hot electron tunneling

7/31/2019 Hot Electron Tunneling

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doi: 10.1098/rspa.2007.0030, 2907-29274632007Proc. R. Soc. A

Richard G Forbes and Jonathan H.B DeaneemissionNordheim tunnelling and cold field electron

Reformulation of the standard theory of Fowler

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Reformulation of the standard theory ofFowlerNordheim tunnelling and cold field

electron emission

BY RICHARD G. FORBES1,* AN D JONATHAN H. B. DEANE2

1Advanced Technology Institute (BB), and 2Department of Mathematics,School of Electronics and Physical Sciences, University of Surrey, Guildford,

Surrey GU2 7XH, UK

This paper presents a major reformulation of the standard theory of FowlerNordheim(FN) tunnelling and cold field electron emission (CFE). Mathematical analysis andphysical interpretation become easier if the principal field emission elliptic function v isexpressed as a function v(l0) of the mathematical variable l0hy2, where y is theNordheim parameter. For the SchottkyNordheim (SN) barrier used in standard CFEtheory, l0 is equal to the scaled barrier field f, which is the ratio of the electric field thatdefines a tunnelling barrier to the critical field needed to reduce barrier height to zero.The tunnelling exponent correction factor nZv(f). This paper separates mathematicaland physical descriptions of standard CFE theory, reformulates derivations to be interms of l0 and f, rather than y, and gives a fuller account of SN barrier mathematics.v(l0) is found to satisfy the ordinary differential equation l0(1Kl0)d2v/dl02Z(3/16)v; anexact series solution, defined by recurrence formulae, is reported. Numericalapproximation formulae, with absolute error j3j!8!10K10, are given for v and dv/dl0.The previously reported formula vz1Kl0C(1/6)l0 ln l0 is a good low-order approxi-mation, with j3j!0.0025. With l0Zf, this has been used to create good approximateformulae for the other special CFE elliptic functions, and to investigate a more universal,scaled, form of FN plot. This yields additional insights and a clearer answer to thequestion: what does linearity of an experimental FN plot mean? FN plot curvature ispredicted by a new function w. The new formulation is designed so that it can easily begeneralized; thus, our treatment of the SN barrier is a paradigm for other barrier shapes.We urge widespread consideration of this approach.

Keywords: field emission; FowlerNordheim tunnelling;

field emission elliptic functions

1. General introduction

Nearly 80 years ago, Fowler and Nordheim (FN) published in these Proceedingstheir seminal paper ( Fowler & Nordheim 1928) on cold field electron emission(CFE) from metal surfaces. Their paper has shaped much subsequent work. Thename FowlerNordheim tunnelling is now used for any field-induced electron

Proc. R. Soc. A (2007) 463, 29072927

doi:10.1098/rspa.2007.0030

Published online 21 August 2007

* Author for correspondence ([email protected]).

Received 13 May 2007Accepted 24 July 2007 2907 This journal is q 2007 The Royal Society

http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/http://rspa.royalsocietypublishing.org/


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tunnelling through a roughly triangular (in practice, always rounded) barrier.The main modern contexts are: (i) vacuum breakdown in high-voltage apparatusof all kinds, where one needs to prevent electron emission from asperities, (ii)cold-cathode electron sourcestheir many applications include bright pointsources (for high-resolution electron microscopes and other machines), X-ray

generators, electronic displays and space vehicle neutralizers, and (iii) internalelectron transfer in some electronic devices.

The original 1928 equation used an unrealistic barrier model, which seriouslyunderpredicts CFE current densities. Various modified equations have beenintroduced, which we call FN-type equations. Data analysis has often used theso-called standard FN-type equation (2.10), derived from Murphy & Goods(1956) work (MG). However, the MG paper is not easy to follow, confusionabounds in the literature, and this standard CFE theory has gained areputation for being obscure and difficult. Perhaps as a result, some recentexperimental research papers use simplified FN-type equations, as found in

undergraduate textbooks. These also significantly underpredict CFE currentdensities, typically by a factor of order 100, and give scope for error.The standard FN-type equation uses a mathematical function v, well known in

field emission, and normally evaluated as a function of the Nordheim parameter ydefined by (2.15). Forbes (2006) reported a good simple approximation for v( y).However, it seemed better to take v as a function of a new parameter scaledbarrier field (equal to y2 in standard theory), and then write v as a function ofbarrier field F. This gave, for the first time, a simple, reliable algebraicapproximation for the exponent in the standard FN-type equation. This makesthe equations behaviour easier to investigate, but background theory needspresenting differently.

Specific aims here are to show the approximations mathematical origin, anduse it to gain fuller understanding of FN plots. But, more important, we presenta major reformulation of standard CFE theory itself. This seems timely, for twomain reasons.

First, FN plots are the commonest tool used to analyse experimental CFEdata. Elementary CFE theory, sometimes employed, can explain slopes. But, as6 shows, standard CFE theory is probably the simplest theory able to clarifytheir detailed behaviour. Users of FN plots need to be able to understandstandard theory; improved formulation should help.

Second, the standard FN-type equation was derived for free-electron metals

with planar surfaces and has well-known deficiencies, including limitedapplicability to atomically sharp emitters. Reformulation makes it easier togeneralize standard theory to treat more realistic tunnelling barriers.

In standard theory, clearer conceptual distinctions are needed between purelymathematical aspects and physical aspects. To replace the use of y, we introducetwo related theoretical descriptions, one mathematical and the other physical,each (in principle) with its own names, symbols and definitions. The formerinvolves mathematics that applies primarily to the SchottkyNordheim (SN)tunnelling barrier (Schottky 1914; Nordheim 1928); the latter involves physicaldefinitions and equations that apply (or are easily generalized to apply) to many

different barriers. There are also physical equations specific to the SN barrier.Both descriptions are valid in their own right. For the SN barrier they interact;other barriers have their own specific mathematical analyses.

R. G. Forbes and J. H. B. Deane2908

Proc. R. Soc. A (2007)



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Thus, the SN barrier mathematical description uses a variable l0 defined in thecontext of elliptic function theory (by (4.7)); but the physical description usesthe scaled barrier field fh defined using physical fields (by (2.12)). The SN barrieranalysis has l0Zfh, but other barriers normally do not.

Similarly, the mathematical description uses (for example) the principal field

emission elliptic function v(l0) given by (4.14), but the physical description usesthe tunnelling exponent correction factor n(fh) defined by (2.5). For the SNbarrier, but normally not for other barriers, nZv(fh).

All this is normal scientific practice when using mathematical functions, withv(l0) behaving like cos(q). But standard CFE theory has been different. The samesymbol has been used for both the physical variable and the relatedmathematical function, and the name of the symbol representing a function(e.g. vee) has been used as the name of the function. This unfamiliar conventionand resulting lack of clarity seem to have impeded both wider understanding andtheoretical development.

In writing this paper, a particular problem has been to decide whether theletters r, s, t, u, vand wshould represent mathematical functions (i.e. be part ofthe mathematical description, and normally applicable only to the SN barrier),or should represent general physical quantities applicable to many differentbarriers. Current practice usually treats t, u and v as mathematical functions,but r and s more as physical parameters. The solution adopted here is to treatall as mathematical functions. This makes notation more uniform but meansthat, especially in 2, we need separate symbols (Greek letters are used) for thequantities in the physical description. For simplicity, here, we introduce only n(nu) and the correction factor t defined by (2.6); others will be needed infuture work.

In summary, this paper aims to reduce confusion, make standard CFE theorymore complete and more accessible, and permanently change the way that basicCFE theory is discussed. Analysis is from basic principles, but relies on results inForbes (1999b) and in Forbes (2004).

The papers structure is as follows: 2 presents additional background; 3derives revised definitions for the main functions used in standard CFE theory;4 discusses mathematical expressions for v(l0); 5 establishes simple algebraicapproximations for the other functions; 6 presents new results relating to FNplots; 7 discusses implications; and appendix A presents the exact seriesexpansion for v(l0).

2. Theoretical background

The following notations are used: let edenote the elementary positive charge; methe electron mass in free space; hP Plancks constant; 30 the electric constant; andf the local work function of the emitting surface. The field at the emitter surfaceis denoted by Fand called the barrier field, and the emission current density isdenoted by J; in CFE theory, these positive quantities are the negative of thelike-named quantities used in conventional electrostatics. This field Fdetermines

the tunnelling barrier. Unreduced barrier height h is the height of a tunnellingbarrier when FZ0. Universal constants are evaluated using the 2002 values ofthe fundamental constants and given to seven significant figures.

2909Reformulation of standard CFE theory




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CFE theory involves, first, calculation of the escape probability D for anelectron approaching the emitter surface in a given internal electronic state, andthen summation over all occupied states to give J. Many different levels oftheoretical approximation exist. The basic free-electron CFE treatmentsdiscussed here: (i) ignore atomic structure and assume a Sommerfeld free-

electron model, (ii) assume that the electron distribution is in thermodynamicequilibrium and obeys FermiDirac statistics, (iii) take temperature as zero, and(iv) assume a flat planar emitter surface, of constant uniform local work functionf (i.e. df/dFZ0), with a uniform electric field F outside.

Choices are then needed about modelling the tunnelling barrier and abouttheoretical method. The Schrodinger equation can be solved exactly for FNsoriginal triangular barrier, but has no ordinary analytical solutions for mostmodel barriers of physical interest, including the SchottkyNordheim barrierused in standard CFE theory. So, normally, some approximate method must beused. Usually this is a JWKB-type approximation (Jeffreys 1925, also see

Froman & Froman 1965) or the closely related Miller & Good (1953)approximation. There are several different JWKB-type formulae, applicable tobarriers of different kinds.

For strong barriers, the escape probability D can be written (Landau &Lifschitz 1958) as

DZP expKG; 2:1where G is the so-called JWKB exponent defined by

Ghge M1=2 dz; 2:2and Pis a tunnelling pre-factor. Here, ge [h4p(2me)

1/2/hPz10.24624eVK1/2nmK1]

is the JWKB constant for an electron, and z is the distance measured from theemitters electrical surface (Lang & Kohn 1973; Forbes 1999c). The function M(z)defines the barrier, with integration performed between the classic turning points,i.e. the zeros of M(z). By definition, a strong barrier has G sufficiently large thatexp[KG]/1.

The pre-factor P is included in (2.1) for conceptual completeness. Normally, itis tacitly assumed that P differs from unity by a presumed unimportantmultiplying factor (between 1/5 and 5, say), and is slowly varying with barrier

height hin comparison with exp[KG]. So normal practice sets PZ1 and uses theso-called simple JWKB approximation,

DzexpKG: 2:3The MillerGood approximation also reduces to (2.3) when G is large.

For the elementary triangular barrier of height h and slopeKeF, MZhKeFz;(2.2) then yields the quantity Gel given by

Gel Zbh3=2

F; 2:4

where the second FowlerNordheim constant bh2ge/3eh(8p/3)(2me)1/2/

ehPz6.830890 eVK3/2 V nmK1. For other barriers, a tunnelling exponent





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correction factor n is defined generally by

GhnGel; 2:5and a decay-rate correction factor t is defined generally by

vG

vh

Fh

t

vGel

vh

F; 2:6where partial derivatives are taken at constant barrier field F. It is easily shownthat

tZ nC2

3

h

vn

vh

F: 2:7

For the elementary triangular barrier, the correction factors n and t areboth unity.

Choice of method also occurs when summing tunnelling current contributionsfrom the internal electron states. Forbes (2004) summed over states on a

spherical constant-total-energy surface, and then integrated with respect to totalelectron energy. This approach resembles that used for non-free-electron bandstructures (e.g. Gadzuk & Plummer 1973; Modinos 1984). It can be applied toany well-behaved barrier model, and leads to the current-density equation

JZ tK2F afK1F2

expKnFbf

3=2=Fh i

; 2:8where the first FowlerNordheim constant ahe3/8phPz1.541434!10

K6

A eV VK2, and nF and tF are the values of n and t that apply to a barrier ofunreduced height h equal to the local work function f. For clarity, later,equations of this kind are called curve equations. The emission current Iis then

given by IZ

AJ, where A is a notional emission area that is often field dependent.The subscript F on a quantity shows that it applies to the particular barrierencountered by a Fermi-level electron that is moving forwards (i.e. towardsand normal to the emitting surface). nF and tF enter the theory because deriving(2.8) involves Taylor expansion of the JWKB exponent G (ZnGel) about theFermi level.

The symbols n, nF, t and tF represent general correction factors, and appear inequations that apply to any well-behaved barrier model. Detailed analysesrequire a specific barrier model, and the general quantities in (2.5)(2.8) mustthen be replaced by correction factors specific to this model. Mathematicalevaluations (by computer if necessary) must then be performed for these specific

correction factors. Since all factors specific to a barrier model can be derived fromthe n-like factor in the specific version of (2.5), detailed analysis concentrates onthis factor.

The SchottkyNordheim (SN) barrier is defined (for z greater than someminimum value zmin) by

MSNzZ hKeFzKe2=16p30z: 2:9Putting (2.9) in (2.2) leads to a specific correction factor nSN. Burgess et al.

(1953) showed that nSN is given by a mathematical function vthat they specified1

and MG subsequently used. v is best understood as a function of mathematical

physics in its own right, albeit a very specialized one, and can be called the1 A function specified by Nordheim (1928) is not a correct mathematical expression for nSN, owingto a mistake in defining the argument of a complete elliptic integral.





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principal field emission elliptic function (or, better, the principal SN barrierfunction). A function t is then derived via a specific version of (2.7), namely(3.1b). The final outcome is the so-called standard FN-type curve equation (forthe emission current Ist predicted by standard theory),

Ist ZAtK

2F afK

1F2 expKvFbf3=2=Fh i

; 2:10where vF and tF are the values ofvand tthat apply to the barrier encountered bya forward-moving Fermi-level electron (when this barrier is modelled as a SNbarrier).

The mathematical functions v and t, with s, u and w below, are knowncollectively as the special field emission elliptic functions.2 They depend on asingle mathematical variable, hitherto taken as the Nordheim parameter y (see(2.15)). All can be derived from v and its derivative (see Forbes (1999b) for pastdefinitions using y).

These special elliptic functions can be evaluated accurately by variousmethods, and (except for w) the results are tabulated (Burgess et al. 1953;Good & Muller 1956; Miller 1966; Forbes & Jensen 2001). vcan also be expressedin terms of the complete elliptic integrals Kand E(see MG and Forbes (1999b),3

for definitions using y). However, a need exists for simple, reliable, algebraicapproximations for these functions, especially v. Jensen & Ganguly (1993) andJensen (2001) have derived formulae for v(y), but these are complex andawkward to use in further analysis. Fitting procedures have generated simpleempirical formulae, for example the Spindt et al. (1976) approximation, but thesedo not represent v accurately over the whole range 0%y%1.

As already noted, Forbes (2006) reported a simple good approximation for v,but argued that y2 is the natural variable to use. In mathematics, it seems best toput l0hy2, call l0 a complementary elliptic variable4 and write

vl0z1Kl0C16l0 ln l0: 2:11

The discovery that v(l0) satisfies a simple-looking differential equation in l0,(4.13), supports using l0 as the independent variable. If y is used, the resultingdifferential equation is more complex.

For the real physical situation, we can define a scaled barrier field fh by

fhZF

Fh; 2:12

where Fh is the real field that reduces barrier height from h to zero. For theSchottkyNordheim (SN) barrier, Schottky (1914) showed that a field Flowers thebarrier by an energyDSZcF

1/2, where cZ(e3/4p30)1/2z1.199985 eV VK1/2 nm1/2.

2 But a better collective name might be the SchottkyNordheim barrier functions.3 Typographic errors occur in Forbes (1999b): (i) in the definition of K(m), both brackets shouldbe raised to the power (K1/2) and (ii) the value of coefficient a4 in eqn (32c) should be

a4Z0.01451196212.4 The prime indicates that it is a complementary variable, i.e. l0/0 as the elliptic parameterm/1.





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So, for the SN barrier (but not for other barriers), we estimate Fh and fh by

FSNh Z cK2h2z0:6944617 eVK2 V nmK1h2; 2:13

fSNh Z FFSNh

Z c2

Fh2Z e

3

4p30

FhK2: 2:14

The Nordheim parameter y (Nordheim 1928) is defined as

yhDS

hZ

cF1=2

hZOfSNh : 2:15

In the JWKB integral for the SN barrier, the term (e3/4p30)FhK2 has

previously been replaced with y2. Here, it is replaced with l0 (see 4). Equation

(2.14) shows that the mathematical variable l0 can be identified with the SNbarrier parameter fSNh , and so has a physical interpretation. Probably, theconcept of scaled barrier field will prove more readily understandable thanthe Nordheim parameter has been. These are further advantages of using l0 as theindependent mathematical variable.

When h is the local work function f, then (2.13) yields the critical SN barrierfield (FSNf ) at which the SN barrier for a forward-moving Fermi-level electronvanishes (Schottky 1923). The corresponding scaled SN barrier field fSNf is

fSNf ZF

FSNf : 2:16

So we reach the Forbes (2006) result (but in more careful notation),

vFz1KF

FSNf

C

1

6

F

FSNf

ln

F

FSNf

: 2:17

All these things make it desirable to reformulate standard CFE theory to be interms of the mathematical variable l0 and its physical partner the scaled barrier

field. For simplicity in this paper, we now mostly drop the label SN and just usef: it is always clear from context whether fSNh or fSNf is meant. Since in standard

theory l0Zf, there is some choice as to which is used in formulae. We use l0 instrictly mathematical contexts, f when the formula relates more to experiment.

3. Parameters for FowlerNordheim analysis

This section creates f-based definitions for the mathematical functions used instandard CFE theory. They could equally well be presented using l0, but using f

will make them easier to generalize in future work. Apart from (3.1b), thedefinitions here are specific mathematical versions of general physicalrelationships valid for any well-behaved barrier model.





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In standard theory, n is given by v, t by t. So from (2.7), and then the SNbarrier formula (2.14),

tfZ vC23

hvv

vh

FZ vC

2

3h

dv

df

vf

vh

F; 3:1a

tfZ vK 43

f

dv

df: 3:1b

The functions r, s, uand wrelate to FowlerNordheim plots. We first converttunnelling exponents to be in terms of 1/f. Define a dimensionless parameter h by

hZbf3=2

Ff; 3:2

where Ff is the critical field at which a tunnelling barrier of unreduced height f

vanishes. This leads to the general resultbf3=2

FZ

bf3=2

Ff

FfF

Z

h

f: 3:3

So, in standard theory, from (2.10),

lnfIst=F2gZ lnftK2F AafK1gK vFh=f: 3:4Equation (3.4) is said to be in FN coordinates of type [ln{I/F2} versus 1/f].Our notation for logarithms follows the rule (ISO 1992) that placing an

expression in curly brackets means take the numerical value of this expression,when evaluated in the specified units. Elsewhere these brackets are usednormally. In SI units, (I/F2) and (AafK1) would be in A VK2 m2. Since both Fand fappear, (3.4) is a mixed FN plot form; but this form is useful for discussingbasic theory.

Equation (3.4) is a curve equation, expressed in FN coordinates. If any fielddependence in f, tK2F or A is ignored, then its slope with respect to 1/f at anypoint may be written as Ksh, where s is the standard slope correction function5

introduced by Houston (1952). Letting xh1/f, we have

s

f

Zd

vFx

=dxZ vFCxdvF=dxZ vFC

1=f

dvF=df

=

dx=df

Z vFKfdvF=df: 3:5

Forbes (1999a) argued that the most convenient theoretical model for anexperimental FN plot is the tangentto the chosen FN-type curve equation, whenthis curve equation is expressed in FN coordinates. In standard theory, againignoring any field dependence in f, tK2F or A, a suitable form for this FN-typetangent equation is

lnIst

F2& 'Z lnfrAafK1gKsh

f; 3:6

5 But note his calculations ofsare in error, owing to the mistake in Nordheim (1928); Burgess et al.(1953) gave corrected results.





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where ln{rAafK1} is the intercept the tangent makes with the ln{I/F2} axis.The standard intercept correction function denoted here by r is the function rNintroduced by Forbes (1999a). Both r and s vary along the curve.

Figure 3b shows plots of the quantity DYst [Zln{Ist/F2}Kln{AafK1}]against 1/f. For any specific value 1/fP, DY

st(fP) can be found either from the

curve equation via line V or from the tangent equation via line T. Subtracting(3.4) from (3.6), and using (3.5), yields

ln r=tK2F

Z vFKsh=fZKh dvF=df: 3:7In terms of f, the function u in Forbes (1999b) is u(f)hKdvF/df. Hence,

sfZ vFKfdvF=dfZ vFCuf; 3:8

rf; hZ tK2F expKh dvF=dfZ tK2F exphu: 3:9

Finally, a new function w is introduced to describe the curvature of a FN plotmade against 1/f,

wfZ ds=d1=fZKf2ds=dfZf3d2vF=df2: 3:10For example, wZ0.02 means the FN plot slope changes by 2% when 1/fchangesby 1.

Clearly, all these functions can be obtained from vF(f) and its first twoderivatives.

4. Expressions for the mathematical function v(l0

)

(a) ODE in the complementary elliptic variable l0

We now write vas a function of the purely mathematical complementary ellipticvariable l0, and derive the ordinary differential equation (ODE) that v(l0)satisfies. Following MG, Forbes (1999b) showed how to express v(y) in terms ofthe complete elliptic integrals K and E. His equation numbers are here prefixedF. We can replace (F12) by defining

l0hy2 Z

e3=4p30

F=h2:

4:1

Comparison with (2.14) shows l0ZfSNh . Hence, for the SN barrier (but not forother barriers), l0 has a physical interpretation as the scaled barrier field.Equation (4.1) also means we can treat y in Forbes (1999b) as a convenient

notation for Ol0. Noting bZ(8p/3)(2me)1/2/ehP, we can use (F17a) to provide

definitions convenient for numerical integration,

vl0h 3=4ffiffiffi

2p

1Cffiffiffiffiffiffiffi1Kl0p

1K

ffiffiffiffiffiffiffi1Kl0

p Kx2C2xKl01=2xK1=2 dx; 4:2

dv=dl0ZK3=8ffiffiffi

2p

1Cffiffiffiffiffiffiffi

1Kl0p

1Kffiffiffiffiffiffiffi

1Kl0p x2C2xKl0K1=2xK1=2 dx: 4:3





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In MG and Forbes (1999b), the transformation applied to (F24b) yields thefollowing results. Kand Eare defined in terms of the elliptic parameter mas usedby Abramowitz & Stegun (1965 (AS)),

Km

h

1

0 1Kz2

K1=2

1Kmz2

K1=2 dz;

4:4

Emh1

01Kz2K1=21Kmz2C1=2 dz: 4:5

If we put, as in (F26b),

mZ 1Kl01=2=1C l01=2; 4:6then, noting (F26a) and (F29),

l0Z 1Km=1Cm2; 4:7

vl0Z 1C l01=21=2EmKl01=2Km; 4:8

dv=dl0ZK341C l01=2K1=2Km and 4:9

d2v=dl02 ZK 3161C l01=2K1=24 dKm=dl0Kl0K1=21C l01=2K1=2Km: 4:10

Equation (4.7) establishes the link between l0 and elliptic function theory. From(F27), (4.6), and then (4.10),

dK

m

=dmZ

fE

m

K

1Km

K

m

g=m

1Km

;

4:11

dv=dl0ZKl0K1=21C l01=2K2 and 4:12

l01Kl0d2v=dl02K34vZ 0: 4:13

Thus, v(l0) obeys the ODE (4.13). Note that no factors in l01/2 appear. ThisODE appears to be new both in elliptic function theory and in mathematicalphysics. It is simpler in form than the ODEs associated with K and E (Cayley1876).

(b ) Exact series expansionTwo independent solutions for ODE (4.13) have been found using the method

of Frobenius to develop series expansions. The boundary conditions on v anddv/dl0, as l0/0, then determine an exact series expansion for v(l0). The first fewterms are

vl0Z 1Kf98

ln 2C 316

gl0Kf 27256

ln 2K 511024

gl02Kf 3158192

ln 2K 1778192

gl03C/

C 316C

9

512l0C 105

16 384l02C/l0 ln l0: 4:14

Recurrence relations for the coefficients are given in appendix A. Series (4.14)

was found earlier using MAPLE (Forbes 2006).The derivation of the recurrence relations is lengthy and is presentedseparately elsewhere (Deane et al. 2007). It shows that the ln l0 terms are an





12/23

intrinsic part of the expansion, and that terms involving non-integral powers of l0are not needed. Appendix A contains a shorter alternative derivation; thisdirectly confirms the lower order terms but does not bring out the underlying

mathematics.Evaluating coefficients to five decimal places, we obtain

vl0z1K0:96729l0K0:02330l02K0:0050l03K/C ln l00:18750l0C0:01758l02C0:00641l03C/ 4:15

z1Kl01C0:03271l0C0:00941l02C/C l0 ln l00:18750

C0:01758l0C0:00641l02C/:

4:16

Form (4.16) is explicitly exact at l0Z0 and 1 and has good convergence.

(c) Approximations and numerical evaluations

The two simplest ways to obtain high accuracy values for v(l0) are to evaluate(4.8) or (4.14) using a mathematical package, or to integrate (2.2) or (4.2)numerically. We have checked that different methods give the same numericalresult to at least 12 decimal places.

For some applications, including spreadsheet calculations, approximation

formulae are useful. For a degree-jformula always exact at l0Z0 and 1, we write

vjl0z1Kl0XjiZ0

pil0iC l0 ln l0

XjiZ1

qil0iK1: 4:17

Best-fit values for the coefficients pi and qi were chosen by least squares

minimization of numerical approximations to1

0 vjl0K vEl02 dl0, where vE isthe exact value as determined numerically. Choosing jZ4 yields formulae withabsolute error j3j%8!10K10, without using error spreading techniques. Table 1shows coefficient values. This performance exceeds, by a factor of approximately

25, the Hastings (1955) result j3j%2!10K8

for E(m), which did use error-spreading techniques. As the universal constants are known only to about 1 partin 107, this precision is far more than needed physically.

Table 1. Coefficients for use in connection with (4.17) and (4.18), for degree-4 formulae. (Note:u1z0.8330405509.)

I pi qi si ti

0 1 0.053 249 972 7 0.187 51 0.032 705 304 46 0.187 499 344 1 0.024 222 259 59 0.035 155 558 742 0.009 157 798 739 0.017 506 369 47 0.015 122 059 58 0.019 127 526 803 0.002 644 272 807 0.005 527 069 444 0.007 550 739 834 0.011 522 840 094 0.000 089 871 738 11 0.001 023 904 180 0.000 639 172 865 9 0.003 624 569 4275 K0.000 048 819 745 89





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Similarly, a formula for dv/dl0, with j3j%7!10K10, usesdv

dl0zKu1C1Kl0

XjC1iZ0

sil0iC ln l0

XjiZ0

til0i; 4:18

where u1Z3p=8O2, s0Z[u1K(9/8)ln 2], t0Z3/16, and the other coefficientscome from minimization of absolute errors. This formula goes to infinity in thecorrect way as l0/0 and is exact at l0Z1. As compared with (4.17), extracoefficients are needed to achieve similar precision. A spreadsheet using theseformulae to calculate the standard theory functions is available from RGF.

For analytical explorations and preliminary data analysis, a very simpleformula is best. As already reported, (2.11) has a relative accuracy of 0.33% orbetter, over the whole range 0%l0%1. This outperforms earlier approximationsof equivalent complexity, due to Andreev (1952), Charbonnier & Martin (1962),Dobretsov & Gomoyunova (1966), Miller (1966), Beilis (1971), Spindt et al.(1976), Eupper (1980) (quoted by Hawkes & Kasper (1989)) and Miller (1980).The reason is that, unlike the other formulae, (2.11) resembles the low-orderterms in the exact expansion. However, some formulae do outperform (2.11) overlimited ranges, because they were optimized to perform well there.

For comparison with (2.11), we searched numerically for the bestapproximations of form

vl0z1Kl0Cql0 ln l0; 4:19where q is an adjustable constant. For 0%l0%1, least squares minimization ofabsolute and relative errors leads to q-values of 0.1715 and 0.1691, respectively,so qZ1/6(z0.1667) is close to optimum for both. Figure 1 compares all threeformulae. For qZ1/6, the largest absolute error 3 in v is 0.0024 and occurs nearl0Z0.19; the largest relative error is 0.33% and occurs near l0Z0.3.

The use of qZ1/6 in (4.19) is not an analytical result but a goodapproximation that is convenient for its algebraic simplicity, and also fit for

purpose. The important thing is that (4.19), with qZ1/6, performs sufficientlywell over the whole range 0%l0%1 that we can trust it to reliably model themathematical behaviour of the exact function v(l0).

0.2 0.4 0.6 0.8 1.00

0.001

0.002

0.003

absoluteerror

q=1/6

q=0.1715

q=0.1691

l

(a)

0.2 0.4 0.6 0.8 1.0

0.4

0.2

0

0.2

0.4

relativeerror(%

)

l

q=1/6

q=0.1691

q=0.1715

(b)

Figure 1. (a) Absolute error 3 (approximate valueK

exact value) and (b) relative error (absoluteerror/exact value) for the function v(l0) defined by (4.19), for the three values shown for q.





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5. Explicit approximate expressions for the special elliptic functions

Using (4.19), earlier definitions yield

ul0ZK dvdl0z1KqCq ln l0; 5:1

sl0z1Kql0; 5:2tl0z1C l0=3 1K4qKq ln l0; 5:3

wl0z 316

l02 1C

ql0 ln l0

1Kl0 !

; 5:4

rl0; hztK2 eh1Kql0Kqh: 5:5With (5.4), (4.11) has been used to replace d2v/dl02. Numerics below take qas 1/6.

Figure 2 shows exact values for s, t, u, vand w, and absolute errors 3 when usingthese formulae. All are shown as functions of both l0 (0%l0%1) and x (h1/l0)(1%x%10). Note that urises steeply as l0 falls below approximately 0.2 and goes to

infinity as ln(1/l0), as l0/0. For s, t, vand w, maximum values of j3j are 0.0035,0.0042, 0.0024 and 0.00009, respectively; maximum magnitudes of relative errors(found separately) are 0.37, 0.39, 0.33 and 0.33%, respectively. For u, over the

(b)

l (= fhSN)

u

v

s t

0 0.2 0.4 0.6 0.8 1.00.006

0.004

0.002

0

0.002

0.004

0.006

absolu

teerror

10w

(d)

0.2 0.4 0.6 0.8 1.0

l (= fhSN)

0

0.5

1.0

1.5

2.0

s,t,u

,v,w

t

10w v

u

s

21 14 6 8 10

x(= 1/l =1/fhSN)

0

0.5

1.0

1.5

2.0

s,t,u,v,w

10w

u

v

s

t

0 2 4 6 8 10

x(= 1/l =1/fhSN)

0.006

0.004

0.002

0

0.002

0.004

0.006

absoluteerror v

10w

u

st

(a)

(c)

Figure 2. Exact values of s, t, u, v and w, and absolute errors (as defined for figure 1) associatedwith formulae (4.19) and (5.1)(5.4), taking qZ1/6: (a,b) plotted against l0; (c,d) plotted againstx (h1/l0).





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range 0.2%l0%1, the corresponding figures are 0.0043 and 0.45%; errors getprogressively worse as l0 falls below 0.2 and become serious below 0.1, but this isnot important because l0-values of 0.1 or lower are rarely of practical interest.

The established accuracy in predicting v(l0) is thus reflected in the accuracy inpredicting s(l0), t(l0) and w(l0) for 0%l0%1, and u(l0) for l0T0.12. In these ranges,the approximations can be used reliably in most algebraic manipulations (thoughuse in exponents needs caution).

6. Applications

We now put l0Zf and investigate the standard FN-type equation and associatedtheoretical FN plots. For simplicity, this section drops the suffix from vF, tF andnF. Scaled forms, with the exponent written as h/f, are used because they aremore general. In standard theory,

hZbf3=2

FSNfZ bc2fK1=2z9:836239 eV1=2fK1=2: 6:1

h varies slowly with f. The range 2.7!(f/eV)!6 is 6OhO4. The typical valuefZ4.5 eV is hZ4.64. From (3.3) and (4.19) (with l0 replaced with f), note that

expKvbf3=2

F

" #Z exp

Kvh

f

!zexp

K1KfCqf ln fhf

!

Z eh fKqh exp Khf

!zeh fKqh exp Kbf

3=2

F

" #: 6:2

(a) Predicted straight-line semi-logarithmic plot

From (2.10), (6.2) and fZF/Ff, the emission current Ist predicted by standard

theory is

Istz tK2AafK1 ehFqhfh iF2Kqh exp

Kbf3=2

F" #: 6:3The expanded form of tK2(f) is difficult to manipulate. Since tz1 and has weakfield dependence, it is left unexpanded. In the first bracket in (6.3), the onlyassumed field dependence is in t, and this may be ignored. So, in principle, anexact straight-line plot is given by ln{I/F(2Kqh)} versus 1/F, not ln{I/F2} versus1/F. This is a new prediction; typically, (2Kqh) is approximately 1.2.

However, real emitters often have field dependence in the emission area (e.g.Abbott & Henderson 1939), which opposes theKqh term. Field dependence in f,and hence h (e.g. Jensen 1999), might also exist. For real emitters, the success ofthe conventional experimental FN plot could be partly due to mutual

cancellation of opposing effects. Experiments to measure the true power of Fin FN-type equation pre-exponentials would be of considerable interest, but verydifficult. Change from using the traditional FN plot is not justified.





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(b ) Relationship between standard and elementary CFE theory

Related to (2.10), there is an elementary FN-type equation obtained byreplacing vF and tF with 1. In scaled form, the predicted emission current I

el is

Iel ZAafK1

F2 exp

Kh

f !; ln Iel

F2& 'Z lnfAafK1gK

h

f:

6:4

From (2.10), (6.4) and (6.2), since tK2z1,

Ist

Ielzeh

1

f

qh: 6:5

Typically, f is approximately 0.150.45, h approximately 4.6, q close to 1/6 andqh approximately 3/4, so (6.5) shows that typically the current density andcurrent predicted by standard theory are roughly 100 times greater than therelated elementary theory predictions. The ratio increases as f decreases.

Using t1 to denote t(fZ1), (2.10), (6.2) and (6.4) give

ln Ist

F2

& 'Z ln I

el

F2

& 'C hK2 ln t1C qh ln 1f

K2 ln t

t1

!: 6:6

To show behaviour on the semi-logarithmic FN plot, define

YZ lnI

F2

& '; Y0 Z lnfAafK1g and DYZYKY0: 6:7

The elementary FN equation (6.4) then takes either of the forms,

Yel ZY0Kh

f; DYel ZK

h

f: 6:8

The elementary FN plot intersects the y-axis at Y0; this value Y

0serves as the

reference zero for DY. Figure 3 has the y-axes labelled in this way.Figure 3aand (6.6) show, more clearly than earlier discussions, how a standard

FN plot (curve S) is derived from the related elementary plot (line E). First, lineL is drawn parallel to line E, above it by [hK2 ln t1]. Point B is then marked on Lat 1/fZ1. (For 1/f!1, the SchottkyNordheim barrier is below the Fermi level,so curve S starts from 1/fZ1.) As 1/f increases, S lies increasingly above L, by[qhln(1/f)K2 ln (t/t1)]; for large 1/f, the slopes tend to become equal.

If we ignore small terms in t, the shift from line E to line L relates to the eh termin (6.5) and the shift from line L to curve S relates to the (1/f)qh term. Figure 3ashows that the eh term has the larger effect, for f-values of practical interest.

Experimental data points lie in a small range of f-values (typically in part of0.15!f!0.45), and often seem to lie on a straight line (the experimental FNplot). As already noted, the most convenient theoretical model (for a line fittedby linear regression to the data points) is the tangent to S, taken at a value of1/f among the data points. The intercept of this tangent with the ln{I/F2} axisis the theoretical prediction of the regression line intercept. In standard CFEtheory, the slope and intercept of the tangent relate to those of line E via themathematical correction functions s and r. A tangent taken at 1/fZ5 is shownas line T5.

The shape of S makes sand r vary with 1/f. At 1/fZ1, (5.5) shows that rhas

the value tK21 e

h1Kq

, typically approximately 40. As 1/f increases and thetangent point P moves to the right along S, three effects occur: P movesdownwards; P becomes increasingly distant from line E; and the magnitude of





17/23

the slope of S increases. The result is to increase r. Equation (5.5) shows, perhapscounterintuitively, that this increase is due, almost exclusively, to the increasing

difference between S and L, effectively by the factor (1/f)qh

. For example, for1/fZ5, (1/f)qh is typically approximately 3.3 and r is typically approximately140. As is well known, if one attempts to extract emission area from anexperimental FN plot by putting the Y-intercept equal to ln{AafK1} rather thanln{rAafK1}, overestimation by a factor r occurs.

This analysis provides a clearer picture of how the FN plot works. It is also areminder of the disadvantages of using elementary rather than standard theory:predicted currents will be too low and extracted areas too high, typically byfactors of 100 or more.

(c) The underlying mathematics of the standard FN plotThe underlying mathematics of the standard FN plot deserves comment. In

(6.6), ignore the terms in t, write xh1/f, and use (6.7) to give

DYstxZKvxhx: 6:9Figure 2cshows clearly that v(x) varies quite sharply with x; but figure 3ashowsthat line S, i.e. DYst(x), is almost straight. So a vital mathematical question is:why does (6.9) generate a straight line? At first sight, this behaviour is counter-intuitive. Its mathematical origin is as follows.

From (4.19), with xh1/l0, ln xhKln l0,

vxz1KxK1KxK1q ln x; 6:10DYst ZhCqh ln xKhx: 6:11

0 1 2 3 4 5 6

x = 1/f

1.0 1.1 1.2 1.3 1.4 1.5Y

0

(a)

E

L

B

P

T5T5

SB

S

L

ln{r}

(b)

P

T

V

B

slope=vPh

slope=

sPh

ln{tP2}

ln{rP}

xP = 1/fP

x= 1/f

Y

0

0

S

(Y)P

Figure 3. Scaled FN plots: the horizontal axis shows x (h1/f), and the vertical axis DYZln{I/F2}Kln{AafK1}. (a) To show how a curve S and tangent T derived from the standardFN-type equation relate to the line E given by the related elementary FN-type equation (see text).The plot is drawn to scale, for fZ4.5 eV. The insert shows values near point B at larger scale.(b) Schematic showing the relationship of r, s, t and v for a theoretical FN plot. This is based on(a), but curvature of line S between B and P is exaggerated. vP, rP sP and tP are values taken atpoint P, for the specific value xP (Z1/fP), but the argument is true for any point P.





18/23

Equation (6.11) is another equation for S. Clearly, the reason for its near-linearity is the form of (6.10). The terms in xK1 generate a constant term (h) andthe slowly varying term qh ln x; the 1 simply generates the linear term Khx.

(d) The implications of FN plot linearity

This argument also operates in reverse. A legitimate question is: what doesobserved linearity in a FN plot mean? The argument above implies that, if anexperimental FN plot of type [ln{I/E2} versus 1/E], where E is any of f, F,voltage V or macroscopic field FM, is effectively linear, then, to predict this, thetunnelling exponent correction factor n must have the form n(E) ZBCCE (orreduce to it in the relevant range of E), where B and C are constants or slowlyvarying with E.

Thus, FN plot curvature relates to v2n/vE2. Edgcombe & de Jonge (2006)reach an equivalent conclusion. Observed linearity implies small v2n/vE2. Forstandard theory, this is confirmed qualitatively by figure 2a, which shows that

v(l0) is nearly linear, and quantitatively by the small values of the curvaturefunction w. At l0Z1/3 (equivalent to Jz109 A mK2 for a fZ4.5 eV emitter),w(1/3)z0.02.

It would be no theoretical surprise if more realistic barriers had generallysimilar behaviour, so it is no surprise that FN plots have often been nearly linear.Even for emitters with tip radius of order 1 nm, FN plot curvature can berelatively small, as the work of Edgcombe & de Jonge (2006) brings out. So,where marked curvature occurs in experimental plots, usually some other effectmust be operating, such as the presence of vacuum space charge, electron supplylimitation inside the emitter, or statistical effects associated with a many-

emission-sites electron source.

7. Discussion

This work was stimulated by finding a new approximation for v, and thenrealizing that y2 was better as the independent variable and consequentlyderivations needed reformulating. It seemed best to separate the mathematicaland physical descriptions inherent in standard CFE theory, and generalize thephysical description to apply to all well-behaved barriers. This paper has laidsome foundations and has given a fuller account of SchottkyNordheim

barrier mathematics. By providing approximation formulae with absolute errorj3j!8!10K10, we hope to make v(l0) almost as readily accessible as cos(q).In our view, JWKB-type approaches to linking CFE current density to barrier

field will take the following form in future. There will be general physicalquantities and equations that relate to a general physical form of FN-typeequation and to FN plots. These quantities will include an independent variable(the scaled barrier field of (2.12)), correction factors for the exponent and pre-exponential of the curve equation, and factors that relate to the intercept, slopeand curvature of the FN plot. The exponent correction factor n will be defined by(2.2) and (2.5), and the others by more general, physical, versions of the 3

equations. dn/df will be needed as part of this.In standard CFE theory, with the SN barrier, the dependence of these physicalquantities on the real value of scaled barrier field (2.12) is modelled via the





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specific mathematical functions discussed earlier. We suggest that, in future,they could be known collectively as the SchottkyNordheim barrier functions.Other barriers can be defined by different expressions for M(z), particularly thoseassociated with sharply curved emitters. For each barrier B (different for eachmodel of emitter shape, etc.), new quantities Ff

B and ffB would be determined,

and the dependent physical quantities modelled by new mathematical functionsor (equivalently) value sets generated numerically. ff

B will again lie in the range0%ff

B%1, so the treatment of the SN barrier becomes a paradigm for the

treatment of more realistic barriers.There remain, even for standard theory, awkward issues over how to relate the

physical quantities discussed to the actual behaviour of experimental FN plotsbased on measurements of current versus voltage. These include: how tocalibrate barrier field precisely; how best to define the notional emission area A;the effects of field dependence in f, tF

K2 and A; nonlinearity in the dependence ofbarrier field on applied voltage; and how to establish the f-value you are

operating at. There are also the purely statistical difficulties of fitting to noisyexperimental data and the problems of poorly characterized experiments. Instandard theory, formula (2.17) should help the investigation of some of these. Asin Forbes (1999a), the problem is how to extract results and error limits underconditions of physical uncertainty.

Overall, this paper provides renewal of standard CFE theory, and under-pinning for future developments. We hope many will find this theoreticalapproach more complete, more fruitful and easier than some older literature. Westrongly urge that clearer distinctions be made between mathematical entitiesand physical quantities, and that the special field emission elliptic functions betreated as functions of l0(Zy2), rather than y, and be thought of as the SNbarrier functions. We also commend the scaled form of FN plot, which exhibitsCFE theory in a more universal form.

We wish to acknowledge an initial stimulus provided by Dr C. J. Edgcombes work on the theory ofCFE from curved emitters (e.g. Edgcombe 2005), in particular his use of dimensionless variables.We also thank the referees for their constructive comments.

Appendix A. Series expansion for v(l0)

The function v(l0) has an exact series expansion,

vl0Z 1CXNiZ0

AaiCCnbiC1CCnai ln l0l0iC1; A 1

where AZ(9/8) ln 2, CZ1, nZ3/16, a0Z1, and b1ZK1, and the recurrencerelationships below define the remaining coefficients,

aiC1 ZiiC1CniC1iC2 ai; iR0; A 2

biC1 Z2iK1aiK1K2iC1aiCfiK1iCngbi

iiC1

; iR1: A 3

This generates expansion (4.14). Deane et al. (2007) present a detailed formalderivation of these recurrence relationships.





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Form (4.14) can also be derived from the special expansions for K and Ederived by Cayley (1876, see pp. 5255) and cited by Dwight (1965, formulae(773.3) and (774.3)). Cayley uses a parameter k0 given in terms of our m byk0hO1Km. Here, it is simpler to use two new parameters: m0h1Kmhk02 andLhln(4/k0)[Zln 4C(1/2) ln (1/m0)]. The Cayley expansions then take the form

KZLC12

22LK

2

1,2

m0C

12,32

22,42LK

2

1,2K

2

3,4

m02C/ A 4

EZ 1C1

2LK

1

1,2

m0C

12,3

22,4LK

2

1,2K1

3,4

m02C/: A 5

On substituting for L and rearranging,

KZ ln 4C12

22 ln 4K

2

1,2 m0C1

2,32

22,4

2 ln 4K2

1,2

K2

3,4 m02C/

C1

2ln 1=m0 1C1

2

22m0C

12,32

22,42m0C/

!A 6

EZ1C1

2ln 4K

1

1,2

m0C

12,3

22,4ln 4K

2

1,2K

2

3,4

m02C/

C1

2ln1=m0 1

2m0C

12,3

22,4m02C/ !: A 7

Although the coefficients are necessarily different, (A 6) and (A 7) have the samegeneral forms as the Hastings (1955) approximation formulae for K and E citedby AS as formulae (17.3.34) and (17.3.36) (their m1 is our m

0).From (4.6),

m0Z 1KmZ2l01=2

1C l01=2Z l01=21Kl01=2C l0Kl03=2C/ A 8

ln1

m0 ZKln 2K 1

2ln l0C l01=2K

l0

2C

l03=2

3K/:

A 9

Substituting (A 8) and (A 9) into (A 6) and (A 7), and then these into (4.8),eventually yields (4.14). Although low-order terms can be checked manually,algebraic manipulations become lengthy for higher order terms, and computeralgebraic manipulation can be used with advantage. Although half-integralpowers of l0 appear in the derivation, powers of l0 in the result are integral.

References

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Abramowitz, M. & Stegun, I. A. 1965 Handbook of mathematical functions. New York, NY: Dover.



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[Transl. Teplofizika Vysokikh Temperatur. 9, 184186.]Burgess, R. F., Kroemer, H. & Houston, J. M. 1953 Corrected values of FowlerNordheim field

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Proc. R. Soc. A 463, 29072927 (21 August 2007) (doi:10.1098/rspa.2007.0030)

Reformulation of the standard theory of

FowlerNordheim tunnelling and cold fieldelectron emission

BY RICHARD G. FORBES AND JONATHAN H. B. DEANE

The following equations contain typographical errors that have no consequencefor any other equations or results in the above paper.

dm=dl0ZKl0K1=21C l01=2K2 and 4:12

l01Kl0d2v=dl02K 316vZ0: 4:13

The first column of table 1 was incorrect, and should read as follows.

The first sentence of appendix A was incorrect, and should read as follows.

Appendix A. Series expansion for v(l0)

The function v(l0) has an exact series expansion,

vl0Z 1CXN

iZ0

AaiCCnbiC1CCnai ln l0l0iC1; A 1

where AZK(9/8)ln 2, CZ1, nZ3/16, a0Z1, and b1ZK1, and the recurrencerelationships below define the remaining coefficients,

aiC1ZiiC1Cn

iC1iC2ai; iR0; A 2

biC1Z 2iK1aiK1K2iC1aiCfiK1iCngbiiiC1

; iR1: A 3

Table 1. Coefficients for use in connection with (4.17) and (4.18), for degree-4 formulae. (Note:u1z0.8330405509.)

i pi qi si ti

0 1 0.053 249 972 7 0.187 5

1 0.032 705 304 46 0.187 499 344 1 0.024 222 259 59 0.035 155 558 742 0.009 157 798 739 0.017 506 369 47 0.015 122 059 58 0.019 127 526 803 0.002 644 272 807 0.005 527 069 444 0.007 550 739 834 0.011 522 840 094 0.000 089 871 738 11 0.001 023 904 180 0.000 639 172 865 9 0.003 624 569 4275 K0.000 048 819 745 89

Errata3378


hot electron tunneling

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