## Abstract

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 FN-type equations. Thus: (i) the zero-field barrier height *H*_{R} for the highest occupied state at 0 K is not equal to the local thermodynamic work-function *ϕ*, and *H*_{R} rather than *ϕ* appears in equations; (ii) in the exponent, the power dependence on macroscopic field *F*_{M} can be *F*^{−2}_{M} rather than *F*^{−1}_{M}; (iii) in the pre-exponential, explicit power dependences on *F*_{M} and *H*_{R} 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.

## 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 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. ‘FN-type 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 honeycombe-type 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.

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 2008*b*) 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’ *F*_{M} would exist in its absence, a ‘field enhancement factor’ *γ* can be defined by
1.1
where *X*_{a} is the plate separation, *V* _{app} 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 *F*_{M} (Forbes 2004). This elementary equation gives the ECD *J*_{A} (current per unit area) at zero temperature, as a function of *F*_{M} and the local work-function *ϕ*. It can be written in the equivalent forms:
1.2
Here, *a* and *b* are the first and second FN constants, and *z*_{S} the Sommerfeld supply density, as defined in table 1. *d*^{el}_{F} is the elementary approximation for a parameter *d*_{F} 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).

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, *m*_{e} the electron mass, *h*_{P} Planck’s constant and *k*_{B} 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 d*V* /d*x* rather than −d*V* /d*x*. 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 *F*_{M}) is initially used as the independent field-like variable.

Establishing an expression for local ECD (either current per unit area *J*_{A}, or current per unit length *J*_{L}), needs three basic steps. (i) A reference emitter electron state ‘R’ is chosen, and the zero-field height *H*_{R} (see below) of the barrier seen by an electron in this state is established. (ii) The escape probability (transmission coefficient) *D*_{R} and decay width *d*_{R} for this barrier are calculated, usually by an approximate quantum-mechanical method. (iii) The 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
2.1
where *T* is the emitter’s thermodynamic temperature, *J*(*F*,*T*) is the ECD, and *Z*_{R}(*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 *Z*_{R} are different for different CFE equation families, but transmission coefficient (*D*_{R}) 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*)−*E*_{n}, where *U*(*l*,*F*) is the PE that appears in the Schrödinger equation. *E* is the total electron energy, and *E*_{n} its component associated with motion in the tunnelling direction (with *E*, *E*_{n} and *U*(*l*,*F*) all referred to the same energy-zero). We call *E*_{n} 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
2.2
where *g*_{e} 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
2.3
where *ν* (‘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
2.4
where *P* is a ‘transmission prefactor’ discussed in Forbes (2008*b*). 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’ *c*_{d} or a ‘decay width’ *d*( ≡ 1/*c*_{d}) defined by
2.5
The literature uses both decay rate (e.g. *c*_{0} in Modinos (1984), or *β*_{F} in Jensen (2007)) and decay width (e.g. *d* in Gadzuk & Plummer (1973)). Decay width is used here because the dimensionless quantity (*πk*_{B}*T*/*d*), involving the Boltzmann factor *k*_{B}*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, *ν* → 1, *G* → *bH*^{3/2}/*F* and obtain the ‘elementary’ variants of *d* and *c*_{d} as
2.6
where the derivation of *b* has been used (table 1). In more general situations, the outcome of definition (2.5) is written
2.7
where *τ* is a ‘decay-rate correction factor’ defined by equation (2.7), and obtained by evaluating definition (2.5). The factor *τ* 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 *τ* 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* = *γF*_{M} in equations (2.2) and (2.7), we get
2.8
and
2.9
For states with *H* close to *H*_{R}, a Taylor-type expansion allows *D*(*F*,*H*) to be approximated. If only the linear term is used (which is the customary approximation), then
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

**approaching the emitter surface makes a contribution**

*Q**j*

_{Q}to the total ECD

*J*, where 2.11 Here,

*eΠ*

_{Q}is the electron–current–density component (for state

**) approaching the emitting surface in a direction normal to it, and**

*Q**f*(

*E*

_{Q},

*T*) is the occupancy of state

**. The total ECD**

*Q**J*is obtained by summing-up the contributions from all relevant states: 2.12 Using equation (2.10), this can be put in the form of equation (2.1), with 2.13 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 *ϕ*. 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 *d*^{2}_{F} into the result. Forbes (2004) shows that (for the simpler, commonly used, forms of FN-type equation, at 0 K):
2.14
where *Z*_{F}(*F*) is the effective supply for state ‘F’, *z*_{S} is the Sommerfeld supply density (table 1), and *d*_{F} the decay width for state F. For the elementary FN-type equation, is obtained by replacing *d*_{F} by *d*^{el}_{F}.

For FN-type equations, the results for the ET barrier and the elementary FN-type 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
3.1
where *W*_{i} (*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
3.2
where *p*_{x} and *p*_{y} are the momenta in the *x* and *y* directions, respectively.

The component *W*_{z} is quantized, and its allowed values are given approximately by the values *W*_{zn} for an infinitely deep PE well, namely
3.3
where *w* is the well width, *n* a positive integer that defines the electron’s ‘vibrational level’ and *W*_{zn} and *W*_{z1} are defined by this equation (*W*_{z1} is the value of *W*_{z} associated with the *n* = 1 level). This quantization of *W*_{z} means that the electron states in the nanowall are split into sub-bands SB1,SB2,…SB*n*…, 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
3.4
Here,

*W*

_{F}is the Fermi energy (i.e. the total energy of an electron at the Fermi level). For sub-band SB

*n*, the Fermi–Dirac occupation probability can be written as 3.5

For sub-band SB*n*, the number of electrons from SB*n* 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 *W*_{x}, is called the ‘line supply function for sub-band SB*n*’, and is denoted by *N*_{L,n}(*W*_{x},*T*). This is obtained by integrating over the allowed range of *p*_{y}, taking occupancy of electron states into account. Thus
3.6
*N*_{L,n}(*W*_{x},*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 *F*_{S} at S. This field *F*_{S} is related to the macroscopic field *F*_{M} by equation (1.1), which defines a field enhancement factor *γ*_{S} that is independent of *F*_{M} and is discussed further below. In what follows, it will often be convenient to show field dependences as functions of *F*_{M} rather than *F*_{S}.

Figure 2 shows schematically the variation, along the *x*-axis, of the electron PE *U*(*x*,*F*_{M}) (measured relative to the well base). Because we disregard image-type effects, *U*(*x*,*F*_{M}) is given by
4.1
where *W*_{o} is the nanowall inner PE (or ‘electron affinity’), *ϕ* is the local thermodynamic work-function of its surface and *V* (*x*,*F*_{M}) 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 *F*_{M}.

For states with forwards energy *W*_{x} < *W*_{o}, there exists a tunnelling barrier of zero-field height *H* = *W*_{o}−*W*_{x}. Thus, any formula involving *H* can alternatively be written in terms of *W*_{x}. Owing to disregard of image-type effects, the barrier’s inner edge is at *x* = 0. Its outer edge is at *x* = *L*, where
4.2

In the CFE regime, equation (2.4) is an adequate expression for transmission coefficient. With the approximation *P* → 1, the transmission coefficient *D*(*W*_{x},*F*_{M}) for an electron approaching the nanowall top surface with forwards energy *W*_{x} becomes
4.3a
where *G*(*W*_{x},*F*_{M}) is derived, from equation (2.2) and associated definitions, by the replacements *l* → *x*, *U* → *W*_{o}−*eV* (*x*,*F*_{M}) and *E*_{n} → *W*_{x}. This makes *H* = *W*_{o}−*W*_{x}, and yields
4.3b

*D*(*W*_{x},*F*_{M}) decreases exponentially as *W*_{x} decreases; thus, emission comes preferentially from occupied states with *W*_{x} 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 *W*_{x} is
4.4
which corresponds to a state with *n* = 1 and *p*_{y} = 0. This can be specified as the reference state (‘R1’) for sub-band SB1; *W*_{xR1} is the corresponding forwards energy.

More generally, for states at the Fermi level, the largest possible value of forwards energy *W*_{x} for states in sub-band SB*n* is the forwards energy *W*_{xRn} for the corresponding reference state R*n*:
4.5
The JWKB exponent for the barrier related to state R*n* is denoted by *G*_{n}(*F*_{M}) and given by
4.6
where *x*_{n} is the outer classical turning point for the barrier seen by an electron in state R*n*, and is found by solving equation (4.2), with *L* = *x*_{n}, *H* = *ϕ* + *W*_{zn}.

Thus, in respect of its escape by tunnelling, an electron in state R*n* sees a barrier of zero-field height *H*_{n} given by
4.7
The physical reason why *H*_{n} is greater than the local thermodynamic work-function *ϕ* is that the electron has to ‘use up’ some of its kinetic energy to provide the energy *W*_{zn} associated with its localization in the *z* direction. Obviously, if *W*_{zn} > *W*_{F}, 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 d*H* = −d*W*_{x}, we obtain the decay width *d*_{n} for reference state *Rn* as
4.8

### (b) Characteristic line current density

The present model derives an expression for a quantity *J*_{L}(*F*_{M},*T*) called the ‘characteristic line current density’. *J*_{L}(*F*_{M},*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 *J*_{L,n}(*F*_{M},*T*) from sub-band SB*n* to *J*_{L}(*F*_{M},*T*) is
4.9
The lower limit on *W*_{x} can be taken as , because *D*(*W*_{x},*F*_{M}) gets rapidly smaller as *W*_{x} decreases.

For reference state R*n*, the difference −(*H*−*H*_{R}) in equation (2.10) can alternatively be written
4.10
Combining equations (2.10), (3.5), (3.6), (4.9) and (4.10) yields
4.11
By reversing the order of integration, and defining , the double integral reduces to
4.12
The integral in *u* is a standard form that results in a temperature-dependent correction factor , and the resulting integral over *p*_{y} yields the extra factor (2*πm*_{e}*d*_{n})^{1/2}. Thus, equation (4.11) reduces to
4.13
where *z*^{nw} is an universal constant with the value
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 temperature-dependent 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 4.15

The characteristic line current density *J*_{L}(*F*_{M}) 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 *J*_{L,n}≪*J*_{L,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 (*G*_{2}−*G*_{1}) > 1. In such circumstances, we can neglect emission from higher sub-bands and take *J*_{L}(*F*_{M}) ≈ *J*_{L,1}(*F*_{M}). 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’ *J*^{tnw}_{A}(*F*_{M}) by dividing *J*_{L,1}(*F*_{M}) by the nanowall width *w*, and then using equations (2.8) and (2.9) to expand the resulting expression, giving
4.16a
4.16b
and
4.16c
where *ν*_{1} and *τ*_{1} are the barrier-shape and decay-rate correction factors for reference state R1, and *a*^{nw} is an universal constant given by
4.17

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 zero-field barrier height *H*_{1} is greater than the local thermodynamic work-function *ϕ* by the amount *W*_{1}, and a barrier-shape correction factor is present; in the pre-exponential, the power to which the decay width is raised is 3/2 rather than 2, and the term (*z*^{nw}*w*^{−1}) has replaced *z*_{S}. When, as in equation (4.16c), the term is expanded, *H*_{R} (rather than *ϕ*) appears in the pre-exponential, the powers to which *F*_{M} and *H*_{R} 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 pre-exponential 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 phase-locking’ 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 *G*_{n} and *d*_{n} in equation (4.14) depend on how the electrostatic potential *V* (*x*,*F*_{M}) varies near the nanowall top surface. *V* (*x*,*F*_{M}) could, in principle, be obtained by using a numerical Poisson solver, and *G*_{n} and *d*_{n} 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*,*F*_{M}) by means of the conformal transformation illustrated in figure 3.

The overall transformation *Ξ* combines two stages: *Ξ*_{1} transforms the first quadrant in ‘target space’ (figure 3*a*) into the positive half-plane in ‘virtual space’ (figure 3*b*), and *Ξ*_{2} then transforms this half-plane into a shape in ‘physical space’ (figure 3*c*) that represents the right-hand half of the nanowall. Positions in the three spaces are represented by the complex numbers *ζ*, *η* and *ω*, respectively, using axes as shown, with *i* ≡ √(−1).

In figure 3*c*, *h* is the nanowall height and *r* its half-width (*r* = *w*/2). The *x*-origin is at 0 + i*h*, and 0 + i(*h* + *x*) is an arbitrary point on the *x*-axis. The anode applying the field is at a distance (*x*_{a} = *X*_{a}−*h*) such that *x*_{a}≫*h*. Table 2 shows equivalences between special points in the three spaces.

To simplify equations, we use a mathematical function *E*_{S}(*υ*) defined in appendix A. Transformation *Ξ*_{1} is simply: *η* = *ζ*^{2}. Transformation *Ξ*_{2} is the Schwarz–Christoffel transformation (Driscoll &Trefethen 2002)
5.1
where *η*′ is a dummy variable. The values of *C* in equation (5.1) and *ρ* in figure 3*b* are decided by the special points in figure 3*c*. These values are ultimately determined by the values of *r* and the ratio *r*/*h*, via the relationships (see electronic supplementary material (*b*)):
5.2
and
5.3

From equation (5.2), one sees that *ρ* is an implicit function of (*r*/*h*). For practical field emitters, it is always true that (*r*/*h*)≪1. In this case, *ρ* is a positive number given by
5.4
Here, and in equations below, the form of the relevant expression when *ρ* is small is shown on the right of an arrow. For large values of *r*/*h*, *ρ* becomes unity. For given *ρ*, a parameter *R* is defined by
5.5
In the limit where (*r*/*h*)≪1 and hence *ρ* → 0, *E*_{S}(*ρ*) → *πρ*/4 (appendix A), and
5.6
The point 0 + i*ξ* in target space corresponds to point 0 + i(*x* + *h*) in physical space. The value of *ξ* is given implicitly in terms of *ρ* and the ratio *x*/*r* by (see electronic supplementary material (*c*))
5.7
where *F*(*φ*|*m*) and *E*(*φ*|*m*) are the incomplete elliptic integrals of the first and second kinds, expressed in terms of amplitude *φ* and elliptic parameter *m* (see appendix A for definitions), and *θ* is given by
5.8

In the target space, the anode is taken as represented by a straight line parallel to the real axis and at a distance *ξ*_{a} from it. In the physical space, the anode intersects the imaginary axis at the value i(*h* + *x*_{a}). The relationship between these two parameters is given by (see the electronic supplementary material (*d*))
5.9

For large cathode–anode separation in the physical space, we have *x*_{a}≫*h* and *F*_{M} ≈ *V* _{app}/*x*_{a}. Thus, the uniform ‘field’ *F*_{ξ} along the imaginary axis of the target space is the positive quantity:
5.10
Although *F*_{ξ} 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 + i*ξ* is *F*_{ξ}*ξ*. Thus, in the physical space, the variation of electrostatic potential *V* (*x*) along the *x*-axis is
5.11
where 0 + i*ξ*(*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 d*ξ*/d*x* is needed. Electronic supplementary material (*e*) shows that
5.12
Using equations (5.9) and (5.10), the local field *F*_{loc}(*x*) in the physical space is
5.13
The Schwartz–Cristoffel transformation ensures that the point 0 + i*h* in real space maps into the point 0 + i0 in target space. Thus, the local field (*F*_{S}) at position S (i.e. at *x* = 0) is found by putting *ξ* = 0 into equation (5.13); in the limit of small *ρ* the field enhancement factor *γ*_{S} becomes
5.14

The outer classical turning point at 0 + i(*h* + *x*_{n}) in real space corresponds to the point 0 + i*ξ*_{n} in target space, where *ξ*_{n} is given by
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 G_{n}

This section examines the parameters *G*_{n} and *d*_{n} that appear in expression (4.15) for the contribution *J*_{L,n}(*F*_{M}) from sub-band SB*n* to the total line current density. Starting from equation (4.6), and using equations (4.7), (5.11) and (5.12), we get
6.1
Changing the variable of integration to *t* = *ξ*/*ξ*_{n} yields
6.2
We now introduce a parameter *Γ*_{n} defined by
6.3
Equation (5.15) shows that the factor (*ξ*_{n}*r*/*R*) in equation (6.2) is equal to *H*_{n}/*eF*_{M}. Hence
6.4
where *b* [ ≡ 2*g*_{e}/3*e*] is the second FN constant, as before.

### (b) The parameter d_{n}

From equation (4.8), using substitutions similar to those in §6*a*
and
6.5
Changing the variable of integration to *t* = *ξ*/*ξ*_{n} yields
6.6
A parameter *Θ*_{n} may be introduced by
6.7
Putting (*ξ*_{n}*r*/*R*) = (*H*_{n}/*eF*_{M}), as above, yields
6.8

### (c) Limiting expressions for *Γ*_{n} and *Θ*_{n}

In general, the correction factors *Γ*_{n} and *Θ*_{n} must be calculated numerically. However, useful approximate expressions exist because the term that appears in the integrands in equations (6.3) and (6.7) can take different limiting forms, depending on the relative sizes *ρ* and *ξ*_{n}*t*.

#### (i) The ‘slowly varying field’ approximation: *ξ*_{n} < < *ρ*

In the limit, where *ξ*_{n} < < *ρ* (i.e. when the barrier length *x*_{n} is small when compared with the nanowall half-width *r*), use of a computer algebra package shows that
6.9
and
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 *ν*_{n} ≈ 1, the decay-rate correction factor *τ*_{n} ≈ 1, and the correction factors *Γ*_{n} and *Θ*_{n} both become effectively equal to the field enhancement factor *γ*_{S}.

#### (ii) The ‘sharply varying field’ approximation: *ρ* < < *ξ*_{n}

The opposite limit *ρ* < < *ξ*_{n} corresponds physically to the situation where the nanowall half-width *r* is small in comparison with the barrier length *x*_{n}. We call this the ‘sharply varying field’ approximation. In this limit, the use of a computer algebra package and equation (5.15) shows that
6.11
and
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 *x*_{n}.

### (d) Cold field electron emission equations

If the nanowall is sufficiently thin (i.e. if *w* is sufficiently small that (*G*_{2}−*G*_{1}) > 1), then emission from the *n* = 1 sub-band is dominant, and (by using equation (4.16)) the characteristic area current density *J*_{A}(*F*_{M}) for the nanowall can be approximated by
6.13
Comparisons show that *Γ*_{1} is a generalized replacement for the combination *γ*_{S}/*ν*_{1} in equation (4.16), and *Θ*_{1} a generalized replacement for *γ*_{S}/*τ*_{1}.

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 *Γ*_{1} and *Θ*_{1} by *γ*_{S}:
6.14
This is simply (4.16) with *ν*_{1} = 1, *τ*_{1} = 1; *γ*_{S} 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 *Γ*_{1} and *Θ*_{1} in equation (6.13), leading to
6.15
where *a*^{svf} is an universal constant given by
6.16
Equation (6.15) again has the form of equation (4.16), with *γ*_{S} given by equation (5.14), but here *ν*_{1} and *τ*_{1} are functions of *H*_{1} and *F*_{M} (or *F*_{S}), given by
6.17
and
6.18
Clearly, *ν*_{1} and *τ*_{1} tend to become large as *r* becomes small for constant *h* and *F*_{M}, or as *F*_{S} becomes small for constant *r*, or as *F*_{M} 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 *ν*_{1} and *τ*_{1} approach unity.

### (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 *F*_{M} 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 *F*_{M} (or, equivalently, applied voltage) in the pre-exponential, as proposed elsewhere (Forbes 2008*a*).

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{*J*_{A}} versus 1/*V* ^{2}_{app}].

## 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 *W*_{z}). This requires the nanowall electron states to be grouped into sub-bands defined by a quantum number *n* that determines the allowed values of *W*_{z}. Because part of its total energy must be ‘used up’ to provide this lateral quantization energy, a Fermi-level 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 half-width *r*, and on the relative sizes of the nanowall height *h*, its half-width *r* and the lengths *x*_{n} of the tunnelling barriers for the references states (R*n*) associated with the various sub-bands. Values of *x*_{n} are influenced by the value of the applied macroscopic field *F*_{M} or electrostatic potential *V* _{app}.

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 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 (*G*_{2}−*G*_{1}) > 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 (*G*_{2}−*G*_{1}) > 1 requires
7.1
If, in addition, the ‘slowly decaying field approximation’ applies, then *Γ*_{2} = *Γ*_{1} = *γ*_{S}, and equation (7.1) becomes . Using equation (4.7) to write *H*_{n} = *ϕ* + *n*^{2}*W*_{z1}, expanding this by the binomial theorem, and using equation (3.3) for *W*_{z1}, we get the requirement
7.2
*ϕ* can be taken as 5 eV. It is difficult to guess *F*_{S} accurately, so we assume *F*_{S}∼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 *ρ* > *ξ*_{1}. Using equations (1.1), (5.4), (5.14) and (5.15), we obtain
7.3
Approximating *H*_{1}∼*ϕ* ≈ 5 eV, *F*_{S}∼5 V nm^{−1}, yields *r* > 0.8 nm.

These results suggest that (for given values of *h* and *F*_{S}) 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 *H*_{1} to be greater than the local thermodynamic work-function *ϕ*). 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.

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 quantum-confinement arrangement (nanowall, post, thin-slab, localized-state, etc.). FN-type equations are the family that applies when no significant quantum-confinement effects occur.

The present work also underlines the need to make a clear distinction in CFE theory between the concepts of (i) local thermodynamic work-function *ϕ*, and (ii) zero-field barrier height *H*_{R} for a particular reference state.

## Acknowledgements

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*(*φ*|*m*) and *E*(*φ*|*m*), respectively, are defined in terms of the elliptic parameter *m* and the amplitude *φ* by
A1
and
A2
The complete elliptic integrals of the first and second kinds, *K*(*m*) and *E*(*m*), respectively, are defined by
It is convenient, here, to introduce a special function *E*_{S}(*υ*), related to *E*(*φ*|*m*) and defined by
A3
When *υ* → 0, then arcsin(*υ*) → *υ* and the range of integration in *ϑ* becomes small. Then
A4
Additional mathematical details may be found in the electronic supplementary material.

- Received September 1, 2010.
- Accepted September 15, 2010.

- This journal is © 2011 The Royal Society