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Proc. R. Soc. A (2011) 467, 1029–1051
doi:10.1098/rspa.2010.0460
Published online 13 October 2010
Analytical treatment of cold field electron
emission from a nanowall emitter, including
quantum confinement effects
BY XI-ZHOU QIN1 , WIE-LIANG WANG1 , NING-SHENG XU1 , ZHI-BING LI1, *
AND R ICHARD G. FORBES2
1 State
Key Laboratory of Optoelectronic Materials and Technologies,
School of Physics and Engineering, Sun Yat-Sen University,
Guangzhou 510275, People’s Republic of China
2 Advanced Technology Institute, Faculty of Engineering and Physical Sciences,
University of Surrey, Guildford, Surrey GU2 7XH, UK
An elementary approximate analytical treatment of cold field electron emission (CFE)
from a classical nanowall (i.e. a blade-like conducting structure on a flat surface) is
presented. This paper first discusses basic CFE theory for situations where quantum
confinement occurs transverse to the emitting direction. It develops an abstract CFE
equation more general than Fowler–Nordheim type (FN-type) equations, and then applies
this to classical nanowalls. With sharp emitters, the field in the tunnelling barrier may
diminish rapidly with distance; an expression for the on-axis transmission coefficient for
nanowalls is derived by conformal transformation. These two effects interact to generate
complex emission physics, and lead to regime-dependent equations different from FNtype equations. Thus: (i) the zero-field barrier height HR for the highest occupied state
at 0 K is not equal to the local thermodynamic work-function f, and HR rather than f
appears in equations; (ii) in the exponent, the power dependence on macroscopic field FM
−2
−1
can be FM
rather than FM
; (iii) in the pre-exponential, explicit power dependences on
FM and HR differ from FN-type equations. Departures of this general kind are expected
when nanoscale quantum confinement occurs. FN-type equations are the equations that
apply when no quantum confinement occurs.
Keywords: nanowall field emitters; cold field electron emission; quantum confinement;
wave-mechanical tunnelling
1. Introduction
This paper derives theoretical equations (for line and area emission current
density (ECD)) for cold field electron emission (CFE) from a ‘classical nanowall’.
This is an elementary model for a larger group of quasi-two-dimensional
*Author for correspondence ([email protected]).
Electronic supplementary material is available at http://dx.doi.org/10.1098/rspa.2010.0460 or via
http://rspa.royalsocietypublishing.org.
Received 1 September 2010
Accepted 15 September 2010
1029
This journal is © 2011 The Royal Society
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X.-Z. Qin et al.
material structures, including those built from a few layers of graphite. The
aim here is to understand the basic physics of CFE from nanowalls. A longer
term aim is to develop more sophisticated nanowall equations, by including
atomic structure.
Clarity requires some definitions. ‘Fowler–Nordheim (FN) tunnelling’
(Fowler & Nordheim 1928) is field-induced electron tunnelling through a potential
energy (PE) barrier that is exactly or approximately triangular. ‘CFE’ is the
emission regime in which: (i) the electrons inside and close to the emitting
surface are (very nearly) in thermodynamic equilibrium; and (ii) most emitted
electrons escape by FN tunnelling from states near the emitter Fermi level. ‘FNtype equations’ are a family of approximate equations that describe CFE from
‘bulk metals’.
For CFE theory, the essential characteristics of a ‘bulk metal’ are that (i)
no significant field penetration occurs, (ii) the Fermi level is in the conduction
band, well away from the band edge, and (iii) the electron states in the band
may be treated as travelling wave states for all directions of motion. Hence,
emitter dimensions must be large enough for no significant standing wave
effects (quantum confinement effects) to occur for any direction (although there
will, of course, be interference effects associated with back reflection from the
emitting surface).
A ‘nanowall’ is a blade-like material structure that stands upright on a
substrate. Its width is small absolutely and in comparison with its height, and its
length is large when compared with its width. Carbon nanowalls can be grown in
suitable conditions, and can be efficient field electron emitters (Wu et al. 2002;
Teii & Nakashima 2010). Present emitters tend to grow as irregular honeycombetype structures containing many short interconnected nanowalls. But, if a longer,
uniform, single nanowall could be grown, this might have interesting applications
as a line electron source. Thus, a theory of CFE from nanowalls may have both
scientific and technological relevance.
For CFE theory, the essential feature of a nanowall is that its width is
sufficiently small that the Schrödinger equation solutions for this direction are
standing waves. In consequence, the related energy component is quantized. As
shown below, CFE from a nanowall does not obey the usual FN-type equations
(except in a generalized sense). It seems better to regard the CFE equations
for nanowalls as members of a different (but related) family of theoretical CFE
equations. Our aim is to derive elementary equations belonging to the family of
‘nanowall’ CFE equations.
To help focus on the effects (quantization and barrier-field fall-off) associated
with small nanowall width, and make the problem analytically tractable,
simplifications are used. As shown in figure 1, our nanowall has a simple profile,
and is uniform in the ‘long’ direction. We disregard its atomic structure, assume
its surface has uniform local work-function, and disregard the possibility of field
penetration and band-bending. This enables the nanowall to be treated as a
classical conductor, with shape as in figure 1: we call this a ‘classical nanowall’.
For the nanowall electron states, we use a (confined) free-electron-type model. We
also disregard the correlation-and-exchange (‘image-type’) interaction between
departing electrons and the nanowall: we can then use classical electrostatics to
establish the tunnelling barrier shape. FN made equivalent simplifications in the
original treatment of CFE from a bulk metal with a planar surface.
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CFE from nanowall emitters
(a)
Vapp
(b)
anode plane
anode plane
x
y
FM = Vapp/Xa
Xa
z
Xa
h
h
w
cathode plane
cathode plane
w
Figure 1. Classical nanowall (a) three-dimensional view; (b) projection onto the z–x plane.
FN solved the Schrödinger equation exactly for an exact triangular (ET) PE
barrier. The barrier above the nanowall top surface is not exactly triangular: with
such barriers it is usually mathematically impossible to solve the Schrödinger
equation exactly in terms of the common functions of mathematical physics.
Therefore, we use the simple Jeffreys–Wentzel–Kramers–Brillouin (JWKB)
formula (Jeffreys 1925, also see Forbes 2008b) to generate an approximate
solution. This approach is widely used in CFE theory.
When (as here) the nanowall stands on one of a pair of parallel plates,
between which a uniform ‘macroscopic field’ FM would exist in its absence, a
‘field enhancement factor’ g can be defined by
F = gFM =
gVapp
,
Xa
(1.1)
where Xa is the plate separation, Vapp is the electrostatic potential between them
and F is a local electric field at the emitter surface, called here the ‘barrier field’.
These assumptions mean that the equivalent FN-type equation (for comparison
with results here) is the elementary FN-type equation, expressed in terms of FM
(Forbes 2004). This elementary equation gives the ECD JA (current per unit area)
at zero temperature, as a function of FM and the local work-function f. It can be
written in the equivalent forms:
−bf3/2
−bf3/2
−1 2 2
el 2
JA = af g FM exp
= zS (dF ) exp
.
(1.2)
gFM
gFM
Here, a and b are the first and second FN constants, and zS the Sommerfeld supply
density, as defined in table 1. dFel is the elementary approximation for a parameter
dF called the ‘decay width at the Fermi level’, discussed below. Although less used,
the second form is more fundamental than the first, and arises naturally when
deriving equation (1.2) (Forbes 2004).
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X.-Z. Qin et al.
Table 1. Basic universal emission constants. Values are given in the units often used in
field emission.
name
symbol derivation expression
numerical value units
Sommerfeld supply
densitya
JWKB constant for
electron
first FN constant
second FN constant
zS
—
4peme /hP3
1.618311 × 1014
A m−2 eV−2
ge
—
4p(2me )1/2 /hP
10.24624
eV−1/2 nm−1
a
b
zS (e/ge )2
2ge /3e
e 3 /8phP
1.541434 × 10−6 A eV V−2
(8p/3)(2me )1/2 /ehP 6.830890
eV−3/2 V nm−1
a The
supply density is the electron current crossing a mathematical plane inside the emitter, per
unit area of the plane, per unit area of energy space, when the relevant electron states are fully
occupied. In an unconfined free-electron model for a bulk conductor, the supply density is
isotropic, is the same at all points in energy space, and is given by zS .
The paper’s structure is as follows: §2 outlines a general method for deriving
theoretical CFE equations; §3 then describes our nanowall model, and discusses
issues relating to electron supply; §4 derives a formal expression for nanowall
current density, and compares this with the elementary FN-type equation; §5
derives the electrostatic potential variation in the symmetry plane, by means of
a conformal transformation; §6 derives detailed theoretical equations for CFE
from a nanowall and §7 provides a summary and discussion. Some mathematical
details are provided in appendix A or in the electronic supplementary material.
In this paper, e denotes the elementary positive charge, me the electron mass,
hP Planck’s constant and kB Boltzmann’s constant. We use the universal emission
constants in table 1, and the usual electron emission convention that fields,
currents and current densities are treated as positive, even though negative
in classical electrostatics. However, the quantity V (x) below is conventional
electrostatic potential, so the corresponding ‘CFE field’ is given by dV /dx rather
than −dV /dx. We use the International System of Quantities (BIPM 2006), but
(where appropriate) use the customary units of field emission rather than SI base
units. These customary units simplify formula evaluation and are dimensionally
consistent with SI units.
2. The structure of theoretical cold field electron emission equations
This section extends a previously used approach (Forbes 2004), to allow
description of the common general structure of equations describing CFE. The
barrier field F (rather than macroscopic field FM ) is initially used as the
independent field-like variable.
Establishing an expression for local ECD (either current per unit area
JA , or current per unit length JL ), needs three basic steps. (i) A reference
emitter electron state ‘R’ is chosen, and the zero-field height HR (see below)
of the barrier seen by an electron in this state is established. (ii) The escape
probability (transmission coefficient) DR and decay width dR for this barrier are
calculated, usually by an approximate quantum-mechanical method. (iii) The
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CFE from nanowall emitters
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ECD contributions of all relevant electron states (particularly those energetically
close to R) are calculated, and the result of summing them is written in the form
J (F , T ) = ZR (F ,T )DR (F ),
(2.1)
where T is the emitter’s thermodynamic temperature, J (F ,T ) is the ECD, and
ZR (F ,T ) is called the ‘effective supply of electrons for state R’. Normally, the
reference state would be the occupied electron state at T = 0 K that has the
highest value of D.
Details of calculating ZR are different for different CFE equation families, but
transmission coefficient (DR ) theory is similar for all. When emission comes from a
set of ‘electronic sub-bands’, an analogous procedure is applied to each, and the
total ECD is found by summing-up contributions from the various sub-bands.
Sometimes, emission from one sub-band may dominate.
(a) Transmission coefficients
A tunnelling barrier is described by a function M (l,F ,H ), where l is the
distance measured from some convenient reference point, F is the barrier field
and H (the ‘zero-field barrier height’) is the barrier height seen by an approaching
electron when F = 0. Physically, M (l,F ,H ) is defined by M ≡ U (l,F ) − En , where
U (l,F ) is the PE that appears in the Schrödinger equation. E is the total electron
energy, and En its component associated with motion in the tunnelling direction
(with E, En and U (l,F ) all referred to the same energy-zero). We call En the
‘forwards energy’. The barrier is the region where M ≥ 0.
A parameter G(F ,H ), called here the ‘JWKB transmission exponent’, is then
defined by
M (l, F , H ) dl,
(2.2)
G(F , H ) ≡ ge
where ge is the ‘JWKB constant for an electron’, defined in table 1, and this
‘JWKB integral’ is taken over the region where M ≥ 0. Both M and G, and other
parameters below, may be functions of additional variables that describe emitter
geometry, but these dependences are not shown.
The ET barrier used in deriving equation (1.2) is M ET (l,F ,H ) = H − eFl,
where l is measured from the emitter’s ‘electrical surface’ (Lang & Kohn
1973; Forbes 1999). For a classical conductor, the electrical surface coincides
with the conductor surface. For the ET barrier, the region of integration for
equation (2.2) is 0 ≤ l ≤ H /eF , and G ET (F ,H ) = bH 3/2 /F . For barriers that are
only approximately triangular, the JWKB exponent is written
nbH 3/2
,
(2.3)
F
where n (‘nu’) is a correction factor, relating to barrier shape, derived by
evaluating equation (2.2).
For the CFE regime, a suitable formal expression for the transmission
coefficient D(F ,H ) is
−nbH 3/2
,
(2.4)
D(F , H ) ≈ P exp[−G] = P exp
F
G = nG ET =
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X.-Z. Qin et al.
where P is a ‘transmission prefactor’ discussed in Forbes (2008b). Except in
a few special cases, calculating P requires specialist mathematical/numerical
techniques. These are only just beginning to be applied to the one-dimensional
barriers used in practical CFE applications; thus, a common approximation (also
used here) puts P → 1.
It is important to know how quickly D(F , H ) changes as the zero-field barrier
height H changes. This can be described by either a ‘decay rate’ cd or a ‘decay
width’ d(≡ 1/cd ) defined by
−v ln D −v ln P
vG
1
≡ cd ≡
=
+
.
d
vH
vH
vH
(2.5)
The literature uses both decay rate (e.g. c0 in Modinos (1984), or bF in Jensen
(2007)) and decay width (e.g. d in Gadzuk & Plummer (1973)). Decay width is
used here because the dimensionless quantity (pkB T /d), involving the Boltzmann
factor kB T , is theoretically familiar. The physical interpretation of d is that the
transmission coefficient D decreases by a factor of approximately e (∼
=2.7) when
H increases by d.
When deriving the elementary FN-type equation (1.2), we put P → 1, n → 1,
G → bH3/2 /F and obtain the ‘elementary’ variants of d and cd as
(3/2)bH 1/2 ge H 1/2
1
el
=
c
=
=
,
d
d el
F
eF
(2.6)
where the derivation of b has been used (table 1). In more general situations, the
outcome of definition (2.5) is written
d −1 = cd ≡ tcdel =
tge H 1/2
,
eF
(2.7)
where t is a ‘decay-rate correction factor’ defined by equation (2.7), and obtained
by evaluating definition (2.5). The factor t as used here is a generalization
both of the factor t that appears in the standard FN-type equation (Murphy &
Good 1956; Forbes & Deane 2007), and of the factor t that appears in partially
generalized CFE theories, where P is not taken into account (e.g. Forbes 2004).
When deriving theoretical CFE equations, one needs the values of transmission
coefficient and decay width for the barrier related to the chosen reference state.
Labelling parameters specific to this state/barrier by the subscript ‘R’, and also
substituting F = gFM in equations (2.2) and (2.7), we get
3/2
−nR bHR
DR (F ) ≈ PR exp[−GR ] = PR exp
gFM
and
dR ≈ (tR−1 g)
Proc. R. Soc. A (2011)
e
−1/2
H R FM .
ge
(2.8)
(2.9)
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CFE from nanowall emitters
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For states with H close to HR , a Taylor-type expansion allows D(F ,H ) to
be approximated. If only the linear term is used (which is the customary
approximation), then
D(F ,H ) ≈ DR (F ) exp
−(H − HR )
.
dR
(2.10)
(b) Electron supply
In free-electron-type models, the emitter’s internal electron states are labelled
by a set of three-directional quantum numbers and a spin quantum number. Let
Q label an arbitrary set member. An electron in state Q approaching the emitter
surface makes a contribution jQ to the total ECD J , where
jQ = ePQ f (EQ ,T )D(F , HQ ).
(2.11)
Here, ePQ is the electron–current–density component (for state Q) approaching
the emitting surface in a direction normal to it, and f (EQ , T ) is the occupancy
of state Q. The total ECD J is obtained by summing-up the contributions from
all relevant states:
J=
jQ = e
PQ f (EQ ,T )D(F ,HQ ).
(2.12)
Q
Q
Using equation (2.10), this can be put in the form of equation (2.1), with
(HQ − HR )
PQ f (EQ ,T ) exp −
.
(2.13)
ZR (F ,T ) = e
dR
Q
The summation then has to be evaluated for the system under investigation. The
form of the result will depend on emitter geometry, since (for emitters that are
‘small on the nanoscale’ in any direction) this will determine how the electron
states are quantized.
For bulk metals in thermodynamic equilibrium at zero temperature, the results
are particularly simple (as long known). In the relevant reference state ‘F’
(sometimes called the ‘forwards state at the Fermi level’) an electron at the
Fermi level approaches the emitting surface normally, and the zero-field barrier
height is equal to the local thermodynamic work-function f. The summation
in equation (2.13) can be carried out in several ways. In particular, it can be
reduced to a double integral in a so-called ‘P-T energy space’ (Forbes 2004) in
which the variables of integration are, first, the electron’s kinetic energy parallel
to the emitter surface, and then its total energy. The two integrations introduce
a factor dF2 into the result. Forbes (2004) shows that (for the simpler, commonly
used, forms of FN-type equation, at 0 K):
ZR (F , T = 0) → ZF (F ) ≈ zS dF2 ,
(2.14)
where ZF (F ) is the effective supply for state ‘F’, zS is the Sommerfeld supply
density (table 1), and dF the decay width for state F. For the elementary FN-type
equation, ZFel is obtained by replacing dF by dFel .
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For FN-type equations, the results for the ET barrier and the elementary FNtype equation play a special role. Because FN tunnelling is defined as tunnelling
through an approximately triangular barrier, it is easy to use correction factors
to relate more sophisticated FN-type equations to the elementary equation.
Similarly, it is convenient to treat other CFE equation families by discussing
one or more elementary equations and then introducing correction factors.
3. Electron supply for a classical nanowall
For quantum-mechanical purposes, the classical nanowall in figure 1 can be
modelled (in respect of the z direction) as a deep rectangular PE well. In this
paper, the total energy and energy components are measured relative to the well
base, and are denoted by the basic symbol W . Thus, an electron confined in the
classical nanowall has total energy W given by
W = Wx + W y + W z ,
(3.1)
where Wi (i = x, y, z) is the energy component related to motion along the i-axis.
For the x and y directions the motion is free, with
py2
px2
and Wy =
,
(3.2)
Wx =
2me
2me
where px and py are the momenta in the x and y directions, respectively.
The component Wz is quantized, and its allowed values are given approximately
by the values Wzn for an infinitely deep PE well, namely
nhP 2
1
≡ Wzn = n 2 Wz1 ,
(3.3)
Wz ≈
2me 2w
where w is the well width, n a positive integer that defines the electron’s
‘vibrational level’ and Wzn and Wz1 are defined by this equation (Wz1 is the
value of Wz associated with the n = 1 level). This quantization of Wz means that
the electron states in the nanowall are split into sub-bands SB1, SB2, . . . SBn . . .,
etc., with each sub-band associated with a particular value of n.
Since (in the CFE regime) the emitter is treated as very nearly in
thermodynamic equilibrium, the occupancy of electron state Q is treated as given
by the Fermi–Dirac distribution function
fFD (WQ , T ) =
1
.
1 + exp[(WQ − WF )/kB T ]
(3.4)
Here, WF is the Fermi energy (i.e. the total energy of an electron at the
Fermi level). For sub-band SBn, the Fermi–Dirac occupation probability can be
written as
1
.
(3.5)
fn =
2
1 + exp[{Wx + (py /2me ) + Wzn − WF }/kB T ]
For sub-band SBn, the number of electrons from SBn crossing a plane normal
to the x-axis per unit time, per unit length of the nanowall (in the z direction),
per unit range of Wx , is called the ‘line supply function for sub-band SBn’, and
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CFE from nanowall emitters
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is denoted by NL,n (Wx ,T ). This is obtained by integrating over the allowed range
of py , taking occupancy of electron states into account. Thus
2
NL,n (Wx , T ) =
hP2
+∞
−∞
fn dpy .
(3.6)
NL,n (Wx ,T ) has the SI units (s−1 m−1 eV−1 ), and is not the same as the area
supply function (Nordheim 1928) used when FN-type equations are derived by
integration via the normal energy distribution.
4. Formal expressions for nanowall current density
(a) Tunnelling barrier parameters
The JWKB integral (2.2) has to be evaluated along an appropriate path, which
may in principle be curved and will be influenced by the pattern of electric field
lines in the region through which the electron passes. To avoid the complications
of curved integration paths (Kapur & Peierls 1937), but still provide results
illustrative of nanowall emission, we consider a straight-line integration path that
starts from a symmetry position S on the nanowall top surface and coincides with
the x-axis. The origin of coordinates is taken at the nanowall top surface, at S,
and the barrier field is defined as the local surface field FS at S. This field FS
is related to the macroscopic field FM by equation (1.1), which defines a field
enhancement factor gS that is independent of FM and is discussed further below.
In what follows, it will often be convenient to show field dependences as functions
of FM rather than FS .
Figure 2 shows schematically the variation, along the x-axis, of the electron PE
U (x, FM ) (measured relative to the well base). Because we disregard image-type
effects, U (x, FM ) is given by
U (x, FM ) = Wo − eV (x, FM ) = WF + f − eV (x, FM ),
(4.1)
where Wo is the nanowall inner PE (or ‘electron affinity’), f is the local
thermodynamic work-function of its surface and V (x, FM ) is the conventional
electrostatic potential in the surrounding space, taking the nanowall potential as
zero. Obviously, the ‘shape’ of V and U depends on the nanowall shape, and the
size of V and U depends on its shape and on the value of FM .
For states with forwards energy Wx < Wo , there exists a tunnelling barrier of
zero-field height H = Wo − Wx . Thus, any formula involving H can alternatively
be written in terms of Wx . Owing to disregard of image-type effects, the barrier’s
inner edge is at x = 0. Its outer edge is at x = L, where
eV (L, FM ) = H = Wo − Wx .
(4.2)
In the CFE regime, equation (2.4) is an adequate expression for transmission
coefficient. With the approximation P → 1, the transmission coefficient
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X.-Z. Qin et al.
Wo
H
f
U(x, FM)
WF
Wx
Wx = 0, U = 0
0
L
x
Figure 2. Schematic showing how the electron potential energy U (x, FM ) varies with distance along
the x-axis. An electron with forwards energy Wx sees a barrier of zero-field height H .
D(Wx , FM ) for an electron approaching the nanowall top surface with forwards
energy Wx becomes
D(Wx , FM ) ≈ exp[−G(Wx , FM )],
(4.3a)
where G(Wx , FM ) is derived, from equation (2.2) and associated definitions,
by the replacements l → x, U → Wo − eV (x, FM ) and En → Wx . This makes
H = Wo − Wx , and yields
L (Wo − Wx ) − eV (x, FM )dx.
(4.3b)
G(Wx , FM ) = ge
0
D(Wx , FM ) decreases exponentially as Wx decreases; thus, emission comes
preferentially from occupied states with Wx as high as possible. However, in the
CFE regime, the occupancy of states falls off sharply above the Fermi level. Thus,
most of the emission comes from states near the Fermi level. For states at the
Fermi level, the largest possible value of Wx is
WxR1 = WF − Wz1 = Wo − f − Wz1 ,
(4.4)
which corresponds to a state with n = 1 and py = 0. This can be specified
as the reference state (‘R1’) for sub-band SB1; WxR1 is the corresponding
forwards energy.
More generally, for states at the Fermi level, the largest possible value of
forwards energy Wx for states in sub-band SBn is the forwards energy WxRn
for the corresponding reference state Rn:
WxRn = WF − Wzn .
(4.5)
The JWKB exponent for the barrier related to state Rn is denoted by Gn (FM )
and given by
xn Gn (FM ) = ge
(f + Wzn ) − eV (x, FM )dx,
(4.6)
0
where xn is the outer classical turning point for the barrier seen by an electron
in state Rn, and is found by solving equation (4.2), with L = xn , H = f + Wzn .
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CFE from nanowall emitters
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Thus, in respect of its escape by tunnelling, an electron in state Rn sees a
barrier of zero-field height Hn given by
Hn = f + Wzn ≈ f + n 2 Wz1 .
(4.7)
The physical reason why Hn is greater than the local thermodynamic
work-function f is that the electron has to ‘use up’ some of its kinetic energy
to provide the energy Wzn associated with its localization in the z direction.
Obviously, if Wzn > WF , then the sub-band is empty (except for any electrons
that may get into it as a result of thermal activation), and contributes little or
no emission.
Using equation (2.5), noting that our model puts P → 1, and that dH = −dWx ,
we obtain the decay width dn for reference state Rn as
g xn
1
vG e
−1
dn ≈ −
=
dx.
(4.8)
vWx Wx =WxRn
2 0
f + Wzn − eV (x, FM )
(b) Characteristic line current density
The present model derives an expression for a quantity JL (FM , T ) called
the ‘characteristic line current density’. JL (FM , T ) is the electron current that
would be emitted per unit length of the nanowall if each electron arriving
at its top surface had an escape probability corresponding to the barrier
defined for symmetry position S. For the present paper, this is an adequate
approximation (more sophisticated treatments would introduce a correction
factor). The contribution JL,n (FM , T ) from sub-band SBn to JL (FM , T ) is
+∞
JL,n (FM , T ) = e
D(Wx , FM )NL,n (Wx , T )dWx .
(4.9)
−∞
The lower limit on Wx can be taken as −∞, because D(Wx , FM ) gets rapidly
smaller as Wx decreases.
For reference state Rn, the difference −(H − HR ) in equation (2.10) can
alternatively be written
−(H − Hn ) = Wx − WxRn = Wx + Wzn − WF .
(4.10)
Combining equations (2.10), (3.5), (3.6), (4.9) and (4.10) yields
JL,n (FM , T ) =
2e
exp[−Gn ]
hP2
∞ ∞
exp[dn−1 (Wx + Wzn − WF )]
dpy dWx .
×
−1
2
−∞ −∞ 1 + exp[(kB T ) (Wx + py /2me + Wzn − WF )]
(4.11)
By reversing the order of integration, and defining u = Wx + py2 /2me + Wzn − WF ,
the double integral reduces to
∞
−py2 ∞
exp[dn−1 u]
I=
du dpy .
exp
(4.12)
2me dn −∞ 1 + exp[(kB T )−1 u]
−∞
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X.-Z. Qin et al.
The integral in u is a standard form that results in a temperature-dependent
correction factor (pkB T )/ sin(pkB T /dn ), and the resulting integral over py yields
the extra factor (2pme dn )1/2 . Thus, equation (4.11) reduces to
pkB T /dn
nw
d 3/2 exp[−Gn ],
(4.13)
JL,n (FM , T ) = z
sin(pkB T /dn ) n
where z nw is an universal constant with the value
2e
nw
(2pme )1/2 ∼
z ≡
= 1.119769 × 105 A m−1 eV−3/2 .
2
hP
(4.14)
The proof of equation (4.13) is adequately valid within the CFE emission
regime (though not at higher temperatures), and in this regime, the temperaturedependent correction factor is always relatively small. The error in omitting it is
always far less than the error involved in disregarding image-type effects. Thus,
practical calculations can use the zero-temperature limiting form
JL,n (FM ) ≈ z nw dn3/2 exp[−Gn ].
(4.15)
The characteristic line current density JL (FM ) is obtained by summing-up
the contributions from the various sub-bands. If the width of the nanowall is
sufficiently small, then a working field range should exist where JL,n JL,1 for all
n ≥ 2, and emission from the n = 1 sub-band will be dominant. We call this the
‘thin-nanowall case’. A suitable condition for this to occur (discussed further in
§7) is (G2 − G1 ) > 1. In such circumstances, we can neglect emission from higher
sub-bands and take JL (FM ) ≈ JL,1 (FM ). In what immediately follows, we assume
the nanowall is thin enough to allow this.
At the other limit, as the nanowall width gets larger, the sub-bands with
n > 1 contribute increasingly, and the theory ultimately becomes equivalent to
the elementary FN-type equation.
(c) Formal comparison with elementary Fowler–Nordheim-type equation
For the thin nanowall case, we can derive a ‘characteristic area current
density’ JAtnw (FM ) by dividing JL,1 (FM ) by the nanowall width w, and then using
equations (2.8) and (2.9) to expand the resulting expression, giving
JAtnw (FM ) ≈ z nw w −1 d1 exp[−G1 ]
3/2
bH
−n
1
3/2
1
JAtnw (FM ) = z nw w −1 d1 exp
g S FM
3/2
and
−3/2 −3/4 3/2 3/2
JAtnw (FM ) = a nw w −1 t1 H1 gS FM
(4.16a)
(4.16b)
3/2
−n1 bH1
exp
gS F M
,
(4.16c)
where n1 and t1 are the barrier-shape and decay-rate correction factors for
reference state R1, and a nw is an universal constant given by
3/2
e
∼
a nw ≡ z nw
(4.17)
= 1.079631 × 10−10 A m−1 eV3/4 V−3/2 m3/2 .
ge
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CFE from nanowall emitters
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Comparing equation (4.16b) with the abstract form of the elementary FN-type
equation (1.2) shows the following main differences. In the exponent, the zerofield barrier height H1 is greater than the local thermodynamic work-function f
by the amount W1 , and a barrier-shape correction factor is present; in the preexponential, the power to which the decay width is raised is 3/2 rather than 2, and
3/2
the term (z nw w −1 ) has replaced zS . When, as in equation (4.16c), the d1 term
is expanded, HR (rather than f) appears in the pre-exponential, the powers to
which FM and HR are raised are different, a decay-rate correction factor appears
and the ‘constant’ terms are different.
Of course, equations (4.16) represent an elementary member of the family
of nanowall CFE equations. With advanced members of the family, the preexponential will also include a transmission pre-factor and additional supply-type
correction factors.
We emphasize that the differences between equation (4.16) and the elementary
FN-type equation (1.2) result from differences in the way that the effective
electron supply for the reference state has to be calculated when ‘lateral phaselocking’ of electron waves (and hence quantization of a lateral energy component)
occur. With a thin nanowall, there are also special features related to barrier
shape. These are now explored.
5. Electrostatic potential variation for a classical nanowall
The parameters Gn and dn in equation (4.14) depend on how the electrostatic
potential V (x, FM ) varies near the nanowall top surface. V (x, FM ) could, in
principle, be obtained by using a numerical Poisson solver, and Gn and dn could,
in principle, then be obtained by numerical integration of the JWKB integral.
However, some features of the mathematics of nanowalls are not easily brought
out by numerical solutions; thus, an analytical illustration is provided here.
The first step obtains an expression for V (x, FM ) by means of the conformal
transformation illustrated in figure 3.
The overall transformation X combines two stages: X1 transforms the first
quadrant in ‘target space’ (figure 3a) into the positive half-plane in ‘virtual space’
(figure 3b), and X2 then transforms this half-plane into a shape in ‘physical space’
(figure 3c) that represents the right-hand half of the nanowall. Positions in the
three spaces are represented by
√ the complex numbers z, h and u, respectively,
using axes as shown, with i ≡ (−1).
In figure 3c, h is the nanowall height and r its half-width (r = w/2). The
x-origin is at 0 + ih, and 0 + i(h + x) is an arbitrary point on the x-axis. The
anode applying the field is at a distance (xa = Xa − h) such that xa h. Table 2
shows equivalences between special points in the three spaces.
To simplify equations, we use a mathematical function ES (y) defined in
appendix A. Transformation X1 is simply: h = z2 . Transformation X2 is the
Schwarz–Christoffel transformation (Driscoll &Trefethen 2002)
h h − r2
dh + ih,
(5.1)
u(h) = C
h (h − 1)
0
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X.-Z. Qin et al.
(b)
(c)
x
(a)
X2
X1
ix
i(x + h)
x=0
0
r
•
1
–x 2
0
r2
r + ih
ih
•
1
0 r
z
•
Figure 3. (a) The target (z) space; (b) the virtual (h) space; and (c) the physical (u) space. These
spaces are connected via two conformal transformations X1 and X2 .
Table 2. Equivalent points in the three spaces in figure 3.
target (z) space
virtual (h) space
physical (u) space
0 + ix
–x2 + i0
i(x + h)
0
0
0 + ih
r + i0
r2 + i0
r + ih
1 + i0
1 + i0
r + i0
where h is a dummy variable. The values of C in equation (5.1) and r in
figure 3b are decided by the special points in figure 3c. These values are ultimately
determined by the values of r and the ratio r/h, via the relationships (see
electronic supplementary material (b)):
rES (r)
r
=
(5.2)
h
1 − r 2 ES ( 1 − r 2 )
and
C=
r
.
2rES (r)
(5.3)
From equation (5.2), one sees that r is an implicit function of (r/h). For
practical field emitters, it is always true that (r/h) 1. In this case, r is a positive
number given by
r 1/2
.
(5.4)
r → 2p−1/2
h
Here, and in equations below, the form of the relevant expression when r is small
is shown on the right of an arrow. For large values of r/h, r becomes unity. For
given r, a parameter R is defined by
R = rES (r).
(5.5)
In the limit where (r/h) 1 and hence r → 0, ES (r) → pr/4 (appendix A), and
r R→
.
(5.6)
h
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CFE from nanowall emitters
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The point 0 + ix in target space corresponds to point 0 + i(x + h) in physical
space. The value of x is given implicitly in terms of r and the ratio x/r by (see
electronic supplementary material (c))
1 + x2
Rx
,
(5.7)
− E(q|1 − r2 ) + r2 F (q|1 − r2 ) =
x 2
2
r +x
r
where F (4|m) and E(4|m) are the incomplete elliptic integrals of the first and
second kinds, expressed in terms of amplitude 4 and elliptic parameter m (see
appendix A for definitions), and q is given by
x
.
(5.8)
q = arcsin r2 + x 2
In the target space, the anode is taken as represented by a straight line parallel
to the real axis and at a distance xa from it. In the physical space, the anode
intersects the imaginary axis at the value i(h + xa ). The relationship between
these two parameters is given by (see the electronic supplementary material (d))
r
x
xa
≈ lim = → h.
xa x→∞ x R
(5.9)
For large cathode–anode separation in the physical space, we have xa h and
FM ≈ Vapp /xa . Thus, the uniform ‘field’ Fx along the imaginary axis of the target
space is the positive quantity:
Fx =
Vapp Vapp r
r
=
≈ FM → hFM .
xa
xa R R
(5.10)
Although Fx has the function of a field, it formally has the dimensions of
electrostatic potential.
It follows that in the target space the electrostatic potential at point 0 + ix
is Fx x. Thus, in the physical space, the variation of electrostatic potential V (x)
along the x-axis is
(5.11)
V (x) = Fx x(x),
where 0 + ix(x) is the point in the target space that corresponds to point 0 +
i(h + x) in real space.
To proceed further, an expression for dx/dx is needed. Electronic
supplementary material (e) shows that
R
1 + x2
dx
.
(5.12)
=
dx
r
r2 + x 2
Using equations (5.9) and (5.10), the local field Floc (x) in the physical space is
dV dx
dx
1 + x2
dV
=
= Fz
= FM 2
.
(5.13)
Floc (x) =
dx
dx dx
dx
r + x2
The Schwartz–Cristoffel transformation ensures that the point 0 + ih in real space
maps into the point 0 + i0 in target space. Thus, the local field (FS ) at position S
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X.-Z. Qin et al.
(i.e. at x = 0) is found by putting x = 0 into equation (5.13); in the limit of small
r the field enhancement factor gS becomes
Floc (0) 1
p1/2 h 1/2
gS =
= →
.
(5.14)
FM
r
2
r
The outer classical turning point at 0 + i(h + xn ) in real space corresponds to
the point 0 + ixn in target space, where xn is given by
xn ≡ x(xn ) =
Hn
Hn
RHn
→
.
=
eFx eFM r
eFM h
(5.15)
To support this analytical treatment, solutions of Laplace’s equation for
the classical-nanowall geometry have been obtained by numerical (finite
element) methods (Qin et al. 2010). Analytical and numerical solutions are in
good agreement.
6. Detailed cold field electron emission theory for a classical nanowall
(a) The parameter Gn
This section examines the parameters Gn and dn that appear in expression (4.15)
for the contributionJL,n (FM ) from sub-band SBn to the total line current density.
Starting from equation (4.6), and using equations (4.7), (5.11) and (5.12), we get
xn xn √
1 − x/xn
1/2
Hn − eV (x, FM ) dx = ge Hn
dx
Gn (FM ) = ge
dx/dx
0
0
xn r (1 − x/xn )(r2 + x2 )
=
dx.
(Hn )1/2 ge
R
1 + x2
0
Changing the variable of integration to t = x/xn yields
1 xn r
(1 − t)(r2 + x2n t 2 )
dt.
(Hn )1/2 ge
Gn (FM ) =
R
1 + x2n t 2
0
We now introduce a parameter Gn defined by
1 (1 − t)(r2 + x2n t 2 )
3
dt.
Gn−1 =
2 0
1 + x2n t 2
(6.1)
(6.2)
(6.3)
Equation (5.15) shows that the factor (xn r/R) in equation (6.2) is equal to
Hn /eF M . Hence
Gn (FM ) =
2ge Hn3/2
bHn3/2
=
,
3eGn FM Gn FM
where b [≡ 2ge /3e] is the second FN constant, as before.
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CFE from nanowall emitters
1045
(b) The parameter dn
From equation (4.8), using substitutions similar to those in §6a
g xn
e
dn−1 =
[Hn − eV (x)]−1/2 dx
2 0
and
dn−1
r
= Hn−1/2 ge
R
xn 0
r2 + x 2
dx.
(1 − x/xn )(1 + x2 )
Changing the variable of integration to t = x/xn yields
1 r
r2 + x2n t 2
x
n
Hn−1/2 ge
dt.
dn−1 =
R
(1 − t)(1 + x2n t 2 )
0
A parameter Qn may be introduced by
1 r2 + x2n t 2
1
Q−1
dt.
≡
n
2 0 (1 − t)(1 + x2n t 2 )
Putting (xn r/R) = (Hn /eFM ), as above, yields
e
dn = Qn
Hn−1/2 FM .
ge
(6.5)
(6.6)
(6.7)
(6.8)
(c) Limiting expressions for Gn and Qn
In general, the correction factors Gn and Qn must be calculated numerically.
However, useful approximate expressions exist because the term (r2 + x2n t 2 )1/2
that appears in the integrands in equations (6.3) and (6.7) can take different
limiting forms, depending on the relative sizes r and xn t.
(i) The ‘slowly varying field’ approximation: xn << r
In the limit, where xn << r (i.e. when the barrier length xn is small when
compared with the nanowall half-width r), use of a computer algebra package
shows that
3r 1 (1 − t)
−1
dt ≈ r ≈ g−1
(6.9)
Gn ≈
S
2 0 1 + x2n t 2
and
1
Q−1
n ≈ r
2
1 0
t2
dt ≈ r ≈ g−1
S .
(1 − t)(1 + x2n t 2 )
(6.10)
We call this the ‘slowly varying field’ approximation. In this limit, the decay of
the field strength with distance along the x-axis is relatively slow, and the barrier
is ‘nearly triangular’. Thus, the barrier-shape correction factor nn ≈ 1, the decayrate correction factor tn ≈ 1, and the correction factors Gn and Qn both become
effectively equal to the field enhancement factor gS .
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X.-Z. Qin et al.
(ii) The ‘sharply varying field’ approximation: r << xn
The opposite limit r << xn corresponds physically to the situation where the
nanowall half-width r is small in comparison with the barrier length xn . We call
this the ‘sharply varying field’ approximation. In this limit, the use of a computer
algebra package and equation (5.15) shows that
1 2Hn
(1 − t)t 2
3
2xn
Gn−1 ≈ xn
≈
(6.11)
dt ≈
2
2
2
1 + xn t
5
5eFM h
0
and
Q−1
n ≈
xn
2
1
0
2Hn
t
2xn
≈
.
dt ≈
3
3eFM h
(1 − t)(1 + x2n t 2 )
(6.12)
The issue of which approximation is more appropriate in a given situation depends
both on the nanowall half-width r, and on field strengths near position S, since
the latter determine the tunnelling barrier width xn .
(d) Cold field electron emission equations
If the nanowall is sufficiently thin (i.e. if w is sufficiently small that
(G2 − G1 ) > 1), then emission from the n = 1 sub-band is dominant, and (by using
equation (4.16)) the characteristic area current density JA (FM ) for the nanowall
can be approximated by
3/2
−bH
J1 (FM )
3/2
−3/4
3/2
1
JA (FM ) ≈
.
(6.13)
= a nw w −1 Q1 H1 FM exp
G1 FM
w
Comparisons show that G1 is a generalized replacement for the combination gS /n1
in equation (4.16), and Q1 a generalized replacement for gS /t1 .
In the ‘slowly varying field’ limit, equations (6.9) and (6.10) show that a CFE
equation is obtained from equation (6.13) by replacing both G1 and Q1 by gS :
3/2
−bH1
nw −1 −3/4 3/2 3/2
JA (FM ) ≈ a w H1 gS FM exp
.
(6.14)
gS FM
This is simply (4.16) with n1 = 1, t1 = 1; gS is given by equation (5.14).
In the ‘sharply varying field’ limit, equations (6.11) and (6.12), with n = 1, can
be used to substitute for G1 and Q1 in equation (6.13), leading to
3/2 5/2
−(2/5)bH
h
−9/4
3
1
JA (FM ) ≈ a svf
exp
,
(6.15)
H1 FM
2
w
ehFM
where a svf is an universal constant given by
a svf ∼
= 1.983410 × 10−10 A m−5/2 eV9/4 V−3 m3 .
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CFE from nanowall emitters
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Equation (6.15) again has the form of equation (4.16), with gS given by equation
(5.14), but here n1 and t1 are functions of H1 and FM (or FS ), given by
n1 ≈
(p1/2 /5)H1
(p/10)H1
=
1/2
1/2
eFM r h
eFS r
(6.17)
and
(p1/2 /3)H1
(p/6)H1
.
(6.18)
=
1/2
1/2
eFM r h
eFS r
Clearly, n1 and t1 tend to become large as r becomes small for constant h and
FM , or as FS becomes small for constant r, or as FM becomes small for constant
r and h. In the opposite limit, formulae (6.17) and (6.18) break down as we move
back towards the ‘slowly varying field’ limit, and n1 and t1 approach unity.
t1 ≈
(e) Comment
Obviously, equation (6.14) would generate a nearly straight line in an FN plot
or a Millikan–Lauritsen (ML) plot, but the power of FM in the pre-exponential
is different from that predicted by the elementary FN-type equation. This
difference underlines the merit (when the FN plot or ML plot is nearly straight)
of attempting to measure the empirical power of FM (or, equivalently, applied
voltage) in the pre-exponential, as proposed elsewhere (Forbes 2008a).
On the other hand, equation (6.15) would generate a curved line in an
FN or ML plot, but would generate a straight line in a plot of type [ln{JA }
2
].
versus 1/Vapp
7. Discussion
(a) Summary
This paper sets out (in §2) a general approach for developing theoretical
CFE equations, and has applied it to a thin nanowall. Although simplifying
assumptions are used (equivalent to those of FN and others when analysing CFE
from bulk metals), the resulting emission theory is complex.
A major feature results from the ‘quantum confinement’ in the direction of the
nanowall width (i.e. phase-locking of the relevant wave function component, and
consequent quantization of the associated energy component Wz ). This requires
the nanowall electron states to be grouped into sub-bands defined by a quantum
number n that determines the allowed values of Wz . Because part of its total
energy must be ‘used up’ to provide this lateral quantization energy, a Fermilevel electron sees a tunnelling barrier of zero-field height greater than the local
thermodynamic work-function.
Options within the theory depend on the absolute size of the nanowall halfwidth r, and on the relative sizes of the nanowall height h, its half-width r and
the lengths xn of the tunnelling barriers for the references states (Rn) associated
with the various sub-bands. Values of xn are influenced by the value of the applied
macroscopic field FM or electrostatic potential Vapp .
In respect of electron supply, if r is sufficiently small, then the theory can be
simplified by assuming that only the lowest sub-band contributes significantly to
emission. This has been called the ‘thin nanowall’ case. The physical condition
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X.-Z. Qin et al.
assumed is that the transmission coefficient for reference state R2 should be less
than that for R1 by a factor of at least e (≈2.7). This requires (G2 − G1 ) > 1.
Emission from the n = 1 sub-band would also be dominant if the states in higher
sub-bands were unoccupied, but we consider the transmission coefficient condition
to be more critical.
A second effect is produced by the possible influence of a sharp emitter on
the tunnelling barrier shape, in particular on how quickly the local electric field
decays with distance from the emitting surface, over a distance comparable with
tunnelling barrier length. For a thin nanowall, if this decay is slow (so the barrier
is nearly exactly triangular), then equation (6.14) applies; if this decay is fast (as
illustrated in figure 2), then equation (6.15) applies.
(b) Conditions of applicability
Because of (i) interconnections between different parts of the theory, (ii) the
absence of clear transitions, and (iii) the existence of other constraints, it is
difficult to formulate reliable, precise conditions for the clear existence of the
emission situations discussed above. The following treatment aims to be indicative
rather than exact.
The issue of dominance of emission from the n = 1 sub-band depends mainly
on the nanowall width. The condition (G2 − G1 ) > 1 requires
3/2
3/2
H1
H2
b
−
> 1.
(7.1)
FM
G2
G1
If, in addition, the ‘slowly decaying field approximation’ applies, then G2 = G1 =
3/2
3/2
gS , and equation (7.1) becomes (b/FS )(H2 − H1 ) > 1. Using equation (4.7)
2
to write Hn = f + n Wz1 , expanding this by the binomial theorem, and using
equation (3.3) for Wz1 , we get the requirement
9 hP2 bf1/2
f1/2
≈ (2.889698 eV−1/2 V nm−1 ) ×
,
(7.2)
64 me FS
FS
f can be taken as 5 eV. It is difficult to guess FS accurately, so we assume FS ∼
5 V nm−1 . This yields the condition r < 1.1 nm.
If emission from the n = 1 sub-band is dominant, then the condition for the
‘slowly varying field’ approximation to be the better approximation is r > x1 .
Using equations (1.1), (5.4), (5.14) and (5.15), we obtain
p H1
.
(7.3)
r>
4 eFS
Approximating H1 ∼ f ≈ 5 eV, FS ∼ 5 V nm−1 , yields r > 0.8 nm.
These results suggest that (for given values of h and FS ) there is a narrow
range of r-values near 1 nm where emission from the n = 1 sub-band is dominant,
and where we expect an FN or ML plot to be nearly straight (but where we
expect the zero-field barrier height H1 to be greater than the local thermodynamic
work-function f). For larger r-values, there is a slow transition towards FN-like
behaviour. For smaller r-values, there is a transition towards behaviour described
by equation (6.15). However, one might expect that use of a square PE well (to
describe lateral quantum-confinement) would become an inadequate model when
r approaches atomic dimensions.
r2 <
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CFE from nanowall emitters
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In principle, in such circumstances, one should develop atomic-theory-based
quantum-mechanical models for the band-structure of a thin slab comprising
several layers of atoms, but this is beyond the scope of the present paper. When
the width of the nanowall is a few atoms, then one might expect equation (6.15)
to be unreliable in detail. However, the effects associated with the barrier shape
relate to the electrostatics of the field distribution above the nanowall; thus,
they should physically occur for nanowalls with half-width of order 1 nm or less,
whatever quantum-mechanical model one uses for the thin nanowall. Hence, there
is a more general expectation that, for nanowall emitters of this size, FN and ML
plots may be curved.
8. Conclusions
Our main conclusions are as follows: in the CFE regime, FN-type equations will
not describe the behaviour of a nanowall field emitter of very small width. A
suitable criterion for significant departure from FN-type behaviour might be a
width w of around 2 nm, or below.
With an emitter as thin as this, there are two intrinsically linked possible
consequences: (i) quantum confinement effects; and (ii) barrier effects that occur
because the local field falls off rapidly with distance from the surface (hence the
electrostatic PE variation becomes ‘cusp-like’). Detailed consequences of having
quantum confinement and a cusp-like barrier, in particular effects on the form of
theoretical CFE equations for ECD, have been described.
The best experimental approach for detecting the effects described here may
be to look first for curvature in measured FN or ML plots (which is a known
experimental phenomenon with carbon field emitters, but which may have other
causes), and then for current versus voltage behaviour that specifically conforms
with equation (6.15).
Although there is previous work that investigates either cusp-like barrier effects
(e.g. Cutler et al. 1993; Edgcombe & de Jonge 2006) or quantum-confinement
effects for small emitters (e.g. Liang & Chen 2008), we believe that this paper
is the first to look at the operation of both together, for a particular emitter
geometry (the thin nanowall). We have felt that the conceptual structure of the
physical results would be best brought out by using a relatively naive analytical
model. Clearly, there is scope both for detailed numerical investigations that
extend and amplify the present work, and for related investigations based on
detailed atomic-level models of the band structure.
More generally, we consider that it will be important to co-investigate
quantum confinement and barrier-shape effects for other experimentally plausible
emitter geometries. Expectation is that emission from such emitters will
conform to equations that are both different from FN-type equations, and
different in detail from the equations presented here. What we expect,
eventually, is that CFE will be described by several families of theoretical
equations, with each family corresponding to a particular type of quantumconfinement arrangement (nanowall, post, thin-slab, localized-state, etc.).
FN-type equations are the family that applies when no significant quantumconfinement effects occur.
Proc. R. Soc. A (2011)
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1050
X.-Z. Qin et al.
The present work also underlines the need to make a clear distinction in CFE
theory between the concepts of (i) local thermodynamic work-function f, and (ii)
zero-field barrier height HR for a particular reference state.
The project is supported by the National Natural Science Foundation of China (grant nos 10674182,
10874249 and 90306016) and the National Basic Research Programme of China (2007CB935500
and 2008AA03A314).
Appendix A. Elliptic integrals and related functions
The incomplete elliptic integrals of the first and second kinds, F (4|m) and
E(4|m), respectively, are defined in terms of the elliptic parameter m and the
amplitude 4 by
4
F (4|m) =
(1 − m sin2 w)−1/2 dw
(A 1)
(1 − m sin2 w)+1/2 dw.
(A 2)
0
4
and
E(4|m) =
0
The complete elliptic integrals of the first and second kinds, K (m) and E(m),
respectively, are defined by
p p |m
and E(m) = E
|m .
K (m) = F
2
2
It is convenient, here, to introduce a special function ES (y), related to E(4|m)
and defined by
ES (y) = E(arcsin(y)|y−2 )
arcsin(y)
(1 − y−2 sin2 w)1/2 dw.
=
(A 3)
0
When y → 0, then arcsin(y) → y and the range of integration in w becomes
small. Then
y w2
p
(A 4)
1 − 2 dw = y.
ES (y) ≈
y
4
0
Additional mathematical details may be found in the electronic supplementary
material.
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