Tutorial Lecture on Field Electron Emission Theory

TUTORIAL LECTURE

INTRODUCTION TO FIELD ELECTRON EMISSION AND ITS THEORY

Richard G. Forbes

Advanced Technology Institute &

Department of Electronic Engineering, University of Surrey, UK

High-Electric-Field Nanoscience

Note*

This tutorial lecture was first presented at the International Vacuum Nanoelectronics Conference (IVNC 2013) in Roanoke, USA in July 2013. It was presented again, with small changes, at an international conference in Amman, Jordan, in April 2014. This version has been updated (in March 2015), to take account of more recent developments, and associated improvements in terminology and notation. Slides where changes have been made have an asterisk in the title. The main notational changes have been to:

change αM to αn , and change name to "notional area efficiency"; change λM to αf , and change name to "formal area efficiency"; introduce symbol Af defined by λCAn , and call Af "formal area".

A .pdf file of the original 2013 version of the presentation is available from the American Vacuum Society website, and is most easily found by a web search on the presentation title.

Why discuss theory ?

Theory is understanding, as well as mathematics. Theory can help to

•  Clarify language of discussion (e.g., different types of field) •  interpret/explain results •  establish limits of understanding •  characterize emitters •  suggest how to improve emitters •  avoid technological and financial disasters •  make predictions •  contribute to wider scientific progress (e.g., in tunnelling theory)

Status of field emission (FE) theory

There is a lot of FE theory around.

But, in places, basic FE theory is poorly presented or under investigation. [Historically, it probably lags behind where it ought to be by now.]

Also, FE theory is fragmented e.g., aspects of carbon nanotube FE theory link poorly to mainstream FE

theory.

In general terms, we need to do three things: •  Complete and consolidate basic theory; •  Use existing theory to simulate technologically interesting effects; •  Push out and try to predict/explain new effects and phenomena.

The need for field emission science

Especially with basic theory, need is for "proper field emission science" Need coherent intellectual structure

accommodating basic theory, practical theory and experiment Need better means of comparing theory with experiment

better procedures for existing data new forms of basic experiment.

Need to start from "simplest realistic model" (which is close to where many experimentalists are) & build outwards. This "simplest realistic model" involves

tunnelling through a Schottky-Nordheim (SN) barrier, using semi-classical (JWKB-like) mathematical methods summation over emitter states by using free-electron models technical completeness, but "no details of complications"

This tutorial is mainly about this "simplest realistic model".

An engineering physics approach

The traditional method of science is a "bottom up" approach in which one formulates a detailed hypothesis, calculates the implications, and compares these with experiment. I believe that some problems (including field emission) are too complicated and "messy" for this approach to work well by itself. Rather, I think that a "top down" approach more akin to engineering design (e.g., integrated circuit design) is appropriate. In this approach, one pays considerable attention to getting the overall structure of the problem complete, self-consistent and clearly explained, before making detailed comparisons between theory and experiment. This "top down" thinking influences my approach to field emission science.

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Preliminary issues

Basic terminology Electron emission sign convention Electrical surface and barrier field

Device macroscopic field True electrostatic field enhancement factor

Kernel current densities Emission area definitions & related topics

Conversion lengths Device parameters and circuit parameters

Universal FN variables and universal FN-type equation Scaled variables & equation for the SN barrier

Current-density regimes

Basic terminology

Fowler-Nordheim (FN) tunnelling = Electron tunnelling through an exact or rounded triangular barrier.

Deep tunnelling = Tunnelling well below the top of the energy barrier [which needs simpler tunnelling theory].

Cold field electron emission (CFE) = A statistical emission regime where (i) the electrons in the emitting region are effectively in local thermodynamic equilibrium, and (ii) most electrons escape by deep tunnelling from states close to the emitter's Fermi level.

Basic terminology

Fowler-Nordheim (FN) tunnelling = Electron tunnelling through an exact or rounded triangular barrier.

Deep tunnelling = Tunnelling well below the top of the energy barrier [which needs simpler tunnelling theory].

Cold field electron emission (CFE) = A statistical emission regime where (i) the electrons in the emitting region are effectively in local thermodynamic equilibrium, and (ii) most electrons escape by deep tunnelling from states close to the emitter's Fermi level.

For the last part of this name, there are several alternatives in the literature: Field electron emission Field emission Electron field emission (especially in the carbon literature) It's convenient to use the abbreviation "FE" - interpret it as you prefer.

Basic terminology

Fowler-Nordheim-type (FN-type) equations = A large family of approximate equations that (strictly) describe CFE from metal emitters that are "not too sharp" (apex radius greater than around 10 nm, say). Simple FN-type equations are also used as empirical fitting equations (e.g., for FN-plot analysis), but this has limited validity.

Technically complete equation = A FN-type equation is technically complete if it contains formal correction factors defined so that all physical effects relevant to the dependent variable are encompassed within the equation.

Characteristic value = Fields, current densities, and related parameters vary with position on an emitter surface. Characteristic values relate to some position (such as the emitter apex) considered characteristic of the emitter.

The original & elementary FN-type equations

The original FN-type equation (FN 1928), gives the local emission current density JL in terms of the local work-function φ and the local barrier field FL

JL = (aφ–1FL2) PF exp[–bφ3/2/FL ]

where: a and b are universal constants (sometimes called the first and second FN constants), PF is a tunnelling pre-factor, discussed later.

This equation applies to a large flat planar surface of a free-electron metal, but the derivation is "adequately applicable" to curved surfaces of large radius. In this case, interest is in characteristic values (e.g., emitter-apex values) of barrier field and current density, written FC and JC . So we write:

JC = (aφ–1FC2) PF exp[–bφ3/2/FC]

The elementary FN-type equation is a simplified version that omits PF :

JC = (aφ–1FC2) exp[–bφ3/2/FC]

Fowler-Nordheim & Schottky constants

Fundamental constants in FE customary units

Electron emission sign convention

In electron emission contexts, the convention is to treat fields, currents and current densities as positive, even though they would negative in classical electromagnetism.

"Fields" denoted by the symbol F in mainstream FE theory are the negatives of the corresponding classical electrostatic field E , i.e.

F = – E .

The more-recent FE convention of using E to denote the negative or magnitude of classical electrostatic field can be confusing in some contexts. It may be better to use F : this also allows E to be used for electron energy level (or for classical electrostatic field).

Electrical surface & barrier field

For an emitter, its electrical surface is the surface of a smooth classical conductor that models the real emitter surface. The electrical surface is "where the electrostatic field outside the surface seems to start".

Details of atomic structure are "effectively averaged". In practice, the electrical surface is near the outer electron edges of the surface atoms.

Electrical surface & barrier field

For an emitter, its electrical surface is the surface of a smooth classical conductor that models the real emitter surface. The electrical surface is "where the electrostatic field outside the surface seems to start".

Details of atomic structure are "effectively averaged". In practice, the electrical surface is near the outer electron edges of the surface atoms. The barrier field is the field that determines the tunnelling barrier shape and strength.

The local barrier field is the field at a local position in the electrical surface. The term "local" refers to lateral position across the emitter surface.

As just noted, FN-type equations use quantities defined at some characteristic lateral position.

Barrier field

Electrical Surface

Barrier field is slope here

Characteristic barrier field

characteristic barrier field FC here

surface protrusion

Device (or "true") macroscopic field*

True (or "device") macroscopic field FM is the (true) field in the protrusion's absence.

True (or "device") macroscopic field*

emission voltage Ve

separation of plane-parallel electrodes Δz

True (or "device") macroscopic field FM is the (true) field in the protrusion's absence. In general: FM = Ve / ζM , where ζM is the macroscopic conversion length (applies to both planar AND curved electrodes). In plane-parallel configuration ζM = Δz , so: FM ≈ Ve / Δz .

True (or "device") macroscopic field*

emission voltage Ve

separation of plane-parallel electrodes Δz

Characteristic (true) macroscopic field enhancement factor (FEF) γC is

γC = FC / FM .

["β " has three different meanings in FE literature, so I avoid using it here.]

FC

Electrostatic potential difference vs Voltage*

Strictly, the formula relating field FM to the electrode separation is

FM = ΔΦ /Δz ,

where ΔΦ is the electrostatic potential difference between the electrodes. The difference between ΔΦ and Ve is nearly always less than 1 volt, and the distinction is usually neglected.

Kernel current density*

The elementary FN-type equation for characteristic local current density is

JCel = (aφ–1FC

2) exp[–bφ3/2/FC]

This equation relates to an exactly triangular barrier. BUT: •  it neglects exchange-and-correlation (XC) effects (usually modelled as

image effects); •  it is not adequately valid for highly curved emitters.

Formally, we can include both XC and curved-surface effects with a barrier form correction factor νF

GB ("nu") for the model barrier ("GB") of interest, yielding:

JkCGB = (aφ–1FC

2) exp[–νFGBbφ3/2/FC].

This quantity JkCGB can be calculated exactly for any chosen barrier

model, and I call it the kernel current density for the barrier.

Local pre-exponential correction factor*

To allow for other corrections, it is necessary to include a characteristic local pre-exponential correction factor λ CGB , giving

JCGB = λ CGB JkC

GB = λ CGB (aφ–1FC2) exp[–νF

GBbφ3/2/FC].

The factor λ CGB allows formally for corrections due to: •  improved tunnelling theory that includes a tunnelling pre-factor •  more accurate integration over emitter electron states •  temperature effects •  effects due to the use of atomic-level wave-functions •  effects related to non-free-electron band-structure •  any other operating physical effect not specifically considered.

The equation above is a "technically complete FN-type equation for CFE characteristic local current density".

Local pre-exponential correction factor*

To allow for other corrections, it is necessary to include a characteristic local pre-exponential correction factor λ CGB , giving

JCGB = λ CGB JkC

GB = λ CGB (aφ–1FC2) exp[–νF

GBbφ3/2/FC].

The factor λ CGB allows formally for corrections due to: •  improved tunnelling theory that includes a tunnelling pre-factor •  more accurate integration over emitter electron states •  temperature effects •  effects due to the use of atomic-level wave-functions •  effects related to non-free-electron band-structure •  any other operating physical effect not specifically considered.

The equation above is a "technically complete FN-type equation for CFE characteristic local current density".

For notational simplicity, I now omit the superscript "GB". Relevant parameters apply to a "general barrier" unless otherwise indicated.

Contributions to λC *

Table above shows current best guesses (in 2015) at various contributions to λ C , for emission from a flat planar metal emitter, assuming the Schottky-Nordheim (SN) barrier discussed later.

For comparison, inclusion of νFSN increases current density by 100-500.

Physical origin of correction factor

Symbol for effect Value of multiplier

Tunnelling pre-factor PF ~ (0.4 to 1.1) Correct summation over states λD ~ (0.9 to 1.3) Combination of above effects 1) λDPF ~ (0.5 to 1.0) Temperature effects at 300 K λT (300 K) ~ 1.1 Electronic effects [atomic wave-functions & band-structure effects]

λE ~ (0.01 to 10)

All effects together λCFE ~ (0.005 to 11) 1) Note that high values of λD tend to be associated with low values of PF.

Emission area & related definitions*

Expressions for total emission current i can be written in various forms:

i = ∫ J dA ≡ AnJC = (Anλ C) JkC ≡ Af JkC

where An (the notional emission area), and Af (the formal emission area) are defined by this equation.





For a large-area field emitter (LAFE), the total area (i.e., "footprint") of the device is called the macroscopic area, and is denoted by AM . The macroscopic current density JM is defined by/can be written

JM ≡ i / AM = (An/AM) JC ≡ αnJC

where the notional area efficiency αn is defined by this equation.





For a large-area field emitter (LAFE), the total area (i.e., "footprint") of the device is called the macroscopic area, and is denoted by AM . The macroscopic current density JM is defined by/can be written

JM ≡ i / AM = (An/AM) JC ≡ αnJC

where the notional area efficiency αn is defined by this equation.

JM can also be written JM ≡ αnJC = (αnλ C) JkC ≡ αfJkC

where the formal area efficiency αf is defined by this equation.

Universal dependent variables*

These definitions provide similar formal expressions for all dependent variables of interest:

•  Characteristic local current density JC is given by JC = λC JkC , •  Macroscopic current density JM is given by JM = αf JkC , •  Emission current (device current) ie is given by ie = Af JkC .

Each formula has a part (JkC) that can be calculated exactly (for a given barrier model), and a pre-factor containing “all the residual uncertainty”.

Universal dependent variables*

These definitions provide similar formal expressions for all dependent variables of interest:

•  Characteristic local current density JC is given by JC = λC JkC , •  Macroscopic current density JM is given by JM = αf JkC , •  Emission current (device current) ie is given by ie = Af JkC .

Each formula has a part (JkC) that can be calculated exactly (for a given barrier model), and a pre-factor containing “all the residual uncertainty”. Since JkC has the form:

JkC = aφ–1FC2 exp[-νFbφ3/2/FC],

All the equations above are variants of

Y = CYF FC2 exp[-νFbφ3/2/FC],

where CYF is an appropriately chosen parameter.

Y is the universal dependent variable.

"Conversion lengths" convert field to voltage (or vice-versa), and are denoted generally by the symbol ζ .

For a field emitter, the characteristic barrier field FC is related to the emission voltage Ve between it and the adjacent electrode by

FC = Ve / ζC ,

where ζC is the characteristic local conversion length (LCL). Sharp emitters have relatively small LCLs, but values depend on the whole geometrical environment.

Conversion Lengths*

"Conversion lengths" convert field to voltage (or vice-versa), and are denoted generally by the symbol ζ .

For a field emitter, the characteristic barrier field FC is related to the emission voltage Ve between it and the adjacent electrode by

FC = Ve / ζC ,

where ζC is the characteristic local conversion length (LCL). Sharp emitters have relatively small LCLs, but values depend on the whole geometrical environment.

As noted earlier, the true macroscopic field FM is FM = Ve / ζM .

The corresponding (true) macroscopic field enhancement factor (FEF) γC is

γC = FC / FM = ζM / ζC .

Conversion Lengths*

Emission parameters & measured parameters*

Emission voltage Ve and emission current ie are the parameters that appear in Fowler-Nordheim-type equations as derived from FN theory (usually written without the suffix "e" ).

Measured voltage Vm and measured current im are the measured parameters.


Emission voltage Ve and emission current ie are the parameters that appear in Fowler-Nordheim-type equations as derived from FN theory (usually written without the suffix "e" ).

Measured voltage Vm and measured current im are the measured parameters.

If parallel resistance Rp is made large, then measured current im = ie .

In this case, measured voltage Vm = Vc = Vd ONLY IF series resistance Rs [= Rs1 + Rs2] is negligibly small .


Related parameters are:

•  True macroscopic field FM : FM = Ve / ζM

•  True electrostatic FEF γC : γC = FC / FM = FCζM / Ve

•  Apparent macroscopic field FA : FA = Vm / ζM

•  Apparent FEF [pseudo-FEF] βapp : βapp = FC / FA = FCζM / Vm

As we shall see, pseudo-FEFs can be much greater than true FEFs.

In general, we may define

FC ≡ cXX , BX ≡ bφ3/2/cX ,

X is the universal independent variable. cX is the universal auxiliary constant.

Universal independent variable



X is the universal independent variable. cX is the universal auxiliary constant. Earlier we had the equation

Y = CYF FC2 exp[−νFbφ3/2/FC],

Hence we may write Y = CYX X2 exp[−νFBX /X]

This is the universal FN-type equation.

Universal FN-type equation



X is the universal independent variable. cX is the universal auxiliary constant. Earlier we had the equation

Y = CYF FC2 exp[−νFbφ3/2/FC],

Hence we may write Y = CYX X2 exp[−νFBX /X]

This is the universal FN-type equation.

This equation is both universal and technically complete. It is not the most general CFE equation that could be written, but more-general equations would not be FN-type equations.

The "universal" form is useful because some general results relating to FN plots do not depend on which pair of variables is used.

Universal FN-type equation

Emission-current-density regimes*

An emission-current-density regime (also called an emission regime) is a region of parameter space (either {F,T |φ} or {f,T |φ}) where a particular expression for emission current density is approximately valid.

Thus, for a free-electron metal, the CFE regime is where FN-type equations are approximately valid.

For an emitter with φ = 4.50 eV, the barrier field FC ≈ f ×(14 V/nm), where f is the scaled barrier field (see later for a fuller definition).

Emission-current-density regimes*

There is no consensus on how to define and name emission regimes.

Physically, the theory depends on the relative positions (in energy) of the barrier top, the Fermi level, and the peak of the normal-energy distribution.

One set of options is:

Barrier top significantly above Fermi level:

•  Thermal electron emission [basically no tunnelling]

•  Schottky electron emission [~50% tunnelling, near barrier top] [or "barrier top emission"]

•  Intermediate-temperature emission [tunnelling "intermediate"]

•  Cold field electron emission [~100% tunnelling, near Fermi level]

Barrier top near or below Fermi level:

•  Intermediate-field emission [barrier top near Fermi level]

•  Field-induced ballistic emission [barrier top well below Fermi level]

2a

Outline history of field electron emission Basic/Theoretical topics

First reports

Early history – basic experimental facts

Early history – surface barrier and image potential energy

1920s – the search for linearity & the Fowler-Nordheim 1928 paper

Selected theoretical events – 1928-1960

Other 20th century theoretical topics

Recent theoretical topics

Political maxim:

Those who ignore history are condemned to repeat it.

Maxims

Political maxim:

Those who ignore history are condemned to repeat it.

Scientific maxim:

Those who ignore history are condemned to re-discover it.

Maxims

Early history – first reports

•  A FE-induced gas discharge was apparently first reported by Winkler, in Leipzig, 1744-5.

Priestley, in his 1767 textbook "The history and present state of electricity, with original experiments" has a whole section on

"Experiments which prove a CURRENT OF AIR from the points of bodies electrified either positively or negatively"

Presumably, Priestley was observing field electron emission, field ion emission and related electrical discharge phenomena.

Early history – first reports

Celebrating electrical research in America, in 1748

Joseph Priestley

Benjamin Franklin

•  1745 to ~1910: CFE is part of subject of electrical discharges, and is not recognised as a separate effect.

•  A particular discharge effect was the cathode ray. •  1897: J.J. Thompson identifies cathode rays as a beam of elementary

low-mass particles, later called "electrons".


•  1745 to ~1910: CFE is part of subject of electrical discharges, and is not recognised as a separate effect.

•  A particular discharge effect was the cathode ray. •  1897: J.J. Thompson identifies cathode rays as a beam of elementary

low-mass particles, later called "electrons". •  1910 to 1920: Various researchers, but particularly Lilienfeld (from

Leipzig) establish that field-induced electron emission is an effect in its own right (i.e., not a side-effect of gas in the vacuum or gas adsorbed on the emitter surface). Lilienfeld calls this effect autoelectronic emission.

•  In the 1914-18 war, Lilienfeld was a German military (medical) doctor. A particular driving force for field emitter development was the need for more-portable X-ray machines in military field hospitals.

•  1922: Lilienfeld publishes first accurate report in English of the basic experimental facts of autoelectronic emission.


Early history – surface barrier to electrons

1900s: Due to the work of Einstein (1905) on photo-emission and Richardson (from 1908) on thermal electron emission, it becomes clear that electrons in bulk metals are held into the metal by a surface barrier of height equal to the local work-function φ .

1850s: Kelvin introduces the idea of image methods in general (and applies them to spherical geometry).

1875: Maxwell introduces the idea that a potential hump, due to image force, would exist for an electrified point situated close outside a charged sphere. [But gets mathematical details incorrect.]

1914: Schottky introduces the same image-hump idea for a point electron above a planar surface, and reaches formula:

Image potential energy (PE) = –e2/16πε0x

[Schottky refers directly to Kelvin's ideas.]

1940: Bardeen shows the image PE formula is the correct classical limit (at large x) for quantum-mechanical correlation effects.

Early history – image potential energy

Early history – surface barrier details*

1914: Schottky argues that, when an electric field is applied to a metal, the potential energy (PE) outside surface has two components:

•  electrostatic component (– eFx), •  image PE (– e2/16πε0x).

1923: Hence (if restraining barrier is φ ), Schottky assumes that

Barrier PE = φ – eFx – e2/16πε0x .

This is the well-known Schottky planar image-hump, known in field emission as the Schottky-Nordheim (SN) barrier.

Early history – surface barrier details*

1914: Schottky argues that, when an electric field is applied to a metal, the potential energy (PE) outside surface has two components:

•  electrostatic component (– eFx), •  image PE (– e2/16πε0x).

1923: Hence (if restraining barrier is φ ), Schottky assumes that

Barrier PE = φ – eFx – e2/16πε0x .

This is the well-known Schottky planar image-hump, known in field emission as the Schottky-Nordheim (SN) barrier.

Back to the main history of CFE

In 1922, Lilienfeld published the first accurate report

in English of the experimental behaviour of "autoelectronic emission"

[J.E. Lilienfeld, Am. J. Roentgenol. 9 (1922) 192]

This then became a "hot topic"

1920s – the search for linearity

Experimentally, autoelectronic current i increased rapidly with applied voltage V.

The next step was to find the empirical form of the relationship i(V).

By 1922: i vs V (direct plot) not linear

1920s: ln{i} vs V (semi-logarithmic plot) not linear

1923: ln{i} vs V1/2 (Schottky plot) not linear

1928: ln{i} vs 1/V (Millikan-Lauritsen plot) linear

It was also found experimentally that the temperature dependence of autoelectronic current was negligible near room temperature.

1920s – background theory developments

In the 1920s there were many other relevant developments: •  Schrödinger-type wave-mechanics was under development •  Richardson (and others) were developing understanding of thermal

electron emission (TE). •  Fermi and Dirac introduced a new form of statistics for electrons. •  In 1926, Jefferies developed an improved theory for solving certain

types of non-linear 2nd-order differential equation. [This was based on earlier work by others, including Fowler.]

•  In 1927, Sommerfeld introduced a new theory of electrons in metals, based on the ideas of Fermi and Dirac.

•  In 1927, Nordheim was applying the "new wave-mechanics" to investigate "reflection effects" in TE theory.

•  In 1927, Fowler incorporated electron spin into TE theory. Then, 1928 is the "golden year" for tunnelling theory …

1928 - the golden year for tunnelling theory

1. Oppenheimer proposes that: (a) field-induced ionization of atoms is due to field-induced tunnelling; (b) the Millikan & Lauritsen (ML) linearity finding could be explained as

due to field-induced tunnelling from orbitals in surface atoms. His prediction is: ln{i/V1/4} vs 1/V should be linear. [But he makes mathematical errors.]

2. Fowler & Nordheim (FN) show that the ML finding, and the temperature independence of emission current, can be explained by a theory of cold field electron emission from delocalised electron-band states, involving:

(a) electron emission by wave-mechanical tunnelling; (b) Fermi-Dirac statistics and Sommerfeld's theory of electrons in metals.

Their prediction is: ln{i/V2} vs 1/V should be linear.

3. Gurney & Condon use FN's ideas to propose a theory of radioactivity in which alpha-particles escape from the nucleus by tunnelling. [They also re-interpret Oppenheimer's work.] Gamow independently proposes similar theory.

Basic FE models

In 1928, FN win the the argument about the origin of CFE, because they can explain the temperature independence, and Oppenheimer didn't.

In the 1970s, something like the Oppenheimer model returns, when Modinos argues that in semiconductor CFE, many electrons are emitted from electron-surface-state bands, localised to the emitter surface.

Basic FE models

In 1928, FN win the the argument about the origin of CFE, because they can explain the temperature independence, and Oppenheimer didn't.

In the 1970s, something like the Oppenheimer model returns, when Modinos argues that in semiconductor CFE, many electrons are emitted from electron-surface-state bands, localised to the emitter surface.

Modinos has also suggested that, with metals, electrons are emitted not from plane-wave states (as FN assumed), but from band states "with enhanced amplitude at the surface atoms".

In the context of metal CFE, these issues are unresolved, but it is very clear that surface atoms are electrically polarised at a field emitter surface. This has been shown by several workers (Tsong, Forbes, Inglesfield, in 1970s & 1980s; Z.B. Li, Peng & colleagues, more recently). Also, Plummer found structure in energy distributions that "carried the chemical signature of surface atoms".

Polarisation of surface atoms

Polarisation effects can occur with negative applied fields, as shown by J. Peng et al. in first-principles calculations on closed carbon nanotubes.

The historical importance of the FN paper

The historical importance of Fowler and Nordheim's work is that:

a)  it was the first good explanation of auto-electronic emission;

b)   it was one of the first group of papers to apply the idea of wave-mechanical tunnelling to the interpretation of physical phenomena;

c)  it was one of the first applications of Fermi-Dirac statistics and Sommerfeld's ideas to an experimental effect related to electron behaviour in metals; it was one of the first papers to incorporate the idea of electron spin into metal theory.

d)   it helped justify the idea of a metal conduction band, because it argued that field emitted and thermally emitted electrons could come from a single band, under different field and temperature conditions.

e)  It stimulated the Condon & Gurney work on the theory of radioactivity; their work led on to theories of nuclear physics.

The historical importance of the FN paper

The historical importance of Fowler and Nordheim's work is that:

a)  it was the first good explanation of auto-electronic emission;

b)   it was one of the first group of papers to apply the idea of wave-mechanical tunnelling to the interpretation of physical phenomena;

c)  it was one of the first applications of Fermi-Dirac statistics and Sommerfeld's ideas to an experimental effect related to electron behaviour in metals; it was one of the first papers to incorporate the idea of electron spin into metal theory.

d)   it helped justify the idea of a metal conduction band, because it argued that field emitted and thermally emitted electrons could come from a single band, under different field and temperature conditions.

e)  It stimulated the Condon & Gurney work on the theory of radioactivity; their work led on to theories of nuclear physics.

BUT FN's exact triangular barrier is NOT a realistic barrier model, because it neglects rounding due to image potential energy; thus the original FN-type equation significantly under-predicts true current densities.


1928: Nordheim attempts tunnelling theory for Schottky-Nordheim barrier, but makes mathematical mistake over elliptic functions.

1929: Stern, Gosling and Fowler introduce the idea of FN plots, and develop a first theory of field emitted vacuum space-charge.

1933: Bethe and Sommerfeld write seminal review article on the theory of electrons in metals which (in effect) establishes modern band theory. They discuss CFE in detail, but do not notice Nordheim's mistake.

1953: Burgess, Kroemer and Houston find the error in Nordheim's theory, and use early computers to tabulate correct values of the Schottky-Nordheim (SN) barrier functions v and s .


1956: Murphy and Good (MG) introduce FN-type equations based on SN barrier, including the parameter tF .

[Later, Forbes and Deane re-formulate MG theory by introducing ideas of scaled barrier field f and local pre-exponential correction factor λC, and by finding characteristic equation and a good simple approximation for principal SN barrier function v.]

1956: Murphy and Good also introduce theory of Schottky emission ("barrier-top emission") based on SN barrier.

[Later, Jensen develops this further and introduces general theory of field induced (and photon-induced) electron emission.]

1960: Young realises that, due to conservation of angular momentum in a central field, measured CFE energy distributions are total energy distributions (TEDs), and develops a theory of TEDs.

Other 20th century theoretical topics

•  FN plot interpretation [Stern et al., Houston, Charbonnier, Spindt et al., Forbes, & others]

•  Field emitted vacuum space-charge [Stern et al, Barbour et al, Forbes, Jensen et al. & others]

•  Field electron microscope resolution theory [Rose, Müller] •  Field emission optics (for electron columns)

[Swanson, Schwind & colleagues, Hawkes & Kasper, Kruit & colleagues]

•  Effect of band structure on FE energy distributions [Gadzuk, Plummer, Cutler, Miskovsky & colleagues, & others]

•  Reformulation of FN's proof in terms of Airy functions [Gadzuk, Plummer, Jensen & colleagues, Forbes & Deane]

•  Evaluation of tunnelling pre-factors [Mayer] •  FE from semiconductors [Stratton, Modinos, Xanthakis, & others] •  Triple-junction effects [Cheung & colleagues]

Recent theoretical topics

•  FE from large-area field emitters of various kinds [various] •  FE from carbon nanotubes & nanostructures

[many workers, especially Z.B. Li & colleagues] •  FE from small emitters [various, especially Z.B. Li & colleagues] •  FE from semiconductor nanowires [various] •  Explosive emission [Mesyatts, Fursei]

•  Field emission plasmonics [effects of pulsed lasers on field emitters]

In parallel with the theoretical research

there was much experimental and technological research.

This is far too extensive to summarise easily.

I indicate some of the early work and the main recent topics.

Early history - Field electron microscopy

An important scientist in the early development of field emitters and field electron microscopy, from the 1930s to the 1950s, was Erwin W. Müller. Erwin Müller can also be seen as the grandfather of nanoscience. He saw atoms in 1956, using field ion microscopy, four years before Feynmann suggested nanotechnology.

Erwin W. Müller

Early history - Field electron microscopy

Müller invented the field electron microscope (FEM).

He developed techniques for cleaning pointed wire emitters.

He developed basic understanding of the origin of FEM images, i.e., bright image regions correspond to emitter regions where J is high, because: the local barrier field F is high; or the local work-function φ is low.

W(110) FEM image

Understanding FEM images from metals was key to early FEM use.

A current problem is that there is no consensus over the interpretation of some CNT images.

Selected experimental/technological topics up to around 1980

•  Development of clean single-tip field emitters (STFEs) & field electron microscopy (FEM) [Müller and co-workers]

•  Development of high-brightness CFE sources [Dyke, Swanson & colleagues]

•  FEM as early technique for adsorption studies [Gomer & others] •  Vacuum-breakdown mechanisms [Latham & colleagues & many others] •  Field electron spectroscopy (energy distributions)

as technique for metal surface studies [Plummer, & others] •  Spindt arrays [Spindt & colleagues, & many others] •  Schottky emitters [Swanson, Schwind & colleagues]

Selected experimental/technological topics since around 1980

•  Conductor-Dielectric-Conductor emitters •  Diamond-like-carbon large-area field emitters (LAFEs) •  Carbon nanotube & nanostructures (STFEs & LAFEs) •  Semiconductor nanowires & nanostructures (STFEs & LAFEs) •  Explosive electron emission •  Liquid-metal electron sources •  FEM of individual molecules •  Pulsed-laser field electron sources

What theory do we need ?

For instance (1), the simpler applications of FE would need to include theory relating to:

ALL APPLICATIONS •  Field emission electrostatics •  Prediction of current-voltage characteristics •  Thermal-field shaping effects, and emitter degradation mechanisms

SINGLE-TIP FIELD EMITTERS (STFEs) FOR ELECTRON SOURCES •  Energy distributions •  Field emission optical effects, such as brightness & source size

FIELD ELECTRON MICROSCOPY (FEM) •  FEM image interpretation, FEM image resolution

LARGE-AREA FIELD EMITTERS (LAFEs) •  FN plot interpretation •  Maybe, other characterization parameters •  Optical properties (e.g., in context of microwave generation)


For instance (2), LAFEs require supporting theory relating to LAFE characterization. Properties one may need to characterize include:

•  Emitter geometrical sharpness (by true FEF, or by onset macroscopic field)

•  Emission response to voltage (by characteristic local conversion length, or by onset voltage)

•  Current generating capacity (by macroscopic current density)

•  Effectiveness in terms of fraction of "footprint area" that emits (by notional or formal area efficiency)

•  Uniformity of emission, both in space and in time (by appropriate statistical measures)

•  Whether emission is orthodox, and if not then why not (via orthodoxy test, to begin with)

•  Robustness in industrial vacuum conditions (via lifetime measurements under defined operating conditions)


It is impossible to cover all useful theory in a reasonable time.

Hence, for this tutorial, I have selected some of the most relevant basic theory.

3

Some background theoretical ideas

Sommerfeld model

Origin of work-function

Patch fields

Field penetration and band-bending

Derivation of elementary FN-type equation

Sommerfeld model

Arnold Sommerfeld

Basic Sommerfeld model

χ ... well depth

KF ...... Fermi energy

φ ....... local work-function

occupied electron states

Local vacuum level Evac or Eo

Fermi level EF

Base of conduction band Ec

TOTAL ELECTRON ENERGY

Origin of local work-function

The well depth and the local work-function have TWO components •  a bulk (or "chemical") component •  a surface (or "electric dipole") component.

The bulk component explains why different materials have different work functions; the surface component explains why different faces of a single material have different work-functions.

A simple explanation (Smoluchowski 1941) of the surface dipole explains it by the behaviour of the surface electron-wave-function:

•  spreading (outwards) decreases φ ;

•  smoothing (sideways) increases φ ; (see next slide).

Origin of local work-function

Patch fields

Patch fields are a well known effect with thermionic cathodes.

They can occur with emitters that are neutral overall.

They occur when differences in electrostatic potential exist between points in space immediately outside different surface regions (often as a result of work-function differences).

The field is greatest in the vicinity of the join between the parts of the surface at different electrostatic potentials.

Semiconductor band-bending

4

Derivation of FN-type equations

Motive energy and barrier strength

Barrier form correction factor

Transmission probability

Transmission probability formulae

The tangent method

Intercept correction factor

Levels of approximation

Research state of play

Motive energy & Barrier strength

The one-dimensional Schrödinger equation in direction z can be written

d2Ψ/dz2 = (2m/!2) (Ez–U) Ψ ≡ –κ2M(z)Ψ

where M(z) [≡ U–Ez] is the motive energy, and κ ≡ (2m)1/2/! . The related barrier strength G is defined by

G = g ∫ M1/2 dz

where the JWKB constant g = 2κ, and the integral is taken "across the barrier", i.e. where M>0. [G has also been called the "Gamow exponent" and the "JWKB exponent".]

Two well-known special barrier forms exist:

Two special barrier forms

Exactly triangular (ET) barrier

H

M(z) = H – eFz

used by Fowler & Nordheim

0 z slope = –eF

Schottky-Nordheim (SN) barrier

H

M(z) = H – eFz – 1/16πε0z

used by Murphy & Good

0 z slope = –eF

Barrier form correction factor

For electron tunnelling through an exactly triangular (ET) barrier of zero-field height φ, at barrier field FC, the barrier strength GF

ET evaluates to

GFET = bφ3/2/FC .

For an arbitrary but well-behaved "general barrier" (GB), the integral can be evaluated numerically to give the result GF

GB . For this barrier GB, the barrier form correction factor νF

GB ("nuFGB") is

defined by νF

GB ≡ GFGB / GF

ET .

In the special case of the SN barrier, νFSN is given by the mathematical

expression v(f) ("vee(f)", where v(f) is the principal SN barrier function, expressed as a function of scaled barrier field. The subscript "F" indicates that the parameter relates to a barrier of height φ, as seen by an Fermi-level electron moving "forwards".

From earlier slides, the universal FN-type equation is:

Y = CYX X2 exp[−νFBX /X] .

In the case of the Schottky-Nordheim (SN) barrier, one can put:

X→ FC; Y→ JC; BX→ bφ3/2; νF→ v(f); CYX→ λCSNaφ–1;

and write the result in the form of the linked equations

JCSN(FC) = λC

SNJkCSN(FC),

JkCSN(FC) = aφ–1exp[−v(f)bφ3/2

/FC] .

This is the JC(FC) form of the Orthodox FN-type Equation.

Orthodox FN-type equations*

For the SN barrier, it is useful to apply scaling ideas. Consider SN barrier of zero-field height φ .

The reference field FR that reduces the barrier height to zero is

FR = c–2φ2 ≈ (0.694 4616 eV−2 V nm−1) φ2 ,

where c is the Schottky constant. For illustration, an emitter with φ = 4.50 eV has FR≈ 14.1 V/nm. The scaled barrier field f is related to the barrier field FC by

f = FC/FR .

Scaled barrier field for the SN barrier*

Derivation of scaled form for JkSN *

Define φ-dependent parameters η(φ) and θ (φ) by

η(φ) = bφ3/2/FR = bc2φ–1/2 ,

θ(φ) = aφ–1FR2 = ac–4φ3 ,

where bc2 and ac–4 are universal constants (see table earlier).

In scaled form, the equation for JkCSN becomes

JkCSN = θ f2 exp[–η v(f)/f] .

For example, for φ= 4.50 eV, then η≈ 4.64 and θ≈ 6.77×1013 A/m2. This scaled equation contains only one field-like variable (f), and a good simple approximation exists for v(f). Hence JkC

SN is well approximated by JkC

SN ≈ θ f2 exp[η·{1 – (1/6)lnf – 1/f}].

This formula is useful because it can be evaluated on a spreadsheet.

More terminology:

Transmission = General name for wave-mechanical passage across an energy barrier.

Tunnelling = Wave-mechanical transmission through the barrier.

Flyover = Wave-mechanical transmission over the barrier.

Deep tunnelling = Tunnelling well below the top of the barrier.

High flyover = Flyover well above the top of the barrier.

Transmission, tunnelling and flyover


In CFE, electrons escape by wave-mechanical tunnelling through a field-lowered energy barrier.

Tunnelling is not mysterious: it is a property of wave-theories. Tunnelling occurs with light, with sound, and also with waves on strings.

Consider electron approaching barrier in specific electron state k :

there is a tunnelling probability (also called transmission probability) Dk .

e− (in state k) Tunnelling probability Dk

Reflection probability (1–Dk)

energy barrier


To calculate the transmission probability D, we need to solve the one-electron one-dimensional Schrödinger equation. For the exact triangular barrier, the Schrödinger equation was (apparently) solved exactly by Fowler & Nordheim (in 1928). Their treatment (based on Bessel functions) is difficult to follow and contains several typos. The result is correct (within the approximations used), but it is unclear whether the mathematical argument is strictly valid. Modern treatments (Jensen, Forbes & Deane), based on Airy functions, are more transparent. These generate a general formula for DET (valid for both tunnelling and flyover), as follows.

Wave-matching at the emitter surface leads to the exact general formula:

where A, B are the values of the Airy functions Ai(x), Bi(x); and A', B' are the values of the derivatives of Ai, Bi with respect to x, all evaluated at the emitter surface, where x = xL .

The dimensionless parameter ω is given by (where cκ is an universal constant, and W is "total forwards kinetic energy"):

ω = cκ W1/2/F1/3 = [1.723903 eV–1/2 (V/nm)1/3] (W1/2 / F1/3) .

The dimensionless parameter x is related to real-distance X, and its value (xL) at the emitter surface is given by

xL = –cκ 2 w/F2/3

= [–2.971842 eV–1 (V/nm)2/3] (w / F2/3) ,

where w is the "forwards energy level relative to the barrier peak".

Exact universal formula for DET *

DET = 1

12 + 1

4 πω ( A2 + B2 )+ 14 πω

−1( A '2+ B '2 )

In this case of the ET barrier, the general formula reduces to very different approximate/asymptotic forms in different regions of the parameter space defined by the variables {F2/3, w}.

This leads to different transmission (-probability) regimes, as shown on the next slide.

Common theory applies to "shallow tunnelling" and "low flyover", which together constitute the barrier top regime.

Transmission regimes for ET barrier

ET barrier - transmission regime diagram

HF = high flyover; LF = low flyover; BT = barrier-top regime; ST = shallow tunnelling; DT = deep tunnelling.

1. Start from exact universal formula:

2. Neglect A and A' :

3. Use only the leading term in each series expansion, obtaining:

4. When G is large, neglect exponent in denominator, giving:

5. Use definition of u (not provided here) and re-arrange:

F&D derivation of FN asymptotic formula

DET = 1

12 + 1

4 πω ( A2 + B2 )+ 14 πω

−1( A '2+ B '2 )

DET = 1

12 + 1

4 πωB2 + 14 πω

−1B '2

DET {4 / (u−1 + u)}e−G

DET ≈ {4W 1/2 H 1/2 / (W + H )} exp[−bH 3/2 /F]

DET 1/ [ 1

2 +14 eG{u−1 + u}] = e−G / [ 1

2 e−G+ 14{u−1 + u}]

The above is an expression for transmission probability.

To obtain the original FN-type equation for local current density:

6. Put H = φ , and W equal to the Fermi energy KF , obtaining an expression DF

ET for the transmission probability for a Fermi-level electron approaching the emitter surface at right angles:

DFET ≈ {4KF

1/2φ1/2 / (KF+φ)} exp[−bφ3/2/F] .

7. Carry out integration over emitter electron states (see later) and write result (for local current density JL ) in form

JL ≈ (aF2φ−1) {4KF

1/2φ1/2 / (KF+φ)} exp[−bφ3/2/F] .

This is the "original FN-type equation".

F&D derivation of FN asymptotic formula

DET ≈ {4W 1/2 H 1/2 / (W + H )} exp[−bH 3/2 /F]


For the exact triangular barrier, the Schrödinger equation was (apparently) solved exactly by Fowler & Nordheim (in 1928), as just indicated.

But this barrier shape is not physically realistic. The FN formula under-predicts current density by a factor usually around 100-500.

The simplest realistic barrier model (for traditional emitters) is the Schottky-Nordheim barrier, discussed earlier.

Transmission probability*

For the exact triangular barrier, the Schrödinger equation was (apparently) solved exactly by Fowler & Nordheim (in 1928), as just indicated.

But this barrier shape is not physically realistic. The FN formula under-predicts current density by a factor usually around 100-500.

The simplest realistic barrier model (for traditional emitters) is the Schottky-Nordheim barrier, discussed earlier. 200 years of mathematics have shown that, for the SN barrier, it is impossible in principle to solve the Schrödinger equation analytically in terms of the ordinary functions of mathematical physics. Solutions can be found by so-called semi-classical methods. These are sometimes called path-integral methods or JWKB-type methods (JWKB = Jefferies-Wentzel-Kramers-Brillouin), but strictly JWKB-type methods are one type of semi-classical method.

The most satisfactory simple-but-reasonably-general semi-classical formula is derived from the work of Fröman and Fröman.

The Fröman & Fröman formula

For a smooth barrier of arbitrary shape, Fröman & Fröman (FF) obtained a JWKB-type formula that can be put in the form:

D = Pe–G / (1 + Pe–G)

where: G is barrier strength, as before P is a tunnelling pre-factor derived from FF's work.

G can be evaluated exactly, by numerical integration if necessary. P is very difficult to calculate exactly; but we can usually put P ≈ 1 .

The Fröman & Fröman formula

For a smooth barrier of arbitrary shape, Fröman & Fröman (FF) obtained a JWKB-type formula that can be put in the form:

D = Pe–G / (1 + Pe–G)

where: G is barrier strength, as before P is a tunnelling pre-factor derived from FF's work.

G can be evaluated exactly, by numerical integration if necessary. P is very difficult to calculate exactly; but we can usually put P ≈ 1 .

The FF formula can can be compared with the exact ET barrier formula:

It is not clear why there are three terms in the denominator of this formula, but only two in the denominator of the FF formula.

It is thus debatable whether FF theory is exact and correct.

We assume that in practice it is "very nearly correct".

DET = 1

12 + 1

4 πω ( A2 + B2 )+ 14 πω

−1( A '2+ B '2 )

Summation over emitter states

A general procedure (for finding an analytical current-density equation) is:

•  Choose a "reference electron state" "R" in the emitter for which the tunnelling probability is particularly high.

•  Calculate transmission probability Dk for each emitter electron state.

•  Sum contributions to local emission current density JL from all states.

•  Write result in form: JL = ZR DR

where: DR is transmission probability for the reference state; ZR is a parameter called the effective electron supply.

For CFE and FN-type equations, the reference state is the "forwards state at the Fermi level" ("state F"), as earlier, and we write:

JL = ZF DF

[Numerical or analytical procedures may be used to do the summation.]

Summation over emitter states

In a free-electron model, the summation is particularly straightforward.

As well as the total energy ("T"), there are two componenst of energy: parallel to the surface ("P") and normal to the surface ("N").

Any two of these can be used, so there are several methods of doing the summation.

In the "PT" method, one first sets up a "PT energy diagram" …

PT-type energy-space diagram

K = TOTAL KINETIC ENERGY Eo

Kp

ε

ε=–KF

ε=0

ε=φ

EF

Ec

Kp=Ktot

Ktot=0

Ktot

KF

COMPONENT Kp OF KINETIC ENERGY PARALLEL TO EMITTER SURFACE

fFD∼0

ε = TOTAL ENERGY relative to Fermi level

PT-type= Parallel-energy on x-axis, Total-energy on y-axis

fFD∼1

More terminology - decay width

For some given barrier field FC, let the tunnelling probability for a barrier of zero-field height H be D(H). The decay width d is a positive quantity (usually given in eV) defined via

d−1 = − ∂lnD / ∂H .

When H = φ, the decay width is denoted by dF . In the simplest approximation, −lnD ≈ bH3/2/FC , and dF ≈ 2FC/3φ1/2 .

More terminology - supply density

Inside a conductor, the supply density is the electron current crossing a mathematical surface in real space, per unit area of the surface, per unit area of energy-space. In a free-electron model, the supply density is constant in energy-space (i.e., it's the same at all energies), and is given by the so-called Sommerfeld supply density zS . Hence, when the electron distribution is in thermodynamic equilibrium, the contribution from the energy-space-element dKpdε to the current density crossing the mathematical surface is: fFDzS dKpdε , where fFD is the Fermi-Dirac distribution function. Hence the contribution to the emission current density is fFDzS D dKpdε .

Integration in energy-space

Kp

ε

ε=–KF

ε=0

ε=φ

δKp

δε

Consider electron states in element dKpdε

Contribution to local emission current density (ECD) is zS fFD D dKpdε where zS is Sommerfeld supply density

fFD is Fermi-Dirac distribution function D is tunnelling probability.

So ECD is: J = ∫ ∫ zS fFD D dKpdε

Integration over energy-space

At 0 K, integration over energy-space yields the characteristic local emission current density (ECD) as:

JC = ∫ ∫ zS fFDD dKpdε = (zSdF2) DF .

In the simplest approximation (the so-called elementary approximation), it can be shown that

(zSdF2) = aφ−1FC

2

DF = exp[−bφ3/2/FC]

Hence we retrieve the JC(FC) form of the elementary FN-type equation:

JCel = aφ−1FC

2 exp[−bφ3/2/FC] .

More sophisticated treatments introduce correction factors into the exponent and/or the pre-exponential.

Result of integration over energy-space

Eo

Kp

ε

ε=–KF

ε=0

ε=φ

EF

Ec

"F"

dF√2

JC = ∫ ∫ zS fFD D dKpdε

συµ

At zero temperature, the integration yields the ECD JC as

JC = zSdF2DF [ZF = zSdF

2]

This is a general abstract form, at 0 K, for all FN-type equations derived using the Landau & Lifschitz formula for D and a free-electron model for the emitter. The emission mostly comes from a relatively small region of energy space near state F.

4

Analysis of FN plots

Conventions

The task

Concept of slope correction factor

The tangent method

Intercept correction factor

Alternative approaches

The "curly brackets" in ln{z} and log10{z} mean: "Express the quantity z in specified units, and take the logarithm of its numerical value."

If needed, units can be specified separately. SI units are often best. Theoretical discussions may not need units to be explicitly stated. [This convention is part of an International Standard.]

The "curly-brackets" convention

Consider a function Y(X). The mathematical operation of "writing Y(X) in Fowler-Nordheim coordinates" is the following:

•  Define a function L ≡ ln{Y/X2} ; •  Express L in the form L(X–1) .

A Fowler-Nordheim plot is a plot of the type L(X–1) vs X–1 .

Fowler-Nordheim Coordinates

Analysing FN plots – the task

The universal FN-type equation is:

Y = CYX X2 exp[–νF BX / X]

In FN coordinates this becomes L(X–1) = ln{CYX} – BXνF X–1

In real situations, CYX , BX and νF may depend on barrier field. This makes the theoretical FN plot curved.

Also, if exchange-and-correlation (XC) effects are in the barrier model, the theoretical FN plot stops at the value XR

–1 at which the barrier height goes to zero.

However, FN plots are analyzed by fitting straight lines. The issue is:

"How do we relate the parameters derived from the fitted straight line to the parameters that appear in the theoretical expression for L(X–1) ?"

The concept of slope correction factor

The theoretical FN plot is L(X–1) = ln{CYX} – BXνF X–1

Its slope SYX is SYX = ∂L/∂(X–1)

= ∂ln{CYX}/∂(X–1) – BX νF – BX X–1 ∂νF/∂(X–1) – νF X–1 ∂BX/∂(X–1)

The related slope correction factor σYX is σYX = – SYX / BX

= –[∂ln{CYX}/∂(X–1)]/BX + [νF + X–1 ∂νF/∂(X–1)] + [νF X–1 ∂BX/∂(X–1)]/BX

Hence the FN-plot slope can be written:

SYX = – σYX BX = – σYXbφ3/2/cX

Discussions in the literature often omit any correction factor similar to σYX. This results in a defective equation.


If voltage drop (due to series resistance) occurs, then the theory is more complicated. But it remains true that a correction factor analogous to σYX is needed, and that its omission leads to defective data-analysis equations.


If voltage drop (due to series resistance) occurs, then the theory is more complicated. But it remains true that a correction factor analogous to σYX is needed, and that its omission leads to defective data-analysis equations. Now continue with FN-plot analysis procedure. Historically, this has been done in several ways, but the most useful approach is the "tangent method".

Analysing FN plots - the tangent method

The tangent method has the following steps. •  Model the fitted straight line by the tangent to theoretical FN plot. •  Assume fitted line is parallel to the tangent at some horizontal-axis

value Xt–1 (usually not known).

•  Write equation for theoretical tangent at Xt–1 as

L(X–1) = ln{Itan} + Stan X–1 = ln{ρtCYX} – σt BXX–1 , where Stan and Itan are the tangent's slope and intercept, and σt and ρt are slope and intercept correction factors, evaluated at Xt

–1.

"L" is the theoretical FN plot (but with curvature exaggerated). "T" is its tangent, taken at "t".

The meaning of σt and ρt

"L" is the theoretical FN plot (but with curvature exaggerated). "T" is its tangent, taken at "t". For "L": L(Xt

–1) = ln{CYX} – νFtBXXt–1

For "T": L(Xt

–1) = ln{ρtCYX} – σt BXXt–1

Hence (using cXXt = FCt)

lnρt = [σt–νFt] (bφ3/2/FCt)

where FCt corresponds to Xt .


"L" is the theoretical FN plot (but with curvature exaggerated). "T" is its tangent, taken at "t". For "L": L(Xt

–1) = ln{CYX} – νFtBXXt–1

For "T": L(Xt

–1) = ln{ρtCYX} – σt BXXt–1

Hence (using cXXt = FCt)

lnρt = [σt–νFt] (bφ3/2/FCt)

where FCt corresponds to Xt .


Before estimating ρt we need to estimate σt . But, before that, we need an expression for σYX .

Analysing FN plots - the tangent method

The tangent method has the following steps. •  Model the fitted straight line by the tangent to theoretical FN plot. •  Assume fitted line is parallel to the tangent at some horizontal-axis

value Xt–1 (usually not known).

•  Write equation for tangent at Xt–1 as

L(X–1) = ln{Itan} + Stan X–1 = ln{ρtCYX} – σt BXX–1 , where Stan and Itan are the tangent's slope and intercept, and σt and ρt are slope and intercept correction factors, evaluated at Xt

–1. •  Use an appropriate model of the emitting device (or of the device

operating in the context of the measuring circuit) to make theoretical estimates of σt and ρt .

•  Write equation for fitted line in the form L(X–1) = ln{Ifit} + Sfit X–1

where Sfit and Ifit are the fitted slope and intercept.

Analysing FN plots - the tangent method*

[The tangent method has the following steps (cont.)] •  Hence, identify Sfit = – σt BX , and Ifit = ρtCYX and use estimated σt and ρt

values to extract BX and CYX . •  Use BX and CYX to find values of relevant characterization parameters,

in particular cX . Obviously, a common application is to take X as some form of true or apparent macroscopic field; in this case cX is some form of field enhancement factor (a true FEF or a pseudo-FEF, depending on circumstances).

The tangent method - comments*

The tangent method has the following steps (cont.) •  Hence, identify Sfit = – σt BX , and Ifit = ρtCYX and use estimated σt and ρt

values to extract BX and CYX . •  Use BX and CYX to find values of relevant characterization parameters,

in particular cX . Obviously, a common application is to take X as some form of true or apparent macroscopic field; in this case cX is some form of field enhancement factor (a true FEF or a pseudo-FEF, depending on the circumstances). Strictly, the fitted line is a chord, not a tangent. A chord correction can in principle be made. This correction is small and is usually neglected. In the general case, determining σYX and σt is difficult. This is an active research topic - will now briefly summarize progress.

As just shown, the slope correction factor σYX is given by: σYX = –[∂ln{CYX}/∂(X–1)]/BX + [νF + X–1 ∂νF/∂(X–1)] + [νF X–1 ∂BX/∂(X–1)]/BX

The basic approximation (a) assumes that no series-resistance effects are operating, and hence

that the emission voltage Ve is equal to the measured voltage Vm ; ** (b) disregards the first and last terms in the equation above, and uses

the expression given by the remaining terms in this equation.

** This point was not sufficiently emphasised in a recent paper by Forbes, Fischer & Mousa.

The basic approximation*

As just shown, the slope correction factor σYX is given by: σYX = –[∂ln{CYX}/∂(X–1)]/BX + [νF + X–1 ∂νF/∂(X–1)] + [νF X–1 ∂BX/∂(X–1)]/BX

The basic approximation (a) assumes that no series-resistance effects are operating, and hence

that the emission voltage Ve is equal to the measured voltage Vm ; ** (b) disregards the first and last terms in the equation above, and uses

the expression given by the remaining terms in this equation.

In the basic approximation, the basic slope correction factor σB is given by σB = νF – F ∂νF/∂F ,

and the chosen barrier model determines both νF and σB . In particular - •  for the Schottky-Nordheim (SN) barrier: σB → s , •  for the exactly triangular (ET) barrier: σB → 1 ,

where s is the usual mathematical SN barrier function.

Much modern literature uses the above elementary approximation σ B = 1 .

The basic approximation

To do better than existing treatments, one approach is to make better estimations of σt . Specifying σt is itself is particularly difficult. First, we need results for σYX , to see what range of values is plausible. Options are:

(1) Explore how to deal with series resistance [as described in a separate talk at IVNC13] General theory using apparent macroscopic fields and pseudo-FEFs is confusing. It is easier to use measured and emission voltages. Best way forwards may be to explore how to extract emission current- voltage characteristics from measured data.

(2) Deal with field-dependent geometry.

Looks to be difficult to model reliably. Emitters based on field-dependent-geometry effects may be unattractive to industry if difficult to manufacture reliably.

Improved approaches to σYX and σt *

(3) Examine influence of emitter shape, using the basic approximation Few correction-factor calculations yet exist. The author and colleagues are exploring the spherical model and the sphere-on-orthogonal-cone model (the aim being to assess the influence of small emitter apex radius). Other models (e.g. existing CNT models, hemisphere-on-post models, Xanthakis-type models) are of interest. [Note added, Mar 15: However, very recent investigations seem to have uncovered fundamental difficulties with the quasi-classical quantum mechanics of tunnelling in non-planar geometry.]

(4) Examine influence of other neglected factors

In particular, the influence of field dependence in φ and CYX . Since CYX is a composite correction factor describing (up to) seven different physical effects, an exhaustive investigation may be time-consuming (although it is clear that some effects have little influence).

Improved approaches to σYX and σt *

5

The spurious FEF-value problem


The test for lack of field emission orthodoxy

In the universal treatment, if we let: Y represent macroscopic current density JM , X represent apparent macroscopic field FA ,

then cX represents apparent field enhancement factor βapp ,

and the extracted value of βapp is given by

(βapp)extr = σt bφ3/2 / Sfit , where σt is an appropriately calculated slope correction factor.


In the universal treatment, if we let: Y represent macroscopic current density JM , X represent apparent macroscopic field FA ,

then cX represents apparent field enhancement factor βapp ,

and the extracted value of βapp is given by

(βapp)extr = σt bφ3/2 / Sfit , where σt is an appropriately calculated slope correction factor. In reality, the situation is more complicated than this, because FA and βapp

are defined in terms of the measured circuit voltage rather than the emission voltage. To get the true electrostatic FEF γ , an additional correction factor κ may be required, and the "true" extraction formula would be:

γ extr = (κσt) bφ3/2 / Sfit

Clearly, the common approximation of setting (κσt) = 1 disregards many correction terms.

The spurious FEF-value problem*

γ extr = (κσt) bφ3/2 / Sfit

IF (κσt) is taken as equal to 1 for both green and red lines, THEN:

IF green line is expected to give a meaningful FEF-value, THEN red line is expected to give a spuriously large FEF-value.

Spuriously high enhancement factors

[Diagram: courtesy: Zhu, Baumann & Bower (2001)]

In the ideal world, one would calculate κσt (or find a better approach).

But, especially in series-resistance situations, this appears to be very difficult.

It will not happen quickly.

So we need to understand the scale of the spurious-FEF-value problem.

Hence the creation of a test for lack of field emission orthodoxy. [Details have appeared separately, and have been published in Proc. R. Soc. Lond., entitled: "Development of a simple quantitative test for lack of field emission orthodoxy". (doi: 10.1098/rspa.2013.0271)]


The orthodox emission hypothesis is a set of five physical and mathematical assumptions that permit well-specified analysis of measured current-voltage data relating to field electron emission (FE). These are:

(1) The emission voltage Ve (between the emitter and a counter-electrode) is the same at all points on the emitter surface (i.e., no Fermi level variations across the surface), and is equal to the measured voltage Vm (i.e., no voltage drop due to series resistance).

(2) The measured current im is equal to the emission current ie (i.e., no leakage currents), and is controlled by CFE at the emitter/vacuum interface.

(3) The emission mechanism is deep tunnelling through a SN barrier, and consequently (from 1 and 2 above) the applicable equation is the ie(Ve) form of the orthodox FN-type equation, but with the suffices "e" replaced by "m".

The orthodox emission hypothesis*

(4) In addition, in the orthodox FN-type equation: the only quantities that depend on the measured voltage Vm are the independent variable and the barrier-form correction factor.

(5) In particular, the emitter local work-function φ is independent of both measured current and voltage, and has a value close to that theoretically assumed.

The orthodox emission hypothesis*

Assuming orthodox emission means that all auxiliary parameters can be treated as constants; all independent variables become linearly related; all dependent variables become linearly related; and theory becomes much simpler.

However, with real emitters, many things can happen that are incompatible with orthodox emission, in particular:

•  voltage drop in the measuring circuit, or other forms of "saturation"; •  leakage currents; •  patch fields; •  field-emitted vacuum space charge; •  current-induced changes in emitter temperature; •  field penetration and band-bending; •  strong field fall-off; •  quantum confinement associated with small apex-radius emitters; •  field-related changes in emitter geometry or emission area or local

work-function.

Orthodox emission

The orthodoxy test is based on extracting f-values from FN plots by assuming the orthodox emission hypothesis.

The present rules for the orthodoxy test are as follows.

•  If the extracted f-value range is completely within the "apparently reasonable" range" (0.15<f<0.45) then the test is "passed".

•  If any part of the extracted f-value range is in one of the "clearly unreasonable" ranges (f<0.10, or f>0.75), then the test is "failed".

•  In other situations (when some part of the extracted f-value range is outside the "apparently reasonable" range, but not in the "clearly unreasonable" range), then the test is "undecided", and further investigation is needed.

If the test is failed or undecided, then an extracted FEF-value is unreliable. If the test is failed then very probably an extracted FEF-value is spuriously large.

Test Criteria

Application of test to some literature data

•  When researchers are looking for high-FEF materials, the test should always be applied to any apparently promising results.

•  There would be merit in applying the test systematically to published literature, to look for trends.

•  There would be merit in a convention that (certainly in materials contexts) the orthodoxy test is always applied to FN plots before publication, and that the result is reported (perhaps in the plot caption).

•  We need to look for alternative methods of extracting true FEF-values.

Provisional conclusions

6

What next ?

Proposals for improved practice in reporting CFE results are: •  Make clear what your terms "field" and "current density" mean, and

distinguish if necessary between local and macroscopic quantities. •  It may often be preferable to make FN plots using raw data (voltages

and currents) without pre-conversion to "apparent macroscopic field". •  If macroscopic fields and current densities are used, state quantitatively

how these have been obtained from the raw experimental data. •  For ease of comparison, give current densities in A/m2. •  With FN plots, give the units in which the argument of the logarithm is

measured [e.g., (im/Vm2 in A/V2)].

•  Do not use defective equations, because these mislead non-experts. (1) Use a technically complete equation for current or current density,

even if you subsequently approximate correction factors as unity. (2) Include a slope correction factor if you state an equation for

extracting a field enhancement factor or a local conversion length. [Based on: Nanotechnology 23 (2012) 095706.]

Proposals for improved reporting practice*

Topics in basic theory that need further work include:

•  Orthodoxy tests on existing literature.

•  Alternative methods of dealing with series resistance.

•  Effects of emitter shape and barrier form on FN plots & related parameters (including correction factors).

•  Analysis of other potential causes of non-orthodoxy.

•  Theory of tunnelling, especially near top of SN barrier.

•  FEM resolution.

•  Power of the field in the CFE equation pre-exponential.

•  New methods of reliably comparing experiment and theory.

•  More on quantum confinement.

•  Behaviour of non-metallic field emitters, particularly CNTs.

Where next (in basic theory)

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