − L s (cid:2) ( ρ ) + ( h − s ) (cid:3) / ds + (cid:90) L − L s ( h − s ) ds (cid:105) + E h = 0 (21)so that { λ i } can be determined by simultaneously solvingthe set of N equations. We shall instead merely express λ i in terms of the ratio λ j /λ i and use it in Eq. 20 toexpress the field enhancement factor γ ( i ) a = 2 h/R a ln (cid:0) h/R a (cid:1) − (cid:80) j (cid:54) = i λ j λ i α S ij (22)= 2 h/R a ln (cid:0) h/R a (cid:1) − α S i (23)for the i th emitter.Eq. 23 as such does not help us in computing γ ( i ) a with-out actually solving the electrostatic problem. However,there are clearly two aspects in the shielding term thatmust draw our attention. The first is a geometric factor α S ij which merely depends on the ratio of the height h and the mutual distance ρ ij and does not require a so-lution of the Poisson equation. The second is the ratio λ j /λ i which does require knowledge about the line chargedensity. As a first approximation, for average separations ρ ij comparable to or larger than the height h , we set theratio λ j /λ i = 1. Thus ˜ α S i = α S i so that γ ( i ) a (cid:39) h/R a ln (cid:0) h/R a (cid:1) − (cid:80) j (cid:54) = i α S ij (24)= 2 h/R a ln (cid:0) h/R a (cid:1) − α S i . (25)A comparison of the discrepancy between the AFEFvalues computed using Eq. 23 and 25 is shown in Fig. 1 fora randomly distributed N-emitter system. The emitterspositions are drawn from a uniform distribution using astandard random number generator. It is apparent thatthe error on ignoring the variation in { λ i } is acceptablewhen the mean separation exceeds the emitter height.Moreover even when the separation is half the emitterheight, the average error is only 6%. Thus, the shieldingprocess is pre-dominantly a geometric effect and the fieldenhancement factor can be computed quite accuratelyonly from a knowledge of the positions, height and apexradius of curvature of the emitters. We have also deter-mined the mean error of only those emitters for which theAFEF exceeds the mean value of the AFEF since thesepredominantly contribute to the field-emission current atlow to medium local field strengths ( < III. THE FIELD-ENHANCEMENT-FACTORDISTRIBUTION FOR RANDOMLY PLACED EMITTERS
As seen above, the field enhancement factor can becomputed for individual emitters in a LAFE purely from % M ean E rr o r ( γ a ) Mean separation ( µ m) FIG. 1. The mean error in apex field enhancement factor(AFEF) is shown for different mean spacings. A total of 2500emitters is considered in each case. The field enhancementfactor is computed using Eq. 23 (denoted by (cid:32) ), and, Eq. 25(denoted by (cid:4) ). The error in the second case decreases asthe mean spacing exceeds the height h = 1500 µm . Theemitters have a base radius b = 10 µ m and an apex radius R a = 66 .
67 nm. For Eq. 23, the error is uniformly small( ≤ . geometrical considerations. While this information isuseful, a distribution of field enhancement factors is de-sirable so that the total current emitted from a LAFE canbe computed based on only a few parameters such as themean and standard deviation of the AFEF distribution.We shall thus explore the existence of a universal AFEFdistribution when the emitters are distributed uniformlyon a rectangular patch.Our studies on the field-enhancement-factor distribu-tion for various mean separations, height and apex radiusshow that the distribution is closer to a Gaussian whenthe mean separation is equal to or somewhat smallerthan the emitter height. However, as the mean separa-tion increases, the field-enhancement-factor distributionis skewed to the left for emitters distributed uniformlyon a rectangular patch as seen in Fig. 2. The skew-ness persists for separations beyond twice the emitterheight. Also, its mean and standard deviation dependon the mean separation of emitters. While the meanAFEF increases with separation, the standard deviationdecreases.An analytical expression for the AFEF distribution isdifficult to derive but fits to various left-skewed distribu-tions show that the Gumbel minimum distribution bestdescribes the field enhancement factor for mean separa-tions exceeding the emitter height. This is also the regionof interest since as the optimal separation at which thecurrent density is highest lies here.The Gumbel distribution has a probability densityfunction F r equen cy d i s t r i bu t i on γ a FIG. 2. The normalized frequency distribution of the fieldenhancement factors together with the Gumbel (minimum)distribution (solid curve). The LAFE has a mean separation2000 µ m containing 5000 emitters, each of height h = 1500 µ m. The parameters of the Gumbel distribution are fixed usingEqns. 28 and 29 with the mean µ and standard deviation σ calculated using the exact numerical values of AFEF whichare obtained by numerical differentiation of the potential. f γ ( x ) = 1 β e x − αβ e − e x − αβ (26)and its cumulative density function is F γ ( x ) = 1 − e − e x − αβ . (27)Here α and β are parameters in terms of which, the mean µ and standard deviation σ are µ = α − βγ EM (28) σ = πβ/ √ γ EM (cid:39) . γ a , µ and σ can be determined. TheGumbel parameters α and β can thus be evaluated us-ing Eqns. 28 and 29. A comparison of the normalizedfrequency distribution with the Gumbel distribution isshown in Fig. 2. The agreement is good for mean sepa-rations exceeding the emitter height.The Gumbel distribution thus shows good agreementwhen the AFEF values are determined by numerical dif-ferentiation after solving for the electrostatic potential(Eq. 23 may instead be used since the errors are small atall separations but requires knowledge of the line chargedensity). We can alternately study the AFEF distribu-tion when individual AFEF values are determined using CD F γ aAnalyticalGumbelGaussian CD F γ aAnalyticalGumbelGaussian CD F γ aAnalyticalGumbelGaussian FIG. 3. A comparison of the cumulative density function(CDF) with the Gaussian and Gumbel minimum distributionsfor 3 values of mean separation. The values of γ a are obtainedusing the analytical expression Eq. 25. The mean separationsare 1500 µ m (left), 2000 µ m (middle) and 2500 µ m (right)while the height of the emitters is 1500 µ m. The parametersfor the Gaussian and Gumbel distributions are obtained usingthe mean and standard deviation. Eq. 25 which does not require knowledge of the electro-static problem. The Gumbel distribution again providesa good approximation when the mean separation betweenemitters exceeds the emitter height (see Fig. 3) while theGaussian distribution is a better approximation for meanseparation around the emitter height. These conclusionshold for other height and apex radius combinations.
A. The harmonic mean and standard deviation using thepair-wise distance distribution
So far, we have determined the Gumbel parameters byfirst computing γ a (using Eq. 23 or 25) and determining µ and σ . In principle, it should be possible to determine themean and standard deviation using Eq. 25 (but withoutevaluating individual γ ( i ) a ) and noting that the emittersare distributed uniformly. It thus requires knowledge ofthe probability density function (PDF) of the distance ρ ij where the i th emitter is fixed and the other N − i th emitter andhence various cases need to be listed. A simpler and wellknown PDF is that of the pair-wise distance between any2 points i and j located on a rectangular patch. This canbe used to calculate the harmonic mean by noting thatEq. 24 can be rewritten as2 hR a γ i = ln (cid:16) hR a (cid:17) − (cid:88) j (cid:54) = i α S ij (30)so that2 hR a (cid:88) i γ i = N (cid:104) ln (cid:16) hR a (cid:17) − (cid:105) + (cid:88) i (cid:88) j (cid:54) = i α S ij . (31)Thus, the harmonic mean, µ h is (cid:16) N (cid:80) i γ i (cid:17) − = hR a ln( hR a ) − N (cid:80) i (cid:80) j (cid:54) = i α S ij (32)= hR a ln( hR a ) − N − (cid:82) f ρ ( x ) α S ( x ) (33)where α S ( x ) = xh (cid:104) − (cid:114) (cid:16) hx (cid:17) (cid:105) + ln (cid:12)(cid:12)(cid:12)(cid:114) (cid:16) hx (cid:17) + 2 hx (cid:12)(cid:12)(cid:12) (34)and the probability density function, f ρ ( x ), for the pair-wise distance between any two points distributed uni-formly on a square patch of length L is related to f S ( s ) = − √ sL + πL + sL < s ≤ L ; − L + L sin − (cid:0) L √ s (cid:1) + L √ s − L − πL − sL L < s ≤ L (35)where s = x and f ρ ( x ) = 2 xf S ( s ). Note that (cid:82) f ρ ( x ) dx = 1 so that Eq. 33 has the factor ( N − for an ex-pression for f S ( s ) when the area is rectangular) patch.A similar expression can be derived for µ h = (cid:16) N (cid:88) i γ i (cid:17) − (36)using the probability density function f ρ ( x ) and hencethe ‘harmonic’ standard deviation, defined as σ h = (cid:104)(cid:16) N (cid:88) i γ i (cid:17) − − (cid:16) N (cid:88) i γ i (cid:17) − (cid:105) / (37)can be evaluated. In terms of f ρ ( x ), µ h (cid:39) (cid:0) hR a (cid:1) ∆ + 2 ∆ N (cid:80) i (cid:80) i (cid:54) = j α S ij + N (cid:80) i (cid:80) j (cid:54) = i (cid:80) k (cid:54) = i α S ij α S ik (38)where ∆ = ln( hR a ) −
2. The summations can be expressedas 2 ∆ N (cid:88) i (cid:88) i (cid:54) = j α S ij = 2∆( N − (cid:90) f ρ ( x ) α S ( x ) dx (39)and1 N (cid:88) i (cid:88) j (cid:54) = i (cid:88) k (cid:54) = i α S ij α S ik = ( N − (cid:90) f ρ ( x ) α S ( x ) dx +( N − N − (cid:104) (cid:90) f ρ ( x ) α S ( x ) dx (cid:105) (40)and hence µ h and σ h can be evaluated in terms of thepair-wise distance distribution f ρ ( x ). B. Gumbel parameters using the harmonic mean andstandard deviation
It is thus possible to evaluate µ h and σ h using the pair-wise distribution. The Gumbel parameters α and β canbe determined if µ h and σ h can be found for the Gumbeldistribution as well.Noting that β is generally small compared to α for thefield enhancement distribution, approximate expressionsfor µ h and σ h can be derived as µ h (cid:39) α − βγ EM − β α π σ h (cid:39) πβ/ √ β (cid:39) √ σ h /π (43) α (cid:39) (cid:0) µ h + βγ EM (cid:1) + (cid:113)(cid:0) µ h + βγ EM (cid:1) + β π µ h evaluated using Eq. 33 and σ h using Eq. 37.Thus, in principle, the Gumbel parameters for N uni-formly distributed emitters can be evaluated using thedistance distribution f ρ . Note that the expressions for µ h and σ h above are only approximate and hence theGumbel parameters computed this way are not expectedto be accurate. However, they can be calculated usingthe pair-wise distance distribution alone. Table I showsa comparison of the Gumbel parameters for 3 differentmean separations and in each case, 3 different methodsare adopted to determine the parameters α and β . It isclear that a reasonably good approximation to the Gum-bel distribution parameters can be obtained using thepairwise distribution function. Separation Method Gumbel α Gumbel β µ m numerical with µ and σ µ h and σ h µ h and σ h µ m numerical with µ and σ µ h and σ h µ h and σ h µ m numerical with µ and σ µ h and σ h µ h and σ h µ or µ h and standard deviation σ or σ h are evaluated using approximate analytical values of γ a ob-tained using Eq. 25 for a given uniform distribution of emit-ter positions. These are then equated to the correspondingexpressions for the Gumbel minimum distribution in orderto determine α and β . The third method uses the pairwisedistance distribution to evaluate µ h and σ h directly using nu-merical integration. Note that the second and third methodsuse approximate expressions for µ h (Eqn. 41) and σ h (Eq. 42). IV. CURRENT FROM A LAFE
In the previous sections, we have derived a formula forthe apex field enhancement factor (AFEF) γ a of individ-ual emitters in a LAFE. In addition, we have found thatthe Gaussian distribution approximates the AFEF dis-tribution well when the mean separation is close to theemitter height while the Gumbel minimum distribution isa better approximation, for mean separation larger thanthe height of individual emitters. Assuming recent re-sults on the variation of field enhancement factor aroundthe apex of individual emitters in a LAFE, the netcurrent emitted can be expressed as I LAFE = (cid:88) i I i = 2 πR a (cid:88) i J a i G i (45)where I a i is the current from the i th emitter and thecorresponding apex current density is J a i = 1 t F i A FN φ E a i e − B FN ν Fi φ / /E ai . (46)while the area-factor is -0.08-0.04 0 0.04 0.08-0.08 -0.04 0 0.04 0.08 Y ( m ) X (m)
FIG. 4. A uniformly distributed LAFE. Each point denotesthe position of an emitter. -0.08-0.04 0 0.04 0.08-0.08 -0.04 0 0.04 0.08 Y ( m ) X (m) 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35
FIG. 5. The corresponding current map of the LAFE. Thedark regions have low electron emission. The current is mea-sured in amperes. G i = E a i B FN φ / − f i / . (47)In the above, E a i = γ a i E is the local fieldat the apex of the i th emitter, γ a i is the apex en-hancement factor and E is the asymptotic electricfield. Here, A FN (cid:39) . µ A eV V − and B FN (cid:39) . − / V nm − are the conven-tional FN constants, φ is the work function while ν F i (cid:39) − f i + f i ln f i and t F i (cid:39) f i / − f i ln f i are correction factors due to the image potential with f i (cid:39) c S E a i /φ and c S = 1 . V − nm. Unlessotherwise stated, the work function φ = 4 . µ m. The corre-sponding current map is shown in Fig. 5. It can beseen that sparse regions have high electron emission whiledenser regions have lower emission. -80-60-40-20 0 1 2 3 4 5 l og ( J / E ) l og ( J / E ) l og ( J / E ) l og ( J / E ) l og ( J / E ) l og ( J / E ) FIG. 6. A comparison of the FN plots for the LAFE currentdensity for 6 mean separations found by (a) summing indi-vidual emitter currents using exact AFEF (continuous curve)(b) using Gumbel distribution and approximate analytical γ ( i ) a (Eq. 25) to find µ and σ and hence the Gumbel parameters( (cid:4) ) (c) using Gumbel distribution and approximate analyti-cal γ ( i ) a (Eq. 25) to find µ h and σ h using pair-wise distribution( (cid:78) ) (d) using Gaussian distribution and approximate analyt-ical γ ( i ) a (Eq. 25) to find µ and σ ( (cid:32) ). The mean separationsare 1500 µ m (top-left), 2000 µ m (top-middle), 2500 µ m (top-right), 1500 µ m (bottom-left), 2000 µ m (bottom-middle),2500 µ m (bottom-right). Cases (b) and (c) are virtually in-distinguishable. The X-axis has units of µ m/V, while J/E has units of A/V . The Gaussian distribution over-estimatesthe current for mean separations greater than 2000 µ m. The net current from the LAFE can alternately becalculated using the apex field enhancement factor dis-tribution. Using an AFEF distribution, the summationin Eq. 45 can be expressed as I LAFE = (cid:88) i I i = 2 πR a ∞ (cid:90) J a ( x ) G ( x ) f γ ( x ) dx (48)where f γ ( x ) is the Gumbel distribution and J a ( x ) = 1 t F ( x ) A FN φ E a ( x ) e − B FN ν F ( x ) φ / /E a ( x ) (49) G ( x ) = E a ( x ) B FN φ / − f ( x ) /
6) (50) ν F ( x ) = 1 − f ( x ) + 16 f ( x ) ln f ( x ) (51) t F ( x ) (cid:39) f ( x ) − f ( x ) ln f ( x ) (52) f ( x ) (cid:39) c S E a ( x ) /φ . (53)
1 1.2 1.4 1.6 1.8 2 J L A F E E FIG. 7. A comparison of the exact LAFE current density(A/m ) for different mean separations is shown as a functionof the asymptotic field E (V/ µ m). The mean separations inunits of µ m are 1000 ( (cid:78) ), 1500 ( × ), 2000 ( ∗ ), 2500 ( (cid:50) ), 3000( (cid:4) ), 4000 ( (cid:13) ), 4500 ( (cid:32) ) and 5000 ( (cid:52) ). In Fig. 6, the net current obtained by summing overindividual pins (continuous curve) is compared with thecurrent obtained using Gaussian/Gumbel AFEF distri-butions, as a Fowler-Nordheim (FN) plot for differentmean separations. The parameters of the Gumbel distri-bution are obtained in 2 ways. In the first ( (cid:4) ), the meanand standard deviation of { γ ( i ) a } (obtained from Eq. 25for a given realization of uniform distribution), are usedto evaluate α and β (Eqns. 28 and 29). In the second( (cid:78) ), α and β are obtained using the harmonic mean andstandard deviation which in turn are evaluated using thepairwise distribution. Also shown is the current obtainedusing a Gaussian distribution with µ and σ obtained us-ing { γ ( i ) a } ( (cid:32) ). The agreement with the Gumbel distribu-tion is reasonably good at all the separations except whenthe mean separation equals the emitter height where theGaussian distribution performs much better especially atlower field strengths. The Gumbel distribution performswell even at low field strengths and notably even whenthe parameters are obtained using the pairwise distribu-tion. Note that even though a wide range of externalasymptotic fields E has been investigated, for practicalpurposes, values of ln( J/E ) exceeding -50 are relevant. J L A F E mean separation FIG. 8. Variation of the exact LAFE current density (A/m )with mean separation ( µ m) at E (cid:39) .
96 V/ µ m. Finally, we investigate the optimal mean separationat which the emission current density for the randomLAFE considered here is maximum. Fig. 7 shows thecurrent density plotted against the asymptotic electro-static field E for various mean separations. The maxi-mum current density peaks sharply after the mean sepa-ration crosses the emitter height (1500 µ m) and plateausat around 2500-3000 µ m for all field strengths (see Fig. 8for the variation of current density with mean separationat E (cid:39) .
96 V/nm). This is similar to the trend seenfor an infinite array . V. SUMMARY AND CONCLUSIONS
We have studied shielding effects in a random largearea field emitter starting with a 2-emitter system. Meth-ods similar to one used in recently were used to firstarrive at a formula for the apex field enhancement factor(AFEF) in a 2-emitter system and this was subsequentlygeneralized for an arbitrary N-emitter system where theemitter placements may be in a line, a 2-dimensional ar-ray or even randomly distributed. It was found that forpurposes of field emission where emitters with the largestAFEFs contribute, the shielding effect can be consideredto be purely geometric and the AFEFs can be deter-mined, within acceptable limits, purely from the emitterlocations without solving the full electrostatic problem.The question of AFEF distribution was subsequentlyinvestigated. It was found that the distribution is closerto a Gaussian when the mean separation is close to orsomewhat less than the emitter height, but is better ap-proximated by a Gumbel minimum distribution for spac-ings larger than emitter height. It is in this regime thatthe maximum LAFE current density is found to lie. These results are supported by computation of theemission current, both, by directly summing over indi-vidual pins after solving the full electrostatic problem,and at the other extreme by using the expression for ap-proximate (geometric) analytical AFEFs together withthe Gumbel and Gaussian distributions. The comparisonshows that the latter method with Gumbel distributioncan be used profitably for a large range of field strengthsfor mean separations larger than the emitter height but atmean separations close to the emitter height, the Gaus-sian distribution performs much better.Finally, for the uniform distribution of emitters, wehave evaluated the Gumbel parameters using the pair-wise distance distribution by calculating the harmonicmean and standard deviation. The evaluation of currentusing these parameters gives excellent results (for meanseparation greater than emitter height) that are virtuallyindistinguishable from results with parameter obtainedfrom a given realization of the emitter pins. VI. ACKNOWLEDGEMENTS
The authors thank Raghwendra Kumar, Gaurav Singhand Rajasree for useful discussions.
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