IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 65, NO. 6, JUNE 2018 2089

Fractional Fowler–Nordheim Law for FieldEmission From Rough Surface With

Nonparabolic Energy DispersionMuhammad Zubair , Member, IEEE, Yee Sin Ang, and Lay Kee Ang, Senior Member, IEEE

Abstract— The theories of field electron emission fromperfectly planar and smooth canonical surfaces are wellunderstood, but they are not suitable for describing emis-sion from rough, irregular surfaces arising in modernnanoscale electron sources. Moreover, the existing mod-els rely on Sommerfeld’s free-electron theory for thedescription of electronic distribution, which is not a validassumption for modern materials with nonparabolic energydispersion. In this paper, we derive analytically a gener-alized Fowler–Nordheim (FN)-type equation that considersthe reduced space-dimensionality seen by the quantummechanically tunneling electron at a rough, irregular emis-sion surface. We also consider the effects of nonparabolicenergy dispersion on field emission from narrow-gap semi-conductors and few-layer graphene using Kane’s bandmodel. The traditional FN equation is shown to be a limitingcase of our model in the limit of a perfectly flat surfaceof a material with parabolic dispersion. The fractional-dimension parameter used in this model can be experimen-tally calculated from appropriate current–voltage data plot.By applying this model to experimental data, the standardfield-emission parameters can be deduced with better accu-racy than by using the conventional FN equation.

Index Terms— Electron emission, electron guns, fieldemission, fractional calculus, non-parabolic energy disper-sion, rough surface.

I. INTRODUCTION

THE field electron emission from the surface of a materialis a well-known quantum mechanical process, which has

found a vast number of technological applications, includingfield-emission displays, electron microscopes, electron guns,and vacuum nanoelectronics [1]. Fowler–Nordheim (FN) the-ory describes the electric field-induced electron tunneling

Manuscript received October 30, 2017; revised December 15, 2017;accepted December 15, 2017. Date of publication May 1, 2018; dateof current version May 21, 2018. This work was supported in partby USA AFOSR AOARD under Grant FA2386-14-1-4020, in part bySingapore Temasek Laboratories under Grant IGDS S16 02 05 1, andin part by A*STAR IRG under Grant A1783c0011. The review of thispaper was arranged by Editor D. R. Whaley. (Corresponding author:Muhammad Zubair.)

M. Zubair was with the SUTD-MIT International Design Center,Singapore University of Technology and Design, Singapore 487372.He is now with the Department of Electrical Engineering, Infor-mation Technology University, Lahore 54000, Pakistan (e-mail:muhammad.zubair@itu.edu.pk).

Y. S. Ang and L. K. Ang are with the SUTD-MIT Interna-tional Design Center, Singapore University of Technology andDesign, Singapore 487372 (e-mail: yee_sin@sutd.edu.sg; ricky_ang@sutd.edu.sg).

Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.

Digital Object Identifier 10.1109/TED.2017.2786020

from a flat perfectly conducing planar surface through anapproximately triangular potential-energy barrier [2], [3]. Thisleads to an FN-type equation given here as

JFN = aFNF2

φexp

(−νbFNφ

3/2

F

). (1)

This gives the local emission current density JFN interms of local work function φ and surface electricfield F . The symbols aFN (≈1.541434 μeVV−2) andbFN (≈6.830890 eV−3/2Vnm−1) denote the first and secondFN constants [3], and ν is a correction factor associated withthe barrier shape. For exactly triangular (ET) barrier, we takeν = 1. Taking the tunneling barrier as ET is not alwaysphysically realistic, so the barrier seen by tunneling electronis conventionally modeled as an image-force-rounded modelbarrier known as a “Schottky–Nordheim (SN)” barrier [3].Typically, the barrier shape correction factor (νSN) has avalue of 0.7, which enhances the predicted current densityby approximately two orders of magnitude [4]. However, latermathematical developments [5] have yielded more accurateapproximation for νSN.

There have been many analytical or semianalytical worksso far that address emission from specific geometry emit-ters. For instance, the field emission from sharp-tip [6],spherical [7], and hyperbolic [8] emitters has also beenstudied recently. At present, the situation can be viewedfrom a different standpoint. A mathematical treatment forslightly irregular or rough planar surface is required forbetter understanding of surface topographical effects on fieldemission. Also, the original FN equation assumes quasi-freeelectron with parabolic energy dispersion. This assumptionis no longer correct for materials with nonparabolic energydispersion such as a narrow-gap semiconductor and a graphenemonolayer [9]–[11].

Motivated by above, the purpose of this paper is threefold.In the first place, we rederive the FN equation using a

fractional-dimensional-space approach [12], [13] by assumingthat the effect of surface irregularity on the tunnelingprobability is captured by the space fractional-dimensionparameter (α). The underlying approach relies on thefractional-dimensional system of spatial coordinates to beused as an effective description of complex and confinedsystems (see [14] and references therein for details). Somesuccessful applications of this approach are in the areasof a quantum field theory [15], general relativity [16],

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2090 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 65, NO. 6, JUNE 2018

thermodynamics [17], mechanics [18], hydrodynamics [19],electrodynamics [13], [20]–[28], thermal transport [29],mechanics of anisotropic fractal materials [30], and highcurrent emission from rough cathode [14].

Second, we compare the presented generalized FN theorywith the conventional FN equations, showing its advantages.Based on the conventional FN equations, many experi-ments (some examples listed in [31]) have reported field-enhancement factor (FEF) in the range of few thousands toover hundred thousands. Our model proposes a new field-enhancement correction factor which solves the issue of suchspuriously high FEFs. The historical FN theory [2] sug-gests that the empirical current–voltage (i–V ) relationshipsof field emission should obey i = CVk exp(−B/V ), whereB , C , and α are effectively constants and k = 2. However,later developments [32] showed that the values of k maydiffer from 2 due to voltage dependence of various parametersin FN equations under nonideal conditions. Based on ourmodel, we propose that the experimental i–V relationships offield emission from practical rough surfaces should approxi-mately obey the empirical law i = CV2α exp(−B/V α), where0 < α ≤ 1. Since, lim(1/V )→min(d ln(i/V 2)/d ln(V )) =2α − 2, the fractional parameter α for a given emissionsurface can be experimentally established from sufficientlyaccurate i–V measurements. The knowledge of experimen-tal α values would lead to a better physical understanding offield emission and its variations due to surface irregularitiesand material properties. It is emphasized that the feasibilityof such measurements should be carefully explored by theexperimentalists.

Finally, using Kane’s nonparabolic dispersion [33], we alsogeneralize the fractional FN theory for materials with highlynonparabolic energy dispersions. This extension will be impor-tant to model the field emission from rough surfaces ofmaterials, like narrow-gap semiconductors and graphene,where surface morphology is expected to affect the emissionproperties.

II. FRACTIONAL GENERALIZATION OF FOWLER–NORDHEIM FIELD-EMISSION EQUATION

Field emission involves the extraction of electrons from asolid by tunneling through the surface potential barrier. The FNmethod expresses the emission current density (JFN) in termsof the product of supply function N(E⊥) and transmissioncoefficient D(E⊥) as follows:

JFN =∫ ∞

0N(E⊥)D(E⊥)d E⊥ (2)

where E⊥ is the normal energy, measured relative to thebottom of the conduction band. From free-electron theory,the supply function N(E⊥) is given as

N(E⊥) = eme

2π2h̄3 ln

[1 + exp

(− E⊥ − EF

kB T

)](3)

where kB is Boltzmann’s constant, T is the temperature, h̄ isthe reduced Plank’s constant, and EF is the Fermi energy.It is to be noted that this form of supply function is valid onlyfor conducting emitters, and ignores the quantum confinementeffects which may arise in nanoscale wire emitters [34]. Fornarrow-gap semiconductors and Dirac materials, we need other

Fig. 1. Schematic for fractional generalization of field emission in solids.Here, α = 1 corresponds to the ET barrier at perfectly smooth planarinterface, while 0 < α ≤ 1 corresponds to reduced dimensionality seenby the tunneling electrons at rough surface. The shape of barrier energypotential Ub(x) varies with parameter α.

forms of supply function due to the anisotropic distributionsof electrons [9], [35] (see Section III for more details). It isknown that, in general, (3) provides an adequate descriptionof the supply function for varying shape of emitters, as theeffect of emitter shape is included in external potential barrieronly [7].

In quantum mechanics, the transmission coefficient througha potential barrier is defined by D(E⊥) = jt/ji , wheretransmitted, incident ( jt , ji ) probability current density isrelated to respective wave functions (ψt , ψi ) by

jt,i = i h̄

2me

(ψt,i

∂

∂xψ∗

t,i − ψ∗t,i∂

∂xψt,i

).

FN’s elementary theory uses the physical simplificationthat tunneling takes place from a flat planar surface, throughan ET barrier given by Vb(x) = E0 − eFx , whereE0 = EF +φ and φ is the work function at zero bias (F = 0).Thus, the Schrödinger equation in the x-direction

− h̄2

2me

∂2ψi (x)

∂x2 = E⊥ψi (x), x ≤ 0, (4)

− h̄2

2me

∂2ψt (x)

∂x2 + Vb(x)ψt (x) = E⊥ψt (x), x ≥ 0. (5)

From Wentzel–Kramers–Brillouin (WKB) solution [3], thisstandard formulation leads to the FN equation given in (1).However, the effect of irregular, anisotropic solid surfacecan be effectively modeled by assuming that the potentialbarrier lies in a fractional-dimensional space with dimension0 < α ≤ 1 (see Fig. 1). In doing so, the effect of roughnessis accounted instead of traditional approach of assumingF = βapp F0 in (1), where βapp is the apparent FEF and F0is the macroscopic field. Hence, (5) can be replaced withfractional-dimensional Schrödinger equation given by

− h̄2

2me∇2αψt (x)+ Vb(x)ψt (x) = E⊥ψt (x), x ≥ 0 (6)

where ∇2α is the modified Laplacian operator in fractional-

dimensional space given by [14], [36]

∇2α = 1

c2(α, x)

(∂2

∂x2 − α − 1

x

∂

∂x

)(7)

with

c(α, x) = πα/2

(α/2)|x |α−1. (8)

It can be seen that the effect of the noninteger (fractional)dimensions in (6) is to modify the wave function and hence

ZUBAIR et al.: FRACTIONAL FN LAW FOR FIELD EMISSION FROM ROUGH SURFACE 2091

Fig. 2. Exactly triangular and fractional potential energy barriers withvarying fractional dimension α for φ = 5.3 eV, EF = 6 eV, andF = 3 eV/m. α = 1 corresponds to the ET barrier.

the probability current density so that it considers the measuredistribution of the space. This approach can effectively modelthe nonideal emission surfaces by considering the spacedimension α as description of complexity, assuming α = 1represents an ideal planar surface. From the solution of (6),we can calculate the new form of transmission coefficientD(E⊥, α) which will represent the tunneling probability andhence the emission current as a function of space dimension α.We can write (6) as[∂2

∂x2 − α − 1

x

∂

∂x− 2me

h̄2 f 2(α)x2α−2(Vb(x)− E⊥)]ψt (x)=0

(9)

where f (α) = (πα/2/(α/2)).Equation (9), through substitution of ψt (x) = ξt (x)x (α−1/2),

can be reduced to standard WKB form [37] given by[∂2

∂x2 − 2me

h̄2

(Ub(x)− f 2(α)x2α−2 E⊥

)]ξt (x) = 0 (10)

where

Ub(x) = − h̄2

2me

(α − 1

2x2 + (α − 1)2

4x2

)+ f 2(α)x2α−2Vb(x)

≈ f 2(α)x2α−2Vb(x). (11)

The shape of this effective fractional potential barrier in (11)is shown in Fig. 2 together with the exactly triangular barrier(corresponding α = 1).

By solving (10) through the WKB method, and matchingthe wave function [3], we get the transmission coefficient ofthis barrier as

D(E⊥) = exp

(−ge

∫ x2

x1

√V0(x)dx

)(12)

with V0(x) = f 2(α)x2α−2(Vh − eFx), where Vh = φ +EF − E⊥ is the zero-field height of the potential barrier,ge = 2

√(2me/h̄2), x1 = 0, and x2 = Vh/eF . Using the

binomial expansion and taking eFx/Vh � 1, the aboveintegral is approximated as∫ x2

x1

√V0(x)dx = f (α)

∫ x2

x1xα−1

√(Vh − eFx)dx

≈ 0.44 f (α)

(α + 2

α2 + α

)V α+1/2

h

(eF)α. (13)

Inserting (13) into (12), and following proper algebraicsteps, we get:

D(E⊥) ≈ DFα exp

[E⊥ − EF

dFα

](14)

where

DFα = exp

[−bα

φα+1/2

Fα

](15)

bα = 0.44 f (α)

(α + 2

eα(α2 + α)

)ge (16)

1

dFα= 0.22 f (α)

(2α2 + 5α + 2

α2 + α

)geφ

α−1/2

(eF)α(17)

where φ is measured in eV and F in V/nm. The transmissioncoefficient depends on work function (φ), electric field (F),and the fractional-dimension parameter (α).

Substituting (3) and (14) into (2), the fractional form ofemission current density can be written as

JFNα = eme

2π2h̄3 DFα

∫ ∞

0exp

[E⊥ − EF

dFα

]

× ln

[1 + exp

(− E⊥ − EF

kB T

)]d E⊥. (18)

At room temperature, where electrons near Fermi energydominate the tunneling, we can use approximation

kB T ln

[1 + exp

(− E⊥ − EF

kB T

)]= EF − E⊥

to get

JFNα = eme

2π2h̄3 DFα×∫ EF

0exp

[E⊥ − EF

dFα

](EF − E⊥)d E⊥.

.(19)

In solving (19), the generalized fractional field-emissiondensity can be given in the FN form as

JFNα = aFNαF2α

φ2α−1 exp

(−bFNαφ

α+1/2

Fα

)(20)

where

aFNα = e2α+1me

2π2h̄3g2e

1

0.0484 f 2(α)

((α2 + α)2

(2α2 + 5α + 2)2

)(21)

bFNα = 0.44 f (α)

(α + 2

eα(α2 + α)

)ge. (22)

Note that for α = 1 (ideal planar emission surface),(20)–(22) reduce exactly to the original FN equation (1)and the standard FN constants, respectively. It is importantto establish a correct value of parameter α while fittingthe experimental data in order to extract the standard field-emission parameters like FEF and emission area.

In the traditional approach, F = βapp F0 is assumed inthe exponential part of (1), where βapp is the apparent FEF,F0 is the macroscopic field, and F is the surface field ofthe emitter. This βapp is chosen by using the slope (Sfit) ofexperimental FN (JFN/F2 versus 1/F) plot in the expressionβapp = −bFNφ

3/2/Sfit [32]. However, in many experiments,spuriously high values of βapp have been reported (see [31]

2092 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 65, NO. 6, JUNE 2018

Fig. 3. Field-enhancement correction factor (βcorr) versus fractional-dimension parameter α at varying values of F and φ = 5.3 eV using (24).

for examples) ranging from few thousands to above hun-dred thousand in some cases, and are less comprehensibleif regarded as values for true electrostatic FEFs. Our modelof (20) suggests that such spurious values of FEF could resultfrom the traditional assumption of α = 1, which is not truefor irregular, rough surfaces. Taking F = βapp F0 in (20) andcomparing with original FN in (1), we can see that the actualvalue of FEF (called βactual) may differ from βapp for varyingvalues of α and can be given written as

βactual = βcorrβapp (23)

where the correction factor can be written as

βcorr = bFN Fα−10

bFNαφα−1 . (24)

Fig. 3 shows a field-enhancement correction factor (βcorr)versus fractional-dimension parameter α at varying values ofF and φ = 5.3 eV. At α = 1, the correction factor is unity forperfectly smooth planar surface. It is shown in analytical cal-culations for various cathode geometries that the electrostaticFEF varies from unity to a few hundred [38]–[41]. However,in many reported experiments (see [31] for examples), thereported FEFs (βapp) have spuriously high values. In practicalapplications, it is assumed that βactual will always be greaterthan 1 up to a few hundred as suggested by electrostatic fieldcalculations at various geometries. This means that there isalways a lower bound on βcorr depending on βapp and correctlycalculated values of α. Using this correction factor (βcorr), thespuriously high values of FEF can be corrected, given thatthe parameter α is correctly established using the procedurediscussed at the end of this section. It is to be noted thatthe βcorr considers the voltage or field dependence of FEFthrough Fα−1

0 term. Such voltage-dependent FEF has alsobeen studied recently [42].

Now, assuming F = γ V , where γ is a constant, andthat the emission area A is constant in the relation i =AJFN, (1) suggests that the empirical field-emission i–Vcharacteristics should obey

i = CVk exp(−B/V ) (25)

where B , C , and k are constants and k = 2. Due to (25),it is a modern customary experimental practice to plot field-emission results as an FN-plot [i.e., ln(i/V k) versus 1/V ],assuming that this should generate an exact straight line fork = 2. The 1928 FN theory assumes an exact triangulartunneling barrier, which is not physically realistic for morepractical problems, where emission area A and barrier-shapecorrection factor ν may become field-dependent [43]. Thus,it is believed that the true value of k in (25) might bedifferent than 2. Several experimentalists have tried fittingtheir experimental data with arbitrary values of k rangingfrom −1 to 4, usually with a goal to achieve straight-lineFN plot. A complete history of various attempts to find thecorrect value of k has been summarized in [32]. Some ofthese results had no obvious theoretical explanation and havelargely been ignored, including k = 4 suggested by Abbottand Henderson [44] and k = 0.25 by Oppenheimer [45].

The most notable theoretical explanation for k �= 2 is basedon the approximate calculation of SN barrier-shape correctionfactor νSN by Forbes [43], which suggests k = 2−η/6, whereη = bFNφ

3/2/Fφ with Fφ = (4π 0/e3)φ2. Hence, the valueof k varies for different values of work function φ. For φ =4.5 eV, we get Fφ = 14 V/nm, η = 4.64, and k = 2 −η/6 = 1.2 [43]. The main factor that determines the valueof k so far is the explicit field dependence that appears in thepreexponential part of the FN equation. However, the shape ofpotential-barrier and field enhancement at the surface of theemitter may also effect the exponential part in the FN equation.

In our reported model, we have modeled these realisticeffects in terms of “effective space-dimensionality” throughwhich the electrons move prior to emission. From our resultsin (20), we suggest a more generalized form of empiricallaw given as

i = CV2α exp(−B/V α) (26)

where α is the fractional space-dimensionality parameter with0 < α ≤ 1. Equation (26) is clearly a more generalized formof (25) with an addition parameter in the exponential termwhich can capture the field-enhancement effect of nonplanarsurfaces. This suggests a fitting of experimental data with (26),taking α as an unknown. One could try plotting ln(i/V 2α)versus 1/V α in search of a straight-line plot, but, this is nota good method when α is unknown. Alternatively, from (26)

ln(i/V 2) = ln(C)+ (2α − 2) ln(V )− B/V α

(27)d ln(i/V 2)

d ln(V )= 2α − 2 + αB/V α (28)

lim 1V →min

d ln(i/V 2)

d ln(V )= 2α − 2. (29)

Using (29), the value of α can be calculated fromy-intercept of “[d ln(i/V 2)/d ln(V )] versus 1/V ” or“[d ln(J/F2)/d ln(F)] versus 1/F” plot. This plottingmethod was tested on three experimental data sets shownin Fig. 4. In Fig. 4(a), the field-emission data from [46] wereextracted and processed using the above-mentioned procedure.It is shown that the extracted value of α is 0.4. This impliesthat classical FN equation (assuming α = 1) is not suitable to

ZUBAIR et al.: FRACTIONAL FN LAW FOR FIELD EMISSION FROM ROUGH SURFACE 2093

Fig. 4. Extraction of fractional-dimension parameter α from appropriatecurrent–voltage or current–field plots. (a) Data set for field emission ofZNO nanopencils taken from [46]. (b) Data set for field emission of carbonnanotubes from [47]. (c) Data set for field emission of nanoshuttles takenfrom [48], here the required form of data is too noisy to extract theα parameter. We emphasize the need for direct measurement of di/dVin order to correctly establish the α parameter.

extract emission parameters in this case. Following the sameprocedure on data taken from [47], we show that the extractedvalue of α parameter is 1, so the classical FN equation seemsvalid in this case. Finally, Fig. 4(c) shows the data processedfrom [48]. In this case, the resulting plot is too noisy foruseful conclusions. It is clear that this method requirescurrent–voltage data form carefully crafted experiments withminimal noise. The similar problem of noisy data has alreadybeen encountered in [43]. An alternate form of (29) would bebetter in which “(d ln(i/V 2)/d ln(V )) = (V/ i)(di/dV )− 2.”The direct measurement of di/dV would be more accurateby using some phase-sensitive detection techniques suggestedin [43].

It should be noted that the fractional FN model developedabove does not consider the space-charge-limited (SCL) effect,which is expected to become dominant at a high bias regime.In a previous work [14], we show that the fractional Child–Langmuir (CL) model predicts a stronger current for a roughsurface when compared with the classical CL law. It ispossible to combine the fractional FN law with the fractionalCL law in a future work to study the continuous transitionbetween the two transport regimes. Nonetheless, a conservativeestimation can be obtained by calculating the transitional biasvoltage, VT , at which the fractional FN current is equal tothe fractional CL current. We found that VT is related withvacuum gap D and parameter α through the relation Dα =ln((9/4

√2)V (2α−3/2)

T )[VT ]α . This VT serves as a conservativeupper limit for the fractional FN law to be valid. For biasvoltage larger than VT , the SCL effect needs to be explicitlyconsidered. We further note that, for the emerging class ofnovel 2-D materials, it is well known that their surfacesoften exhibit corrugations and crumples [49]. The fractionalapproach developed here may also be extended to model the

current–voltage characteristics of 2-D-material-based planarfield emitter, which has recently been shown to exhibit non-FNbehavior and a complete absence of SCL effect [50].

III. DERIVATION OF FIELD EMISSION WITH KANE’SNONPARABOLIC DISPERSION

In this section, Kane’s nonparabolic dispersion is usedto derive a general field-emission model that covers bothnonrelativistic and relativistic charge carrier regimes. Suchmodel was developed to describe the finite coupling betweenconduction and valence band in narrow-bandgap semiconduc-tor and also in the high energy regime of semiconductorwhere band nonparabolicity becomes nonnegligible. Kane’sdispersion [33] can be written as

E (1 + γ E ) = h̄2k2 2m∗ (30)

where γ is a nonparabolic parameter and m∗ is the electroneffective mass. For γ → 0, the conventional parabolic energydispersion, E (γ → 0) = h̄2k2 /2m∗, is obtained. Forγ → ∞, Kane’s dispersion leads to a k -linear form of

E (γ → ∞) = h̄vF k (31)

where vF ≡ 1/√

2m∗γ . Equation (30) can be explicitlysolved as

E =

√1 + 4γ h̄2k2

2m∗ − 1

2γ. (32)

By differentiating the left-hand side of (30) with respect to k ,we obtain the following relation:

k dk = 2m∗

h̄2 (1 + 2γ E )d E . (33)

Equation (33) shall play the critical role of determining theelectron supply function density for the field-emission process.

In general, the field-emission current from a bulk materialcan be obtained by summing all k-modes

JFN,Kane = egs,v

∑k

v(E⊥)D(E⊥) f (k) (34)

where gs,v is the spin-valley degeneracy, v(E⊥) and D(E⊥)are the carrier velocity and transmission probability along theemission direction, and f (k) is the Fermi–Dirac distributionfunction. Here, it is assumed that the total energy of thecarrier, E(k), can be partitioned as E(k) = E (k )+ E⊥(k⊥),where k and k⊥ are the wave vectors transverse to and alongthe emission direction. In the continuum limit, (34) becomes

JFN,Kane = gs,ve

(2π)3

∫dk

∫v(E⊥)D(E⊥) f (k)dk⊥ (35)

which can be simplified as follows: 1) the integral∫ ∞0 (· · · )v(E⊥)dk⊥ can be replaced by h̄−1 ∫ ∞

0 (· · · )d E⊥since v(E⊥) = h̄−1d E⊥/dk⊥ and 2) for field emission,only carriers up to the Fermi level take part in thetransport process, and thus the Fermi–Dirac distributioncan be represented by a two-variable Heaviside function,i.e., f (k) → �(E + E⊥ − EF )—this further limits the

E⊥-and E -integral to∫ EF −E

0 (· · · )d E⊥ and∫ EF

0 (· · · )d E ,respectively. Correspondingly, (35) is simplified as

JFN,Kane = gs,ve

h̄(2π)2

∫ EF

0k dk

∫ EF −E

0d E⊥D(E⊥). (36)

2094 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 65, NO. 6, JUNE 2018

By using the semiclassical WKB approximation ofD(E⊥) = DF exp [(E⊥ − EF )/dF ] for a triangular barrier,where DF and dF are constants defined in (15) and (17) inthe limit of α = 1, respectively, and considering (33), (36)becomes

JFN,Kane = gs,ve

(2π)22m∗

h̄3 DF

∫ EF

0d E (1 + 2γ E )

×∫ EF −E

0e

E⊥−EFdF d E⊥

= gs,ve

(2π)22m∗

h̄3 DF dF [A + B] (37)

where

A ≡∫ EF

0d E

(e− E

dF − e− EF

dF

)

= dF

[1 −

(1 + EF

dF

)e− EF

dF

](38)

and

B ≡ 2γ∫ EF

0d E E

(e− E

dF − e− EF

dF

)

= 2γ d2F

{1 −

[1 + EF

dF+ 1

2

(EF

dF

)2]

e− EF

dF

}. (39)

For typical conducting solid, EF � dF , and hence

A ≈ dF (40)

and

B ≈ 2γ d2F . (41)

From (37), we obtained the field-emission current density as

JFN,Kane ≈ gs,ve

(2π)22m∗

h̄3 DF[d2

F + 2γ d3F

](42)

which is composed of two parts: a conventional FN componentand a correction factor that accounts for band nonparabolicity.

Using (14) in (37), the fractional generalization of (42) canbe written as

JFNα,Kane ≈ gs,ve

(2π)22m∗

h̄3 DFα[d2

Fα + 2γ d3Fα

]. (43)

Equation (43) is the most generalized from of FN fieldemission, where band nonparabolicity is captured by the non-parabolicity parameter (γ ) and the interface irregularity, andtopography effect is captured by a fractional-dimension para-meter (α). Note that for α = 1 (ideal planar emission surface)and γ = 0 (conventional parabolic dispersion), (43) reducesexactly to the standard FN equation (1). This fractional gener-alization of FN theory for materials with highly nonparabolicdispersion will be very useful to study the field emission atsurfaces of novel materials like narrow-gap semiconductorsand graphene.

IV. CONCLUSION

The standard FN theory-based existing formulas for thefield-induced emission of electrons from ideal planar surfacesare not always suitable for practical rough, irregular surfacesand may lead to significant order-of-magnitude errors whenapplied to experimental data incorrectly. Also, the standardmodels rely on free-electron theory assuming an isotropicdistribution of electrons with parabolic energy dispersion.Such an assumption is no longer valid for narrow-gap semi-conductors and few layer graphene, where energy dispersionbecomes nonparabolic due to band structure.

In this paper, accurate equations for the field-emissioncurrent densities from rough planar surfaces are derived usingKane’s nonparabolic dispersion model. The reported expres-sions are applicable to metallic, narrow-gap semiconductorsand a few-layer graphene planar electron emitters with roughsurfaces in many modern applications. Based on our model,we have proposed a new form of field-enhancement correc-tion factor βcorr to avoid spuriously high FEFs in recentexperiments [31]. We propose that the experimental current–voltage (i–V ) relationships of field emission from practicalrough surfaces should approximately obey the empirical lawi = CV2α exp(−B/V α), where 0 < α ≤ 1. We haveshown that the fractional-dimension parameter α can beestablished from the y-intercept of “[d ln(i/V 2)/d ln(V )] or(V/ i)(di/dV )−2” versus “1/V ” plot. This requires an accu-rate phase-detection measurement of (di/dV ), previously pro-posed by Forbes [43] in his work. Determination of α valuescould be very useful for better understanding of field emissionand its variation as between different materials and surfaceroughness profiles. We strongly reemphasize Forbes’s callof experiment to directly measure (di/dV ) in field-emissionexperiments. The electronic availability of raw i–V measure-ment data from experimentalists would be extremely usefulfor further explorations and improvements in the existingtheoretical models.

REFERENCES

[1] N. Egorov and E. Sheshin, Field Emission Electronics. Cham,Switzerland: Springer, 2017, doi: 10.1007/978-3-319-56561-3.

[2] R. H. Fowler and L. Nordheim, “Electron emission in intense electricfields,” Proc. Roy. Soc. Lond. A, Math. Phys. Sci., vol. 119, no. 781,pp. 173–181, May 1928.

[3] S.-D. Liang, Quantum Tunneling and Field Electron Emission Theories.Singapore: World Scientific, 2013.

[4] R. G. Forbes, “Simple good approximations for the special ellip-tic functions in standard Fowler–Nordheim tunneling theory for aSchottky–Nordheim barrier,” Appl. Phys. Lett., vol. 89, no. 11,p. 113122, 2006.

[5] R. G. Forbes and J. H. B. Deane, “Reformulation of the standardtheory of Fowler–Nordheim tunnelling and cold field electron emission,”Proc. Roy. Soc. London A, Math. Phys. Sci., vol. 463, no. 2087,pp. 2907–2927, 2007.

[6] S. Sun and L. K. Ang, “Analysis of nonuniform field emission from asharp tip emitter of Lorentzian or hyperboloid shape,” J. Appl. Phys.,vol. 113, no. 14, p. 144902, 2013.

[7] J. T. Holgate and M. Coppins, “Field-induced and thermal electroncurrents from earthed spherical emitters,” Phys. Rev. Appl., vol. 7, no. 4,p. 044019, 2017.

[8] K. L. Jensen, D. A. Shiffler, M. Peckerar, J. R. Harris, and J. J. Petillo,“Current from a nano-gap hyperbolic diode using shape-factors: Theory,”J. Appl. Phys., vol. 122, no. 6, p. 064501, 2017.

[9] Y. S. Ang, S.-J. Liang, and L. K. Ang, “Theoretical modeling of electronemission from graphene,” MRS Bull., vol. 42, no. 7, pp. 505–510, 2017.

ZUBAIR et al.: FRACTIONAL FN LAW FOR FIELD EMISSION FROM ROUGH SURFACE 2095

[10] Y. S. Ang and L. K. Ang, “Current-temperature scaling for a Schottkyinterface with nonparabolic energy dispersion,” Phys. Rev. Appl., vol. 6,no. 3, p. 034013, 2016.

[11] Y. S. Ang, M. Zubair, and L. K. Ang, “Relativistic space-charge-limitedcurrent for massive dirac Fermions,” Phys. Rev. B, Condens. Matter,vol. 95, no. 16, p. 165409, 2017.

[12] F. H. Stillinger, “Axiomatic basis for spaces with noninteger dimension,”J. Math. Phys., vol. 18, no. 6, pp. 1224–1234, 1977.

[13] M. Zubair, M. J. Mughal, and Q. A. Naqvi, Electromagnetic Fields andWaves in Fractional Dimensional Space. New York, NY, USA: Springer,2012, doi: 10.1007/978-3-642-25358-4.

[14] M. Zubair and L. K. Ang, “Fractional-dimensional Child–Langmuir lawfor a rough cathode,” Phys. Plasmas, vol. 23, no. 7, p. 072118, 2016.

[15] C. Palmer and P. N. Stavrinou, “Equations of motion in a non-integer-dimensional space,” J. Phys. A, Math. Gen., vol. 37, no. 27, p. 6987,2004.

[16] M. Sadallah and S. I. Muslih, “Solution of the equations of motion forEinstein’s field in fractional d dimensional space-time,” Int. J. Theor.Phys., vol. 48, no. 12, pp. 3312–3318, 2009.

[17] V. E. Tarasov, “Heat transfer in fractal materials,” Int. J. Heat MassTransf., vol. 93, pp. 427–430, 2016.

[18] M. Ostoja-Starzewski, J. Li, H. Joumaa, and P. N. Demmie, “Fromfractal media to continuum mechanics,” ZAMM-J. Appl. Math. Mechan-ics/Zeitschrift Angewandte Mathematik Mechanik, vol. 94, no. 5,pp. 373–401, 2014.

[19] A. S. Balankin and B. E. Elizarraraz, “Map of fluid flow in fractal porousmedium into fractal continuum flow,” Phys. Rev. E, Stat. Phys. PlasmasFluids Relat. Interdiscip. Top., vol. 85, no. 5, p. 056314, 2012.

[20] M. J. Mughal and M. Zubair, “Fractional space solutions of antennaradiation problems: An application to Hertzian dipole,” in Proc. IEEE19th Conf. Signal Process. Commun. Appl. (SIU), Apr. 2011, pp. 62–65.

[21] Q. A. Naqvi and M. Zubair, “On cylindrical model of electrostaticpotential in fractional dimensional space,” Opt.-Int. J. Light Electron,vol. 127, no. 6, pp. 3243–3247, 2016.

[22] M. Zubair, M. J. Mughal, and Q. A. Naqvi, “An exact solution ofthe cylindrical wave equation for electromagnetic field in fractionaldimensional space,” Prog. Electromagn. Res., vol. 114, pp. 443–455,2011, doi: 10.2528/PIER11021508.

[23] H. Asad, M. J. Mughal, M. Zubair, and Q. A. Naqvi, “Electromagneticgreen’s function for fractional space,” J. Electromagn. Waves Appl.,vol. 26, nos. 14–15, pp. 1903–1910, 2012.

[24] H. Asad, M. Zubair, and M. J. Mughal, “Reflection and transmis-sion at dielectric-fractal interface,” Prog. Electromagn. Res., vol. 125,pp. 543–558, 2012, doi: 10.2528/PIER12012402.

[25] M. Zubair, M. J. Mughal, and Q. A. Naqvi, “An exact solution of thespherical wave equation in d-dimensional fractional space,” J. Electro-magn. Waves Appl., vol. 25, no. 10, pp. 1481–1491, 2011.

[26] M. Zubair, M. J. Mughal, Q. A. Naqvi, and A. A. Rizvi, “Differentialelectromagnetic equations in fractional space,” Prog. Electromagn. Res.,vol. 114, pp. 255–269, 2011, doi: 10.2528/PIER11011403.

[27] M. Zubair, M. J. Mughal, and Q. A. Naqvi, “On electromagnetic wavepropagation in fractional space,” Nonlinear Anal., Real World Appl.,vol. 12, no. 5, pp. 2844–2850, 2011.

[28] M. Zubair, M. J. Mughal, and Q. A. Naqvi, “The wave equation andgeneral plane wave solutions in fractional space,” Prog. Electromagn.Res. Lett., vol. 19, pp. 137–146, 2010, doi: 10.2528/PIERL10102103.

[29] F. A. Godínez, O. Chávez, A. García, and R. Zenit, “A space-fractionalmodel of thermo-electromagnetic wave propagation in anisotropicmedia,” Appl. Thermal Eng., vol. 93, pp. 529–536, Jan. 2016.

[30] A. S. Balankin, “A continuum framework for mechanics of fractalmaterials I: From fractional space to continuum with fractal metric,”Eur. Phys. J. B, vol. 88, no. 4, pp. 1–13, 2015.

[31] R. G. Forbes, “Development of a simple quantitative test for lack offield emission orthodoxy,” in Proc. Roy. Soc. London A, Math. Phys.Sci., vol. 469, no. 2158, p. 20130271, 2013.

[32] R. G. Forbes, “Use of Millikan–Lauritsen plots, rather thanFowler–Nordheim plots, to analyze field emission current-voltage data,”J. Appl. Phys., vol. 105, no. 11, p. 114313, 2009.

[33] B. M. Askerov, Electron Transport Phenomena in Semiconductors.Singapore: World Scientific, 1994.

[34] X.-Z. Qin, W.-L. Wang, N.-S. Xu, Z.-B. Li, and R. G. Forbes, “Analyticaltreatment of cold field electron emission from a nanowall emitter,including quantum confinement effects,” Proc. Roy. Soc. Lond. A, Math.Phys. Sci., vol. 467, no. 2128, pp. 1029–1051, 2011.

[35] S.-J. Liang and L. K. Ang, “Electron thermionic emission from grapheneand a thermionic energy converter,” Phys. Rev. Appl., vol. 3, no. 1,p. 014002, 2015.

[36] V. E. Tarasov, “Anisotropic fractal media by vector calculus in non-integer dimensional space,” J. Math. Phys., vol. 55, no. 8, p. 083510,2014.

[37] E. Meissen. (2013). Integral Asymptotics and the WKB Approxima-tion. Accessed: Jul. 4, 2017. [Online]. Available: https://math.arizona.edu/meissen/docs/asymptotics.pdf

[38] Y. B. Zhu and L. K. Ang, “Space charge limited current emission for asharp tip,” Phys. Plasmas, vol. 22, no. 5, p. 052106, 2015.

[39] J. Lin, P. Y. Wong, P. Yang, Y. Y. Lau, W. Tang, and P. Zhang, “Electricfield distribution and current emission in a miniaturized geometricaldiode,” J. Appl. Phys., vol. 121, no. 24, p. 244301, 2017.

[40] Y. B. Zhu, P. Zhang, A. Valfells, L. K. Ang, and Y. Y. Lau, “Novelscaling laws for the Langmuir–Blodgett solutions in cylindrical andspherical diode,” Phys. Rev. Lett., vol. 110, no. 26, p. 265007, 2013.

[41] S. Sun and L. K. Ang, “Onset of space charge limited current for fieldemission from a single sharp tip,” Phys. Plasmas, vol. 19, p. 033107,Mar. 2012.

[42] R. G. Forbes, “The theoretical link between voltage loss, reduction infield enhancement factor, and Fowler-Nordheim-plot saturation,” Appl.Phys. Lett., vol. 110, no. 13, p. 133109, 2017.

[43] R. G. Forbes, “Call for experimental test of a revised mathematical formfor empirical field emission current-voltage characteristics,” Appl. Phys.Lett., vol. 92, no. 19, p. 193105, 2008.

[44] F. R. Abbott and J. E. Henderson, “The range and validity of the fieldcurrent equation,” Phys. Rev., vol. 56, no. 1, p. 113, 1939.

[45] J. R. Oppenheimer, “On the quantum theory of the autoelectric fieldcurrents,” Proc. Nat. Acad. Sci., vol. 14, no. 5, pp. 363–365, 1928.

[46] R. C. Wang, C. P. Liu, J.-L. Huang, S.-J. Chen, Y.-K. Tseng, andS.-C. Kung, “Zno nanopencils: Efficient field emitters,” Appl. Phys. Lett.,vol. 87, no. 1, p. 013110, 2005.

[47] C. Li et al., “Carbon nanotubes as an ultrafast emitter with a narrowenergy spread at optical frequency,” Adv. Mater., vol. 29, no. 30,p. 1701580, 2017, doi: 10.1002/adma.201701580.

[48] J. Li, M. Chen, S. Tian, A. Jin, X. Xia, and C. Gu, “Single-crystal SnO2nanoshuttles: Shape-controlled synthesis, perfect flexibility and high-performance field emission,” Nanotechnology, vol. 22, no. 50, p. 505601,2011.

[49] S. Deng and V. Berry, “Wrinkled, rippled and crumpled graphene:An overview of formation mechanism, electronic properties, and appli-cations,” Mater. Today, vol. 19, no. 4, pp. 197–212, 2016.

[50] Y. S. Ang, M. Zubair, K. J. A. Ooi, and L. K. Ang. (Nov. 2017).“Generalized Fowler–Nordheim field-induced vertical electron emis-sion model for two-dimensional materials.” [Online]. Available:https://arxiv.org/abs/1711.05898

Muhammad Zubair (S’13–M’15) received thePh.D. degree in electronic engineering from thePolitecnico di Torino, Turin, Italy, in 2015.

From 2015 to 2017, he was with the SUTD-MITInternational Design Center, Singapore. Since2017, he has been with Information Technol-ogy University, Lahore, Pakistan. His currentresearch interests include charge transport, elec-tron device modeling, computational electromag-netics, fractal electrodynamics, and microwaveimaging.

Yee Sin Ang received the bachelor’s degreein medical and radiation physics and the Ph.D.degree in theoretical condensed matter physicsfrom the University of Wollongong, Wollongong,NSW, Australia, 2010 and in 2014, respectively.

He is currently a Research Fellow with theSingapore University of Technology and Design,Singapore.

Lay Kee Ang (S’95–M’00–SM’08) received theB.S. degree from the Department of NuclearEngineering, National Tsing Hua University,Hsinchu, Taiwan, in 1994, and the M.S. and Ph.D.degrees from the Department of Nuclear Engi-neering and Radiological Sciences, University ofMichigan, Ann Arbor, MI, USA, in 1996 and 1999,respectively.

Since 2011, he has been with the SingaporeUniversity of Technology and Design, Singapore.