New Type of Intercept Correction Factor

for Fowler-Nordheim Plots

Richard G. Forbes*

Advanced Technology Institute

& Department of Electronic Engineering,

University of Surrey, Guildford, Surrey GU2 7XH, UK

Permanent e-mail alias: R.Forbes@trinity.cantab.net

Andreas Fischer

Department of Physics, Mu'tah University,

Al-Karak, 61710, Jordan

now at: Insitut für Physik, Technische Universität Chemnitz

Chemnitz, Germany

Marwan S. Mousa

Department of Physics, Mu'tah University,

Al-Karak 61710, Jordan

Abstract—This paper defines a new type of intercept correction

factor for use in connection with the tangent method of analyzing

Fowler-Nordheim plots. Unlike the factor previously used, the

new factor is well defined and can be evaluated precisely for

simple barrier models. Theory using the new factor is intended to

replace existing theory. Applications will be presented elsewhere.

Keywords: theory of field electron emission; current-voltage

characteristics; Fowler-Nordheim plots; intercept correction factor.

I. INTRODUCTION

A technically complete FN-type equation for macroscopic

("LAFE-average") current density JM, in terms of macroscopic

field FM and local work-function �, has the form [1]

J

M= �

Ma� �1� 2 F

M

2 exp[��Fb� 3/2 /� F

M] , (1)

where a and b are the FN constants, �F ("nuF") is a correction

factor related to barrier form and �M is a macroscopic pre-

exponential correction factor. � is a macroscopic field

enhancement factor characteristic of strongly emitting LAFE

regions, and is related to a characteristic local surface field

(the "barrier field") FC by FC = �FM. If (1) is written directly

in terms of FC then it takes the simpler form

J

M= �

Ma� �1F

C

2 exp[��Fb� 3/2 /F

C] . (2)

In practice, the barrier is often modeled as a Schottky-

Nordheim (SN) barrier, and �F is then replaced by a particular

value vF ("veeF") of the principal SN barrier function v.

�M is determined mainly by the fraction of the total LAFE

area that emits, and varies greatly as between LAFEs. It is

thought to normally lie between 10–9

and 10–3

. Theoretically,

�M is a composite correction factor that can be decomposed

into factors relating to specific effects. Methods of finding �M

experimentally are of interest. Several exist, including the use

of FN plots. If a FN plot is straight, then one can fit a

regression line, and try to interpret the intercept. For some

materials/situations there are difficulties with this approach.

However, if emission is physically orthodox as defined in [1],

then in principle the linear-regression approach is useful. This

paper makes a small technical improvement to the

mathematics of the tangent method of FN-plot analysis [2].

In practice, regression analysis of FN plots is not

straightforward, due to the mathematical behaviour of FN-type

equations, and slope and intercept correction factors need to

be defined. Existing theory (see [2]) defines the intercept

correction factor in an inconvenient way. The present paper

introduces a new type of intercept correction factor.

Obviously, different forms of FN plot exist, depending on the

variables used. Discussion here uses the emitter local work-

function � and the voltage V as the independent variables, and

the emission current i as the dependent variable, because these

are usually the measured quantities.

II. BACKGROUND THEORY

To determine i(V), two auxiliary equations are needed:

FC=�VV, where �V is a voltage-to-barrier-field conversion; and

i=AMJM, where AM is the measurable LAFE macroscopic area.

Thus, in so-called FN coordinates for i(V):

ln{i / V 2} = ln{A

M�

Ma��1�

V

2 }exp[��Fb�3/2 /�

VV ] . (5)

For simplicity, consider parameters S* and QiV defined by

S* � b� 3/2 /�

V;

Q

iV= A

M�

Ma��1�

V

2. (6)

These definitions allow (5) to be simplified to

ln{i(V )/V 2} = ln{Q

iV(V )}� S *�[�

F(V ) / V ] . (7)

Fig. 1 plots (7) schematically as the broadened line "K".

This line stops at point "B", which corresponds to the voltage

VB at which �F becomes zero and the barrier just vanishes for a

Fermi-level electron moving "forwards", i.e. normal to the

emitter surface. For the sake of clarity, the curvature in line K,

as it approaches point B, is greatly exaggerated.

P2-04

Fig. 1.

To illustrate the

definition of �iV .

For any particular value 1/VP, ln{i(V)/V2} has the value

ln{i(V

P)/V

P

2} = ln{QiV

(VP)}� S *� [�

F(V

P) / V

P] . (8)

This formula is represented by the lower straight line (N) in

Fig. 1, which joins points "Q" and "P".

III. DEFINITON OF CORRECTION FACTORS

The slope SiV(V) of line K (and hence of its tangent "T") is

S

iV(V ) � ��

iV(V )� b�3/2 /�

V = ��

iV(V )� S * , (9)

where the slope correction factor iV is defined by (9). Line T

cuts the vertical axis at "R", where ln{i(V)/V2} has the value

ln{RiV(VP)} and RiV(V) can be written

R

iV(V ) � �

iV(V )�Q

iV(V ) . (10)

The parameter �iV is defined by (10). �iV is the factor by

which the intercept of the tangent to (5), taken at voltage V, is

greater than the value of the pre-exponential QiV, taken at

voltage V. As shown in Fig. 1, one can also write:

ln�iV = ln{RiV} – ln{QiV}. (11)

In this paper and future work, the name intercept correction

factor will be applied to �iV as defined by (10), and similar

factors for other choices of variables. This usage differs from

that in [2] and earlier papers, and the meaning of �iV differs

from that of the symbol � previously used.

It follows that line T is described by

ln{i(V

P)/V

P

2} = ln{RiV

(VP)}� �

iV(V

P)� S * /V

P. (12)

Hence, by geometrical analysis of Fig. 1, or by combining (8)

and (12) to eliminate ln{i(VP)/VP2, and noting that the result is

valid for any voltage V, we obtain

ln�

iV= {�

iV��

F}(b�3/2 /�

VV ) = {�

iV��

F}(b�3/2 /F

C) . (13)

(b�3/2/FC) is the JWKB exponent (or "strength") GF

ET of an

elementary triangular (ET) barrier of zero-field height �; thus

(13) can also be written more concisely, as

ln�

iV= {�

iV��

F}G

F

ET. (14)

Fig. 2. To illustrate how the basic intercept correction factor �B

varies with barrier field, for spherical emitter of radius ra .

If � is known and iV can be found by modelling, then (9)

yields �V . If, in addition, �iV could be found by modelling,

then values for QiV, and hence AM�M and �M, could be found.

IV. BASIC APPROXIMATIONS FOR IV AND �IV

The full a-priori calculation of iV and �iV, if possible at

all, would be very complicated, and has never been done. The

simplest useful approximation takes emission as physically

orthodox, and all of AM, �M, � and �V as independent of field,

voltage, current and temperature. This can be called the basic

approximation: the related slope and intercept correction

factors are denoted by B and �B, respectively.

If, in addition, an SN-barrier model is used, then B is

given by the SN-barrier function s, and �B is given by an

appropriate value of the parameter r2012 defined by

r2012

� exp[(s � vF)b�3/2 /�

VV ] = exp[(s � v

F)b�3/2 /F]. (15)

r2012 is a new SN-barrier function, introduced here. It can be

shown that a concise form of (15) is

r

2012= exp[��SNdv

F/df ] = exp[�SNu

F] , (16)

where f is scaled barrier field, vF is as before, uF � –dvF/df, c is

the Schottky constant, and �SN�bc2�–1/2

. For �=4.50 eV,

�SN=4.64. This result is similar to eq. (35) in [2], but––due to

the new definition of intercept correction factor used here––

(16) does not contain the term tF–2

.

For barriers other than simple barriers, B and �B can be

evaluated numerically. For illustration, Fig. 2 shows how �B

varies with 1/F for a spherical emitter of radius ra. Other

applications of the theory here will be presented elsewhere.

REFERENCES

[1] R. G. Forbes. "Extraction of emission parameters for large area field

emitters, using a technically complete Fowler-Nordheim-type equation,"

Nanotechnology 23, 095706, 2012.

[2] R. G. Forbes and J. H. B. Deane, "Comparison of approximations for the

Principal Schottky-Nordheim Barrier Function v(f), and comments on

Fowler-Nordheim Plots," J. Vac. Sci. Technol. B 28, C2A33, 2010.