Application of a general electron emission equation to ...

Application of a general electron emission equation to surface nonuniformity and current density variation Kevin L. Jensen, John J. Petillo, Eric J. Montgomery, Zhigang Pan, Donald W. Feldman, Patrick G. O’Shea, Nathan A. Moody, M. Cahay, Joan E. Yater, and Jonathan L. Shaw Citation: Journal of Vacuum Science & Technology B 26, 831 (2008); doi: 10.1116/1.2827508 View online: http://dx.doi.org/10.1116/1.2827508 View Table of Contents: http://scitation.aip.org/content/avs/journal/jvstb/26/2?ver=pdfcov Published by the AVS: Science & Technology of Materials, Interfaces, and Processing

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Application of a general electron emission equation to surface nonuniformity and current density variation Kevin L. Jensena兲 Code 6843, ESTD, Naval Research Laboratory, Washington, DC 20375

John J. Petillo Science Applications Int’l Corporation, Burlington, Massachusetts

Eric J. Montgomery, Zhigang Pan, Donald W. Feldman, and Patrick G. O’Shea IREAP, University of Maryland, College Park, Maryland

Nathan A. Moody Los Alamos National Laboratory, Los Alamos, New Mexico

M. Cahay University of Cincinnati, Cincinnati, Ohio

Joan E. Yater and Jonathan L. Shaw Code 6843, ESTD, Naval Research Laboratory, Washington, DC 20375

共Received 4 September 2007; accepted 28 November 2007; published 1 April 2008兲 Electron emission nonuniformity is a cause of intrinsic emittance from the electron source, and is a consequence of work function variation due to crystal faces and coatings such as cesium, field enhancement effects due to surface structure, and temperature. Its investigation using particle-in-cell 共PIC兲 codes such as MICHELLE is hampered due to the lack of an emission model that can treat thermal, field, and photoemission effects particularly in crossover regions where the canonical equations, e.g., the Fowler-Nordheim, Richardson-Laue-Dushman, and Fowler-Dubridge equations are compromised. A recently developed thermal-photo-field emission equation is used here to simulate the consequences of nonuniformity due to work function variation induced by coating variation. The analysis is performed both theoretically using simple models as well as using particle-in-cell codes 共MICHELLE兲 to assess changes in current density and emittance. PIC simulations considering an idealized model of geometric effects and crystal face variation indicate that a flat, grainy surface causes the emittance to increase by a factor of 5 while the addition of hemispherical bumps causes the emittance to increase by an additional factor of 6 even though the current is but 10% larger. © 2008 American Vacuum Society. 关DOI: 10.1116/1.2827508兴

I. INTRODUCTION High frequency 共submillimeter and terahertz兲 microwave power amplifiers,1 particle accelerators, and free electron lasers2 require low emittance, uniform emission electron sources, and are the catalysts for an effort to understand the causes of emittance 共the tendency of a beam to spread as it propagates兲 at the cathode, generally designated “intrinsic emittance.” The understanding of the impact of causes of emittance growth is hampered by the general lack of good emission models in beam simulation codes. In this work, the impact of emission nonuniformity due to work function variation and geometric field enhancement is considered using a combination of simple models in numerical simulations using the particle-in-cell 共PIC兲 code MICHELLE 共Ref. 3兲 that make use of a recently developed thermal-fieldphotoemission model that is able to handle transition regions where the canonical electron emission equations for thermal, field, and photoemission are impaired. A brief account of the general thermal-field 共GTF兲 and photoemission equations4 is given, then applied to nonuniform conditions such as work a兲

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function variation due to crystal face differences and variations in low work function coating 共cesium兲 coverage, followed by field enhancement variation due to surface geometry. The impact on beam emittance is assessed using the MICHELLE PIC code.

II. GENERAL THERMAL-FIELD EQUATION In a separate work,4 the GTF equation has been derived, whereas here, its departure from the conventional FowlerNordheim 共FN兲 and Richardson-Laue-Dushman 共RLD兲 equations of field and thermal emission, respectively, is considered. Under the assumption that the supply function f共E兲 and transmission probability D共E兲 are parametrically represented as D共E兲 ⬇ 1/关1 + exp共␤F共Eo − E兲兲兴,

冉

共1兲

冊

m ln关1 + exp共␤T共␮ − E兲兲兴, f共E兲 = ␲ ␤ Tប 2 where ␤T = 1 / kBT and ␤F共Em兲 are energy slope factors and Eo共Em兲 is a reference energy, whereas Em is the location of

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FIG. 1. GTF equation 共symbols兲 compared to the FN and RLD equations 共solid and dashed lines, respectively兲 for ⌽ = 1.8 eV 共circle兲 and 4.5 eV 共square兲, and T = 300 K 共blue兲 and 1300 K 共red兲. The FN lines are temperature independent and use the form of JFN from Ref. 27; the RLD lines depend on field only through the Schottky factor. Deviation from linearity is evidence of thermal effects that are ignored in the usual FN equation.

the current density integrand maximum 共and therefore field dependent兲, then the GTF equation is

冦

JF + JT 共n ⬍ 1兲 JGTF共F,T兲 = n2 JF + n2JT 共n ⬎ 1兲,

冧

共2兲

where the ratio of energy slope factors is defined as n共F , T兲 ⬅ ␤T / ␤F, and where JT ⬅ ARLD共kB␤T兲−2⌺共n兲exp关− ␤T共Eo − ␮兲兴,

冉冊

共3兲

1 exp关− ␤F共Eo − ␮兲兴, JF ⬅ ARLD共kB␤F兲−2⌺ n ⌺共x兲 ⬇

1 1 − x共1 + x兲 + x3共7x − 3兲 + ␨共2兲x2共1 − x2兲, 4 1−x

where ARLD = 120.173 A / cm2 K2 is the Richardson constant and kB is Boltzmann’s constant. The “conventional” RLD and FN equations 共in a notation following Ref. 5兲 are defined by replacing ⌺共x兲 with ⌺共0兲 = 1 and using Eo = ␮ + ␾ for RLD and Eo = ␮ + 共2␯共y兲 / 3t共y兲兲⌽ for FN, where ␾ = ⌽ − 冑4QF ⬅ 共1 − y兲⌽, ⌽ is the work function, F is the product of the electron charge and field, Q = 0.359 99 eV nm, and ␯共y兲 and t共y兲 are the conventional Fowler-Nordheim functions 共though they will be evaluated using the Forbes approximation6 to ␯共y兲兲. For the GTF equation, the determination of Eo and ␤F are not simple, and the transition region where n is approximately unity is markedly different than the sum of the FN and RLD equations. For conditions of interest here, namely, low work function and intermediate field enhancement values, both thermal and field components contribute and exhibit a more complex behavior than anticipated from conventional FN- and RLD-like plots. By way of example, consider the current density evaluated using the GTF equation compared to both FN and RLD on a FN-type plot, as in Fig. 1, for bare tungsten 共⌽ = 4.5 eV兲 and cesiated tungsten 共⌽ = 1.8 eV兲 work functions at room temperature and an elevated temperature of 1300 K. Departures of the GTF

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FIG. 2. Similar to Fig. 1, but on a Richardson plot. The numbers following the lines correspond to applied field in eV/nm. Deviation from linearity is evidence of field effects 共apart from Schottky barrier lowering兲 that are ignored in the usual RLD equation.

curves from linearity indicate the steadily increasing contribution of thermal effects that are particularly pronounced for low work function. Repeating the analysis for the case of F = 1.8 and fields of 0.1, 0.32, and 1.0 GV/ m, but on a RLDtype plot, as in Fig. 2, shows the steadily increasing contribution of field effects that cause departures from linearity on a RLD plot. A low work function surface, as is typical for a submonolayer of cesium on a tungsten surface, that exhibits geometric surface structure will therefore have a complicated behavior as temperatures and fields change, causing JT and JF to be of varying impact. Clearly, analyzing emission nonuniformity for complex surfaces requires the consideration of both JF and JT as conditions change 共in spite of the subscripts, both depend on field and temperature兲, and is therefore the focus of the present study. III. GTF EQUATION APPLIED TO A SURFACE WITH SIMULTANEOUS WORK FUNCTION AND FIELD ENHANCEMENT VARIATION On a complex surface exhibiting both variation in field enhancement factor and work function due to crystal face variation and coating, the GTF model reveals the behavior of the transition region between thermal and field emission. A numerical 共not experimental兲 example is shown in Fig. 3 for the temperatures 共a兲 800, 共b兲 1000, 共c兲 1200, and 共d兲 1400 K in which the field emission current which dominates at low temperature is seen to be progressively complemented by a thermal component. The field enhancement variation 共that is, the ratio between the field at the emission site and the macroscopic field兲 is randomly distributed, as shown in Fig. 3共e兲, for a macroscopic field of 33 MV/ m with variations in the field enhancement ranging from 1 to 3 共dark blue and red, respectively兲. The work function variation is distributed as in Fig. 3共f兲 among the values 2.0, 2.1, and 2.2 eV 共white, gray, and black, respectively兲 in a pattern meant to mimic a polycrystalline surface. In Figs. 3共a兲–3共d兲, whitish yellow is the maximum current density whereas blackish red is the minimum, and the low work function regions are seen to progressively contribute. On a Richardson plot, if the surface were

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PFD共ប␻兲 ⬅ U关␤T共ប␻ − ␾兲兴 / U关␤T␮兴, where U共x ⬎ 0兲 = ␲2 / 6 + x2 / 2 − e−x is the Fowler-Dubridge function.7 A Momentsbased formulation has modified the form of the probability of emission term PFD considerably and has changed the manner in which the underlying terms are evaluated, particularly the relaxation time and the emission probability.5 While the Moments approach is better for modeling experimental data, a modified Fowler-Dubridge 共MFD兲 model in which PFD is altered performs acceptably to within an approximately constant scale factor for investigations herein, and is given by QEMFD共ប␻兲 = 共1 − R共␻兲兲F␭共ប␻,T兲PMFD共ប␻兲, 共ប␻ − ␾兲2 + 2␨共2兲共␤T−2 + ␤F−2兲 . PMFD共ប␻兲 ⬅ 2ប␻共2␮ − ប␻兲

FIG. 3. GTF equation applied to a surface with varying field and work function for several temperatures: 共a兲 T = 800 K, 共b兲 T = 1000 K, 共c兲 T = 1200 K, 共d兲 T = 1400 K for variations in field enhancement shown 共e兲 such that the maximum field of 10 MV/ m 共red兲 is a factor of 3 larger than the minimum field 共blue兲, in addition to variations work function shown in 共f兲 due to presumed crystal-face dependent changes such that white= 2.0 eV, gray= 2.1 eV, and black= 2.2 eV. Both variations due to the field enhancement in 共e兲 and work function in 共f兲 occur in 共a兲–共d兲: it is the shifting dependence of the GTF equation on temperature and field that gives rise to the behavior shown.

uniform with no field enhancement and at a work function of 2.0 eV, then the current density 共in A / cm2兲 would satisfy ln共Jo / T2兲 = 4.7889− ␾o␤T, where ␾o = 1.7820 eV is the work function lowered by the Schottky factor for the macroscopic field, whereas the average 具J典 satisfies ln共具J典 / T2兲 = 5.5103 − 2.1295␤T + 0.007 575 1␤T2 , which can be rewritten as 4.7889A − ␾共␤T兲␤T where ␾共␤T兲 = 共1.0889− 0.0038733␤T兲␾o and A = 1.2095. Not unexpectedly, the presence of competing contributions makes the work function as determined by the slope on a Richardson plot have the erroneous appearance of temperature dependence. IV. MODIFIED FOWLER DUBRIDGE EQUATION FOR PHOTOEMISSION For a general total electron emission equation, the photoemission current J␭ must be added to JGTF. In past work, a Fowler-Dubridge-like formulation7 was used to find the quantum efficiency by QEFD = 共1 − R共␻兲兲F␭共ប␻ , T兲PFD共ប␻兲, where J␭ is the electron current density, ប␻ is the photon energy, F␭ is a scattering factor accounting for the fraction of photoexcited electrons retaining a favorable energy to be emitted as they migrate to the photocathode surface, and R is the wavelength-dependent reflectivity. The probability factor

共4兲

The reflectivity and scattering terms in Eq. 共4兲 are ancillary to the present focus, are complex to calculate and material dependent, and are discussed elsewhere.7 The variation in PMFD will therefore be the focus of the present treatment, though all terms in Eq. 共4兲 will be evaluated when estimates of photoemission current are required below. The development of a general thermal-field-photoemission equation is motivated by the need to characterize the surfaces of dispenser photocathodes,5,7–14 in particular, when cesium is used as the low work function coating, as the cesium coverage during activation and possibly operation can be nonuniformly distributed, which, by GyftopoulosLevine theory, indicates that the work function can vary from the bulk metal value 共e.g., 4.5 eV for tungsten兲 to the monolayer coverage value 共e.g., 1.6 for Cs on tungsten兲 as the cesium coverage varies over the surface from ␪ = 0 共bare兲 to ␪ = 1 共full monolayer兲. Along the surface of a controlled porosity dispenser 共CPD兲 cathode, cesium should migrate from pores to form a monolayer coating serving to substantially lower the work function thereby significantly enhancing quantum efficiency. When the surface degrades, in principle, the monolayer can be reestablished through the dispensation of more cesium by heating, analogous to the behavior of conventional dispenser cathodes. The questions of rejuvenation time and surface migration are therefore important in the determination of a protocol to restore functionality to a degraded photocathode. Experiments underway to measure the characteristics of, and migration rates across, a dispenser cathode using a PEEM system15 shall be reported separately, though the study of an analogous system of alkali diffusion, namely, the investigation of potassium on the surface of palladium using a PEEM system has been carried out previously16,17 and may therefore anticipate what will be observed for cesium on tungsten, though we note our preliminary findings suggest cesium on antimony has a different character due to the creation of Cs3Sb and its semiconductor nature. The theoretical model here concerns the diffusion of cesium from pores and the related profile of PMFD共ប␻兲. Work function variation dependence on submonolayer coverage is taken to follow Gyftopoulos-Levine theory.7,18,19 A complication is that the surface number density per unit area 共f兲 of the coating atoms 共Cs兲 varies with exposed crystal

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FIG. 5. Mean 共circle兲 and maximum/peak 共square兲 currents for the simulations shown in Fig. 4, showing both single pore 共open兲 and multiple pore 共solid兲 behavior for two wavelengths of 266 nm 共top/blue兲 and 532 nm 共bottom/red兲. Over time, the peak and average converge, as the distribution of Cs becomes uniform, whether or not both continue to decrease as for a single pore or achieve a steady-state value as for multiple pores.

FIG. 4. Behavior of PMFD as a function of time for t = 1to, 4to, and 9to for cesium diffusing radially from 共a兲 an individual pore and 共b兲 from an array of pores on a relative scale where white/yellow denotes the maximum and black/red denotes the minimum: for short times, the maximum and minimum values are well separated, but as time progresses, they converge, a fact obscured on a relative plot. The area shown is 共1.5兲2 times larger than the unit cell.

face of the bulk material, but the assumption here will be to use a single f factor. For parameters studied previously5 and assumed below, a relation between work function ⌽共␪兲 and local submonolayer coverage ␪ 艋 1 is approximately ⌽共␪兲 ⬇ 兺40cn␪n, where the coefficients, to three digit accuracy, are c0 = 4.60, c1 = −24.6, c2 = 55.5, c3 = −49.4, and c4 = 15.5. The monolayer coverage away from a pore and over a surface is a consequence of diffusion20–22 and therefore taken to follow a radial diffusion equation that evaluations of PMFD will follow, as represented in Fig. 4共a兲 for a single pore and Fig. 4共b兲 for an array of pores 共mimicking a CPD兲 for normalized times and distances, in which the characteristic radius is related to the 2D diffusion function u共x,y,t兲 ⬇

冋

册

共x2 + y 2兲 1 exp − , 4␲Dt 4Dt

共5兲

that satisfies ⳵tu = −Dⵜ2u. As with Fig. 3, caution in interpreting the images is required as the colors black/red and white/yellow represent the lowest and highest values of PMFD, respectively, without regard to the absolute magnitude: as time progresses, the coverage becomes more uniform, a fact obscured by relative comparisons. It is important to emphasize that the coverage ␪ starts off highly nonuniform and over time, as per ␪共x , y ; t兲 ⬀ u共x , y ; t兲, it asymptotically approaches uniformity to a submonolayer value dictated by the initial amount of cesium per pore, the pore to pore separation, and the boundary condi-

tions. At early times, ␪ is near unity only near the pores, whereas the majority of the surface is bare. As time progresses, cesium diffuses across the surface as per Eq. 共5兲. Therefore, initially, the surface is characterized by the bulk work function value that is high with a small region characterized by the cesiated work function value that is low, whereas after an asymptotically long time, ␪ has become uniform across the surface at a submonolayer value dictated by the initial amount of cesium, the pore-to-pore separation, and the boundary conditions. Consequently, the initial state consists of a small region that has a work function below the photoemission threshold, but when the surface is asymptotically uniform, if the work function is above the photon energy 共that is, 具⌽共␪兲典 ⬇ ⌽共具␪典兲 ⬎ ប␻兲, the surface taken as a whole produces less photoemission current than the small region originally with a higher concentration of cesium. Thus, depending on the final state, the photoemission current can remain comparable to the initial state if the photon energy 共e.g., for 4.6611 eV, ␭ = 266 nm兲 is higher than the asymptotic work function value, but fall precipitously if the photon energy 共e.g., for 2.3305 eV, ␭ = 532 nm兲 is lower than the asymptotic work function value. Each case is considered below. The behavior of PMFD in Eq. 共4兲 as a function of time for differing wavelengths as a function of both field and temperature is now examined. The maximum value and the mean value of PMFD averaged over a unit cell are considered for a field of 50 MV/ m and a temperature of 300 K for the wavelengths of 266 and 532 nm. The expectation is that the mean and peak values will converge as time progresses as the coverage becomes more uniform: the rapidity at which this happens serves as an estimate of the “rejuvenation time.” Similarly, the departure of the single pore 共n = 0兲 compared to the multiple pores 共n = 1兲 data in time is an indication of when adjacent pores make their influence known via the boundary conditions, and this behavior is shown in Fig. 4. In Fig. 5, using the peaks and averages for various times 共of



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which three cases are shown in Fig. 4兲 the “mean” value of PMFD is the average over the surface, whereas the “peak” value is the maximum that often occurs near the pore, where the caveat “often” indicates that this is not always so. According to Gyftopoulos-Levine theory, the lowest work function value occurs for submonolayer coverage 共e.g., on the order of 60% for Cs on W兲, so the value of ␪ can pass through this minimum, and is the origin of the donut-like appearance of the initial state in Fig. 4 as well as the initial rise of the curves in Fig. 5. The convergence of mean 共circles兲 and peak 共squares兲 as time progresses is a consequence of the minimum value of ⌽ asymptotically approaching the average value, whether or not both are increasing with time 共n = 0 or single pore兲 or approaching a constant value 共n = 1 or multiple pores兲.

V. GENERAL THERMAL-FIELD PHOTOEMISSION In the previous examples, flat surfaces were held at constant temperature and subject to uniform macroscopic field, the microscopic field being given by a randomly distributed field enhancement without reference to an underlying geometrical model in which the surface barrier variation was due to work function changes reflecting a distribution of crystal faces or nonuniform coverage. The next level of modeling complexity is encountered when the surface barrier to emission varies as a consequence of changing conditions, as well as field variation due to surface geometry, temperature variation due to laser intensity, and emission variation due to the combined actions of field, thermal, and photoemission processes. These conditions will affect the quality 共e.g., emittance兲 of the electron beam coming off the surface of the cathode, and therefore have consequences for how these electron sources are used and what is demanded of them. The motivation for this study is that photocathodes in modulated rf guns generate high brightness, high peak current electron bunches such that knowledge of the properties of the beam and how it evolves is critical for the optimization of gun design, as the value of the emittance determines the matched beam radius in the beam envelope equation.23,24 The beams are on the order of millimeters in diameter, smaller than the cavities and beam tunnel, but a great deal larger than the length scale characteristic of the emission surface. Similarly, time scale characteristics of the laser pulse are in tens of picoseconds 共ripples on that pulse can be far smaller兲, whereas the rf signal is in hundreds of picoseconds. All said, such considerations impact the accurate prediction of emittance generation and growth. The environment in which the beam propagates is complex, as waveguides, ports and tuning stubs affect beam dynamics. Therefore, the modeling of electron beam evolution under such complications is beyond analytical methods and is treated using numerical simulations, and, in particular, finite-element PIC codes such as MICHELLE 共Ref. 3兲关and others such as VORPAL 共Ref. 25兲兴. With their ability to resolve fine geometrical details near the surface of the cathode and to follow the electron beam as it forms and propagates, PIC codes can ascertain the impact of

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emission nonuniformity on overall device performance and capabilities. VI. PIC CODE IMPLEMENTATION Models of the general thermal-field emission equation and the time- and temperature-dependent photoemission theory were therefore implemented into the MICHELLE code and used to consider the growth in emittance as a consequence of various hypothetical test surfaces exhibiting work function variation and geometrical sources of field enhancement. Both time and temperature are important as the laser intensity may be sufficient to induce temperature changes in the photocathode material, which then affects thermal-field emission in the GTF model and scattering rates and emission probability in the photoemission model.5,7 Development of emission models that are self-consistent and stable as well as able to transition smoothly from one emission regime to another within a particle simulation code is very important. Though not considered herein, the code can also model the impact of other effects of importance to photoemission, such as laser ripple. MICHELLE is an electrostatic steady-state and timedependent PIC code, and is a good candidate for implementation of surface emissions models because it has a conformal meshing capability which allows it to resolve features that are widely disparate in scale 共i.e., from the micron-scale emitter surface structure to centimeter-scale and larger electron gun and injector dimensions兲, and can therefore address surface irregularities and emission nonuniformities, in an otherwise large volume that orthogonal-mesh electromagnetic PIC codes often find intractable. For standard vacuum electronic amplifier design, MICHELLE has demonstrated its predictive power in enabling first pass design success of a multibeam klystron26 using a conventional thermal emission model. The present work has replaced that emission model with a photoemission model that accounts for the impact of local work function variations across the surface of the cathode on the photoemission current, but which is otherwise used in an analogous way to the thermal model. Thus, space charge fields from the emitted current in addition to the rf fields are dynamically accounted for and the consequent impact on the emitted photocurrent obtained. Copper, being a well-studied metal in terms of work function variation among its crystal faces 共the 关100兴, 关110兴, 关111兴, and 关112兴 crystal faces of copper have work functions of 5.1, 4.48, 4.94, and 4.53 eV, respectively兲 and regularly used as a photocathode source, represented a good test case. In early numerical simulations using the photoemission models in MICHELLE, initial time-dependent studies examined field enhanced photoemission from a hexagonally packed periodic pattern of 6 ␮m spherical bosses with 12 ␮m spacing subject to a high laser intensity of 160 MW/ cm2 and a field of 5 MV/ m. It was found that oscillations in the electron current due to space charge effects exhibited a period of 3 fs concentrated about the apex of the boss where the field enhancement is highest. Subsequent simulations of the bosses were then placed nonuniformly on an area on which multiple crystal faces were presumed present, as shown in Fig. 6 for



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TABLE I. Emittance growth for “bump” and “no bump” surfaces, referring to hemisperical bosses placed atop the polycrystalline model 共as in Fig. 6兲. Run

Grain

Current 共mA兲 E-surface 共MV/m兲 Emittance ratio

No bump From scan Bumpy From scan

2.943 3.319

0–4.88 0–12.9

1 5.6

emission sources as the electron beams from them, unlike thermionic sources in, for example, microwave amplifiers, are not smoothed by space charge limited operation. VII. CONCLUSION

FIG. 6. MICHELLE simulation results of the photoemission model applied crystal face nonuniformities with and without superimposed 6 ␮m diameter hemispherical bosses are shown. Figures on the left are for no bosses and those on the right are with bosses. The colors in the top row show the current density on the emission surface, the second row shows transverse velocity vectors, and the third row superimposes the two. When the bosses are included the emittance is enhanced by a factor of 5.6, as in Table I. It is seen that the bosses introduce a substantial transverse velocity component that causes the emittance to increase precipitously, even though the current increases only by a modest amount due to increased area and field enhancement.

We have presented a general thermal-field-photoemission model and have applied it to several pedagogical situations. First, the general thermal-field emission equation was used to analyze electron emission in the transition region between thermal and field emission where temperatures are warm and intermediate fields encountered. Second, conditions in which work function variation occurs due to differing crystal faces as well as a radially diffusing low work function coating of cesium from a pore were examined by the GTF and photoemission models. Finally, the performance of the models in a particle-in-cell code was used to estimate the relative impact of emittance growth on conditions of work function nonuniformity and surface structure. Such codes have application to modeling advanced rf vacuum electronic and accelerator devices such as free electron lasers, for which the impact of the electron beam generated by a rugged and self-repairing high quantum efficiency photocathode based on the dispenser cathode technology is of interest. ACKNOWLEDGMENTS

conditions made to be compatible with an operational regime of interest 共1 nC of bunch charge extracted in 10 ps from a cathode area of 2 mm in radius under an applied field of 10 MV/ m and subject to a laser intensity of 1 MW/ cm2 which was shown to be below the level causing space charge limited emission and virtual cathode effects兲. The presence and presumed shape of multiple crystal faces was modeled from actual scans of polycrystalline surfaces used in our photoemission studies. In the figure, the left hand images show the result of the crystal face nonuniformity, and those on the right have the added complication of the 6 ␮m bosses. The illustrations on the top row show the current density emitted. The perpendicular velocity component of the emitted charge 共shown as the vector lines in the second row of images兲 is a measure of emittance: therefore, the behavior of the emitted charge in the comparisons between the no boss/boss images is reflected in the numerical evaluation of the relative emittance growth from such a surface shown in Table I, and shown to increase the emittance by a factor of 5.6. The faceting alone causes an increase in the emittance over uniform crystals by a factor of 10–60. The ability of the PIC code to perform this type of small-scale detailed modeling provides insight into the intrinsic emittance of photoemitters and field

The authors gratefully acknowledge funding provided by the Joint Technology Office and the Office of Naval Research. The authors thank many colleagues, in particular, J. Lewellen 共ANL兲, D. Dowell 共SLAC兲, and J. Smedley 共BNL兲 for helpful discussions. 1

R. K. Parker, R. Abrams, B. Danly, and B. Levush, IEEE Trans. Microwave Theory Tech. 50, 835 共2002兲. 2 P. G. O’Shea and H. P. Freund, Science 292, 1853 共2001兲. 3 J. J. Petillo, E. M. Nelson, J. F. Deford, N. J. Dionne, and B. Levush, IEEE Trans. Electron Devices 52, 742 共2005兲. 4 K. L. Jensen, J. Appl. Phys. 102, 024911 共2007兲. 5 K. L. Jensen, N. A. Moody, D. W. Feldman, E. J. Montgomery, and P. G. O’Shea, J. Appl. Phys. 102, 074902 共2007兲. 6 R. G. Forbes, Appl. Phys. Lett. 89, 113122 共2006兲. 7 K. L. Jensen, D. W. Feldman, N. A. Moody, and P. G. O’Shea, J. Appl. Phys. 99, 124905 共2006兲. 8 C. Travier, Nucl. Instrum. Methods Phys. Res. A 304, 285 共1991兲. 9 B. Leblond, Nucl. Instrum. Methods Phys. Res. A 317, 365 共1992兲. 10 B. Leblond, G. Kuznetsov, and M. Batazova, Nucl. Instrum. Methods Phys. Res. A 372, 562 共1996兲. 11 C. Travier, G. Devanz, B. Leblond, and B. Mouton, Nucl. Instrum. Methods Phys. Res. A 393, 451 共1997兲. 12 K. L. Jensen, D. W. Feldman, M. Virgo, and P. G. O’Shea, Phys. Rev. ST Accel. Beams 6, 083501 共2003兲. 13 K. L. Jensen, D. W. Feldman, and P. G. O’Shea, Appl. Phys. Lett. 85, 5448 共2004兲. 14 N. A. Moody, K. L. Jensen, D. W. Feldman, P. G. O’Shea, and E. J.



837


Montgomery, Appl. Phys. Lett. 90, 114108 共2007兲. J. L. Shaw, J. E. Yater, J. E. Butler, D. W. Feldman, N. Moody, and P. G. O’Shea, IEEE International Vacuum Electronics Conference, 2006 held Jointly with 2006 IEEE International Vacuum Electron Sources 共unpublished兲, p. 431. 16 M. Ondrejcek, V. Chab, W. Stenzel, M. Snabl, H. Conrad, and A. Bradshaw, Surf. Sci. 333, 764 共1995兲. 17 M. Snabl, M. Ondrejcek, V. Chab, W. Stenzel, H. Conrad, and A. M. Bradshaw, Surf. Sci. 352, 546 共1996兲. 18 E. P. Gyftopoulos and J. D. Levine, J. Appl. Phys. 33, 67 共1962兲. 19 K. L. Jensen, D. W. Feldman, N. A. Moody, and P. G. O’Shea, J. Vac. Sci. Technol. B 24, 863 共2006兲. 20 I. Langmuir and J. B. Taylor, Phys. Rev. 40, 463 共1932兲. 21 K. Stolt, W. Graham, and G. Ehrlich, J. Chem. Phys. 65, 3206 共1976兲. 15

837

R. Gomer, Rep. Prog. Phys. 53, 917 共1990兲. M. Reiser, Theory and Design of Charged Particle Beams 共Wiley, New York, 1994兲. 24 B. E. Carlsten, L. M. Earley, F. L. Krawczyk, S. J. Russell, J. M. Potter, P. Ferguson, and S. Humphries, Phys. Rev. ST Accel. Beams 8, 062001 共2005兲. 25 D. A. Dimitrov, D. l. Bruhwiler, J. R. Cary, P. Messmer, P. Stoltz, K. L. Jensen, D. W. Feldman, and P. G. O’Shea, Proceedings of the Particle Accelerator Conference, 2005 共unpublished兲, p. 2583. 26 D. K. Abe, D. E. Pershing, K. T. Nguyen, F. N. Wood, R. E. Myers, E. L. Eisen, M. Cusick, and B. Levush, IEEE Electron Device Lett. 26, 590 共2005兲. 27 C. A. Spindt, I. Brodie, L. Humphrey, and E. R. Westerberg, J. Appl. Phys. 47, 5248 共1976兲. 22 23

