TUTORIAL LECTURE, APRIL 2018: FIELD ...

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Electrical & Electronic Engineering

High-Electric-Field Nanoscience

TUTORIAL LECTURE, APRIL 2018: FIELD ELECTRON EMISSION AND THE INTERPRETATION OF FOWLER-NORDHEIM PLOTS Richard G. Forbes Advanced Technology Institute & Dept. of Electrical & Electronic Engineering, University of Surrey [Permanent e-mail alias: [email protected]]

Workshop on "Materials Science & Technology", University of Jordan, Amman, April 2018

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TUTORIAL LECTURE, APRIL 2018: FIELD ELECTRON EMISSION AND THE INTERPRETATION OF FOWLER-NORDHEIM PLOTS METADATA This presentation was made at a Workshop on "Materials Science and Technology", held at the University of Jordan, Amman, Jordan, 8-9 April 2018. The workshop combined the functions of a training workshop in presentation skills for MSc and PhD students, and a scientific workshop on topics in field electron emission. This presentation may be cited as R.G. Forbes, "Tutorial lecture, April 2018: Field electron emission and the interpretation of Fowler-Nordheim plots", Workshop on "Materials Science and Technology", University of Jordan, Amman, April 2018. doi:. This document has been allocated a doi by ResearchGate. At the time of writing, this doi can be found by "clicking" on the image associated with the relevant entry in my contributions list. This will bring up a metadata page with the doi recorded at the top left (but the metadata will not download).

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TUTORIAL LECTURE, APRIL 2018: FIELD ELECTRON EMISSION AND THE INTERPRETATION OF FOWLER-NORDHEIM PLOTS UPDATE STATUS In order to make this file more suitable for non-oral presentation, numerous minor changes have been made to the original presentation (including correction of a typographical error relating to the mathematical definition of a Schottky-Nordheim barrier on slide 17), and a significant amount of extra explanatory material has been added.


Structure of Tutorial

Background theory 1.

Introduction

2.

Conventions and terminology

3.

Ideal and non-ideal devices/systems

4.

Transmission regimes and emission current density regimes

5.

Core emission theory for the Fowler-Nordheim FE regime

Application to FN plot interpretation 6.

Auxiliary-parameter definitions

7.

Interpretation of FN plots

8.

Summary


Some basic terminology

This tutorial is about an electron emission process. It has several different names 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. Personally, I prefer the term "field electron emission", because I also work on the theory of field ion emission, and like to have symmetry in the names.


Basic field electron emission process

Electrons escape from field electron emitters by wave-mechanical tunnelling through a field-lowered potential-energy barrier. Tunnelling is not mysterious: it is a property of a wave-theory; tunnelling occurs with light, and also with waves on strings.

e−

incident reflected

potential-energy barrier

transmitted slope relates to barrier field

For electron tunnelling to be significant, the electrostatic field at the emitter surface (the so-called barrier field) must be of high magnitude, typically in the range 1-10 V/nm.


Two types of field electron emitter

Single-tip field electron emitter (STFE) [Tungsten wire emitter shown]

Large-area field electron emitter (LAFE) [Has many/very-many individual emitters/emission sites.] [Silicon carbide pillar array shown] [Diagram courtesy: M.-G. Kang, H.J. Lezec & F. Sharifi, Nanotechnology 24, 065201 (2013), Fig. 1.]


Main applications and contexts

(A) The main contemporary applications of field electron emitters are: •  STFEs: as electron sources for electron microscopes and related machines, including pulsed-laser machines as the electron emitter in a field electron microscope or a related energy-analysis machine •  LAFEs: as large-area electron sources for X-ray machines and microwave generators possibly as spacecraft neutralisers (B) Another important context is: •  understanding the causes and mechanisms of electrical breakdown in vacuum and in low-pressure gases, and how to prevent breakdown


2

2. Conventions and terminology 2a: The idea of "mainstream theory" 2b: Equation systems and unit systems 2c: The electron emission convention 2d: Terminology: fields 2e: Terminology: emission current densities


Mainstream FE device/system theory

This tutorial deals with mainstream FE device/system theory. Basically, "mainstream FE device/system theory" covers the basic topics that ALL researchers working in FE probably ought to know about. Part of this is mainstream emission theory, which is a "relatively simple" form of emission theory. Strictly, this theory applies to metal field emitters that are "not too sharp" (apex radius greater than 10-20 nm, say). Mainstream theory also covers the interpretation of current-voltage (i-V) characteristics, and basic theory relating to how emitters can be formed and can change/be changed during operation. This tutorial is mainly about how to interpret i-V data.


Specialist emission theory

Outside mainstream emission theory there are many areas of specialist emission theory. Mainstream emission theory does NOT fully cover any of the following: •  FE from very sharp emitters (apex radius less than about 10-20 nm); •  FE from emitters where significant quantum confinement occurs; •  FE from semiconductors; •  FE from graphene, carbon nanotubes, and related materials. However, when discussing emission from these materials, experimentalists often use mainstream emission theory as a first approximation.


Smooth-surface conceptual models

Mainstream emission theory is simple basic theory that a)  uses a smooth-surface conceptual model that •  ignores the existence of atoms •  assumes the emitter has a smooth classical planar surface •  uses the Sommerfeld free-electron model [in order to define electron states and determine their energies] •  uses Fermi-Dirac statistics •  assumes an uniform constant external electrostatic field b)  within the framework of the smooth-surface conceptual model, normally evaluates transmission probabilities by using semi-classical quantum mechanics, usually the Kemble or simple-JWKB formalisms.


Smooth-surface conceptual models

An initial step is to describe the potential energy (PE) models used in mainstream emission theory: first, the Sommerfeld model for a free-electron metal; then, the PE barrier models.

Sommerfeld model using forwards energy

ELECTRON-ENERGY component perpendicular to surface also called "FORWARDS ENERGY" Local vacuum level

φ Fermi level

Conduction band base

....... local work-function

χ occupied electron states

... well depth

KF ...... Fermi energy

Sommerfeld model using forwards energy

ELECTRON-ENERGY component perpendicular to surface also called "FORWARDS ENERGY" Local vacuum level Excited electron state

zero-field barrier height

Fermi level

Conduction band base

H

φ

....... local work-function

χ occupied electron states

... well depth

KF ...... Fermi energy


Two special barrier forms

With planar smooth-surface models, two barrier forms are commonly used to model the electron potential energy (PE) variation normal to the surface: the exactly triangular (ET) barrier H is zero-field barrier height

H slope = –eF

M=0

z

e is the elementary positive charge F is called the local barrier field z is distance measured from emitter's electrical surface

M(z) = H – eFz used by Fowler & Nordheim (1928)

The barrier is described by the quantity M(z) which I call the electron motive energy.


Two special barrier forms

With planar smooth-surface models, two barrier forms are commonly used to model the electron potential energy (PE) variation normal to the surface: the exactly triangular (ET) barrier

H

the Schottky-Nordheim (SN) barrier

H slope = –eF

slope = –eF

M=0

z

M=0

z

M(z) = H – eFz

M(z) = H – eFz – e2/16πε0z

used by Fowler & Nordheim (1928)

used by Murphy & Good (1956)


Scaled barrier field for the SN barrier

With a Schottky-Nordheim (SN) barrier, the barrier top can be pulled down to the Fermi level by applying a sufficiently high local field. We call this high field the barrier removal field FH . For any given barrier height H and local field FL , the scaled barrier field fH is given by fH = FL/FH .



With a Schottky-Nordheim (SN) barrier, the barrier top can be pulled down to the Fermi level by applying a sufficiently high local field. We call this high field the barrier removal field FH . For any given barrier height H and local field FL , the scaled barrier field fH is given by fH = FL/FH . For a SN barrier with H = φ , the barrier removal field is denoted by FR and called the reference field, and the scaled barrier field is denoted simply by f . (For example, if φ= 4.50 eV, then FR≈ 14.1 V/nm.) Then, it can be shown that f = FL/FR = cS2φ-2 FL , where cS is the Schottky constant, and cS2 = (e3/4πε0) ≅ 1.439 965 eV2 (V/nm)–1 [In this tutorial, all universal constants are given to 7 significant figures.]

This dimensionless parameter f is important in modern FE theory.



For example, values of scaled barrier field f can be used as measures of "when things happen", as shown below for a single-tip tungsten emitter: vF,

vF

[Table adapted from: R.G. Forbes, J. Vac. Sci. Technol. B16, 788 (2008).]


2b

2b: Equation systems and unit systems


Equation systems

Many different equation systems have been used in physics and engineering. The SI system of units is based on a system of equations and physical quantities in which the electric constant ε0 appears in Coulomb's Law. This system has been called the "rationalized metre-kilogram-second (rmks) system", and also the "metre-kilogram-second-ampere (mksa) system". However, its official name is now the "International System of Quantities (ISQ)". This name is used here. This tutorial uses the ISQ, but many older FE papers and textbooks use older systems. Changing to use of the ISQ is strongly desirable.


Field emission customary units

Modern FE uses the ISQ, but often does not use SI units. Instead, FE uses an ISQ-compatible system of atomic-level units, sometimes called field emission customary units. These customary units use the electronvolt (eV) as the energy-unit, write the elementary positive charge as 1 eV V–1, and often express electrostatic fields in V/nm. Using these customary units makes certain common calculations easy to carry out. All FE customary units are officially recognized for continued use alongside SI units.


Fowler-Nordheim and Schottky constants


2c

2c: The electron emission convention


The electron emission convention

The fields applied by a high-voltage generator to a field emitter are electrostatic fields. In classical electrostatics, when distance z from the emitter increases positively towards the right (or positively upwards), the fields applied to field electron emitters are negative in sign. Currents and current densities are also negative. The electron emission convention is to treat these field, currents and current densities as if they were positive. This tutorial lecture uses the electron emission convention.


The symbols "E" and "F"

The official symbol for classical electrostatic field is E. In scalar contexts, E denotes the signed magnitude of classical field. In FE this classical field E is negative. Some FE theoreticians use F to denote a positive quantity that is the negative of classical electrostatic field. This is done in this tutorial. This leaves E free to denote energies, which is useful. However, many FE experimentalists use E to mean the positive quantity that is the absolute magnitude of classical electrostatic field. This is acceptable, but using F provides more flexibility.


2d

2d: Terminology: fields


Characteristic local barrier field

(a) The local barrier field FL is the surface field that defines the tunneling barrier at a particular location on the emitter surface. (b) The characteristic (local) barrier field FC is the barrier field at some characteristic point "C" that is representative of the emitter as a whole. [In modeling, "C" is often taken at the emitter apex.] FC When experiments on large area field electron emitters (LAFEs) take place in plane-parallel-plate geometry, the mean field between the plates is also important ….


Macroscopic fields

(c) A macroscopic field FM is a field that characterizes the whole system geometry. There are different types of macroscopic field. In modeling, FM is usually taken as the uniform field (the "plate field" FP) between two plane parallel plates, of lateral extent large compared with their separation. [FP is often simply denoted by FM and called "macroscopic field", particularly in past literature; it is also sometimes called "external field", "applied field" or "ambient field". My change to calling it "plate field" is very recent".] Other types of macroscopic field are also used in the literature, in particular the "gap field" FG (which is the average field between an emitter and a counter-electrode).


Definition of "plate field" voltage between plates Vp ("plate voltage") separation of plane-parallel electrodes dsep

Plate field FP = Vp/dsep


Definition of "plate field enhancement factor" voltage between plates Vp ("plate voltage") separation of plane-parallel electrodes dsep

Plate field FP = Vp/dsep

FC FC is characteristic (local) barrier field Characteristic plate field enhancement factor (FEF) γCP is

γCP = FC / FP .


FEFs as characterization parameters

True field enhancement factors (FEFs) are useful emitter-characterization parameters, because they are a measure of "how pointy" a LAFE is. High FEF-values are technologically desirable, because they can lead to relatively low operating voltages.


3

Ideal and non-ideal devices and systems


FE devices and systems

FE takes place at the active surface of a field emitting device (FE device). Usually, this device is incorporated into an electrical circuit, to form an FE system. As in all electrical engineering, the measured voltage Vm and the measured current im are in principle the properties of the system. As in all electrical engineering, different contexts may require analysis of different circuits of different levels of complexity. Consider a simple FE system ….


Emission parameters and measured parameters

Note: This diagram uses the electron emission sign convention.

Vm and im are the measured voltage and current (at the voltage supply); Ve is the emission voltage, i.e., the voltage between the emitting region at the emitter tip, and the counter-electrode; ie is the emission current. The emission resistance Re is given by: Re = Ve/ie .


Why sometimes Vm ≠ Ve

If the parallel resistance Rp is made large, then measured current im = ie . In this situation, the measured voltage Vm = Ve ONLY IF the series resistance Rs [= Rs1 + Rs2] is negligibly small, - which is not always the case.


The ideal FE device/system

An ideal FE device/system is one where: (a) The behaviour of the device/system is determined by emission theory alone. This implies that: (i) emission voltage Ve is equal to the measured voltage Vm ; (ii) emission current ie is equal to the measured current im . (b) There are no "complicating effects" that need to be described as part of the emission theory (e.g., no space-charge effects, or fielddependent geometry). This implies that: •  all relevant auxiliary parameters (see later) can be treated as constants.


Non-ideal devices/systems

A non-ideal FE device/system is one that is not ideal. Well-recognised causes of non-ideality include: •  current-dependence in field enhancement factors; •  series resistance in the measurement circuit; •  field-dependent geometry (e.g., due to Maxwell-stress effects); •  heating-dependent changes in work function (due to adsorbate removal); •  effects due to field emitted vacuum space charge (FEVSC). But there may be others, such as field penetration effects into non-metals, or quantum confinement effects. Many FE devices/systems are non-ideal. We will see later that the usual methods of analysing Fowler-Nordheim plots apply only to ideal emitters. If you try to apply the usual methods to non-ideal emitters, then spurious results may be generated. It is thought that many experimental FE papers (maybe many hundreds of papers) have reported spurious results.


4

Transmission regimes and emission-current-density (ECD) regimes


Terminology about regimes

A transmission regime = A region of parameter-space (typically field and forwards energy) where particular effects determine transmission, or a particular formula for transmission probability D is an adequate mathematical approximation. An emission [or emission-current-density (ECD)] regime = A region of parameter-space (typically field and temperature, for given work-function) where a particular formula for local ECD J is an adequate mathematical approximation. These regimes can be depicted visually on regime diagrams. The details of the regime diagram depend on the assumption made about barrier form.


The FNFE ECD-regime for the SN barrier

For the SN barrier there is a low-temperature ECD regime variously called the Fowler-Nordheim field electron emission (FNFE) regime, or the "cold field electron emission regime (CFE)", or just the "field emission regime". For the SN barrier, for φ = 4.50 eV, the FNFE ECD-regime is shown alongside. [Data originally from: E.L. Murphy & R.H. Good, Phys. Rev. 102, 1464 (1956).]

[For an emitter with φ = 4.50 eV, barrier field F ≈ f ×(14.1 V/nm).]


The FNFE ECD-regime for the SN barrier

For the SN barrier there is a low-temperature ECD regime variously called the Fowler-Nordheim field electron emission (FNFE) regime, or the "cold field electron emission regime (CFE)", or just the "field emission regime". For the SN barrier, for φ = 4.50 eV, the FNFE ECD-regime is shown alongside. [Data originally from: E.L. Murphy & R.H. Good, Phys. Rev. 102, 1464 (1956).]

[For an emitter with φ = 4.50 eV, barrier field F ≈ f ×(14.1 V/nm).]

In the FNFE regime, the ECD is sometimes described by the (1956) MurphyGood (MG) finite-temperature Fowler-Nordheim-type (FN-type) equation. [But the so-called "new standard" FN-type equation (see later) provides a better formal description.] In the FNFE regime, FN plots are used to analyze current-voltage data. So today's tutorial is about data analysis for this emission regime.


Relating practical devices to SN ECD regimes Commercial items

Transmission regimes

ECD regimes (after Swanson/Bell/Forbes)

Classical transmission regime

High-T (low-F) limit = Classical thermal electron emission (CTE)

Surface-reflection regime

Quantum-mechanical thermal electron emission (QMTE)

Thermionic emitter

Barrier-top regime

Barrier-top electron emission (BTE) [or "extended Schottky"]

Schottky emitter

Deep tunnelling

Fowler-Nordheim FE (FNFE) [or "cold FE" (CFE)] Low-T limit = Zero-T FNFE

No devices

Field emitter


5

5. Core emission theory for the FNFE regime 5a: The idea of complexity level 5b: Scaled form for JkSN 5c: Comments on the SN barrier functions 5d: Comments on the Nordheim parameter y


5a

5a: The idea of complexity level


Fowler-Nordheim-type equations

A Fowler-Nordheim-type (FN-type) equation is any FNFE equation with the mathematical form Y = CYX X2 exp[−BX /X] , where: X is any FE independent variable (e.g., a field or a voltage); Y is any FE dependent variable (e.g., a current or current density); BX is a function related to choice of X and barrier form; CYX is a function related to other choices, including X, Y, and BX . BX and CYX are NOT constants (except in the most elementary models). The core theoretical forms of FN-type equation (those derived directly from theory) give local emission current density (ECD) J in terms of local work function φ and (the absolute magnitude F of) the local barrier electrostatic field.


The elementary J(F)-form FN-type equation

The simplest core FN-type equation is the elementary J(F)-form equation: Jel = aφ–1F2 exp[–bφ3/2/F] . where a and b are the FN constants [see table earlier]. This is based on assuming an exactly triangular (ET) barrier, and is a simplification of the original equation derived by FN in 1928. The equation above is good for undergraduate teaching, but is too simple to describe real situations. Hence, it has to be generalised, in TWO ways.


The concept of "kernel current density"

(1) The elementary equation relates to an ET barrier. BUT: •  it neglects exchange-and-correlation (XC) effects (usually modelled as image effects); •  it is not adequately valid for highly curved emitters. We formally include both effects with a barrier form correction factor, here for a general barrier (GB). Thus, the general barrier of zero-field height φ has a correction factor νFGB ("nuFGB "), and the resulting equation is JkGB =

aφ–1F2 exp[–νFGBbφ3/2/F].

JkGB is a mathematical quantity that can be calculated exactly for a given barrier form, when values of φ and F are given. I call JkGB the kernel current density for the chosen barrier form "GB".


Local pre-exponential correction factor

(2) To allow for other corrections, it is necessary to include a local pre-exponential correction factor λGB. Thus the physical local ECD JGB is given by JGB

=

λGB JkGB

=

λGB (aφ–1F2) exp[–νFGBbφ3/2/F].

The factor λGB allows formally for corrections due to all of: •  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; •  any unrecognized theoretical inadequacy. The equation above is the core general-barrier FN-type equation.


Equation complexity levels

Historically, many different assumptions/models have been used to obtain expressions for νFGB and λGB . The complexity level of a FN-type equation is decided by the choices of: (a) barrier form (which determines νF); and (b) what effects/approximations to include in λ . For planar emitters, the main complexity levels used historically and currently are shown in the following table. I give each of the main complexity levels a specific name.


Most significant planar complexity levels

TABLE 1. Complexity levels of core planar Fowler-Nordheim-type equations. Name Date λGB → Barrier form νFGB → Note a Elementary 1995? 1 ET 1 b Original 1928 PFFN ET 1 Fowler-1936 1936 4 ET 1 ET Extended elementary 2015 ET 1 λ c Dyke-Dolan 1956 1 SN vF c Murphy-Good (zero temperature) 1956 tF–2 SN vF d vF Murphy-Good (finite temperature) 1956 λT tF–2 SN e Orthodox 2013 SN vF λSN0 vF New-standard 2015 λSN SN e "Barrier-effects-only" 2013 λGB0 GB νFGB General ("Modinos-Forbes") 1999 GB λGB νFGB a

Earlier imprecise versions exist, but the first clear statements seem to be in 1995 and 1999. PFFN is the Fowler-Nordheim tunnelling pre-factor. c vF and tF are appropriate particular values of the SN barrier functions v and t. d λT is the Murphy-Good temperature correction factor e The superscript " 0 " indicates that the factor is to be treated mathematically as constant. b

For details, see: R.G. Forbes et al., "Fowler-Nordheim plot analysis: a progress report", Jordan J. Phys. 8 (2015) 125; arXiv:1504.06134v7 . Historically, around 15-20 different mathematical approximations have been used for the particular value vF of the principal SN barrier function (or "field emission v-function") v.


5b

5b: Scaled form for JkSN


Derivation of scaled form for JkSN

The kernel current density for the SN barrier (JkSN) can be written in socalled scaled form, as follows. Define φ-dependent scaling parameters η(φ) and θ (φ) by

η(φ) = bφ3/2/FR = bcS2φ–1/2 , θ (φ) = aφ–1FR2 = acS–4φ3 , where bcS2 [≈ 9.836239 eV1/2] and acS–4 [≈ 7.433979×1011 A m–2 eV–3] are universal constants. In scaled form, the equation for JkSN becomes JkSN = θ f2 exp[–η v(f)/f] . For example, for φ= 4.50 eV, then FR≈ 14.1 V/nm, η≈ 4.64, θ ≈ 6.77×1013 A/m2.


Derivation of scaled form for JkSN JkSN = θ f2 exp[–η v(f)/f] .

This scaled equation contains only one field-like variable (f), and it is now known that a simple good approximation exists for v(f). Hence JkSN is well approximated by JkSN ≈ θ f2 exp[η {1 – (1/6)lnf – 1/f}]. This formula is useful because it can be evaluated on a spreadsheet. Having the exponent of JkSN in scaled form is very useful in a number of FE contexts, in particular when discussing the orthodoxy test, or the extraction of formal emission areas from FN plots.


Derivation of scaled form for JkSN JkSN = θ f2 exp[–η v(f)/f] .

This scaled equation contains only one field-like variable (f), and it is now known that a simple good approximation exists for v(f). Hence JkSN is well approximated by JkSN ≈ θ f2 exp[η {1 – (1/6)lnf – 1/f}]. This formula is useful because it can be evaluated on a spreadsheet. Having the exponent of JkSN in scaled form is very useful in a number of FE contexts, in particular when discussing the orthodoxy test, or the extraction of formal emission areas from FN plots. In any context in which the auxiliary parameter cX can be treated as constant (which includes the "orthodox emission situation" discussed below), we can write FR=cXXR, F=cXX, and hence that f=F/FR=X/XR. Hence the scaled equation for JkSN can alternatively be written in the hybrid form JkSN = θ f2 exp[–v(f)⋅ηXR/X] . This form is sometimes useful.


5c

5c: Comments on the SN barrier functions


The special mathematical function v(x)

A special mathematical function is a named mathematical function with a well-defined mathematical form. Simple examples are: sin(x) and exp(x). More complicated examples are Bessel functions and Airy functions. A special mathematical function denoted here by v(x) has an important role in emission theory for the SN barrier. We can separate the "purely mathematical" aspects of v(x) from its role in FE theory, and discuss these mathematical aspects as "pure mathematics".


The special mathematical function v(x)

A special mathematical function is a named mathematical function with a well-defined mathematical form. Simple examples are: sin(x) and exp(x). More complicated examples are Bessel functions and Airy functions. A special mathematical function denoted here by v(x) has an important role in emission theory for the SN barrier. We can separate the "purely mathematical" aspects of v(x) from its role in FE theory, and discuss these mathematical aspects as "pure mathematics". My preferred name for v is the "principal SN barrier function" (or it could be called the "field emission v-function".) [Older names for v are "Nordheim function", "special field emission elliptic function", or simply "vee", but these are now thought less suitable.]


Barrier form correction factor νFSN

From detailed mathematical analysis, it can be shown that for a SchottkyNordheim barrier of zero-field height φ , the physical barrier form correction factor νFSN is obtained from the special mathematical function v(x) by setting x=f (where f , as before, is scaled barrier field, for a barrier of zerofield height φ ). That is:

νFSN = v(x→f) = v(f) . When x is replaced by f, all mathematical formulae for v(x), expressed in terms of x, become modelling formulae applicable to the SN barrier, expressed in terms of f. For example, the mathematical formula v(x) ≈ 1 – x + (x/6)lnx converts to become the modelling formula v(f) ≈ 1 – f + (f/6)lnf .


Mathematical functions closely related to v(x)

FE theory also makes use of several special mathematical functions closely related to v(x) . These are defined as follows, and – in relation to tunnelling through the SN barrier – have the roles shown (when x→f). u(x) ≡

– dv/dx

[u(f) is –dv/df]

t(x) ≡

v – (4/3) x dv/dx

[t(f) is decay-rate correction function]

s(x) ≡

v – x dv/dx

[s(f) is slope correction function]

w(x) ≡

ds/d(1/x)

[w(f) is curvature correction function]

r(x) ≡

exp(η u)

[r(f) is 2012 intercept correction function]

where η [≅ 9.836 239 (eV/φ)1/2] is the scaling parameter defined earlier. At present, the mathematical functions of x do not have well-defined individual names, but can be called the field emission u-function (etc.). Collectively, they can be called the field emission special mathematical functions or the SN-barrier functions. I now typeset these special mathematical functions upright (e.g., v not v).


6

6. Auxiliary parameters

6a: Area-like auxiliary parameters 6b: Auxiliary parameters for the independent variable


Auxiliary equations and auxiliary parameters

Core equations, as developed from basic emission theory, provide formulae for the local ECD (JL, here), in terms of the local work function φ and the local barrier field (FL, here). To link these core equations to quantities/parameters of experimental and practical interest, such as the measured current (im) and voltage (Vm), we need auxiliary equations. These auxiliary equations will contain one or more auxiliary parameters that help specify the nature of the link. In many cases, these auxiliary parameters can serve as characterization parameters for the FE device/ system under discussion.


Auxiliary equations and auxiliary parameters

The detailed theory of these auxiliary equations is much wider than the emission theory involved in the core equations. Often it involves aspects of device physics/engineering and of electronic circuit theory, and in many situations the theory is neither properly understood nor clearly formulated. Thus the interpretation of FE current-voltage measurements can be very complicated, is not fully understood for many "non-ideal" cases, and is still a topic of active research.


Emission area definitions

The local barrier field FL and local ECD JL vary with position on the emitter surface. To find a formula for total emission current ie , first choose a characteristic point "C" on the surface (usually at the emitter apex). A formula for ie is found by integrating over the surface and writing the result as ie = ∫ JL dA ≡ AnJC = AnλCJkC , where An is the notional emission area. For all emitters, the value of λC is uncertain, and for LAFEs the value of An is also uncertain. Having two parameters of uncertain value in an equation is unhelpful, so define a new parameter, the formal emission area Af , by Af ≡ AnλC .


Area efficiency definitions

For large-area field electron emitters (LAFEs) the notional area efficiency αn is defined by αn = An/AM , where AM is the LAFE macroscopic area or "footprint". And the formal area efficiency αf is defined by

αf = Af/AM = AnλC/AM = λCαn .


The different roles of formal and notional parameters

We need both formal and notional "theoretical" area-like parameters, because: •  in appropriate circumstances ("where emission is orthodox"), good values of the formal parameters can deduced from experiment, via FN plots; •  but the notional parameters appear in some existing theory. In principle, the notional parameters are probably closer to "geometrical" area estimates, but (due to present uncertainty in the value of λC), accurate values of the notional parameters cannot be deduced from experiment.


6b

6b: Auxiliary parameters for independent variables


The auxiliary parameter cX

Earlier we had the core general FN-type equation JCGB = λCGB (aφ–1FC2) exp[–νFGBbφ3/2/FC] . To relate the characteristic local barrier field FC to an empirical independent variable "X" used in making predictions or interpreting experiments (usually X is a voltage V or a macroscopic field FM ), we need an equation of the form FC = cX X . This is the auxiliary equation for the independent variable, and cX is the auxiliary parameter for the independent variable . The specific form for cX depends on the independent variable of interest. This tutorial usually takes this to be the measured voltage Vm .


Voltage-related characterization parameters

If X is a measured voltage (Vm), then the auxiliary equation for local barrier field FL can be written in any of the alternative forms: FL = βVVm = Vm/ζ = Vm/kra , where ra is the emitter's apex radius of curvature, and: •  βV is the local voltage conversion factor (VCF) (unit: nm–1) ; •  ζ is the local voltage conversion length (VCL) ** (unit: nm) ; •  k is called the shape factor or field factor (dimensionless) . Because β is widely used in FE literature to denote field enhancement factors (see below), and confusion is possible, I consider it better practice to use VCLs. Note that VCLs are formal parameters, not physical lengths. Thus, for the characteristic barrier field FC , I now prefer to write FC = Vm/ζC , where ζC is the characteristic voltage conversion length. [** This parameter ζ was previously called "local conversion length", with acronym "LCL".]


VCLs as characterization parameters

For a given plane-parallel-plate system, with a LAFE on one plate, the characteristic voltage conversion length (VCL) ζC and the characteristic plate field enhancement factor (FEF) γPC are linked by the approximate relation ζC ~ dsep/γPC [For ideal devices/systems, replace "~" by "=".] The FEF and the VCL are both system characterization parameters. The plate FEF γPC characterizes the emitter alone, but the VCL ζC characterizes the whole system geometry. A small VCL means that the emitter will turn on and operate at relatively low voltages (which is technologically desirable), but the plate separation dsep must not be "too small", because unwanted electrostatic effects and unwanted vacuum breakdown effects can occur if dsep is too small.


7

7: Interpretation of FE current-voltage characteristics 7a:

Basic theory of Fowler-Nordheim plots

7b:

Analysis of FN-plot slope for ideal devices/systems

7c:

The "spurious-characterization" problem, and the test for lack of field emission orthodoxy

7d:

Phenomenological adjustment

7e:

Understanding why some FE devices/systems (particularly LAFEs) are non-ideal

7f:

Extraction of formal emission area for ideal devices/systems


The aim of Fowler-Nordheim-plot analysis

FE current-voltage characteristics can in principle provide useful emittercharacterization information. The commonest method of analyzing such characteristics is by means of a Fowler-Nordheim (FN) plot. Before describing the so-called orthodox method of FN plot analysis, I provide some general information. This may help explain why FN plot analysis can encounter significant practical difficulties. The discussion here uses natural logarithms.


7a

7a: Basic theory of Fowler-Nordheim plots


The "curly-brackets" convention

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.]


Fowler-Nordheim coordinates

Consider a function Y(X). The mathematical operation of "writing Y(X) in Fowler-Nordheim coordinates" is: •  Define a function L ≡ ln{Y/X2} ; •  Express L in the form L(X–1) . A Fowler-Nordheim plot is any plot of the type L(X–1) vs X–1 .


Theoretical equations in FN coordinates

A FN-type equation is any FE equation with the form Y = CYX X2 exp[−BX /X]. Let us write BX /X ≡ νFBXel /X , where νF is the barrier form correction factor for the general-barrier (denoted earlier by νFGB), and BXel is defined as BXel ≡ bφ3/2/cX . In FN coordinates, the top equation then becomes L(X–1) = ln{CYX} – νFBXel X–1 . Because FN plots appear in the literature in various different forms, it is convenient to use the non-specific independent variable X in the theory initially given here.


Analyzing FN plots – the task

The theoretical FN plot is L(X–1) = ln{CYX} – νFBXel X–1 . In real situations, CYX , νF and BXel may all depend on X. This makes the theoretical FN plot curved. Also, if image effects are in the barrier model, then (as X–1 gets smaller) the theoretical FN plot stops at the value XR–1 at which the barrier height goes to zero (i.e., the top of the barrier goes below the Fermi level). However, experimental 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} – νFBXelX–1 . Its slope SYX is SYX = ∂L/∂(X–1) = ∂ln{CYX}/∂(X–1) – νFBXel – BXel X–1 ∂νF/∂(X–1) – νF X–1 ∂BXel/∂(X–1) . The related slope correction factor σYX is defined to be

σYX ≡ – SYX / BXel = –[∂ln{CYX}/∂(X–1)]/BXel + [νF + X–1 ∂νF/∂(X–1)] + [νF X–1 ∂BXel/∂(X–1)]/BXel . Hence the slope of the theoretical FN plot can be written: SYX = – σYX BXel = – σYX bφ3/2/cX . Discussions in the literature often omit any correction factor similar to σYX . This results in defective equations.


Making theory and experiment compatible

Because experimental FN plots are slightly curved, the line fitted to an experimental plot is, strictly, a chord. One now needs to relate the measured properties of this fitted chord to those of the theoretical FN plot. Historically, this has been done in several ways, but my preferred approach is the so-called tangent method.


Analyzing FN plots - the tangent method

The tangent method has the following steps. •  Model the fitted straight line by the tangent to the theoretical FN plot. •  Assume fitted line is parallel to the tangent at some horizontal-axis value Xt–1 (called the fitting point, but usually its value is not initially known). •  Write the equation for the tangent at Xt–1 as L(X–1) = ln{Rttan} + Sttan X–1 ≡ ln{ρtCYX} – σt BXel X–1 , where Sttan and ln{Rttan} are the slope and intercept of the tangent as taken at Xt–1 , and σt and ρt are slope and intercept correction factors, evaluated at Xt–1 [i.e., σt = σYX(Xt–1), and ρt is then defined via the above equation, as explained below.]


The meaning of σt and ρt , and derivation of ρt "L" is the theoretical FN plot (but with curvature exaggerated). "T" is its tangent, taken at "t".


The meaning of σt and ρt , and derivation of ρt "L" is the theoretical FN plot (but with curvature exaggerated). "T" is its tangent, taken at "t". From curve "L": L(Xt–1) = ln{CYX} – νFBXel Xt–1 From tangent "T": L(Xt–1) = ln{ρtCYX} – σt BXel Xt–1 Hence (using cXXt = FCt) lnρt = [σt–νFt] (bφ3/2/FCt) where FCt corresponds to Xt .


The meaning of σt and ρt , and derivation of ρt "L" is the theoretical FN plot (but with curvature exaggerated). "T" is its tangent, taken at "t". From curve "L": L(Xt–1) = ln{CYX} – νFBXel Xt–1 From tangent "T": L(Xt–1) = ln{ρtCYX} – σt BXel Xt–1 Hence (using cXXt = FCt) lnρt = [σt–νFt] (bφ3/2/FCt) where FCt corresponds to Xt .

Clearly, before we can estimate ρt, we need to estimate σt . But, to estimate σt , we need an expression for σYX . We shall return to this issue shortly.


Analyzing FN plots – the tangent method

If valid theoretical estimates of σt and ρt can be made, the tangent method in principle continues as follows: •  Write equation for fitted line in the form L(X–1) = ln{Rfit} + Sfit X–1 where Sfit and ln{Rfit} are the fitted slope and intercept. •  Hence, identify Sfit = Sttan = – σt BXel , and Rfit = Rttan = ρt CYX , and use estimated σt and ρt values to extract values for BXel and CYX . •  Use values of BXel and CYX , and other relevant information (e.g., value of relevant local work function) to find values of relevant characterization parameters, in particular cX . I first discuss how to extract characterization information from a FN-plot slope, and then the issue of extracting information about emission area.


7b

7b: Analysis of FN-plot slope (for ideal devices/systems)



Given that BXel=bφ3/2/cX , we proceed as follows. We have Sfit = – σt bφ3/2/cX . From the measured value Sfit, determine the value of a slope characterization parameter (or "apparent auxiliary parameter" cXapp) by cXapp = –bφ3/2/Sfit . The true value cXtrue of the auxiliary parameter is then given by cXtrue = σt cXapp = – σtbφ3/2/Sfit .



Given that BXel=bφ3/2/cX , we proceed as follows. We have Sfit = – σt bφ3/2/cX . From the measured value Sfit, determine the value of a slope characterization parameter (or "apparent auxiliary parameter" cXapp) by cXapp = –bφ3/2/Sfit . The true value cXtrue of the auxiliary parameter is then given by cXtrue = σt cXapp = – σtbφ3/2/Sfit . For example, if the FE device/system is ideal, and we take X to be the measured voltage Vm, then cX→(ζC)–1, and the characteristic voltage conversion length ζC is given via (ζC)–1 = – σtbφ3/2 / (SVm)fit , where " (SVm)fit " means the slope of a FN plot made against 1/Vm .


The general approach usually can't be applied

The above is a “general approach”. Often, we cannot use it because we cannot reliably calculate σt . Hence most FE data analyses use mathematically restricted versions of the tangent method – either the orthodox approach (which assumes a SN barrier and takes σt≈0.95) or the elementary approach (which sets σt=1). These approaches are adequate approximations ONLY IF the FE device/ system is ideal. We return to this point later. The orthodox data-analysis approach is based on the so-called orthodox emission hypothesis ….


The orthodox emission hypothesis

The orthodox emission hypothesis is a set of physical and mathematical assumptions/approximations that allow well-specified analysis of measured current-voltage data relating to field electron emission (FE). These are: (a) the emission voltage Ve between the emitting regions and a surrounding counter-electrode can be treated as uniform across the emitting surface and equal to the measured voltage Vm; (b) the measured current im is equal to the emission current ie and is controlled by Fowler-Nordheim FE at the emitter/vacuum interface; (c) emission can be taken to involve tunneling though a SN barrier; further, the emission current ie is given by a related FN-type equation in which the only quantities that depend on the measured voltage Vm are the independent variable [best taken as Vm itself] and the barrier-form correction factor; (d) the emitter local work-function φ is constant (i.e., constant across the surface, and independent of time and of ie), and has a value close to that theoretically assumed.


Slope correction factor for orthodox approach

In the orthodox approach, the slope correction factor σt is given by an appropriate value st of the field emission s-function s, which serves as the slope correction function for the SN barrier. The orthodoxy hypothesis also requires that all relevant auxiliary parameters cX be treated as constant; this implies that X can be taken as strictly proportional to the scaled barrier field f . Thus we can write X = f XR , where XR is the value of X at which a barrier of zero-field height φ vanishes, with the fitting value ft given by ft = Xt/XR .


Orthodox approach vs. elementary approach

It follows that the fitting value σt of the slope correction factor given by

σt = st ≡ s(ft) . In practice, s(f) is a weakly varying function of f, and it is nearly always adequate to make the approximation

σt = st ≈ 0.95 .


Orthodox approach vs. elementary approach

It follows that the fitting value σt of the slope correction factor given by

σt = st ≡ s(ft) . In practice, s(f) is a weakly varying function of f, and it is nearly always adequate to make the approximation

σt = st ≈ 0.95 . By contrast, the elementary data-analysis approach is based on the elementary FN-type equation, and uses the approximation σt = 1. Most FE theoreticians believe that (for an ideal device/system) the orthodox approach is more realistic that the elementary approach, and hence that an error of around 5% is involved when applying the elementary approach to a FN plot from an ideal device/system. [Use of the elementary approach is very common in FE experimental literature.]


Derivation of f-values in orthodox theory

For the SN barrier, we can write vFbφ3/2/FC = vFη/f = vFηXR/X , where η [≡ bcS2φ –1/2] is the scaling parameter defined earlier. Hence, in orthodox data-analysis theory, the slope Stan of a theoretical FN plot can alternatively be written Stan = – sηXR , and the slope Stan(Xt–1) at the fitting point Xt–1 is given by Stan(Xt–1) = – stηXR . Identifying Stan(Xt–1) with the slope Sfit of the line fitted to empirical data, as before, leads to the formula 1/XR = – stη/Sfit , where st can be taken equal to 0.95. Hence, the f-value corresponding to a given X-value can be found as f = X/XR = (–stη/Sfit) X .


7c

7c:

The "spurious-characterization" problem, and the test for lack of field emission orthodoxy


The problem with non-ideal devices/systems

The orthodox and elementary data-analysis approaches are adequate approximations ONLY IF the FE device/system is ideal. As indicated earlier, there are many possible physical causes of nonideality. Many real FE devices/systems are non-ideal, and may in fact have σt-values substantially less than 1. Use of the elementary or orthodox approaches on data from non-ideal devices/systems is likely to generate spurious characterization data. It is thought that a large proportion (perhaps as much as 40%) of published field-enhancement-factor (FEF) values may be spuriously high. One way in which this spurious-characterization problem arises is as follows …..


Spurious VCL values

[Diagram courtesy: W. Zhu, P.K. Baumann & C.A. Bower, Chap. 6 in: W. Zhu (ed.) Vacuum Microelectronics (Wiley, New York, 2001), Fig. 6.13(b).]

Consider the formula:

ζC–1 = – σt bφ3/2 / SVfit .

IF σt is taken as equal to st=0.95 for both green and red lines, THEN: IF green line is expected to give a meaningful VCL-value, THEN red line [with its lower value of |SVfit| ] is expected to give a spurious VCL-value.


Spurious FEF-values, too

In planar-parallel-plate geometry, the apparent field enhancement factor (γPC)app extracted from a FN plot is related to the apparent VCL-value ζCapp extracted for the same raw experimental data, by: (γPC)app ~ dsep / ζCapp . Thus, if the apparent VCL-value is spurious (that is, if it differs significantly from the true small-current VCL-value), then the related plate-FEF value will also be spurious. The apparent gap-FEF values derived in some other system geometries will also be spurious if the related apparent VCL-value is spurious.


Treating the spurious-characterization problem

In the ideal world, one would calculate the slope correction factor σt (or find a better approach). But, for non-ideal devices, this appears to be very difficult and complicated. It will probably not happen quickly. So we need to understand the scale of the spurious-characterization problem, and find temporary means to deal with it. Hence: (1) The creation of a test for lack of field emission orthodoxy. (2) The introduction of a procedure for "phenomenological adjustment".


Test Criteria

The orthodoxy test is based on extracting, from FN plots, f-values that describe the range used for the independent variable X. The SN-barrier formula derived earlier [namely f = (–0.95 η/Sfit) X ] is used. The test rules are as follows: •  If the extracted f-value range is completely within an "apparently reasonable" range" (0.15

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