IEEE TRANSACTIONS ON DEVICES, VOL. 65, NO. 6, JUNE 2018 2089 Fractional Fowler–Nordheim Law for Field Emission From Rough Surface With Nonparabolic Dispersion

Muhammad Zubair , Member, IEEE, Yee Sin Ang, and Lay Kee Ang, Senior Member, IEEE

Abstract— The theories of field electron emission from from a flat perfectly conducing planar surface through an perfectly planar and smooth canonical surfaces are well approximately triangular potential-energy barrier [2], [3]. This understood, but they are not suitable for describing emis- leads to an FN-type equation given here as sion from rough, irregular surfaces arising in modern   nanoscale electron sources. Moreover, the existing mod- 2 3/2 F νbFNφ els rely on Sommerfeld’s free-electron theory for the JFN = aFN exp − . (1) description of electronic distribution, which is not a valid φ F assumption for modern materials with nonparabolic energy dispersion. In this paper, we derive analytically a gener- This gives the local emission JFN in alized Fowler–Nordheim (FN)-type equation that considers terms of local φ and surface electric −2 the reduced space-dimensionality seen by the quantum field F. The symbols aFN (≈1.541434 μeVV )and mechanically tunneling electron at a rough, irregular emis- −3/2 −1 bFN (≈6.830890 eV Vnm ) denote the first and second sion surface. We also consider the effects of nonparabolic ν energy dispersion on field emission from narrow-gap semi- FN constants [3], and is a correction factor associated with conductors and few-layer graphene using Kane’s band the barrier shape. For exactly triangular (ET) barrier, we take model. The traditional FN equation is shown to be a limiting ν = 1. Taking the tunneling barrier as ET is not always case of our model in the limit of a perfectly flat surface physically realistic, so the barrier seen by tunneling electron of a material with parabolic dispersion. The fractional- is conventionally modeled as an image-force-rounded model dimension parameter used in this model can be experimen- tally calculated from appropriate current–voltage data plot. barrier known as a “SchottkyÐNordheim (SN)” barrier [3]. By applying this model to experimental data, the standard Typically, the barrier shape correction factor (νSN)hasa field-emission parameters can be deduced with better accu- value of 0.7, which enhances the predicted current density racy than by using the conventional FN equation. by approximately two orders of magnitude [4]. However, later Index Terms— Electron emission, electron guns, field mathematical developments [5] have yielded more accurate emission, fractional calculus, non-parabolic energy disper- approximation for νSN. sion, rough surface. There have been many analytical or semianalytical works I. INTRODUCTION so far that address emission from specific geometry emit- HE field electron emission from the surface of a material ters. For instance, the field emission from sharp-tip [6], Tis a well-known quantum mechanical process, which has spherical [7], and hyperbolic [8] emitters has also been found a vast number of technological applications, including studied recently. At present, the situation can be viewed field-emission displays, electron microscopes, electron guns, from a different standpoint. A mathematical treatment for and nanoelectronics [1]. FowlerÐNordheim (FN) the- slightly irregular or rough planar surface is required for ory describes the electric field-induced electron tunneling better understanding of surface topographical effects on field emission. Also, the original FN equation assumes quasi-free Manuscript received October 30, 2017; revised December 15, 2017; electron with parabolic energy dispersion. This assumption accepted December 15, 2017. Date of publication May 1, 2018; date of current version May 21, 2018. This work was supported in part is no longer correct for materials with nonparabolic energy by USA AFOSR AOARD under Grant FA2386-14-1-4020, in part by dispersion such as a narrow-gap and a graphene Singapore Temasek Laboratories under Grant IGDS S16 02 05 1, and monolayer [9]Ð[11]. in part by A*STAR IRG under Grant A1783c0011. The review of this paper was arranged by Editor D. R. Whaley. (Corresponding author: Motivated by above, the purpose of this paper is threefold. Muhammad Zubair.) In the first place, we rederive the FN equation using a M. Zubair was with the SUTD-MIT International Design Center, fractional-dimensional-space approach [12], [13] by assuming Singapore University of Technology and Design, Singapore 487372. He is now with the Department of Electrical Engineering, Infor- that the effect of surface irregularity on the tunneling mation Technology University, Lahore 54000, Pakistan (e-mail: probability is captured by the space fractional-dimension [email protected]). parameter (α). The underlying approach relies on the Y. S. Ang and L. K. Ang are with the SUTD-MIT Interna- tional Design Center, Singapore University of Technology and fractional-dimensional system of spatial coordinates to be Design, Singapore 487372 (e-mail: [email protected]; ricky_ang@ used as an effective description of complex and confined sutd.edu.sg). systems (see [14] and references therein for details). Some Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. successful applications of this approach are in the areas Digital Object Identifier 10.1109/TED.2017.2786020 of a quantum field theory [15], general relativity [16],

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thermodynamics [17], mechanics [18], hydrodynamics [19], electrodynamics [13], [20]Ð[28], thermal transport [29], mechanics of anisotropic fractal materials [30], and high current emission from rough [14]. Second, we compare the presented generalized FN theory with 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 to over hundred thousands. Our model proposes a new field- Fig. 1. Schematic for fractional generalization of field emission in . enhancement correction factor which solves the issue of such Here, α = 1 corresponds to the ET barrier at perfectly smooth planar interface, while 0 <α≤ 1 corresponds to reduced dimensionality seen spuriously high FEFs. The historical FN theory [2] sug- by the tunneling at rough surface. The shape of barrier energy gests that the empirical currentÐvoltage (iÐV ) relationships potential Ub(x) varies with parameter α. of field emission should obey i = CVk exp(−B/V ),where B, C,andα are effectively constants and k = 2. However, forms of supply function due to the anisotropic distributions later developments [32] showed that the values of k may of electrons [9], [35] (see Section III for more details). It is differ from 2 due to voltage dependence of various parameters known that, in general, (3) provides an adequate description in FN equations under nonideal conditions. Based on our of the supply function for varying shape of emitters, as the model, we propose that the experimental iÐV relationships of effect of emitter shape is included in external potential barrier field emission from practical rough surfaces should approxi- only [7]. mately obey the empirical law i = CV2α exp(−B/V α),where In , the transmission coefficient through 2 a potential barrier is defined by D(E⊥) = j /j ,where 0 <α≤ 1. Since, lim(1/V )→min(d ln(i/V )/d ln(V )) = t i 2α − 2, the fractional parameter α for a given emission transmitted, incident ( jt, ji ) probability current density is ψ ψ surface can be experimentally established from sufficiently related to respective  functions ( t , i )by ¯ ∂ ∂ accurate iÐV measurements. The knowledge of experimen- = ih ψ ψ∗ − ψ∗ ψ . jt,i t,i t,i t,i t,i tal α values would lead to a better physical understanding of 2me ∂x ∂x field emission and its variations due to surface irregularities FN’s elementary theory uses the physical simplification and material properties. It is emphasized that the feasibility that tunneling takes place from a flat planar surface, through of such measurements should be carefully explored by the an ET barrier given by Vb(x) = E0 − eFx,where experimentalists. E0 = EF +φ and φ is the work function at zero bias (F = 0). Finally, using Kane’s nonparabolic dispersion [33], we also Thus, the Schrödinger equation in the x-direction generalize the fractional FN theory for materials with highly 2 2 h¯ ∂ ψi (x) nonparabolic energy dispersions. This extension will be impor- − = E⊥ψ (x), x ≤ , 2 i 0 (4) 2me ∂x tant to model the field emission from rough surfaces of 2 2 h¯ ∂ ψt (x) materials, like narrow-gap and graphene, − + V (x)ψ (x) = E⊥ψ (x), x ≥ 0. (5) ∂ 2 b t t where surface morphology is expected to affect the emission 2me x properties. From WentzelÐKramersÐBrillouin (WKB) solution [3], this standard formulation leads to the FN equation given in (1). II. FRACTIONAL GENERALIZATION OF FOWLER– However, the effect of irregular, anisotropic surface NORDHEIM FIELD-EMISSION EQUATION can be effectively modeled by assuming that the potential Field emission involves the extraction of electrons from a barrier lies in a fractional-dimensional space with dimension solid by tunneling through the surface potential barrier. The FN 0 <α≤ 1(seeFig. 1). In doing so, the effect of roughness method expresses the emission current density (JFN)interms is accounted instead of traditional approach of assuming of the product of supply function N(E⊥) and transmission F = βapp F0 in (1), where βapp is the apparent FEF and F0 coefficient D(E⊥) as follows:  ∞ is the macroscopic field. Hence, (5) can be replaced with fractional-dimensional Schrödinger equation given by JFN = N(E⊥)D(E⊥)dE⊥ (2) 2 0 h¯ 2 − ∇αψt (x) + Vb(x)ψt (x) = E⊥ψt (x), x ≥ 0 (6) where E⊥ is the normal energy, measured relative to the 2me bottom of the conduction band. From free-electron theory, 2 where ∇α is the modified Laplacian operator in fractional- the supply function N(E⊥) is given as    dimensional space given by [14], [36] eme E⊥ − EF   ( ⊥) = + − 1 ∂2 α − 1 ∂ N E 3 ln 1 exp (3) 2 2π2h¯ kB T ∇α = − (7) c2(α, x) ∂x2 x ∂x where k is Boltzmann’s constant, T is the temperature, h¯ is B with E the reduced Plank’s constant, and F is the . πα/2 It is to be noted that this form of supply function is valid only c(α, x) = |x|α−1. (α/ ) (8) for conducting emitters, and ignores the quantum confinement 2 effects which may arise in nanoscale wire emitters [34]. For It can be seen that the effect of the noninteger (fractional) narrow-gap semiconductors and Dirac materials, we need other dimensions in (6) is to modify the and hence ZUBAIR et al.: FRACTIONAL FN LAW FOR FIELD EMISSION FROM ROUGH SURFACE 2091

  α+ / α + 2 V 1 2 ≈ 0.44 f (α) h . (13) α2 + α (eF)α Inserting (13) into (12), and following proper algebraic steps, we get:   E⊥ − EF D(E⊥) ≈ DFα exp (14) dFα where   φα+1/2 = − DFα exp bα α (15)  F  α + 2 bα = 0.44 f (α) g (16) α(α2 + α) e e  1 2α2 + 5α + 2 g φα−1/2 = . f (α) e 0 22 2 α (17) dFα α + α (eF) Fig. 2. Exactly triangular and fractional barriers with where φ is measured in eV and F in V/nm. The transmission α φ = . = varying fractional dimension for 5 3eV,EF 6 eV, and coefficient depends on work function (φ), electric field (F), F = 3eV/m.α = 1 corresponds to the ET barrier. and the fractional-dimension parameter (α). the probability current density so that it considers the measure Substituting (3) and (14) into (2), the fractional form of distribution of the space. This approach can effectively model emission current density can be written as  ∞   the nonideal emission surfaces by considering the space eme E⊥ − EF J α = D α exp α α = FN π2 3 F dimension as description of complexity, assuming 1 2 h¯ 0  dFα  represents an ideal planar surface. From the solution of (6), E⊥ − EF × ln 1 + exp − dE⊥. (18) we can calculate the new form of transmission coefficient kB T D(E⊥,α) which will represent the tunneling probability and At room temperature, where electrons near Fermi energy hence the emission current as a function of space dimension α. dominate the tunneling, we can use approximation We can write (6) as      − ∂2 α − ∂ E⊥ EF 1 2me 2 2α−2 kB T ln 1 + exp − = EF − E⊥ − − f (α)x (Vb(x) − E⊥) ψt (x)=0 k T ∂x2 x ∂x h¯ 2 B (9) to get    EF − f (α) = (πα/2/ (α/ )) eme E⊥ EF where 2 . JFNα = DFα × exp (EF − E⊥)dE⊥. (α−1/2) π2 3 Equation (9), through substitution of ψt (x) = ξt (x)x , 2 h¯ 0 dFα can be reduced to standard WKB form [37] given by (19).   ∂2   2me 2 2α−2 In solving (19), the generalized fractional field-emission − U (x) − f (α)x E⊥ ξ (x) = 0 (10) ∂x2 ¯ 2 b t density can be given in the FN form as h   2α α+1/2 where F bFNαφ   J α = a α exp − (20) 2 2 FN FN φ2α−1 α h¯ α − 1 (α − 1) α− F U (x) =− + + f 2(α)x2 2V (x) b 2 2 b 2me 2x 4x where   2 2α−2 2α+1 2 2 ≈ f (α)x Vb(x). (11) e me 1 (α + α) a α = (21) FN π2 ¯ 3 2 0.0484 f 2(α) (2α2 + 5α + 2)2 The shape of this effective fractional potential barrier in (11) 2 h ge   α + is shown in Fig. 2 together with the exactly triangular barrier 2 bFNα = 0.44 f (α) ge. (22) (corresponding α = 1). eα(α2 + α) By solving (10) through the WKB method, and matching Note that for α = 1 (ideal planar emission surface), the wave function [3], we get the transmission coefficient of (20)Ð(22) reduce exactly to the original FN equation (1) this barrier as    and the standard FN constants, respectively. It is important x 2 to establish a correct value of parameter α while fitting D(E⊥) = exp −ge V0(x)dx (12) x1 the experimental data in order to extract the standard field- emission parameters like FEF and emission area. ( ) = 2(α) 2α−2( − ) = φ + with V0 x f x Vh eFx ,whereVh F = β F − In the traditional approach, app 0 is assumed in EF E√⊥ is the zero-field height of the potential barrier, β 2 the exponential part of (1), where app is the apparent FEF, ge = 2 (2me/h¯ ), x1 = 0, and x2 = Vh/eF.Usingthe F0 is the macroscopic field, and F is the surface field of binomial expansion and taking eFx/Vh 1, the above the emitter. This βapp is chosen by using the slope (Sfit)of integral is approximated as 2   experimental FN (JFN/F versus 1/F) plot in the expression x2 x2 β =− φ3/2/ ( ) = (α) α−1 ( − ) app bFN Sfit [32]. However, in many experiments, V0 x dx f x Vh eFx dx β x1 x1 spuriously high values of app have been reported (see [31] 2092 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 65, NO. 6, JUNE 2018

where B, C,andk 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 for k = 2. The 1928 FN theory assumes an exact triangular tunneling barrier, which is not physically realistic for more practical problems, where emission area A and barrier-shape correction factor ν may become field-dependent [43]. Thus, it is believed that the true value of k in (25) might be different than 2. Several experimentalists have tried fitting their experimental data with arbitrary values of k ranging from −1 to 4, usually with a goal to achieve straight-line FN plot. A complete history of various attempts to find the correct value of k has been summarized in [32]. Some of these results had no obvious theoretical explanation and have largely been ignored, including k = 4 suggested by Abbott Fig. 3. Field-enhancement correction factor (βcorr) versus fractional- dimension parameter α at varying values of F and φ = 5.3 eV using (24). and Henderson [44] and k = 0.25 by Oppenheimer [45]. The most notable theoretical explanation for k = 2 is based for examples) ranging from few thousands to above hun- on the approximate calculation of SN barrier-shape correction dred thousand in some cases, and are less comprehensible factor νSN by Forbes [43], which suggests k = 2−η/6, where 3/2 3 2 if regarded as values for true electrostatic FEFs. Our model η = bFNφ /Fφ with Fφ = (4π 0/e )φ . Hence, the value of (20) suggests that such spurious values of FEF could result of k varies for different values of work function φ.Forφ = from the traditional assumption of α = 1, which is not true 4.5eV,wegetFφ = 14 V/nm, η = 4.64, and k = 2 − η/ = . for irregular, rough surfaces. Taking F = βapp F0 in (20) and 6 1 2 [43]. The main factor that determines the value comparing with original FN in (1), we can see that the actual of k so far is the explicit field dependence that appears in the value of FEF (called βactual) may differ from βapp for varying preexponential part of the FN equation. However, the shape of values of α and can be given written as potential-barrier and field enhancement at the surface of the emitter may also effect the exponential part in the FN equation. βactual = βcorrβapp (23) In our reported model, we have modeled these realistic effects in terms of “effective space-dimensionality” through where the correction factor can be written as which the electrons move prior to emission. From our results α− b F 1 in (20), we suggest a more generalized form of empirical β = FN 0 . corr α−1 (24) law given as bFNαφ α α i = CV2 exp(−B/V ) (26) Fig. 3 shows a field-enhancement correction factor (βcorr) α versus fractional-dimension parameter at varying values of where α is the fractional space-dimensionality parameter with φ = . α = F and 5 3eV.At 1, the correction factor is unity for 0 <α≤ 1. Equation (26) is clearly a more generalized form perfectly smooth planar surface. It is shown in analytical cal- of (25) with an addition parameter in the exponential term culations for various cathode geometries that the electrostatic which can capture the field-enhancement effect of nonplanar FEF varies from unity to a few hundred [38]Ð[41]. However, surfaces. This suggests a fitting of experimental data with (26), in many reported experiments (see [31] for examples), the taking α as an unknown. One could try plotting ln(i/V 2α) β α reported FEFs ( app) have spuriously high values. In practical versus 1/V in search of a straight-line plot, but, this is not β applications, it is assumed that actual will always be greater a good method when α is unknown. Alternatively, from (26) than 1 up to a few hundred as suggested by electrostatic field 2 α calculations at various geometries. This means that there is ln(i/V ) = ln(C) + (2α − 2) ln(V ) − B/V always a lower bound on βcorr depending on βapp and correctly (27) 2 calculated values of α. Using this correction factor (β ), the d ln(i/V ) α corr = 2α − 2 + αB/V (28) spuriously high values of FEF can be corrected, given that d ln(V ) the parameter α is correctly established using the procedure d (i/V 2) ln = α − . lim 1 →min 2 2 (29) discussed at the end of this section. It is to be noted that V d ln(V ) the βcorr considers the voltage or field dependence of FEF α− α through F 1 term. Such voltage-dependent FEF has also Using (29), the value of can be calculated from 0 [ ( / 2)/ ( )] / been studied recently [42]. y-intercept of “ d ln i V d ln V versus 1 V ”or [ ( / 2)/ ( )] / Now, assuming F = γ V ,whereγ is a constant, and “ d ln J F d ln F versus 1 F” plot. This plotting that the emission area A is constant in the relation i = method was tested on three experimental data sets shown in Fig. 4.InFig. 4(a), the field-emission data from [46] were AJFN, (1) suggests that the empirical field-emission iÐV characteristics should obey extracted and processed using the above-mentioned procedure. It is shown that the extracted value of α is 0.4. This implies i = CVk exp(−B/V ) (25) that classical FN equation (assuming α = 1) is not suitable to ZUBAIR et al.: FRACTIONAL FN LAW FOR FIELD EMISSION FROM ROUGH SURFACE 2093

currentÐvoltage characteristics of 2-D-material-based planar field emitter, which has recently been shown to exhibit non-FN behavior and a complete absence of SCL effect [50]. III. DERIVATION OF FIELD EMISSION WITH KANE’S NONPARABOLIC DISPERSION In this section, Kane’s nonparabolic dispersion is used to derive a general field-emission model that covers both nonrelativistic and relativistic charge carrier regimes. Such model was developed to describe the finite coupling between conduction and valence band in narrow-bandgap semiconduc- tor and also in the high energy regime of semiconductor where band nonparabolicity becomes nonnegligible. Kane’s dispersion [33] can be written as 2 2 h¯ k E (1 + γ E ) = (30) 2m∗ where γ is a nonparabolic parameter and m∗ is the electron effective mass. For γ → 0, the conventional parabolic energy 2 2 ∗ dispersion, E (γ → 0) = h¯ k /2m , is obtained. For γ →∞, Kane’s dispersion leads to a k -linear form of Fig. 4. Extraction of fractional-dimension parameter α from appropriate current–voltage or current–field plots. (a) Data set for field emission of E (γ →∞) = h¯ vF k (31) ZNO nanopencils taken from [46]. (b) Data set for field emission of carbon √ ∗ nanotubes from [47]. (c) Data set for field emission of nanoshuttles taken where vF ≡ 1/ 2m γ . Equation (30) can be explicitly from [48], here the required form of data is too noisy to extract the solved as α parameter. We emphasize the need for direct measurement of di/dV α γ 2 2 in order to correctly establish the parameter. + 4 h¯ k − 1 2m∗ 1 E = . (32) extract emission parameters in this case. Following the same 2γ procedure on data taken from [47], we show that the extracted By differentiating the left-hand side of (30) with respect to k , value of α parameter is 1, so the classical FN equation seems we obtain the following relation: valid in this case. Finally, Fig. 4(c) shows the data processed 2m∗ from [48]. In this case, the resulting plot is too noisy for k dk = (1 + 2γ E )dE . (33) h¯ 2 useful conclusions. It is clear that this method requires currentÐvoltage data form carefully crafted experiments with Equation (33) shall play the critical role of determining the minimal noise. The similar problem of noisy data has already electron supply function density for the field-emission process. been encountered in [43]. An alternate form of (29) would be In general, the field-emission current from a bulk material better in which “(d ln(i/V 2)/d ln(V )) = (V/i)(di/dV) − 2.” can be obtained by summing all k-modes The direct measurement of di/dV would be more accurate JFN,Kane = egs,v v(E⊥)D(E⊥) f (k) (34) by using some phase-sensitive detection techniques suggested k in [43]. where gs,v is the spin-valley degeneracy, v(E⊥) and D(E⊥) It should be noted that the fractional FN model developed are the carrier velocity and transmission probability along the above does not consider the space-charge-limited (SCL) effect, emission direction, and f (k) is the FermiÐDirac distribution which is expected to become dominant at a high bias regime. function. Here, it is assumed that the total energy of the In a previous work [14], we show that the fractional Child– carrier, E(k), can be partitioned as E(k) = E (k )+ E⊥(k⊥), Langmuir (CL) model predicts a stronger current for a rough where k and k⊥ are the wave vectors transverse to and along surface when compared with the classical CL law. It is the emission direction. In the continuum limit, (34) becomes possible to combine the fractional FN law with the fractional gs,ve J , = dk v(E⊥)D(E⊥) f (k)dk⊥ (35) CL law in a future work to study the continuous transition FN Kane (2π)3 between the two transport regimes. Nonetheless, a conservative which can be simplified as follows: 1) the integral estimation can be obtained by calculating the transitional bias ∞(···)v( ) ¯ −1 ∞(···) 0 E⊥ dk⊥ can be replaced by h 0 dE⊥ voltage, VT , at which the fractional FN current is equal to − since v(E⊥) = h¯ 1dE⊥/dk⊥ and 2) for field emission, the fractional CL current. We found that V is related with T only carriers up to the take part in the vacuum gap D and parameter α through the relation Dα = √ ( α− / ) transport process, and thus the FermiÐDirac distribution ln((9/4 2)V 2 3 2 )[V ]α.ThisV serves as a conservative T T T can be represented by a two-variable Heaviside function, upper limit for the fractional FN law to be valid. For bias ( ) → ( + − ) i.e., f k E E⊥ EF —this further limits the voltage larger than VT , the SCL effect needs to be explicitly EF −E E E⊥ E (···)dE⊥ F (···)dE considered. We further note that, for the emerging class of -and -integral to 0 and 0 , respectively. Correspondingly, (35) is simplified as novel 2-D materials, it is well known that their surfaces   E E −E often exhibit corrugations and crumples [49]. The fractional gs,ve F F J , = k dk dE⊥ D(E⊥). FN Kane 2 (36) approach developed here may also be extended to model the h¯ (2π) 0 0 2094 IEEE TRANSACTIONS ON ELECTRON DEVICES, VOL. 65, NO. 6, JUNE 2018

By using the semiclassical WKB approximation of IV. CONCLUSION ( ) = ( − )/ D E⊥ DF exp [ E⊥ EF dF ] for a triangular barrier, The standard FN theory-based existing formulas for the where DF and dF are constants defined in (15) and (17) in field-induced emission of electrons from ideal planar surfaces α = the limit of 1, respectively, and considering (33), (36) are not always suitable for practical rough, irregular surfaces becomes and may lead to significant order-of-magnitude errors when  ∗ EF applied to experimental data incorrectly. Also, the standard gs,ve 2m J , = D dE (1 + 2γ E ) FN Kane ( π)2 ¯ 3 F models rely on free-electron theory assuming an isotropic 2 h 0 − − distribution of electrons with parabolic energy dispersion. EF E E⊥ EF × e dF dE⊥ Such an assumption is no longer valid for narrow-gap semi- 0 ∗ conductors and few layer graphene, where energy dispersion gs,ve 2m becomes nonparabolic due to band structure. = DF dF [A + B] (37) (2π)2 h¯ 3 In this paper, accurate equations for the field-emission current densities from rough planar surfaces are derived using where Kane’s nonparabolic dispersion model. The reported expres-    EF − E − EF sions are applicable to metallic, narrow-gap semiconductors A ≡ dE e dF − e dF and a few-layer graphene planar electron emitters with rough 0     surfaces in many modern applications. Based on our model, E − EF F d = dF 1 − 1 + e F (38) we have proposed a new form of field-enhancement correc- dF tion factor βcorr to avoid spuriously high FEFs in recent experiments [31]. We propose that the experimental currentÐ and    voltage (iÐV ) relationships of field emission from practical E E F − − EF rough surfaces should approximately obey the empirical law ≡ γ d − d B 2 dE E e F e F i = CV2α exp(−B/V α),where0 <α≤ 1. We have 0      α 2 E shown that the fractional-dimension parameter can be EF 1 EF − F 2 = γ 2 − + + dF . [d (i/V )/d (V )] 2 dF 1 1 e (39) established from the y-intercept of “ ln ln or dF 2 dF (V/i)(di/dV)−2” versus “1/V ” plot. This requires an accu- rate phase-detection measurement of (di/dV), previously pro-  For typical conducting solid, EF dF , and hence posed by Forbes [43] in his work. Determination of α values could be very useful for better understanding of field emission ≈ A dF (40) and its variation as between different materials and surface roughness profiles. We strongly reemphasize Forbes’s call and of experiment to directly measure (di/dV) in field-emission experiments. The electronic availability of raw iÐV measure- B ≈ 2γ d2 . (41) F ment data from experimentalists would be extremely useful for further explorations and improvements in the existing From (37), we obtained the field-emission current density as theoretical models. ∗   gs,ve 2m 2 3 JFN,Kane ≈ DF d + 2γ d (42) REFERENCES (2π)2 h¯ 3 F F [1] N. Egorov and E. Sheshin, Field Emission Electronics.Cham, which is composed of two parts: a conventional FN component Switzerland: Springer, 2017, doi: 10.1007/978-3-319-56561-3. and a correction factor that accounts for band nonparabolicity. [2] R. H. Fowler and L. Nordheim, “Electron emission in intense electric fields,” Proc. Roy. Soc. Lond. A, Math. Phys. Sci., vol. 119, no. 781, Using (14) in (37), the fractional generalization of (42) can pp. 173Ð181, May 1928. be written as [3] S.-D. Liang, Quantum Tunneling and Field Electron Emission Theories. Singapore: World Scientific, 2013. ∗   [4] R. G. 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Zubair, “On cylindrical model of electrostatic nanoshuttles: Shape-controlled synthesis, perfect flexibility and high- potential in fractional dimensional space,” Opt.-Int. J. Light Electron, performance field emission,” Nanotechnology, vol. 22, no. 50, p. 505601, vol. 127, no. 6, pp. 3243Ð3247, 2016. 2011. [22] M. Zubair, M. J. Mughal, and Q. A. Naqvi, “An exact solution of [49] S. Deng and V. Berry, “Wrinkled, rippled and crumpled graphene: the cylindrical wave equation for electromagnetic field in fractional An overview of formation mechanism, electronic properties, and appli- dimensional space,” Prog. Electromagn. Res., vol. 114, pp. 443Ð455, cations,” Mater. Today, vol. 19, no. 4, pp. 197Ð212, 2016. 2011, doi: 10.2528/PIER11021508. [50] Y. S. Ang, M. Zubair, K. J. A. Ooi, and L. K. Ang. (Nov. 2017). [23] H. Asad, M. J. Mughal, M. Zubair, and Q. A. Naqvi, “Electromagnetic “Generalized FowlerÐNordheim field-induced vertical electron emis- green’s function for fractional space,” J. Electromagn. Waves Appl., sion model for two-dimensional materials.” [Online]. Available: vol. 26, nos. 14Ð15, pp. 1903Ð1910, 2012. https://arxiv.org/abs/1711.05898 [24] H. Asad, M. Zubair, and M. J. Mughal, “Reflection and transmis- sion at -fractal interface,” Prog. Electromagn. Res., vol. 125, pp. 543Ð558, 2012, doi: 10.2528/PIER12012402. Muhammad Zubair (S’13–M’15) received the [25] M. Zubair, M. J. Mughal, and Q. A. Naqvi, “An exact solution of the Ph.D. degree in electronic engineering from the spherical wave equation in d-dimensional fractional space,” J. Electro- Politecnico di Torino, Turin, Italy, in 2015. magn. Waves Appl., vol. 25, no. 10, pp. 1481Ð1491, 2011. From 2015 to 2017, he was with the SUTD-MIT [26] M. Zubair, M. J. Mughal, Q. A. Naqvi, and A. A. Rizvi, “Differential International Design Center, Singapore. Since electromagnetic equations in fractional space,” Prog. Electromagn. Res., 2017, he has been with Information Technol- vol. 114, pp. 255Ð269, 2011, doi: 10.2528/PIER11011403. ogy University, Lahore, Pakistan. His current [27] M. Zubair, M. J. Mughal, and Q. A. Naqvi, “On electromagnetic wave research interests include charge transport, elec- propagation in fractional space,” Nonlinear Anal., Real World Appl., tron device modeling, computational electromag- vol. 12, no. 5, pp. 2844Ð2850, 2011. netics, fractal electrodynamics, and [28] M. Zubair, M. J. Mughal, and Q. A. Naqvi, “The wave equation and imaging. general plane wave solutions in fractional space,” Prog. Electromagn. Res. Lett., vol. 19, pp. 137Ð146, 2010, doi: 10.2528/PIERL10102103. [29] F. A. Godínez, O. Chávez, A. García, and R. Zenit, “A space-fractional Yee Sin Ang received the bachelor’s degree model of thermo-electromagnetic wave propagation in anisotropic in medical and radiation physics and the Ph.D. media,” Appl. Thermal Eng., vol. 93, pp. 529Ð536, Jan. 2016. degree in theoretical condensed matter physics [30] A. S. Balankin, “A continuum framework for mechanics of fractal from the University of Wollongong, Wollongong, materials I: From fractional space to continuum with fractal metric,” NSW, Australia, 2010 and in 2014, respectively. Eur. Phys. J. B, vol. 88, no. 4, pp. 1Ð13, 2015. He is currently a Research Fellow with the [31] R. G. Forbes, “Development of a simple quantitative test for lack of Singapore University of Technology and Design, field emission orthodoxy,” in Proc. Roy. Soc. London A, Math. Phys. Singapore. Sci., vol. 469, no. 2158, p. 20130271, 2013. [32] R. G. Forbes, “Use of MillikanÐLauritsen plots, rather than FowlerÐNordheim 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. Lay Kee Ang (S’95–M’00–SM’08) received the [34] X.-Z. Qin, W.-L. Wang, N.-S. Xu, Z.-B. Li, and R. G. Forbes, “Analytical B.S. degree from the Department of Nuclear treatment of cold field electron emission from a nanowall emitter, Engineering, National Tsing Hua University, including quantum confinement effects,” Proc. Roy. Soc. Lond. A, Math. Hsinchu, Taiwan, in 1994, and the M.S. and Ph.D. Phys. Sci., vol. 467, no. 2128, pp. 1029Ð1051, 2011. degrees from the Department of Nuclear Engi- [35] S.-J. Liang and L. K. Ang, “Electron from graphene neering and Radiological Sciences, University of and a thermionic energy converter,” Phys.Rev.Appl., vol. 3, no. 1, Michigan, Ann Arbor, MI, USA, in 1996 and 1999, p. 014002, 2015. respectively. [36] V. E. Tarasov, “Anisotropic fractal media by vector calculus in non- Since 2011, he has been with the Singapore integer dimensional space,” J. Math. Phys., vol. 55, no. 8, p. 083510, University of Technology and Design, Singapore. 2014.