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Proc. R. Soc. A (2007) 463, 2907–2927
doi:10.1098/rspa.2007.0030
Published online 21 August 2007
Reformulation of the standard theory of
Fowler–Nordheim tunnelling and cold field
electron emission
B Y R ICHARD G. F ORBES 1, *
AND
J ONATHAN H. B. D EANE 2
1
Advanced 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 Fowler–Nordheim
(FN) tunnelling and cold field electron emission (CFE). Mathematical analysis and
physical interpretation become easier if the principal field emission elliptic function v is
expressed as a function v(l 0 ) of the mathematical variable l 0 hy2, where y is the
Nordheim parameter. For the Schottky–Nordheim (SN) barrier used in standard CFE
theory, l 0 is equal to the ‘scaled barrier field’ f, which is the ratio of the electric field that
defines a tunnelling barrier to the critical field needed to reduce barrier height to zero.
The tunnelling exponent correction factor nZv( f ). This paper separates mathematical
and physical descriptions of standard CFE theory, reformulates derivations to be in
terms of l 0 and f, rather than y, and gives a fuller account of SN barrier mathematics.
v(l 0 ) is found to satisfy the ordinary differential equation l 0 (1Kl 0 )d2v/dl 0 2Z(3/16)v; an
exact series solution, defined by recurrence formulae, is reported. Numerical
approximation formulae, with absolute error j3j!8!10K10, are given for v and dv/dl 0 .
The previously reported formula vz1Kl 0 C(1/6)l 0 ln l 0 is a good low-order approximation, with j3j!0.0025. With l 0 Zf, this has been used to create good approximate
formulae 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 the
question: ‘what does linearity of an experimental FN plot mean?’ FN plot curvature is
predicted by a new function w. The new formulation is designed so that it can easily be
generalized; thus, our treatment of the SN barrier is a paradigm for other barrier shapes.
We urge widespread consideration of this approach.
Keywords: field emission; Fowler–Nordheim tunnelling;
field emission elliptic functions
1. General introduction
Nearly 80 years ago, Fowler and Nordheim (FN) published in these Proceedings
their seminal paper (Fowler & Nordheim 1928) on cold field electron emission
(CFE) from metal surfaces. Their paper has shaped much subsequent work. The
name ‘Fowler–Nordheim tunnelling’ is now used for any field-induced electron
* Author for correspondence ([email protected]).
Received 13 May 2007
Accepted 24 July 2007
2907
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R. G. Forbes and J. H. B. Deane
tunnelling through a roughly triangular (in practice, always rounded) barrier.
The main modern contexts are: (i) vacuum breakdown in high-voltage apparatus
of all kinds, where one needs to prevent electron emission from asperities, (ii)
cold-cathode electron sources—their many applications include bright point
sources (for high-resolution electron microscopes and other machines), X-ray
generators, electronic displays and space vehicle neutralizers, and (iii) internal
electron transfer in some electronic devices.
The original 1928 equation used an unrealistic barrier model, which seriously
underpredicts CFE current densities. Various modified equations have been
introduced, which we call ‘FN-type equations’. Data analysis has often used the
so-called ‘standard FN-type equation’ (2.10), derived from Murphy & Good’s
(1956) work (MG). However, the MG paper is not easy to follow, confusion
abounds in the literature, and this ‘standard CFE theory’ has gained a
reputation for being obscure and difficult. Perhaps as a result, some recent
experimental research papers use simplified FN-type equations, as found in
undergraduate textbooks. These also significantly underpredict CFE current
densities, 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 y
defined 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 ‘scaled
barrier field’ (equal to y2 in standard theory), and then write v as a function of
barrier field F. This gave, for the first time, a simple, reliable algebraic
approximation for the exponent in the standard FN-type equation. This makes
the equation’s behaviour easier to investigate, but background theory needs
presenting differently.
Specific aims here are to show the approximation’s mathematical origin, and
use it to gain fuller understanding of FN plots. But, more important, we present
a major reformulation of standard CFE theory itself. This seems timely, for two
main reasons.
First, FN plots are the commonest tool used to analyse experimental CFE
data. Elementary CFE theory, sometimes employed, can explain slopes. But, as
§6 shows, standard CFE theory is probably the simplest theory able to clarify
their detailed behaviour. Users of FN plots need to be able to understand
standard 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 limited
applicability to atomically sharp emitters. Reformulation makes it easier to
generalize standard theory to treat more realistic tunnelling barriers.
In standard theory, clearer conceptual distinctions are needed between purely
mathematical aspects and physical aspects. To replace the use of y, we introduce
two related theoretical descriptions, one mathematical and the other physical,
each (in principle) with its own names, symbols and definitions. The former
involves mathematics that applies primarily to the Schottky–Nordheim (SN)
tunnelling barrier (Schottky 1914; Nordheim 1928); the latter involves physical
definitions 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.
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Reformulation of standard CFE theory
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Thus, the SN barrier mathematical description uses a variable l 0 defined in the
context of elliptic function theory (by (4.7)); but the physical description uses
the scaled barrier field fh defined using physical fields (by (2.12)). The SN barrier
analysis has l 0 Zfh, but other barriers normally do not.
Similarly, the mathematical description uses (for example) the ‘principal field
emission elliptic function’ v(l 0 ) given by (4.14), but the physical description uses
the ‘tunnelling exponent correction factor’ n(fh) defined by (2.5). For the SN
barrier, but normally not for other barriers, nZv(fh).
All this is normal scientific practice when using mathematical functions, with
v(l 0 ) behaving like cos(q). But standard CFE theory has been different. The same
symbol has been used for both the physical variable and the related
mathematical function, and the name of the symbol representing a function
(e.g. ‘vee’) has been used as the name of the function. This unfamiliar convention
and resulting lack of clarity seem to have impeded both wider understanding and
theoretical development.
In writing this paper, a particular problem has been to decide whether the
letters r, s, t, u, v and w should represent mathematical functions (i.e. be part of
the mathematical description, and normally applicable only to the SN barrier),
or should represent general physical quantities applicable to many different
barriers. 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 treat
all as mathematical functions. This makes notation more uniform but means
that, especially in §2, we need separate symbols (Greek letters are used) for the
quantities 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 in
future work.
In summary, this paper aims to reduce confusion, make standard CFE theory
more complete and more accessible, and permanently change the way that basic
CFE theory is discussed. Analysis is from basic principles, but relies on results in
Forbes (1999b) and in Forbes (2004).
The paper’s structure is as follows: §2 presents additional background; §3
derives revised definitions for the main functions used in standard CFE theory;
§4 discusses mathematical expressions for v(l 0 ); §5 establishes simple algebraic
approximations for the other functions; §6 presents new results relating to FN
plots; §7 discusses implications; and appendix A presents the exact series
expansion for v(l 0 ).
2. Theoretical background
The following notations are used: let e denote the elementary positive charge; me
the electron mass in free space; hP Planck’s constant; 30 the electric constant; and
f the local work function of the emitting surface. The field at the emitter surface
is denoted by F and called the ‘barrier field’, and the emission current density is
denoted by J; in CFE theory, these positive quantities are the negative of the
like-named quantities used in conventional electrostatics. This field F determines
the tunnelling barrier. ‘Unreduced barrier height’ h is the height of a tunnelling
barrier when FZ0. Universal constants are evaluated using the 2002 values of
the fundamental constants and given to seven significant figures.
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R. G. Forbes and J. H. B. Deane
CFE theory involves, first, calculation of the escape probability D for an
electron approaching the emitter surface in a given internal electronic state, and
then summation over all occupied states to give J. Many different levels of
theoretical approximation exist. The basic ‘free-electron’ CFE treatments
discussed here: (i) ignore atomic structure and assume a Sommerfeld freeelectron model, (ii) assume that the electron distribution is in thermodynamic
equilibrium and obeys Fermi–Dirac statistics, (iii) take temperature as zero, and
(iv) assume a flat planar emitter surface, of constant uniform local work function
f (i.e. df/dFZ0), with a uniform electric field F outside.
Choices are then needed about modelling the tunnelling barrier and about
theoretical method. The Schrödinger equation can be solved exactly for FN’s
original triangular barrier, but has no ordinary analytical solutions for most
model barriers of physical interest, including the Schottky–Nordheim barrier
used in standard CFE theory. So, normally, some approximate method must be
used. Usually this is a JWKB-type approximation (Jeffreys 1925, also see
Fröman & Fröman 1965) or the closely related Miller & Good (1953)
approximation. There are several different JWKB-type formulae, applicable to
barriers of different kinds.
For ‘strong’ barriers, the escape probability D can be written (Landau &
Lifschitz 1958) as
D Z P exp½KG;
where G is the so-called ‘JWKB exponent’ defined by
ð
G h ge M 1=2 dz;
ð2:1Þ
ð2:2Þ
and P is a ‘tunnelling pre-factor’. Here, ge [h4p(2me)1/2/hPz10.24624 eVK1/2nmK1]
is the JWKB constant for an electron, and z is the distance measured from the
emitter’s 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 that
exp[KG]/1.
The pre-factor P is included in (2.1) for conceptual completeness. Normally, it
is tacitly assumed that P differs from unity by a presumed unimportant
multiplying factor (between 1/5 and 5, say), and is slowly varying with barrier
height h in comparison with exp[KG]. So normal practice sets PZ1 and uses the
so-called ‘simple JWKB approximation’,
D zexp½KG:
ð2:3Þ
The Miller–Good approximation also reduces to (2.3) when G is large.
For the elementary triangular barrier of height h and slope KeF, MZhKeFz;
(2.2) then yields the quantity G el given by
G el Z
bh 3=2
;
F
ð2:4Þ
where the ‘second Fowler–Nordheim constant’ bh2ge/3eh(8p/3)(2me)1/2/
ehPz6.830890 eVK3/2 V nmK1. For other barriers, a tunnelling exponent
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Reformulation of standard CFE theory
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correction factor n is defined generally by
ð2:5Þ
G h nG el ;
and a ‘decay-rate correction factor’ t is defined generally by
el vG
vG
ht
;
ð2:6Þ
vh F
vh F
where partial derivatives are taken at constant barrier field F. It is easily shown
that
2
vn
t ZnC
:
ð2:7Þ
h
3
vh F
For the elementary triangular barrier, the correction factors n and t are
both unity.
Choice of method also occurs when summing tunnelling current contributions
from the internal electron states. Forbes (2004) summed over states on a
spherical constant-total-energy surface, and then integrated with respect to total
electron energy. This approach resembles that used for non-free-electron band
structures (e.g. Gadzuk & Plummer 1973; Modinos 1984). It can be applied to
any well-behaved barrier model, and leads to the current-density equation
h
i
J Z tFK2 af K1 F 2 exp KnF bf3=2 =F ;
ð2:8Þ
where the ‘first Fowler–Nordheim constant’ ahe3/8phPz1.541434!10K6
A eV VK2, and nF and tF are the values of n and t that apply to a barrier of
unreduced height h equal to the local work function f. For clarity, later,
equations of this kind are called ‘curve equations’. The emission current I is then
given by IZAJ, 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 barrier
encountered by a Fermi-level electron that is moving ‘forwards’ (i.e. towards
and normal to the emitting surface). nF and tF enter the theory because deriving
(2.8) involves Taylor expansion of the JWKB exponent G (ZnG el) about the
Fermi level.
The symbols n, nF, t and tF represent general correction factors, and appear in
equations that apply to any well-behaved barrier model. Detailed analyses
require a specific barrier model, and the general quantities in (2.5)–(2.8) must
then be replaced by correction factors specific to this model. Mathematical
evaluations (by computer if necessary) must then be performed for these specific
correction factors. Since all factors specific to a barrier model can be derived from
the ‘n-like’ factor in the specific version of (2.5), detailed analysis concentrates on
this factor.
The Schottky–Nordheim (SN) barrier is defined (for z greater than some
minimum value zmin) by
M SN ðzÞ Z hKeFz Ke2 =16p30 z:
ð2:9Þ
SN
Putting (2.9) in (2.2) leads to a specific correction factor n . Burgess et al.
(1953) showed that n SN is given by a mathematical function v that 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 the
1
A function specified by Nordheim (1928) is not a correct mathematical expression for nSN, owing
to a mistake in defining the argument of a complete elliptic integral.
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R. G. Forbes and J. H. B. Deane
principal field emission elliptic function (or, better, the ‘principal SN barrier
function’). 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 (for
the emission current I st predicted by standard theory),
h
i
I st Z At FK2 af K1 F 2 exp KvF bf3=2 =F ;
ð2:10Þ
where vF and tF are the values of v and t that apply to the barrier encountered by
a forward-moving Fermi-level electron (when this barrier is modelled as a SN
barrier).
The mathematical functions v and t, with s, u and w below, are known
collectively as the ‘special field emission elliptic functions’.2 They depend on a
single 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 past
definitions using y).
These special elliptic functions can be evaluated accurately by various
methods, and (except for w) the results are tabulated (Burgess et al. 1953;
Good & Müller 1956; Miller 1966; Forbes & Jensen 2001). v can also be expressed
in terms of the complete elliptic integrals K and E (see MG and Forbes (1999b),3
for definitions using y). However, a need exists for simple, reliable, algebraic
approximations for these functions, especially v. Jensen & Ganguly (1993) and
Jensen (2001) have derived formulae for v(y), but these are complex and
awkward to use in further analysis. Fitting procedures have generated simple
empirical formulae, for example the Spindt et al. (1976) approximation, but these
do 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 to
put l 0 hy2, call l 0 a ‘complementary elliptic variable’4 and write
vðl 0 Þ z1Kl 0 C ð16 Þl 0 ln l 0 :
ð2:11Þ
The discovery that v(l 0 ) satisfies a simple-looking differential equation in l 0 ,
(4.13), supports using l 0 as the independent variable. If y is used, the resulting
differential equation is more complex.
For the real physical situation, we can define a scaled barrier field fh by
fh Z
F
;
Fh
ð2:12Þ
where Fh is the real field that reduces barrier height from h to zero. For the
Schottky–Nordheim (SN) barrier, Schottky (1914) showed that a field F lowers the
barrier by an energy DSZcF1/2, where cZ(e3/4p30)1/2z1.199985 eV VK1/2 nm1/2.
2
But a better collective name might be the ‘Schottky–Nordheim barrier functions’.
Typographic errors occur in Forbes (1999b): (i) in the definition of K(m), both brackets should
be 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. l 0 /0 as the elliptic parameter
m/1.
3
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Reformulation of standard CFE theory
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So, for the SN barrier (but not for other barriers), we estimate Fh and fh by
FhSN Z c K2 h2 zð0:6944617 eV K2 V nm K1 Þh2 ;
fhSN
3 F
c2 F
e
Z SN Z 2 Z
Fh K2 :
4p30
h
Fh
ð2:13Þ
ð2:14Þ
The Nordheim parameter y (Nordheim 1928) is defined as
yh
DS
cF 1=2
Z
Z OfhSN :
h
h
ð2:15Þ
In the JWKB integral for the SN barrier, the term (e3/4p30)FhK2 has
previously been replaced with y2. Here, it is replaced with l 0 (see §4). Equation
(2.14) shows that the mathematical variable l 0 can be identified with the SN
barrier parameter fhSN , and so has a physical interpretation. Probably, the
concept of scaled barrier field will prove more readily understandable than
the Nordheim parameter has been. These are further advantages of using l 0 as the
independent mathematical variable.
When h is the local work function f, then (2.13) yields the ‘critical SN barrier
field’ (FfSN ) at which the SN barrier for a forward-moving Fermi-level electron
vanishes (Schottky 1923). The corresponding scaled SN barrier field ffSN is
!
F
:
ð2:16Þ
ffSN Z
FfSN
So we reach the Forbes (2006) result (but in more careful notation),
! !
!
F
1
F
F
vF z1K
C
ln
:
6
FfSN
FfSN
FfSN
ð2:17Þ
All these things make it desirable to reformulate standard CFE theory to be in
terms of the mathematical variable l 0 and its physical partner the scaled barrier
field. For simplicity in this paper, we now mostly drop the label ‘SN’ and just use
f: it is always clear from context whether fhSN or ffSN is meant. Since in standard
theory l 0 Zf, there is some choice as to which is used in formulae. We use l 0 in
strictly mathematical contexts, f when the formula relates more to experiment.
3. Parameters for Fowler–Nordheim analysis
This section creates f-based definitions for the mathematical functions used in
standard CFE theory. They could equally well be presented using l 0 , but using f
will make them easier to generalize in future work. Apart from (3.1b), the
definitions here are specific mathematical versions of general physical
relationships valid for any well-behaved barrier model.
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R. G. Forbes and J. H. B. Deane
In standard theory, n is given by v, t by t. So from (2.7), and then the SN
barrier formula (2.14),
2
vv
2
dv
vf
tðf Þ Z v C h
ZvC h
;
ð3:1aÞ
3
vh F
3
df
vh F
4 dv
:
ð3:1bÞ
tðf Þ Z vK
f
3 df
The functions r, s, u and w relate to Fowler–Nordheim plots. We first convert
tunnelling exponents to be in terms of 1/f. Define a dimensionless parameter h by
hZ
bf3=2
;
Ff
ð3:2Þ
where Ff is the critical field at which a tunnelling barrier of unreduced height f
vanishes. This leads to the general result
! Ff
bf3=2
bf3=2
h
ð3:3Þ
Z
Z :
f
F
Ff
F
So, in standard theory, from (2.10),
lnfI st =F 2 g Z lnft FK2 Aaf K1 gK vF h=f :
ð3:4Þ
Equation (3.4) is said to be ‘in FN coordinates of type [ln{I/F 2} 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 used
normally. In SI units, (I/F 2) and (AafK1) would be in A VK2 m2. Since both F
and f appear, (3.4) is a ‘mixed’ FN plot form; but this form is useful for discussing
basic theory.
Equation (3.4) is a ‘curve equation, expressed in FN coordinates’. If any field
dependence in f, t FK2 or A is ignored, then its slope with respect to 1/f at any
point may be written as Ksh, where s is the standard slope correction function5
introduced by Houston (1952). Letting xh1/f, we have
sðf Þ Z d½vF x=dx Z vF C x dvF =dx Z vF C ð1=f ÞðdvF =df Þ=ðdx=df Þ
Z vF Kf dvF =df :
ð3:5Þ
Forbes (1999a) argued that the most convenient theoretical model for an
experimental FN plot is the tangent to the chosen FN-type curve equation, when
this curve equation is expressed in FN coordinates. In standard theory, again
ignoring any field dependence in f, t FK2 or A, a suitable form for this ‘FN-type
tangent equation’ is
st I
sh
ln
Z lnfrAaf K1 gK ;
ð3:6Þ
2
f
F
5
But note his calculations of s are in error, owing to the mistake in Nordheim (1928); Burgess et al.
(1953) gave corrected results.
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Reformulation of standard CFE theory
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where ln{rAafK1} is the intercept the tangent makes with the ln{I/F 2} axis.
The ‘standard intercept correction function’ denoted here by r is the function rN
introduced by Forbes (1999a). Both r and s vary along the curve.
Figure 3b shows plots of the quantity DY st [Zln{I st/F 2}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=t FK2 Z ðvF KsÞh=f ZKh dvF =df :
ð3:7Þ
In terms of f, the function u in Forbes (1999b) is u( f )hKdvF/df. Hence,
sðf Þ Z vF Kf dvF =df Z vF C uf ;
ð3:8Þ
rðf ; hÞ Z t FK2 exp½Kh dvF =df Z t FK2 exp½hu:
ð3:9Þ
Finally, a new function w is introduced to describe the curvature of a FN plot
made against 1/f,
wðf Þ Z ds=dð1=f Þ ZKf 2 ds=df Z f 3 d2 vF =df 2 :
ð3:10Þ
For example, wZ0.02 means the FN plot slope changes by 2% when 1/f changes
by 1.
Clearly, all these functions can be obtained from vF( f ) and its first two
derivatives.
4. Expressions for the mathematical function v(l 0 )
(a ) ODE in the complementary elliptic variable l 0
We now write v as a function of the purely mathematical complementary elliptic
variable l 0 , and derive the ordinary differential equation (ODE) that v(l 0 )
satisfies. Following MG, Forbes (1999b) showed how to express v(y) in terms of
the complete elliptic integrals K and E. His equation numbers are here prefixed
F. We can replace (F12) by defining
l 0 h y2 Z ðe3 =4p30 ÞF=h 2 :
ð4:1Þ
Comparison with (2.14) shows l 0 ZfhSN . Hence, for the SN barrier (but not for
other barriers), l 0 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 Ol 0 . Noting bZ(8p/3)(2me)1/2/ehP, we can use (F17a) to provide
definitions convenient for numerical integration,
pffiffiffiffiffiffiffi
pffiffiffi ð 1C 1Kl 0
0
vðl Þ h ½3=ð4 2Þ pffiffiffiffiffiffiffi ðKx2 C 2xKl 0 Þ1=2 x K1=2 dx;
ð4:2Þ
0
1K 1Kl
pffiffiffiffiffiffiffi
pffiffiffi ð 1C 1Kl 0
dv=dl ZK½3=ð8 2Þ pffiffiffiffiffiffiffi ð–x2 C 2xKl 0 Þ K1=2 x K1=2 dx:
0
1K 1Kl 0
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In MG and Forbes (1999b), the transformation applied to (F24b) yields the
following results. K and E are defined in terms of the elliptic parameter m as used
by Abramowitz & Stegun (1965 (AS)),
ð1
KðmÞ h ð1Kz 2 Þ K1=2 ð1Kmz 2 ÞK1=2 dz;
ð4:4Þ
0
ð1
EðmÞ h
ð1Kz 2 Þ K1=2 ð1Kmz 2 ÞC1=2 dz:
ð4:5Þ
0
If we put, as in (F26b),
m Z ð1Kl 01=2 Þ=ð1 C l 01=2 Þ;
then, noting (F26a) and (F29),
l 0 Z ½ð1KmÞ=ð1 C mÞ2 ;
ð4:6Þ
ð4:7Þ
vðl 0 Þ Z ð1 C l 01=2 Þ1=2 ½EðmÞKl 01=2 KðmÞ;
ð4:8Þ
dv=dl 0 ZKð34 Þð1 C l 01=2 ÞK1=2 KðmÞ
ð4:9Þ
and
2
d2 v=dl 0 ZKð163 Þð1 C l 01=2 Þ K1=2 ½4 dKðmÞ=dl 0 Kl 0K1=2 ð1 C l 01=2 Þ K1=2 KðmÞ: ð4:10Þ
Equation (4.7) establishes the link between l 0 and elliptic function theory. From
(F27), (4.6), and then (4.10),
dKðmÞ=dm Z fEðmÞKð1KmÞKðmÞg=mð1 KmÞ;
ð4:11Þ
dv=dl 0 ZKl 0K1=2 ð1 C l 01=2 ÞK2
ð4:12Þ
and
l 0 ð1Kl 0 Þd2 v=dl 02 Kð34 Þv Z 0:
0
ð4:13Þ
0 1/2
appear. This
Thus, v(l ) obeys the ODE (4.13). Note that no factors in l
ODE appears to be new both in elliptic function theory and in mathematical
physics. It is simpler in form than the ODEs associated with K and E (Cayley
1876).
(b ) Exact series expansion
Two independent solutions for ODE (4.13) have been found using the method
of Frobenius to develop series expansions. The boundary conditions on v and
dv/dl 0 , as l 0 /0, then determine an exact series expansion for v(l 0 ). The first few
terms are
27
51
315
177
vðl 0 Þ Z 1Kf98 ln 2 C 163 gl 0 Kf256
ln 2K1024
gl 02 Kf8192
ln 2K8192
gl 03 C/
9
l 0 C 16105
l 02 C/l 0 ln l 0 :
C ½163 C 512
384
ð4:14Þ
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 presented
separately elsewhere (Deane et al. 2007). It shows that the ln l 0 terms are an
Proc. R. Soc. A (2007)
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Reformulation of standard CFE theory
Table 1. Coefficients for use in connection with (4.17) and (4.18), for degree-4 formulae. (Note:
u 1z0.8330405509.)
I
pi
qi
si
0
1
2
3
4
5
1
0.032 705 304 46
0.009 157 798 739
0.002 644 272 807
0.000 089 871 738 11
—
—
0.187 499 344 1
0.017 506 369 47
0.005 527 069 444
0.001 023 904 180
—
ti
0.053 249 972 7
0.024 222 259 59
0.015 122 059 58
0.007 550 739 834
0.000 639 172 865 9
K0.000 048 819 745 89
0.187 5
0.035 155 558 74
0.019 127 526 80
0.011 522 840 09
0.003 624 569 427
—
intrinsic part of the expansion, and that terms involving non-integral powers of l 0
are not needed. Appendix A contains a shorter alternative derivation; this
directly confirms the lower order terms but does not bring out the underlying
mathematics.
Evaluating coefficients to five decimal places, we obtain
vðl 0 Þ z1K0:96729l 0 K0:02330l 02 K0:0050l 03 K/C ln l 0 ð0:18750l 0
C 0:01758l 02 C 0:00641l 03 C/Þ
ð4:15Þ
zð1Kl 0 Þð1 C 0:03271l 0 C 0:00941l 02 C/Þ C l 0 ln l 0 ð0:18750
C 0:01758l 0 C 0:00641l 02 C/Þ:
ð4:16Þ
Form (4.16) is explicitly exact at l 0 Z0 and 1 and has good convergence.
(c ) Approximations and numerical evaluations
The two simplest ways to obtain high accuracy values for v(l 0 ) 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 numerical
result to at least 12 decimal places.
For some applications, including spreadsheet calculations, approximation
formulae are useful. For a degree-j formula always exact at l 0 Z0 and 1, we write
0
0
vj ðl Þ zð1Kl Þ
j
X
iZ0
0i
0
pi l C l ln l
0
j
X
qi l 0iK1 :
ð4:17Þ
iZ1
Best-fit values for the coefficients pi and qÐi were chosen by least squares
minimization of numerical approximations to 01 ½vj ðl 0 ÞK vE ðl 0 Þ2 dl 0 , where vE is
the exact value as determined numerically. Choosing jZ4 yields formulae with
absolute error j3j%8!10K10, without using error spreading techniques. Table 1
shows coefficient values. This performance exceeds, by a factor of approximately
25, the Hastings (1955) result j3j%2!10K8 for E(m), which did use errorspreading techniques. As the universal constants are known only to about 1 part
in 107, this precision is far more than needed physically.
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2918
R. G. Forbes and J. H. B. Deane
(a) 0.003
(b)
0.4
q=
0.2
relative error (%)
6
l′
0.2
0.4
0.6
0.8
1.0
1
69
0.1
1
–0.2
q=
1.0
–0.4
5
0.8
71
0.6
0.1
l′
0.4
0
1/
q=
69
0.2
1/6
0.1
715
0.1
0
q=
q=
0.001
q=
absolute error
0.002
Figure 1. (a) Absolute error 3 (approximate valueKexact value) and (b) relative error (absolute
error/exact value) for the function v(l 0 ) defined by (4.19), for the three values shown for q.
Similarly, a formula for dv/dl 0 , with j3j%7!10K10, uses
jC1
j
X
X
dv
0
0i
0
zKu 1 C ð1Kl Þ
si l C ln l
ti l 0i ;
dl 0
iZ0
iZ0
ð4:18Þ
where u 1 Z 3p=ð8O2Þ, s0Z[u 1K(9/8)ln 2], t 0Z3/16, and the other coefficients
come from minimization of absolute errors. This formula ‘goes to infinity in the
correct way’ as l 0 /0 and is exact at l 0 Z1. As compared with (4.17), extra
coefficients are needed to achieve similar precision. A spreadsheet using these
formulae to calculate the standard theory functions is available from RGF.
For analytical explorations and preliminary data analysis, a very simple
formula is best. As already reported, (2.11) has a relative accuracy of 0.33% or
better, over the whole range 0%l 0 %1. This outperforms earlier approximations
of 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-order
terms in the exact expansion. However, some formulae do outperform (2.11) over
limited ranges, because they were optimized to perform well there.
For comparison with (2.11), we searched numerically for the best
approximations of form
vðl 0 Þ z1Kl 0 C ql 0 ln l 0 ;
ð4:19Þ
0
where q is an adjustable constant. For 0%l %1, least squares minimization of
absolute 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 three
formulae. For qZ1/6, the largest absolute error 3 in v is 0.0024 and occurs near
l 0 Z0.19; the largest relative error is 0.33% and occurs near l 0 Z0.3.
The use of qZ1/6 in (4.19) is not an analytical result but a good
approximation that is convenient for its algebraic simplicity, and also fit for
purpose. The important thing is that (4.19), with qZ1/6, performs sufficiently
well over the whole range 0%l 0 %1 that we can trust it to reliably model the
mathematical behaviour of the exact function v(l 0 ).
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Reformulation of standard CFE theory
(a) 2.0
(b) 0.006
0.004
u
absolute error
s, t, u, v, w
1.5
t
1.0
s
0.5
0
0.2
0.4 0.6 0.8
l′ (= fhSN )
0
10w
–0.006
1.0
v
–0.002
–0.004
v
10w
t
s
0.002
u
0
0.2
0.4 0.6 0.8
l ′ (= fhSN)
1.0
(d) 0.006
(c) 2.0
0.004
absolute error
s, t, u, v, w
1.5
u
t
1.0
s
v
0.5
1 2
4
v
0
6
x (= 1/l ′ = 1 / fh
8
SN )
10
t
10w
–0.002
u
–0.004
10w
0
s
0.002
–0.006
0 1 2
4
6
8
SN
x (= 1/l′ = 1 / fh )
10
Figure 2. Exact values of s, t, u, v and w, and absolute errors (as defined for figure 1) associated
with formulae (4.19) and (5.1)–(5.4), taking qZ1/6: (a,b) plotted against l 0 ; (c,d ) plotted against
x (h1/l 0 ).
5. Explicit approximate expressions for the special elliptic functions
Using (4.19), earlier definitions yield
dv
uðl 0 Þ ZK 0 zð1KqÞ C q ln l 0 ;
dl
sðl 0 Þ z1Kql 0 ;
tðl 0 Þ z1 C l 0 =3 ½1K4qKq ln l 0 ;
ql 0 ln l 0
0
02
3
wðl Þ z 16 l 1 C
;
ð1Kl 0 Þ
Kqh
ð5:1Þ
ð5:2Þ
ð5:3Þ
ð5:4Þ
ð5:5Þ
rðl 0 ; hÞ zt K2 ehð1KqÞ l 0 :
2
02
With (5.4), (4.11) has been used to replace d v/dl . Numerics below take q as 1/6.
Figure 2 shows exact values for s, t, u, v and w, and absolute errors 3 when using
these formulae. All are shown as functions of both l 0 (0%l 0 %1) and x (h1/l 0 )
(1%x%10). Note that u rises steeply as l 0 falls below approximately 0.2 and goes to
infinity as ln(1/l 0 ), as l 0 /0. For s, t, v and 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
Proc. R. Soc. A (2007)
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2920
R. G. Forbes and J. H. B. Deane
range 0.2%l 0 %1, the corresponding figures are 0.0043 and 0.45%; errors get
progressively worse as l 0 falls below 0.2 and become serious below 0.1, but this is
not important because l 0 -values of 0.1 or lower are rarely of practical interest.
The established accuracy in predicting v(l 0 ) is thus reflected in the accuracy in
predicting s(l 0 ), t(l 0 ) and w(l 0 ) for 0%l 0 %1, and u(l 0 ) for l 0 T0.12. In these ranges,
the approximations can be used reliably in most algebraic manipulations (though
use in exponents needs caution).
6. Applications
We now put l 0 Zf and investigate the standard FN-type equation and associated
theoretical FN plots. For simplicity, this section drops the suffix from vF, tF and
nF. ‘Scaled’ forms, with the exponent written as h/f, are used because they are
more general. In standard theory,
hZ
bf3=2
Z bc2 f K1=2 zð9:836239 eV1=2 Þf K1=2 :
FfSN
ð6:1Þ
h varies slowly with f. The range 2.7!(f/eV)!6 is 6OhO4. The typical value
fZ4.5 eV is hZ4.64. From (3.3) and (4.19) (with l 0 replaced with f ), note that
"
#
Kvbf3=2
Kvh
Kð1Kf C qf ln f Þh
Z exp
zexp
exp
f
f
F
"
#
Kh
Kbf3=2
h Kqh
h Kqh
Ze f
exp
exp
ze f
:
ð6:2Þ
f
F
(a ) Predicted straight-line semi-logarithmic plot
From (2.10), (6.2) and fZF/Ff, the emission current Ist predicted by standard
theory is
"
#
h
i
Kbf3=2
st
K2
K1 h qh
ð2KqhÞ
:
ð6:3Þ
exp
I z t Aaf e Ff F
F
The expanded form of tK2( f ) is difficult to manipulate. Since tz1 and has weak
field dependence, it is left unexpanded. In the first bracket in (6.3), the only
assumed field dependence is in t, and this may be ignored. So, in principle, an
exact straight-line plot is given by ln{I/F (2Kqh)} versus 1/F, not ln{I/F 2} versus
1/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 the Kqh term. Field dependence in f,
and hence h (e.g. Jensen 1999), might also exist. For real emitters, the success of
the conventional experimental FN plot could be partly due to mutual
cancellation of opposing effects. Experiments to measure the true power of F
in FN-type equation pre-exponentials would be of considerable interest, but very
difficult. Change from using the traditional FN plot is not justified.
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Reformulation of standard CFE theory
2921
(b ) Relationship between standard and elementary CFE theory
Related to (2.10), there is an elementary FN-type equation obtained by
replacing vF and tF with 1. In scaled form, the predicted emission current I el is
el Kh
I
h
el
K1
2
ð6:4Þ
Z lnfAaf K1 gK :
I Z ðAaf ÞF exp
; ln
f
f
F2
From (2.10), (6.4) and (6.2), since tK2z1,
qh
I st
h 1
ze
:
ð6:5Þ
f
I el
Typically, f is approximately 0.15–0.45, h approximately 4.6, q close to 1/6 and
qh approximately 3/4, so (6.5) shows that typically the current density and
current predicted by standard theory are roughly 100 times greater than the
related elementary theory predictions. The ratio increases as f decreases.
Using t1 to denote t(fZ1), (2.10), (6.2) and (6.4) give
st el I
I
1
t
Z ln
C ½hK2 ln t1 C qh ln
:
ð6:6Þ
ln
K2 ln
2
2
f
t1
F
F
To show behaviour on the semi-logarithmic FN plot, define
I
ð6:7Þ
; Y0 Z lnfAaf K1 g and DY Z Y K Y0 :
Y Z ln
F2
The elementary FN equation (6.4) then takes either of the forms,
h
h
ð6:8Þ
Y el Z Y0 K ; DY el ZK :
f
f
The elementary FN plot intersects the y-axis at Y0; this value Y0 serves as the
reference zero for DY. Figure 3 has the y-axes labelled in this way.
Figure 3a and (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, line
L is drawn parallel to line E, above it by [hK2 ln t1]. Point B is then marked on L
at 1/fZ1. (For 1/f!1, the Schottky–Nordheim 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 term
in (6.5) and the shift from line L to curve S relates to the (1/f )qh term. Figure 3a
shows 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 of
0.15!f!0.45), and often seem to lie on a straight line (the ‘experimental FN
plot’). As already noted, the most convenient theoretical model (for a line fitted
by linear regression to the data points) is the tangent to S, taken at a value of
1/f among the data points. The intercept of this tangent with the ln{I/F 2} axis
is the theoretical prediction of the regression line intercept. In standard CFE
theory, the slope and intercept of the tangent relate to those of line E via the
mathematical correction functions s and r. A tangent taken at 1/fZ5 is shown
as line T5.
The shape of S makes s and r vary with 1/f. At 1/fZ1, (5.5) shows that r has
the value t 1K2 ehð1KqÞ , typically approximately 40. As 1/f increases and the
tangent point P moves to the right along S, three effects occur: P moves
downwards; P becomes increasingly distant from line E; and the magnitude of
Proc. R. Soc. A (2007)
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2922
R. G. Forbes and J. H. B. Deane
(a)
ln{r}
(b)
T5
T5
0
ln{rP}
B
S
S
B
T
slo
pe
∆Y
L
1.0 1.1 1.2 1.3 1.4 1.5
ln{tP– 2}
0
=
–s
Ph
B
V
L
S
slop
e=–
(∆ Y )P
P
∆Y
P
vP h
E
xP = 1/ fP
0
1
2
3
x = 1/f
4
5
6
0
x = 1/f
Figure 3. ‘Scaled’ FN plots: the horizontal axis shows x (h1/ f ), and the vertical axis DYZ
ln{I/F2}Kln{AafK1}. (a) To show how a curve S and tangent T derived from the standard
FN-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 at
point P, for the specific value xP (Z1/fP), but the argument is true for any point P.
the slope of S increases. The result is to increase r. Equation (5.5) shows, perhaps
counterintuitively, that this increase is due, almost exclusively, to the increasing
difference between S and L, effectively by the factor (1/f )qh. For example, for
1/fZ5, (1/f )qh is typically approximately 3.3 and r is typically approximately
140. As is well known, if one attempts to extract emission area from an
experimental FN plot by putting the Y-intercept equal to ln{AafK1} rather than
ln{rAafK1}, overestimation by a factor r occurs.
This analysis provides a clearer picture of how the FN plot works. It is also a
reminder of the disadvantages of using elementary rather than standard theory:
predicted currents will be too low and extracted areas too high, typically by
factors of 100 or more.
(c ) The underlying mathematics of the standard FN plot
The 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
DY st ðxÞ ZKvðxÞhx:
ð6:9Þ
Figure 2c shows clearly that v(x) varies quite sharply with x; but figure 3a shows
that line S, i.e. DY st(x), is almost straight. So a vital mathematical question is:
‘why does (6.9) generate a straight line?’ At first sight, this behaviour is counterintuitive. Its mathematical origin is as follows.
From (4.19), with xh1/l 0 , ln xhKln l 0 ,
vðxÞ z1Kx K1 Kx K1 q ln x;
ð6:10Þ
DY st Z h C qh ln x Khx:
Proc. R. Soc. A (2007)
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Reformulation of standard CFE theory
2923
Equation (6.11) is another equation for S. Clearly, the reason for its nearlinearity is the form of (6.10). The terms in xK1 generate a constant term (h) and
the 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 does
observed linearity in a FN plot mean?’ The argument above implies that, if an
experimental FN plot of type [ln{I/E 2} versus 1/E ], where E is any of f, F,
voltage V or macroscopic field FM, is effectively linear, then, to predict this, the
tunnelling exponent correction factor n must have the form n(E ) ZBCCE (or
reduce to it in the relevant range of E ), where B and C are constants or slowly
varying with E.
Thus, FN plot curvature relates to v2n/vE 2. Edgcombe & de Jonge (2006)
reach an equivalent conclusion. Observed linearity implies small v2n/vE 2. For
standard theory, this is confirmed qualitatively by figure 2a, which shows that
v(l 0 ) is nearly linear, and quantitatively by the small values of the curvature
function w. At l 0 Z1/3 (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 generally
similar 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 be
relatively small, as the work of Edgcombe & de Jonge (2006) brings out. So,
where marked curvature occurs in experimental plots, usually some other effect
must be operating, such as the presence of vacuum space charge, electron supply
limitation inside the emitter, or statistical effects associated with a manyemission-sites electron source.
7. Discussion
This work was stimulated by finding a new approximation for v, and then
realizing that y 2 was better as the independent variable and consequently
derivations needed reformulating. It seemed best to separate the mathematical
and physical descriptions inherent in standard CFE theory, and generalize the
physical description to apply to all well-behaved barriers. This paper has laid
some foundations and has given a fuller account of Schottky–Nordheim
barrier mathematics. By providing approximation formulae with absolute error
j3j!8!10K10, we hope to make v(l 0 ) 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 physical
quantities and equations that relate to a general physical form of FN-type
equation and to FN plots. These quantities will include an independent variable
(the scaled barrier field of (2.12)), correction factors for the exponent and preexponential of the curve equation, and factors that relate to the intercept, slope
and 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 physical
quantities on the real value of scaled barrier field (2.12) is modelled via the
Proc. R. Soc. A (2007)
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2924
R. G. Forbes and J. H. B. Deane
specific mathematical functions discussed earlier. We suggest that, in future,
they could be known collectively as the ‘Schottky–Nordheim barrier functions’.
Other barriers can be defined by different expressions for M(z), particularly those
associated with sharply curved emitters. For each barrier ‘B’ (different for each
B
model of emitter shape, etc.), new quantities F B
f and f f would be determined,
and the dependent physical quantities modelled by new mathematical functions
or (equivalently) value sets generated numerically. f fB will again lie in the range
0%f fB%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 plots
based on measurements of current versus voltage. These include: how to
calibrate barrier field precisely; how best to define the notional emission area A;
the effects of field dependence in f, t K2
F and A; nonlinearity in the dependence of
barrier 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 noisy
experimental data and the problems of poorly characterized experiments. In
standard theory, formula (2.17) should help the investigation of some of these. As
in Forbes (1999a), the problem is how to extract results and error limits under
conditions of physical uncertainty.
Overall, this paper provides renewal of standard CFE theory, and underpinning for future developments. We hope many will find this theoretical
approach more complete, more fruitful and easier than some older literature. We
strongly urge that clearer distinctions be made between mathematical entities
and physical quantities, and that the special field emission elliptic functions be
treated as functions of l 0 (Zy 2), rather than y, and be thought of as the ‘SN
barrier functions’. We also commend the scaled form of FN plot, which exhibits
CFE theory in a more universal form.
We wish to acknowledge an initial stimulus provided by Dr C. J. Edgcombe’s work on the theory of
CFE 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(l 0 )
The function v(l 0 ) has an exact series expansion,
N
X
iC1
vðl 0 Þ Z 1 C
ðAai C CnbiC1 C Cnai ln l 0 Þl 0 ;
ðA 1Þ
iZ0
where AZ(9/8) ln 2, CZ1, nZ3/16, a0Z1, and b1ZK1, and the recurrence
relationships below define the remaining coefficients,
iði C 1Þ C n
a ; iR 0;
aiC1 Z
ðA 2Þ
ði C 1Þði C 2Þ i
biC1 Z
ð2i K 1ÞaiK1 Kð2i C 1Þai C fði K1Þi C ngbi
;
iði C 1Þ
iR 1:
ðA 3Þ
This generates expansion (4.14). Deane et al. (2007) present a detailed formal
derivation of these recurrence relationships.
Proc. R. Soc. A (2007)
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Reformulation of standard CFE theory
2925
Form (4.14) can also be derived from the special expansions for K and E
derived by Cayley (1876, see pp. 52–55) and cited by Dwight (1965, formulae
(773.3) and (774.3)). Cayley uses a parameter k 0 given in terms of our m by
k 0 h Oð1KmÞ. Here, it is simpler to use two new parameters: m 0 h1Kmhk 0 2 and
Lhln(4/k 0 )[Zln 4C(1/2) ln (1/m 0 )]. The Cayley expansions then take the form
12
2
12 ,32
2
2
K
K Z L C 2 LK
ðA 4Þ
m 0 C 2 2 LK
m 02 C/
1,2
1,2 3,4
2
2 ,4
1
1
12 ,3
2
1
0
E Z1C
LK
m C 2
K
m 02 C/:
LK
2
1,2
ð1,2Þ 3,4
2 ,4
On substituting for L and rearranging,
12
2
12 ,32
2
2
0
K Z ln 4 C 2 ln 4K
m C 2 2 ln 4K
K
m 02 C/
1,2
1,2
3,4
2
2 ,4
1
12 0 12 ,32 0
0
C ln ð1=m Þ 1 C 2 m C 2 2 m C/
2
2
2 ,4
1
1
12 ,3
2
2
0
ln 4K
K
E Z1C
ln 4K
m C 2
m 02 C/
2
1,2
1,2 3,4
2 ,4
1
1
12 ,3
C lnð1=m 0 Þ m 0 C 2 m 02 C/ :
2
2
2 ,4
ðA 5Þ
ðA 6Þ
ðA 7Þ
Although the coefficients are necessarily different, (A 6) and (A 7) have the same
general forms as the Hastings (1955) approximation formulae for K and E cited
by AS as formulae (17.3.34) and (17.3.36) (their m1 is our m 0 ).
From (4.6),
m 0 Z 1Km Z
2l 01=2
Z l 01=2 ð1Kl 01=2 C l 0 Kl 03=2 C/Þ
ð1 C l 01=2 Þ
1
1
l 0 l 03=2
K/:
ln
ZKln 2K ln l 0 C l 01=2 K C
0
m
2
2
3
ðA 8Þ
ðA 9Þ
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 computer
algebraic manipulation can be used with advantage. Although half-integral
powers of l 0 appear in the derivation, powers of l 0 in the result are integral.
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3378
Errata
Proc. R. Soc. A 463, 2907–2927 (21 August 2007) (doi:10.1098/rspa.2007.0030)
Reformulation of the standard theory of
Fowler–Nordheim tunnelling and cold field
electron emission
B Y R ICHARD G. F ORBES
AND
J ONATHAN H. B. D EANE
The following equations contain typographical errors that have no consequence
for any other equations or results in the above paper.
dm=dl 0 ZKl 0K1=2 ð1 C l 01=2 ÞK2
and
ð4:12Þ
l 0 ð1Kl 0 Þd2 v=dl 0 2 Kð163 Þv Z 0:
ð4:13Þ
The first column of table 1 was incorrect, and should read as follows.
Table 1. Coefficients for use in connection with (4.17) and (4.18), for degree-4 formulae. (Note:
u 1z0.8330405509.)
i
pi
0
1
2
3
4
5
1
0.032
0.009
0.002
0.000
—
qi
705
157
644
089
304
798
272
871
46
739
807
738 11
si
—
0.187
0.017
0.005
0.001
—
499
506
527
023
344
369
069
904
1
47
444
180
0.053
0.024
0.015
0.007
0.000
K0.000
ti
249
222
122
550
639
048
972
259
059
739
172
819
7
59
58
834
865 9
745 89
0.187
0.035
0.019
0.011
0.003
—
5
155
127
522
624
558
526
840
569
74
80
09
427
The first sentence of appendix A was incorrect, and should read as follows.
Appendix A. Series expansion for v(l 0 )
The function v(l 0 ) has an exact series expansion,
N
X
vðl 0 Þ Z 1 C
ðAai C CnbiC1 C Cnai ln l 0 Þl 0iC1 ;
ðA 1Þ
iZ0
where AZK(9/8)ln 2, CZ1, nZ3/16, a 0Z1, and b1ZK1, and the recurrence
relationships below define the remaining coefficients,
aiC1 Z
biC1 Z
Proc. R. Soc. A (2008)
iði C 1Þ C n
a;
ði C 1Þði C 2Þ i
iR 0;
ð2i K 1ÞaiK1 Kð2i C 1Þai C fði K1Þi C ngbi
;
iði C 1Þ
ðA 2Þ
iR 1:
ðA 3Þ