Semiconductor Science and Technology

PAPER

A model study of the role of workfunctionvariations in cold field emission frommicrostructures with inclusion of fieldenhancementsTo cite this article: H Qiu et al 2015 Semicond. Sci. Technol. 30 105038

View the article online for updates and enhancements.

Related contentEvaluation of high field and/or localheating based material degradation ofnanoscale metal emitter tips: a moleculardynamics analysisZ Zhang, M Giesselmann, J Mankowski etal.

-

Nanotube field electron emission:principles, development, and applicationsYunhan Li, Yonghai Sun and J T W Yeow

-

Carbon nanotube-based electron fieldemittersAleksandr V Eletskii

-

Recent citationsDynamic analysis of material ejection fromcathodic metal nano-tips due to localheating and field generated stressX. Qiu and R. P. Joshi

-

Comparison between Single-Walled CNT,Multi-Walled CNT, and Carbon Nanotube-Fiber Pyrograf IIIMarwan S. Mousa

-

Low pressure hydrogen sensing based oncarbon nanotube field emission:Mechanism of atomic adsorption inducedwork function effectsYangyang Zhao et al

-

This content was downloaded from IP address 129.118.3.25 on 05/09/2018 at 18:15

https://doi.org/10.1088/0268-1242/30/10/105038http://iopscience.iop.org/article/10.1088/1361-6463/aa6497http://iopscience.iop.org/article/10.1088/1361-6463/aa6497http://iopscience.iop.org/article/10.1088/1361-6463/aa6497http://iopscience.iop.org/article/10.1088/1361-6463/aa6497http://iopscience.iop.org/article/10.1088/0957-4484/26/24/242001http://iopscience.iop.org/article/10.1088/0957-4484/26/24/242001http://iopscience.iop.org/article/10.3367/UFNe.0180.201009a.0897http://iopscience.iop.org/article/10.3367/UFNe.0180.201009a.0897http://dx.doi.org/10.1063/1.5018441http://dx.doi.org/10.1063/1.5018441http://dx.doi.org/10.1063/1.5018441http://iopscience.iop.org/1757-899X/305/1/012025http://iopscience.iop.org/1757-899X/305/1/012025http://iopscience.iop.org/1757-899X/305/1/012025http://dx.doi.org/10.1016/j.carbon.2017.09.032http://dx.doi.org/10.1016/j.carbon.2017.09.032http://dx.doi.org/10.1016/j.carbon.2017.09.032http://dx.doi.org/10.1016/j.carbon.2017.09.032http://oas.iop.org/5c/iopscience.iop.org/982127857/Middle/IOPP/IOPs-Mid-SST-pdf/IOPs-Mid-SST-pdf.jpg/1?

A model study of the role of workfunctionvariations in cold field emission frommicrostructures with inclusion of fieldenhancements

H Qiu1, R P Joshi2, A Neuber2 and J Dickens2

1Department of Electronic Engineering Technology, Fort Valley State University, Fort Valley, GA31030, USA2Department of Electrical and Computer Engineering, Texas Tech University, Lubbock, TX 79409, USA

E-mail: ravi.joshi@ttu.edu

Received 5 June 2015, revised 5 August 2015Accepted for publication 24 August 2015Published 21 September 2015

AbstractAn analytical study of field emission from microstructures is presented that includes position-dependent electric field enhancements, quantum corrections due to electron confinement andfluctuations of the workfunction. Our calculations, applied to a ridge microstructure, predictstrong field enhancements. Though quantization lowers current densities as compared to thetraditional Fowler–Nordheim process, strong field emission currents can nonetheless be expectedfor large emitter aspect ratios. Workfunction variations arising from changes in electric fieldpenetration at the surface, or due to interface defects or localized screening, are shown to beimportant in enhancing the emission currents.

Keywords: field emission, workfunction variations, model analyses, fowler nordheim, quantumconfinement

(Some figures may appear in colour only in the online journal)

1. Introduction

Electron emission from cathodes can provide high currentdensities (several kA cm−2) for a variety of applications suchas high power microwave generation [1–3], free electronlasers [4], pumping of excimer lasers [5], in the plasmaphysics arena [6], and for surface modifications and materialprocessing [7, 8]. Field emitters also play an important role invacuum electronics [9], and as sources for neutron generators[10, 11]. Field emission is attractive (e.g. over the thermionicprocess) since no heating is required and the emission currentis almost solely controlled by the external field. Cold cathodetechnology has employed Spindt-type field emitter arrays[12–14] or the more recent carbon fiber cathodes with CsIcoating [15, 16]. The Spindt-type emitters, however, sufferfrom high manufacturing cost and limited lifetime, withfailure often caused by ion bombardment from the residualgas species that blunt the emitter cones. Diamond emitting

structures, though considered attractive in the past, areunstable at high current densities [17]. Electron field emissionfrom carbon nanotubes (CNTs), first demonstrated in 1995[18, 19], is becoming increasingly popular and has beenstudied extensively. CNTs offer advantages of nanometer-sizediameter, structural integrity, high electrical and thermalconductivity, possibility of large aspect ratios, and chemicalstability [20].

Field emission is often characterized by Fowler–Nord-heim (FN)-type equations describing the process of electrontunneling from the conduction band in terms of the work-function f and the local electric field FL, as reviewed byGomer [21]. The local field can be significantly higher thanthe average externally applied field F0, and this increase isconventionally denoted by the enhancement factor β=FL/F0. Several reports in the literature have presented theoreticaldiscussions on electric field increases [22–27] using methodof images or series expansion techniques to solve Laplace’s

Semiconductor Science and Technology

Semicond. Sci. Technol. 30 (2015) 105038 (8pp) doi:10.1088/0268-1242/30/10/105038

0268-1242/15/105038+08$33.00 © 2015 IOP Publishing Ltd Printed in the UK1

mailto:ravi.joshi@ttu.eduhttp://dx.doi.org/10.1088/0268-1242/30/10/105038http://crossmark.crossref.org/dialog/?doi=10.1088/0268-1242/30/10/105038&domain=pdf&date_stamp=2015-09-21http://crossmark.crossref.org/dialog/?doi=10.1088/0268-1242/30/10/105038&domain=pdf&date_stamp=2015-09-21

equation for different geometries. The β factor depends on thegeometric shape [28–31], though a constant enhancementfactor has often been used, or simply chosen as a fittingparameter to be extracted from experimental data. To avoidconfusion, it may be mentioned that the enhancement factoralluded to here is separate from a corrective prefactor thatoften becomes necessary for FN field emission. Such a pre-factor is associated with spatial details of potential barriers forelectronic tunneling [32]. On the experimental side, therehave also been developments that have looked at novel waysto represent the electric field profile around sharp emitters[33]. These have included general methods for mapping theequipotential profile for CNT or Spindt tip structures havinghigh aspect ratios, to obtain the electric field enhancementfactor.

A complete and accurate treatment of field emission iscomplicated and a variety of physics-based issues need to beconsidered. Emission relies on both electron tunnelingthrough a potential barrier, which yields the escape prob-ability, and the electron ‘supply’ function, which requiresknowledge of the band structure and any defect states near theemitting surface. Typical model analyses assume: (i) negli-gible field penetration occurs in the cathode and so electronsinside the emitter are nearly in thermodynamic equilibrium.However, this is strictly incorrect since the very application ofan electric field (especially of high magnitude) breaks thethermodynamic equilibrium to then establish an electron flowthrough the system; (ii) tunneling emission of electronsoccurs through states near the emitter Fermi level (metals) orthe conduction band edge; (iii) electron states within thecathode can essentially be treated as plane waves so thatcarrier confinement and effects on electron wavefunctions canbe neglected; (iv) the workfunction and barrier height areconstant and independent of either position or external field.Yet another simplification often used is near-triangular barrierpotentials [34] that ignore corrections to the simple shape[32, 35, 36] including dipole and image potentials, andexchange-correlation effects [37]. In addition, the role ofspace-charge created by electron emission [38] or the effectsof incoming ions that might distort the local fields [39] areusually overlooked. More complicated situations can arise inthe case of multiple field-emitting arrays [40, 41] for whichthe different local field enhancement factors as well asscreening effects become operative [42–44]. The interplaybetween field enhancements and screening for emitter arraysdepends on the degree of charge sharing. For example, a fixedcharge density associated with the emitter side of an emittingarray has to be collectively shared between individual emit-ters. Sharing between a large number of emitters per unit arearesults in low field enhancement, whereas sharing between asmall number of emitters per unit area yields much strongerfield enhancement factors.

From the above, it is clear that the physical detailsassociated with field emission are quite complex and manyprocesses are at work. Many of the complexities have alreadybeen analyzed in the literature, though only a few importantcontributions have been cited. Here, we attempt to focus on arelatively modest aspect that, though known, has not been

explicitly evaluated to the best of our knowledge. The role ofspatial variation and fluctuations of the workfunction areexamined here through numerical simulations for nanoscaleridge emitters. These variations are included in calculations ofthe field emission currents. For completeness, the electric fieldenhancements as a function of position and emitter aspectratio are probed analytically based on conformal mappingtechniques. Quantum corrections to the electron current aris-ing from reductions in the emitter width are included alongwith workfunction variability.

1.1. Workfunction variability in field emitters

The original FN approach modeled field emission as one-electron tunneling through a one-dimensional triangularpotential barrier. This picture was subsequently modified toinclude an ‘image-force’ correction to account for the elec-trostatic effect of the tunneling electron that lead to a‘rounding’ of the triangular barrier [45, 46]. The image cor-rection, however, represents an idealization, introducing anunbounded force close to the surface and overemphasizing theelectrostatic effect. As is well known, conduction electronsfrom the emitter can quantum mechanically spillover into thevacuum up to very short distances, producing Freidel oscil-lations [47, 48]. In the immediate vicinity of the surface, thespillover electrons have been shown to create an interfacedipole barrier that influences the local workfunction [49] andmodifies the surface field. In addition, penetration of theelectric field into the metal can also change the effectiveworkfunction [50], alter the potential and modify electrondensity at the surface. The change in the workfunction for ametal was roughly given to be [50]: 0.5qF0λ/[{1 − qF0λ/(6EF)}{1−qF0λ/(3EF)}

½], with λ={EF/(6πq2n0)}

½, EFdenoting the Fermi level, F0 the electric field and n0 being thefree electron density. For metal electrodes with electrondensity n0 on the order of 8×10

28 m−3 (e.g. copper) and aFermi level of ∼6.8 eV, calculations yield λ=0.89 Å andcan result in a workfunction change above ∼0.5 eV at fieldsof 1010 Vm−1 or higher. However, in practical situationsinvolving substantially lower fields, changes on the order of0.15 eV have been reported [51, 52]. In the general case of anemitter, the free electron density at the surface could also bespatially nonuniform along the lateral plane due to traps ormaterial inhomogeneities. Furthermore, the electron field atthe surface could vary across microstructures due to sharp orpointed geometries, changes in the enhancement factor, ordifferences in plasma densities near the emitter. Hence, theworkfunction would be nonuniform, and this represents acomplicated problem that is also expected to be dynamic.Thus, the usual assumption of a uniform and constantworkfunction is not physically correct. Variation of work-function has been explored to some extent in the past usinginhomogeneous barrier height and patch-field models [53–56]that also probe barrier heights and ideality factors [57, 58].Here, for simplicity the workfunction is simply treated as arandom variable about a mean value fm. Effects of barrierheight fluctuations have previously been probed in the contextof Schottky diodes [59]. Since the fluctuations are the

2

Semicond. Sci. Technol. 30 (2015) 105038 H Qiu et al

collective result of multiple discrete events and local electricfield variations, many-body charge contributions, and anyimpurities or defects in the emitter, one can approximate thedistribution function to have a Gaussian form based on thelaw of large numbers. The normalized distribution for theworkfunction p(f) then takes the following form: p(f)={2πσ2}−½ exp[−(f − fm)

2/(2πσ2)], with σ being thevariance parameter. The use of such a statistically averagedGaussian function has been reported [54, 60], and implicitlyassumes that correlation effects are negligible.

1.2. Model analyses of field enhancement in an emitting ridgemicrostructure

In the context of a protruding ridge as shown schematically infigure 1, the field enhancement factor can easily be obtainedthrough conformal mapping techniques. The protrusion isassumed to have a total width (x-axis) of 2a units, and aheight (y-axis) of h units. Transforming the structure from thez (≡x, y)-plane onto the w (≡u, v)-plane with the vertices A,B, C and D mapping onto A’, B’, C’ and D’, the Schwarz–Christoffel mapping is given by [61–63]

z

wK

w u

wa

d

d 1, 1

202

2( )= -

-

K a wu w

wbwith, d

1. 1

u

0

02 2

2

0

( )ò=-

-/

Integrating from D to C (z-plane) with a correspondingintegration from C′ to D′ (w-plane), and similarly integratingfrom (0, 0) to C (in the z-plane) with a corresponding

integration from (0, 0) to C′ (in the w-plane) yields

h K ww u

wad

1. 2

u

1 202

20

( )ò=-

-

a K wu w

wbd

1. 2

u

0

02 2

2

0

( )ò=-

-

Dividing equation (2a) by (2b) then leads to

h

a

ww u

w

wu w

w

N u

D u

d1

d1

. 3u

u

1 202

2

0

02 2

2

0

0

0

0

( )( )

( )ò

ò=

-

-

-

-

º

The numerator N(u0) in the above equation can beevaluated as

N u ww u

wE u

u F u a

d1 2

, 1

2, 1 , 4

u0

1 202

2 02

02

02

0

( )

( )

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

òp

p

=-

-= -

- -

where E u, 12 0

2( )-p is the complete elliptic integral ofthe second kind. Similarly, the integration of the denominatorD(u0) works out to be

D u wu w

wE u

u F u b

d1 2

,

12

, , 4

u

00

02 2

2 0

02

0

0

( )

( )

( )

⎜ ⎟

⎜ ⎟

⎛⎝

⎞⎠

⎛⎝

⎞⎠

òp

p

=-

-=

- -

Figure 1. The cathode ridge structure. (a) Schematic in the real z (≡x, y)-plane with the electric field lines shown as a guide, and (b) aconformal map of the ridge structure in the complex w (≡u, v)-plane.

Figure 2. Variation of the parameter u0 with the aspect ratio h/a forthe microstructured ridge emitter.

3

Semicond. Sci. Technol. 30 (2015) 105038 H Qiu et al

where F u,2 0

( )p is the complete elliptic integral of the firstkind. Both the complete elliptic integrals in equation (4a) canconveniently be expressed in terms of polynomials [64] forease of computational evaluation. The plot of figure 2 showsthe variation of the parameter u0 with the aspect ratio h/a forthe ridge emitter. For large aspect ratios, the value of u0 ispredicted to approach zero, while in the extreme limit ofa→∞, the value of u0 tends to unity.

The value of the electric fields in the vicinity of the ridgecan now be easily determined. Considering the complexelectrostatic potential fw=jKF0w, the field Fz can beexpressed as

F jFw

w u

1. 5z 0

2

202

( )= - --

Hence, on the top surface BC, the electric field works outto be Fx+j Fy=−j F0 [(1 – u

2)/(u02– u2)]½, i.e. Fy=−F0

[(1 – u2)/(u02– u2)]½ for—u0

Wentzel–Kramers–Brillouin (WKB) approximation [69] forevaluating the tunneling probability. Also, in the aboveexpression, F[π/2, (1−x1/x2)

½] and E[π/2,(1−x1/x2)½] are

the complete elliptic integrals of the first and second kind,respectively.

With quantum confinement, the energy available formobile electrons is reduced by the quantized energy En for thenth subband. This effective decrease in the emission currentappears through the exponential dependence on En as given inequation (8). Thus, with downscaling of the emitter widththere would be the effect of electric field enhancement leadingto a stronger driver for electron emission. However, reduc-tions in transmission coefficient reduce the actual throughput.

Finally, taking account of the workfunction variationsdue to charge, electric field variability, surface traps anddefects, or simply due to surface erosion from repetitivedevice operation, the current density Jn as given by theGaussian probability density function becomes

J J F p p a, d d , 11n n L( ) ( ) ( ) ( )⎡⎣⎢⎤⎦⎥

⎡⎣⎢

⎤⎦⎥ò òf f f f f=

p

b

where, 2 exp 2 .

11

m2 1 2 2 2( ){ } ( )( )

( )

⎡⎣ ⎤⎦f ps f f ps= - --

//

In the above, Jn(FL, f) represents the current density asgiven in equation (11).

For completeness, it may be mentioned that strictlyspeaking the application of an electric field (particularly oneof a large magnitude) would perturb the energy levels,wavefunctions and carrier densities in the vicinity of theemitting surface. As a result the number of energy subbandscontributing to electron emission would change depending onthe local electric field magnitude. This aspect have beenprobed and discussed in the literature [70], and shown to leadto interesting step-like current-voltage characteristics. How-ever, such details have been ignored here for simplicity.

2. Results and discussion

The enhancement factor β ≡|F|/F0, was first evaluated forthe ridge structure of figure 1(a). The results of the calculationshowing the electric field enhancement factor as a function ofthe spatial distance along the ridge perimeter are given infigure 4. The three curves were obtained for an emitter heighth=10 μm, but with three different aspect ratios h/a of 10,100, and 500, respectively. The sharp peak in all three cases isat u=u0 and coincides with the corner point C in figure 1(a).The result underscores the potential for strong electric fieldsand, hence, high emission currents for narrow emitter struc-tures. Though the surface area for emission does reduce withthe half-width a, the exponentially dependent nonlinearenhancements in emission current density at the smaller half-widths would work to produce higher currents overall. Theenhancement factor is predicted to dip well below unity forthe h/a=10 curve of figure 4, in the neighborhood of point

D which is at u=3u0. Physically, this represents an effectiveelectric field screening due to the convergence of electric fieldlines at the corner C. Quite simply, the strong fieldenhancement at the corner increases the local charge densityand works to draw down the electronic charge from nearbyregions around point D.

Next, the two-dimensional current density, as given inequations (10), was obtained as a function of the local electricfield. Emitters with the same 10 μm height but different half-widths of 0.5 nm, 2 nm and 10 nm were used for the calcu-lations. The results are shown in figure 5. For comparison, thecurrent density from bulk FN emission, as computed fromequation (9) is also plotted. As evident from figure 5, the bulkformulation yields the highest current density values. Theappearance of lower current density in figure 5 at highervalues of emitter half-width is not counterintuitive, since theoverall current which scales with emitter width is indeedhigher for a wider emitter. Strictly speaking, a staircase-likestructure with increasing field, as discussed by Filip et al [70],due to progressive additive contributions from the varioussubbands occurs. However, this aspect which depends onboth the emitter dimensions and electric field is not seen onthe linear scale and was not the focus of the present study.

The two-dimensional quantization aspect was nextcombined with the electric field enhancement to obtain acomprehensive assessment of the electron emission. Thecurrent density across the top emitting surface B-C of theridge emitter (figure 1(a)) as a function of position is shown infigures 6(a) and (b) for half-widths of 5 nm and 20 nm,respectively. The emitter height was maintained at 10 μm ineach case. Large current increases are predicted near the

Figure 4. Calculated results for the electric field enhancement factorβ as a function of the spatial distance along the ridge perimeter. Thethree curves are for an emitter height h=10 μm, but three differentaspect ratios h/a of 10, 100, and 500.

5

Semicond. Sci. Technol. 30 (2015) 105038 H Qiu et al

emitter corner, with the largest densities predicted for thenarrowest emitter. For these evaluations, the background fieldF0 (i.e. the background value far away from the emitterstructure) was taken to be 107 V m−1. For completeness itmay be mentioned that the FN current densities at this con-stant F0 value would be negligible.

Finally, the role of workfunction fluctuations was ana-lyzed based on a Gaussian distribution as highlighted inequation (11). Such variations would be associated with thespatial variations in the electric field, and discrete nature ofdefects and/or charge near the surface. Simulation results ofthe current density as a function of the surface electric fieldare shown in figure 7 with workfunction variability taken intoaccount. Values of the variance σ were taken to be 0.05 eV,0.25 eV and 1.15 eV. Enhancements in emission current arepredicted in figure 7 with corresponding increases in thevariance parameter. Thus, in addition to field enhancementstypically associated with localized geometry, the role ofworkfunction fluctuations could well remain an important yetlatent and/or overlooked contribution to high field emission.Since the fluctuations depend on the local electric field and itsshielding, such ‘patchwork’ emission is likely to also add adynamic feedback process. The latter could even conceivablygive rise to noisy emission currents, periodic disruption in theelectron emission flux with time due to space-charge effectsassociated with the emission, and varying bright spots at theemitting surface. However, such details are beyond the pre-sent scope. The increase in figure 7 follows directly from thenonlinear dependence of the current on the workfunction.Physically, electrons would be emitted from localized spotshaving a low workfunction, possibly leading to

disproportionately nonuniform current densities and evenfilamentary emission. As an extreme example of the latter,explosive emission in a highly localized manner as has beenobserved [71, 72] in high current cathodes could occur, andthe workfunction fluctuations studied here could well have acontribution. However, in this context other inherentmechanisms would need to be considered in the context oflocalized emission. These would include localized heating

Figure 5. Current density as a function of the electric field. Resultsfrom the two-dimensional formulation for emitters of height 10 μmand half-widths of 0.5 nm, 2 nm and 10 nm are shown, along withthe current density from bulk FN emission.

Figure 6. Calculations of the current density across the top emittingsurface B-C of the ridge emitter shown in figure 1(a). The spatiallyvarying field enhancement factor was coupled with the two-dimensional current density to yield a comprehensive assessment ofthe electron emission. For a constant 10 μm height, emitter half-widths of (a) 5 nm and (b) 20 nm were used.

6

Semicond. Sci. Technol. 30 (2015) 105038 H Qiu et al

through electron–phonon coupling, and the possibility ofcreating a highly nonequilibrium heated phonon populationthat then leads to localized melting and mass ejection.

3. Conclusions

Cold field emission from microstructures has been studied inthis contribution with an emphasis on three distinct aspects,namely: (i) the position-dependent electric field enhance-ments; (ii) the role of quantum corrections to electronic fieldemission; and (iii) the role of probabilistic variations in theworkfunction for nanoscale emitters. These variations wereincluded through a simple Gaussian probability function. Dueto the strongly nonlinear dependence of electron emission onthe workfunction, significant current enhancements and non-uniformity can result. However, in the literature only thegeometric enhancement factor has been attributed [73] to thisphenomenon. For completeness, the electric field enhance-ments as a function of position and emitter aspect ratio werealso probed based on conformal mapping techniques. Quan-tum corrections to the electron current arising from reductionsin the emitter width were also included in the field emissionformulation with workfunction variability. Our results showthat variations in workfunction are important in enhancing theemission currents, and could well play a role in initiatingexplosive emission from surface protrusions.

Results of the present study could easily be extended toother geometries and emitter tip configurations, such as theLorentzian [26] or hyperboloid [74] shapes often discussed inthe context of field emitters. As a final comment, the possi-bility of localized cathode heating, or material ejection thatmight dynamically alter the emitter aspect ratio, was ignoredsince our calculations apply to a shorter temporal regime and/or less intense emission currents. A more complete and rig-orous analysis would require the self-consistent and time-dependent inclusion of electrothermal aspects. For example,the possibility of material vaporization from cathode tips hasbeen reported in the literature [75, 76]. From a theoreticalstandpoint, this is a dynamical energy-flow process with theelectrons transferring energy gained from the field to thelattice via electron–phonon interactions. This in turn, givesrise to nonequilibrium phonons with effective phonon tem-peratures dictated by wavevector-dependent, coupled elec-tron–phonon and anharmonic phonon–phonon relaxationprocesses. At high currents, one can then have the possibilityof localized melting as phonon effective temperatures exceedthe melting point. However, such aspects are beyond thepresent scope.

Acknowledgments

This research was supported by the Office of Naval Research(Grant # N00014-13-1-0567). Useful discussion with KJensen (NRL) is gratefully acknowledged.

References

[1] Benford J, Swegle J and Schamiloglu E 2007 High PowerMicrowaves (New York: Taylor and Francis)

[2] Barker R J and Schamiloglu E 2001 High-Power MicrowaveSources and Technologies (New York: IEEE Press)

[3] Miller R B, McCullough W F, Lancaster K T andMuehlenweg C A 1992 IEEE Trans. Plasma Sci. 20 332

[4] Ben-Zvi I 2006 Physica C 441 21[5] Okuda I, Takahashi E and Owadano Y 2001 Japan. J. Appl.

Phys. 40 7168[6] Sakagami H, Sunahara A, Johzaki T and Nagatomo H 2012

Laser Part. Beams 30 103[7] Hao S, Dong C, Li M, Zhang X and Wu P 2009 Int. J. Mod.

Phys. B 23 1713[8] Lin X X et al 2012 Laser Part. Beams 30 39[9] Gaertner G 2012 J. Vac. Sci. Technol. B 30 060801[10] Ellsworth J L, Tang V, Falabella S, Naranjo B and Putterman S

2013 AIP Conf. Proc. 1525 128[11] Persaud A, Waldmann O, Kapadia R, Takei K, Javey A and

Schenkel T 2012 Rev. Sci. Instrum. 83 02B312[12] Guzenko V A, Mustonen A, Helfenstein P, Kirk E and

Tsujino S 2013 Microelectron. Eng. 111 114[13] Spindt C A, Brodie I, Humphrey L and Westerberg E R 1976

J. Appl. Phys. 47 5248[14] Schwoebel P R and Spindt C A 2009 J. Vac. Sci. Technol. B

12 2414[15] Parson J M, Lynn C F, Mankowski J J, Neuber A and

Dickens J C 2014 IEEE Trans. Plasma Sci. 42 3982[16] Shiffler D, Heidger S, Cartwright K, Vaia R, Liptak D, Price G,

LaCour M and Golby K 2008 J. Appl. Phys. 103 013302

Figure 7. Simulation results of the current density as a function ofthe surface electric field with workfunction variability taken intoaccount. Values of the variance σ were chosen to be 0.05 eV,0.25 eV and 1.15 eV.

7

Semicond. Sci. Technol. 30 (2015) 105038 H Qiu et al

http://dx.doi.org/10.1109/27.142834http://dx.doi.org/10.1016/j.physc.2006.03.052http://dx.doi.org/10.1143/JJAP.40.7168http://dx.doi.org/10.1017/S0263034611000735http://dx.doi.org/10.1142/S0217979209061512http://dx.doi.org/10.1017/S0263034611000668http://dx.doi.org/10.1116/1.4747705http://dx.doi.org/10.1063/1.4802305http://dx.doi.org/10.1063/1.3672437http://dx.doi.org/10.1016/j.mee.2013.02.039http://dx.doi.org/10.1063/1.322600http://dx.doi.org/10.1116/1.587774http://dx.doi.org/10.1109/TPS.2014.2363462http://dx.doi.org/10.1063/1.2821152

[17] Zhu W (ed) 2001 Vacuum Microelectronics (New York:Wiley)

[18] Rinzler A G, Hafner J H, Nikolaev P, Lou L, Kim S G,Tomanek D, Colbert D and Smalley R E 1995 Science269 1550

[19] Heer W A D, Chatelain A and Ugarte D 1995 Science270 1179

[20] Dresselhaus M S, Dresselhaus G and Avouris P (ed) 2001Carbon nanotubes: synthesis, structure, properties, andapplications Topics in Appl. Phys. 80 (Heidelberg: Springer-Verlag)

[21] Gomer R 1994 Surf. Sci. 299 129[22] Dyke W P, Trolan J K, Dolan W W and Barnes G 1953

J. Appl. Phys. 24 570[23] Jensen K L, Kodis M A, Murphy R A and Zaidman E G 1997

J. Appl. Phys. 82 845[24] Kosmahl H G 1991 IEEE Trans. Electron Devices 38 1534[25] Jensen K, Lau Y Y, Feldman D W and O’Sh P G 2008 Phys.

Rev. ST Accel. Beams 11 081001[26] Liu Y F and Lau Y Y 1996 J. Vac. Sci. Technol. B 14 2126[27] Sun S and Ang L K 2012 Phys. Plasmas 19 033107[28] Kusne A G and Lambeth D N 2010 IEEE Trans. Electron

Devices 57 712[29] Yuan G, Song H, Jin Y, Mimura H and Yokoo K 2005 Thin

Solid Films 484 379[30] Wang X Q, Wang M, He P M and Xu Y B J. Appl. Phys.

96 6752[31] Feng Y and Verboncoeur J P 2006 Phys. Plasmas 13 073105[32] Forbes R G 2008 J. of Appl. Phys. 103 114911[33] Filip L D, Carey J D and Silva S R P 2011 J. Appl. Phys. 109

084527[34] Rokhlenko A 2011 J. Phys. A: Math. Theor. 44 055302[35] Murphy E L and Good R H 1956 Phys. Rev. 102 1464[36] Jensen K L 2012 J. Appl. Phys. 111 054916[37] Kiejna A and Wojciechowski K F 1996Metal Surface Electron

Physics (Oxford: Pergamon)[38] Rokhlenko A, Jensen K L and Lebowitz J L 2010 J. Appl.

Phys. 107 014904[39] Chen Y, Deng S Z, Xu N S, Chen J, Ma X C and Wang E G

2002 Mater. Sci. Eng. A 327 16[40] Bonard J M, Weiss N, Kind H, Stockli T, Forro L, Kern K and

Chatelain A 2001 Adv. Mater. 13 184[41] Jo S H, Tu Y, Huang Z P, Carnahan D L, Wang D Z and

Ren Z F 2003 Appl. Phys. Lett. 82 3520[42] Zhbanov A I, Pogorelov E G, Chang Y C and Lee Y G 2011

J. Appl. Phys. 110 114311[43] Forbes R G 2012 J. Appl. Phys. 11 096102[44] Jensen K L 2011 J. Vac. Sci. Technol. B 29 02B101[45] Schottky W 1914 Physik Zeitschr 15 872[46] Nordheim L W 1928 Proc. R. Soc. London A 121 626

[47] Lang N D and Kohn W 1971 Phys. Rev. 3 1215[48] Rudnick J 1972 Phys. Rev. B 5 2863[49] Smith J R 1969 Phys. Rev. 181 522[50] Tsong T T and Muller E W 1969 Phys. Rev. 181 530[51] Yeganeh M A, Rahmatollapur S, Sadighi-Bonabi R and

Mamedov R 2010 Phys. B 405 3253[52] Janardhanam V, Ashok K A, Reddy V R and Reddy P N 2009

J. Alloys Compd. 485 467[53] Tung R T 1992 Phys. Rev. B 43 13509[54] Werner J H and Guttler H H 1991 J. Appl. Phys. 69 1522[55] Bhatnagar M, Baliga B J, Kirk H R and Rozgonyi G A 1996

IEEE Trans. Electron Devices ED-43 150[56] Tung R T 2001 Mater. Sci. Eng. R35 1[57] Huang L, Qin F, Li S and Wang D 2013 Appl. Phys. Lett. 103

033520[58] Yeganeh M A, Mamedov R K and Rahmatallahpur S 2011

Superlattices Microstruct. 50 59[59] Zheng L, Joshi R P and Fazi C 1999 J. Appl. Phys. 85 3701[60] Acar S, Karadeniz S, Tugluoglu N, Selcuk AB and Kasap M

2004 Appl. Surf. Sci. 233 373[61] Tang W, Shiffler D and Cartwright K L 2011 J. of Appl. Phys.

110 034905[62] Miller R, Lau Y Y and Booske J H 2009 J. of Appl. Phys. 106

104903[63] Hildebrand F B 1962 Advanced Calculus for Applications

(Englewood Cliffs: Prentice–Hall)[64] Abramowitz M and Stegun I A 1964 Handbook of

Mathematical Functions with Formulas, Graphs, andMathematical Tables (New York: Dover)

[65] Sommerfeld A 1927 Naturwissenschaften 15 825[66] Qin X Z, Wang W L, Xu N S, Li Z B and Forbes R G 2011

Proc. R. Soc. A: Math. Phys. Eng. Sci. 467 1029[67] Fowler R and Nordheim L 1928 Proc. R. Soc. London, Ser. A

119 173[68] Burgess R E, Kromer H and Houston J M 1953 Phys. Rev.

90 515[69] For example, Liboff R L 2002 Introductory Quantum

Mechanics (Reading: Addison-Wesley)[70] Filip L D, Palumbo M, Carey J D and Silva S R P 2009 Phys.

Rev. B 79 245429[71] Petrin A B 2009 J. Exp. Theor. Phys. 109 314[72] Litvinov E A, Mesyats G A and Proskurovskii D I 1983 Sov.

Phys. Usp. 26 138[73] Sun S and Ang L K 2013 J. Appl. Phys. 113 144902[74] Zuber J D, Jensen K L and Sullivan T E 2002 J. Appl. Phys.

91 9379[75] Anders A 2008 Cathodic Arcs: From Fractal Spots to

Energetic Condensation (New York: Springer)[76] Fursey G N, Polyakov M A, Shirochin L A and Saveliev A N

2003 Appl. Surf. Sci. 215 286

8

Semicond. Sci. Technol. 30 (2015) 105038 H Qiu et al

http://dx.doi.org/10.1126/science.269.5230.1550http://dx.doi.org/10.1126/science.270.5239.1179http://dx.doi.org/10.1016/0039-6028(94)90651-3http://dx.doi.org/10.1063/1.1721330http://dx.doi.org/10.1063/1.365783http://dx.doi.org/10.1109/16.81650http://dx.doi.org/10.1103/PhysRevSTAB.11.081001http://dx.doi.org/10.1116/1.588884http://dx.doi.org/10.1063/1.3695090http://dx.doi.org/10.1109/TED.2009.2039262http://dx.doi.org/10.1016/j.tsf.2005.01.086http://dx.doi.org/10.1063/1.1814439http://dx.doi.org/10.1063/1.2226977http://dx.doi.org/10.1063/1.2937077http://dx.doi.org/10.1063/1.3582141http://dx.doi.org/10.1063/1.3582141http://dx.doi.org/10.1088/1751-8113/44/5/055302http://dx.doi.org/10.1103/PhysRev.102.1464http://dx.doi.org/10.1063/1.3692571http://dx.doi.org/10.1063/1.3272690http://dx.doi.org/10.1016/S0921-5093(01)01871-8http://dx.doi.org/10.1002/1521-4095(200102)13:33.0.CO;2-Ihttp://dx.doi.org/10.1063/1.1576310http://dx.doi.org/10.1063/1.3665390http://dx.doi.org/10.1063/1.4711091http://dx.doi.org/10.1116/1.3523101http://dx.doi.org/10.1098/rspa.1928.0222http://dx.doi.org/10.1103/PhysRevB.3.1215http://dx.doi.org/10.1103/PhysRevB.5.2863http://dx.doi.org/10.1103/PhysRev.181.522http://dx.doi.org/10.1103/PhysRev.181.530http://dx.doi.org/10.1016/j.physb.2010.04.055http://dx.doi.org/10.1016/j.jallcom.2009.05.141http://dx.doi.org/10.1103/PhysRevB.45.13509http://dx.doi.org/10.1063/1.347243http://dx.doi.org/10.1109/16.477606http://dx.doi.org/10.1016/S0927-796X(01)00037-7http://dx.doi.org/10.1063/1.4816158http://dx.doi.org/10.1063/1.4816158http://dx.doi.org/10.1016/j.spmi.2011.05.004http://dx.doi.org/10.1063/1.369735http://dx.doi.org/10.1016/j.apsusc.2004.04.011http://dx.doi.org/10.1063/1.3615846http://dx.doi.org/10.1063/1.3253760http://dx.doi.org/10.1063/1.3253760http://dx.doi.org/10.1007/BF01505083http://dx.doi.org/10.1098/rspa.2010.0460http://dx.doi.org/10.1098/rspa.1928.0091http://dx.doi.org/10.1103/PhysRev.90.515http://dx.doi.org/10.1103/PhysRevB.79.245429http://dx.doi.org/10.1134/S1063776109080184http://dx.doi.org/10.1070/PU1983v026n02ABEH004322http://dx.doi.org/10.1063/1.4798926http://dx.doi.org/10.1063/1.1474596http://dx.doi.org/10.1016/S0169-4332(03)00298-8

1. Introduction1.1. Workfunction variability in field emitters1.2. Model analyses of field enhancement in an emitting ridge microstructure1.3. Quantum corrections to field emission

2. Results and discussion3. ConclusionsAcknowledgmentsReferences