CHARGE TRANSFER UPON CONTACT BETWEEN
METALS AND INSULATORS

by
Donald Ora VanOstenburg

AN ABSTRACT
Submitted to the School for Advanced Graduate Studies of
Michigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

1956

Approved

^ . /M

CHARGE TRANSFER UPON CONTACT BETWEEN METALS AND INSULATORS
The equilibrium charge distribution between sub­
stances in contact has been calculated for several cases in
the one-dimensional approximation, e.g. metal-insulator,
metal-semiconductor, and metal-metal.

In addition to these

cases, we have considered insulator-insulator contacts, and
certain other metal-insulator contacts.

The direction of

charge transfer is, of course, such as to equalize the Fermi
levels.

The order of magnitude of the charge and of the

effective depth of the charged regions depend on the rela­
tive positions of the energy bands in contact.

Thus, for

example, in the metal-insulator case it is possible to have
either a large charge density ( 1 0 ^ statcoulombe/cra ) in a
thin layer (10

cm), or small charge density (10“

lombs/cm^) in a thick layer (10 ~ cm).

statcou-

The insulator-

insulator cases give rise to similar distributions.

The

energy levels in most insulators are too poorly known to
permit strict quantitative comparison with experiment, but
the results permit a semi-quantitative explanation of many
of the observations in static electrification.

The calcula­

tions were carried out assuming the bodies In immediate con­
tact.

No consideration was given the problem of a dielectric

gap separating the bodies.

CHARGE TRANSFER UPON CONTACT BETWEEN
METALS AND INSULATORS

by
Donald Ora VanOstenburg

A THESIS
Submitted to the School for Advanced Graduate Studies
Michigan State University of Agriculture and
Applied Science in partial fulfillment of
the requirements for the degree of

DOCTOR OF PHILOSOPHY
Department of Physics and Astronomy

1956

ProQuest Number: 10008531

All rights reserved
INFORMATION TO ALL USERS
The quality of this reproduction is dependent upon the quality of the copy submitted.
In the unlikely event that the author did not send a complete manuscript
and there are missing pages, these will be noted. Also, if material had to be removed,
a note will indicate the deletion.

uest,
ProQuest 10008531
Published by ProQuest LLC (2016). Copyright of the Dissertation is held by the Author.
All rights reserved.
This work is protected against unauthorized copying under Title 17, United States Code
Microform Edition © ProQuest LLC.
ProQuest LLC.
789 East Eisenhower Parkway
P.O. Box 1346
Ann Arbor, Ml 4 8 1 0 6 - 1346

ACKNOWLEDGMENT
The author wishes to thank Professor
D. J. Montgomery for the suggestion of the
problem and for his invaluable aid in its
solution.

TABLE OF CONTENTS
PAGE

INTRODUCTION................................

1

PROPERTIES OF ELECTRONS IN SOLIDS, CHARGE
DENSITY, AND ELECTRICAL CONTACTS .

3

A.
B.
C.

Electrons in Solids ................
Charge Density......................
Basic Properties of Contacts .

3

METAL-METAL CONTACTS ......................

10

5

7

Charge Density in the Metal
Solution of Poisson’s Equation

10
12

METAL-INSULATOR CONTACTS...................

18

A.
B.

A.
B.

C.

D.
E.

General Behavior . . . . . . .
Case of Large Charge Transfer from
Metal to Insulator ................
1.
Solution of Poisson9s Equation .
2. Evaluation of the Quantities
eAv; , eAV^and 3
.
3. Calculation of the Surface
Charge Density ................
4. Calculation of the Volume
Charge Density ................
Case of Large Charge Transfer from
Insulator to Metal
................
1. Evaluation of the Quantities
eAVJ , eAVL , and j .............
2. Calculation of the Surface and
Volume Charge Density .
The Case of Small Charge Transfer
from Metal to Insulator.............
The Case of Small Charge Transfer
from Insulator to Metal.
.

19
22
24
26
28
29

32
33
34

37

.............

41

General Behavior ...................
Case of Small Charge Transfer.
Case of Large Charge Transfer.

41
41
45

INSULATOR-INSULATOR CONTACTS
A.
B.
C.

18

iv
CHAPTER
VI.

PACE
APPLICATIONS.................................... L8
A.
B.

VII.

Other Features of Contacts.
Comparison with B-.xperiment.

. . .
. . .

C O N C L U S I O N S ..............................

REFERENCES CITED.

.

.

.

.

.

.

.

.

.

.

L8
50
57
58

CHAPTER I
INTRODUCTION
Contact phenomena play an Important role in presentday physics.

It is convenient to make a division into time-

dependent and time-independent processes.
dependent type we have, for example,

Under the time-

the point contact and

the junction transistor where charge flows in more or less
steady fashion.

Under the time-independent type we have the

phenomena of contact potentials and static electrification.
This paper is concerned with only the last process.
We have made an attempt to explain some results of
static electrification as reported by Hersh and Montgomery[l]
and by Harper[2].

Various authors have calculated the equil­

ibrium charge distribution between substances in contact for
several cases in the one-dimensional approximation, e .g. metalinsulatorC3]» metal-semiconductor[3,^ J, metal-metal[4],

Be­

sides these cases we have considered insulator-insulator con­
tacts and certain additional metal-insulator contacts.

The

direction of charge transfer is, of course, such as to equalize
the Fermi levels.

The order of magnitude of the charge and

of the effective depth of the charged region depends on the
relative position of the energy bands in contact.

Assuming

an energy-band structure for the substances, we have determined

2
the sign of the charge transferred the charge density, field
strength, and potential variation with distance in the
materials.

The calculations are carried out by applying

Fermi-Dirac statistics in order to find the charge density.
We then solve Poisson°s equation in each region and match
the solutions at the boundary by means of suitable boundary
conditions.

This procedure gives us the necessary conditions

to determine the constants and, therefore, permits solution
of the problem.

The calculations show that ample charge is

transferred to explain some experimental data and so in these
cases it is unnecessary to invoke surface states or local
heating.

The calculations were carried out assuming the

bodies in immediate contact, the interpenetration of the
bodies being supposed deep enough so that the interior struc­
ture need only be considered.

No consideration was given the

problem of a dielectric gap separating the bodies, although
it is believed that this will have an appreciable effect in
reducing the magnitude of charge transferred.

CHAPTER II
PROPERTIES OF ELECTRONS IN SOLIDS, CHARGE
DENSITY, AND ELECTRICAL CONTACTS
A.

Electrons In Solids [53
The first theory of electrons in solids was given in

1900 by Drude.

He assumed that electrons in a metal were

free and form an electron "gas" in a container.

This

simple model oredicts the law of Wiedemann and Franz that
the ratio of the thermal conductivity K to the electrical
conductivity <T is proportional to the absolute temperature
T.

In addition the value of the Lorentz number L s ^/g-f

was in good agreement with experimental data.

However, the

temperature dependence for the electrical and thermal conduc­
tivities treated separately was in error.

Lorentz extended

the theory using Maxwell - Boltzmann statistics in a more
detailed consideration of the collision phenomena.

The

results explained the Wiedemann— Franz law, but again failed
to give the correct temperature dependence of the conductivi­
ties.

Also the theory indicated that in connection with the

mean free path of electrons there should be one free electron
for each atom of the metal.

If this were true it would lead

to unusually large values for the specific heat of metals as
compared with insulators, in contradiction with the law of

DuLong and Petit.

Eventually Sommerfeld[6] applied Fermi-

Dirac statistics, and resolved the difficulty of the specific
heat, but the predicted value of the conductivity was still
not in good agreement with experiment unless free paths
of the order of a hundred interatomic units were assumed.
At that time there was no reason to expect such large free
paths.

In 1928, W. V. Houston[7] showed that an electron in

a perfect crystallattice can move unhindered throughout the
crystal.
the

In reality there are lattice imperfections, in

form of vibrations and impurities, which cause a finite

free path.

The concepts of quantum mechanics show that when

i solated atoms combine to form a solid the discrete energy level
of an electron broaden into a band structure.
1931 used

WIlson[8] in

these ideas to explain the difference between a

metal and an insulator.

Quantum mechanical calculations

show that for nearly free and for tightly bound electrons the
dependence of energy on the propagation vector k can be placed
in the form[ref.5,p.^3]

where

(1)

■£1 X
E - ■£ h-A/»i

(2)

£ = /7 - -f

/m*

(nearly free)

/>n*

(tightly bound),

is the effective mass of the electron and in (2) fl

is the energy at the top of the band; t) is Planck's
divided by 2_7f .

constant

When the energy of an electron is given by

(l) we have a normal energy band of standard form.

It re­

presents the behavior of nearly free electrons (and in

5
addition tightly bound electrons near the bottom of an
energy band).

When the energy is given by (2), we have an

inverted band of standard form.

It represents the behavior

of tightly bound electrons near the top of an energy band.
If we take into account the exchange energy the term[9J
JL

k

I

+ k

must be subtracted from (1) and (2). k

being the magnitude

of the propagation vector corresponding to maximum energy,
and e the electronic charge.

The exchange energy has this

form when the electrons are perfectly free and are consider­
ed as a degenerate Fermi gas filling all states up to a max­
imum level with two electrons of opposite spin.
The calculations that we wish to make require a know­
ledge of the electron charge density.

This in turn depends

upon the density of electronic states into which electrons
may enter.

Hence, we discuss the density of electronic

states.
B.

Charge Density
The volume of phase space occupied by N electrons

confined to a spatial volume V and occuoying a volume in
momentum space

is

[lOj.

Quantum statistical mechanics

tells us that the volume of the smallest cell in phase space
associated with an electron of given spin direction is equal
to h ? ,

The number of cells is therefore given by \

We define

j

^

to be the number of energy levels per unit

6

volume (for one direction of electron spin) lying in the
range cA £

at E.

This gives

^

■p. in terms of the propagation vector k we have,
(E)^E -

Expressing
since p

f<^%

^ k *

For free electrons k-space fills up with spherical symmetry
and so the volume in k-space contained between two concentric
spheres of radii

k and k + d k is ^ukoik

. , .-

Hnh'Jk

Thus,

/

^ /T

Substituting from(l)and

\ and #4k ^ tylf

k,

, ,

^

(2) the value of kin terms of E, we

have for the density of

states

for bands of standard and in­

verted form respectively,

(3) m(E)j£

w

(e)c>E -- f a (

If each allowed state were occupied by an electron we would
integrate the proper expression for the density of

states

over the required energy region in order to determine the
number density of electrons in that range.

Electrons, how­

ever, obey Fermi-Dirac statistics, and we must multiply the
density of states by the Fermi-Dirac distribution function,
given by

/✓_) _

I_________

7
where E Is the energy of the state in question and J the
Fermi level. Here k is

Boltzmann's constant, and T again

the absolute temperature.

The number density of electrons

in a range (a, b) is given by an expression of the form
r **
J /n (£)"/ (E) (^(E
a
Multiplying the number density by the charge on the electron
gives the charge density.

These concepts will be applied in

the next and later chapters where we consider in detail the
charge distribution at contacts.
C.

Basic Properties of Contacts
As is well known, when two bodies are placed in con­

tact charge tends to flow so as to equalize their Fermi
levels.

This give rise to charge layers of opposite sign in

the two bodies.

In Figures 1, 2, k 9 5» 6, 7, and 8 we see

energy level diagrams of bodies before and after contact.
Before contact the levels are shown constant up to the edge
of the crystal.

After contact, as a result of the transfer­

red charge, the potential energy of an electron within the
crystal will change, the levels shifting accordingly.

The

total shift from one substance to the next is e
where A Vf is the total shift in the electrostatic potential
of an electron In the substance on the left and A Vz

the

corresponding quantity for the substance on the right.
In the insulator we are concerned with the energy Y
released when an electron at rest outside the crystal is

o

8
taken into the lowest level in the uppermost nearly empty
band.

This quantity is the analog of the electron affinity

A for a free molecule.
energy

V 0 necessary

We are concerned also with the

to remove an electron from the highest

level of the uppermost nearly filled band.

This quantity

is the analog of the ionization potential I for a free mole­
cule.

The quantity X Q is usually called "electron affinity"

in solid state

literature* but we prefer some designation

differentiating it from the quantity A for a free molecule.
The quantity

does not seem to have been called "ioniza­

tion potential," however, the reasons for such a designation
are just as good as those for "electron affinity."
sent we do not wish

At pre­

to introduce a new terminology, nor

continue to use what to us is undesirable, and so shall try
to get by with the use of symbols only.
concerned with the energy ^

In the metal we are

released when an electron at

rest outside the crystal is taken to the Fermi level, or
absorbed when an electron Is taken from the Fermi level to
a point outside the crystal.

In a sense

^ may be considered

the analog of either the electron affinity or the ionization
potential for a free molecule.
name, work function.

It has been given the special

We are also concerned with the energy

of an electron at the bottom of the half-filled band in the
metal and designate this by UJQ .
In the figures the energy parametersof the substance
on the left contain the subscript one, and the substance on
the right contain

the subscript two;

For example, Figure 1

9
represents a raetal-metal contact in which the charge flows
from metal 2 to metal 1.

As a result the levels in 1 are

shifted upward by an amount <2 ^ ^
an amount

.

while those in

Near the contact

2 downward

the energy of an electron

is not constant, and so the variation is designated by i/J,
and

in the two metals respectively.

The reference level is

taken with respect to the shifted levels.
volving insulators we take X' and
energy parameters.

In contacts in­

to indicate

variable

Further details are given in the remain­

ing chapters of this paper.

CHAPTER III
METAL-METAL CONTACTS
A,

Charge Density In the Metal
Owing to the large density of states of a metal we

would expect the transferred charge to reside in a small
spatial region.

Suppose the metals placed in contact have

the relative energy band structure shown in Figure 1.
Electrons flow from right to left causing the metal on the
left to become negatively charged, the one on the right
positively charged.
To see in more detail the charge distribution and
potential energy variation we turn to a discussion of the
net charge density and solve PoissonII8 equation for the
potential.
The electron charge density is
oo
14

cO

(^) - ~ l c j / (e)/nj£)cl£
~U)

-2-ejj~ te)m_(C)ei£
~ 00

^ te a l/

and the positive charge density is

P

(#) =

f f£)

f

(s)^j£)ct£

=Zzjf^ (£)/*\J£)cJ£ +

(f)^E)oiEi

~oO
~
0Uf-tC<3^
being a distribution function for the positve charge

It is a step function

_c /r ) - ( 0
I

00

> £ ?T

X > £ y <*>'

The factor 2 accounts forthe two directions of spin.
Assuming that electrons are not excited out of the lower
bands in the metal, the first terms in each expression for
P (*0

are equal but of opposite sign.

Insofar as the

electrons in a metal form a degenerate gas[ll] even at room
temperature, we may replace

(£) by the step function

Combining the two expressions substituting the values of
the net total charge density in the metal Is

/

‘t £AL//
/

The integral

(£)ot£

represents the number density of

j ^*a K

positive charge.

This is constant before and after contact

(no ionic current); therefore, we can change the lower
limit to cAJol+e.AVx.
We get, using (3) P* 6,

12
We take /nJE) =/^_(e) because we

look at /n(£)

density of states for numbers of particles.
variables in each integral to X
X

=

£

- (u j

0|

Changing

=

an(i

1/),respectively, we have
1*
i

+

i t
p, M

as a

eav J

=
h

=

t|W'

This expression represents the net charge density in
metal 1 as an intrinsic function of distance,
occurring in the variable

UJf(V) .

the distance

We are now ready to com­

pute the potential energy variation of an electron in the
metal by means of Poissonfls equation.
B.

Solution of Polsson 8s Equation
We are to solve

cf^
where V is the electrostatic potential and £
constant of free space.
derived in A we see

the dielectric

Substituting the charge density

13

of U>

/>»
'

cU(
Let

7r.

u>,- - e V/

hl )

being the potential energy of an electron.
y,

*

j

+<^\/),

X

= J-('uj»,+e/SV'J.

and then,
= T-r f
31' ~ 1

A

Placing

X

i

_

A

( Ay,

2 ®fy(

cA /v-

/

L /Xy.V

JL

Z

-

k.

Integrating from J/fW

to

- 1:

the value of y f where /%

dJi

, /J y \'

y;

^y, \

y, u)

„

A t )‘ M

(

,

? *

*

*

I* '''*

-y« >-]
7.

- CO

14
£
K

/

L
I s

s

u
1I 1( At
I s
XX

-Jyoi

K
Now

a *

- y , -y oi *

A

= 7'0<
,, » *y_, _

-X,Of

where

^

£

- 1

S

u ,ei - u ; , ,
j +

v 0J

=

j+ ^

o/

= ^'/s
Oi

Our equation becomes

Since for a metal

A y t i^ j j

we may expand the first

y.Oi

terrn^ obtaining
fob
cJ .*■
X-

=

K y (j'i r1- * '
*■

neglecting higher order terms.
But
and

y,

=

<^7. =

yoi

+ a_>,

<7 AJ,
L

so
(

and

7 7

^

=

I K ,

f

=

1

Jf, x

*

15
where we must take the positive sign.
- 0

Integrating from

to rpi
Ay,(*)

s'*
3

/ A tX

T /

- i/M V

k

lu
t

‘

A y/

ajm

A ylc+)
X

A j/'

]/ i K

-

+ ,

&y, f»)
&y, (*)
(o)
Now

Ayto) =

e
C-J W

y/o)

x, X,1' #

~ y o<(v) = e AV,

. a positive

quantity:

y, - y

= Zsjy,
0

= - K

- p

.

/

Thus,

yU.
=

- e

^

C

^

, we have

D,rinl"e ^ ’ J T k T 7 ‘
(5)

J.

w,-w„, = -

(

£ ''/

1^

z. (y)

We proceed in exactly the same manner to find the
charge density and potential energy variation in metal 2.
We find that
Oc>z~ <^>01 ~

where e A V ^

_

it

’

^

^ ^ ^ d.

(A 2- °/

<3h d

£z -

tj
1/ z

/

^
* -X*

is negative^and the negative sign in the expo­

nent is used since /p is positive.

16

To evaluate the unknowns e A Y 3 c-AV * and

5

* we

match the two solutions at the boundary by means of the con­
dition that the field strength must be continuous across the
boundary^ and that at infinity m oi

= J

These conditions give respectively

A V

J,

_
'

and

ujcii

or

4Jo

teAV;

- w lttc

and

.

J.z

+

<-f2 = A-l +eAV/
1.

If we assume the effective mass to be the same in metal 1
as in 2, and
i
AV

and

-- AX
X,

Taking

^

£ AA € . then
/
i. -■
/y ^
^ -\ £±1/ y '6-

^

Xf

2. ^

'

x Z e.v. ,

.

*j

^

70i=Je./..„

- ~^ .l 3 e . v: t

A z ~ ' 9e

_ ,//

c a Vl
c a ^ - eil/t =

e£Vz=
^
eAV

=•

- Poi - 2 . 1 3 ,

-Losy

)

/. /7£

i
i

and

J*

=

^

.6?

*

^

tC hV|

- ~ 51 o i V y.,

If the energy bands are such that the Fermi level in
metal 1 is above that of metal 2 the charge flows in the op­
posite direction.
same manner.

The calculations proceed in exactly the

h~

Ll I

<

hUJ
Energy-Band

<

Schematic

Diagram:

f\l

1.

CONTACT

Metal-Metal

AFTER

Contac

ILU

Fierure

BEFORE

CONTACT

-J

h“
Ui

<

Axxx
//

<

ol

CHAPTER IV
METAL-INSULATOR CONTACTS
A.

G-eneral Behavior
We now consider four cases of metal-insulator con­

tacts.

Figure 2 represents the contact between a metal and

an insulator when before contact the Fermi level in the metal
lies above the Fermi level in the insulator, and when the
Fermi level in the metal lies above the bottom of an empty
band in the insulator.

Then electrons can spill from the

half-filled band in the metal into the empty band in the
insulator.

Hence, large charge is lost by the metal from a

small spatial region, and a large amount gained by the in­
sulator within a small spatial region.
Figure ^ is the case where before contact the Fermi
level in the metal lies below the Fermi level in the insul­
ator and

where the top of a filled band of the insulator

lies above the Fermi level in the metal.

Electrons can

spill from the filled band in the insulator to the half-filled
band in the metal.

Much charge is lost by the Insulator from

a small spatial region, and much gained by the metal within
a small spatial region.
Figure 5 is the case discussed by Mott and Gurney[3]
when in isolation the Fermi level in the metal lies above

the Fermi level in the insulator, and when the bottom of
the empty band of the insulator lies above the Fermi level
in the metal.

Electrons escape by thermal agitation from

the filled band in the metal to the empty band of the insul­
ator.

Small charge is lost by the metal in a small spatial

region and the same amount gained by the insulator in a large
spatial region.
Figure 6 represents the contact when in isolation the
Fermi level in the metal lies below the Fermi level in the
inBulator, and when the top of the filled band of the insul­
ator lies below the Fermi level in the metal.

Electrons

escape from the insulator only by thermal agitation from the
filled band in the
metal.

insulator to the

half-filled bandin the

Small charge is lost by the insulator

from a large

spatial region and an equal charge gained by the metal within
a small spatial region.

Calculations which verify these

statements are made in the remaining portion of this chapter.
B.

Case of Large Charge Transfer from Metal to Insulator
From Figure
in the insulator

yi+cA'/l

2 we see that the electron charge density
is given by

X Z * e & v.
2.

/n (E){f£)<?<£ -2-C- / » . _ ( £ ) 2

(£)yfj£)c*E ,

<
3

o ?
o

UJ
a: _ j
o <

Ll hLd LlJ
CD ^

Schematic Er*r*or*oand PI arrani: Fermi Level of Metal above FerTf Level
Too of filled ''and of Metal above ?ot.ton of Empty Panel of Tr,sv labor

<

P,

cr
o

vr<

Ll_ Ll I

of Insulator

while the positive charge density is given by

Pi/*) - Lc

^

(£)-/j£) ct£.

= 2

In the range zn tc zp/

to

+ e ^ t/_

the energy level

density is zero for electrons and positive charge.

We have

because every possible state for positive charge is occupied
below

ana empty above 1/ f e ^i/ ^

this

being based

on the assumption that the nuclei stay fixed in the process
of charge transfer.
The net total charge density becomes

CJ

)

(P (*) + ^t U )

Ze.J[i _ ^ (£)]

(£)dE - 2

(E)a ^_ (£)#(£

stZ-yc&l/2
where

/

/ - / - (E) = C ^ - £}/kT +/ '

The first integral, which pertains to the lower band, is
concerned with a nearly-filled band, and the density of states
has the inverted form.

The second integral concerning excess

electrons near the bottom of the upper band has the density
function of standard form.

Then we have
oo

Changing variables in the first and second Integrals to
ATV
and

AT*

=

kTf -

;

~ E-(xt-*cAVj

respectively, where

+

; kT I = -(x ^

t <T - tr

and

.-Jj;

ca

^ - s),

^

we have
^ ^ °° _/
^<vj = V/re/z^ l hl) — ■* ^

j 00
_ q7, J z ™ * k r f '( *

r*o „

Defining [12 ]
r

ft

0

(I) - I *
-J A£~ C I I

we finally have

-

O c

(■*I

i

id
if the effective masses of the electrons in the two bands
are equal.

This relation, together with PoissonGs equation,

will lead us to the potential variation in the insulator.
1.

Solution of Poi8sonlls Equation. We have

^

- -a

e

Letting

V[ » - e

^ >

U) - -

(tjlif

f

= fj
Fj_ ft-[) -Fj (f)) .
ll "
"
>

-

Letting

y z - K-*

fl [-

<oj .

where

f\ = lILE^(l'Tlk Z)
f. ' h x J

)

^
and

^3
Integrating, with the boundary condition that aj

^

— > oo

the field strength vanishes (by conservation of charge),
we have

U) (F

tf

J {

where
t^eri ~ ^z.

°l
/Y-'

d£

Introducing the relation

5 cO ,

>7

[iz]

pi (^)

t J F3 (°),

=

and integration from o to x we see
/

,
where

UL

_C^s''J, C z
L t T
U ^ = o ) = ctvl- C^

z-

*■

U»i ' ^

£

^

J

-

From Figure 2 we see that K and (J07_ are negative and
is positive, and that they are in the order y 4 U0 <

.

The function /“^ (
) increases rapidly for positive values
/
/
of
, but is small for negative value of U v . If we stay
sufficiently far from U02_ we may retain only the last term
in the denominator.

We have then
U
[/v, 'z oj

T

where Uz ^ U0l-£

» £

being a positive quantity.

This

gives us an intrinsic expression for the potential as a
function of distance in the insulator.

24

From equation (5) of Chapter II we have the corres­
ponding equation for the metal

(if) /*. . ^

M

{St-Zo)

where & A V { is negative.
Equations (3) and (4) contain the unknowns cAVt
and c a Vl ,

They are determined by the conditions that

at the contact the field strength is continuous and that at
infinity

=J

2.

ir^

-J0l - 3 ,

Evaluation of the Quantities eAV, ,eAViand 5

From equation (3) we find that
\

(5 )

_
=_ ^ /7 #| ,/ h j i 4
1J

+ e A / j / tT]

2

’ 'i

/£- O
where we have taken the positive sign since the slope is
everywhere positive; and from (4)

(6 )

J L-°( I

_

- ca

Vt

From our conditions at infinity
(7 )
(8)

= T

Oi
and

ifO I t e. a

or

£ AV, - e ^ l /

v

, - y„.

= s

SO

(9)

=

'Ui

+ 7 0J. ,

Assuming 3 to lie half way between the bands, we
have <T =

Then (5) becomes

c4Vt

(1 0 )

Substituting (9) into (6) we find

(11)

=

d<M‘\
^

'*<».'ir ~e^i4

L-o

v,

Equating (10) and (11), we get
(

19 ] f r $ & ? ' ’ * % ] } -

.

If /|

p = (<4, -X<“)/(r

then (TfJ{s rl 'I>jV ^

-

(12)

or

,

where

A

(3
= "T
tV t i
/2.n 1.1c

and

B

-

h°).

( ir iU

'

This equation may be solved graphically for A|
and hence e h V ^ .
electron, and set

,

We take f o r ^ t h e normal mass of the
^

- / , and

V01 =

T = 300° K, R value of

assumed for half the band width (as would be
—8
the value for sodium).
This gives
= .673 x 10
cm and
-

6.26

x 10^ cm _1,

whence B = 2.38 and A = 2.38^3.
For a specific example.

26
BO

p = Ztl

and

A

= G><i. S".

Figure 3 shows a plot of (12).
curves intersect at at

We see that the two

= 20, y = 22;

80
= Z e.vx

and

From (8), ^ = 1/^ + e 2^1/ -

.

The gap between the valence

and conduction band of the insulator is CT"= - 3 ev, so
X
and

2

J

=

From (?)

-

5~e.i<.

cAl/

^J-

-

-0. ZJe.i/..

Knowing the shift in potential and the position of
the Fermi level at equilibrium, we are ready to calculate
the surface and volume charge density in the insulator.
3. Calculation of the Surface Charge Density.

By

application of Gauss * law to a plane conductor carrying a
surface charge (T^ , the electric field outside is given by
E

=2/“ ^

.

Since E
dJ i
z7^
where °jjdL
cAy

kLIrrc

is a function of

kkjcf'V'
as seen from equation (2).

80

60

40

20

"

J -Fl

(,)'

0

Figure

o

Sraoh

for

Solution of E q u a t i o n

12,

Sect I o:

r.a p

28
The expression may he simplified as in equation (3)
by defining X

=

, to give

We have then
'

kr
2/T e

where x is determined for corresponding values of
by equation (3).

It should be noted that

-JT

<f^ {o) is an ef­

fective "surface charge" density in the sense that it is a
measure of the charge to the right of the contact.
The following table shows the results;
TABLE I
X

(fl)

0

20

<^6r; (*>“)

}oo <Th (x) /(T% (la)

18300

100$

6050

8

1.5

0

7

720

-k

26

105

-20

76,000

33
3.9
.57
.00019

.035

i see that about 70$ of the charge is located within
1-1/2 angstroms from the interface.
4.

Calculation of the Volume Charge Density.

equation (l) we have

CJ+)*

dz- ( F j ( r - f)' Fi( r)
"
7
1
J

‘/ire. [

.

From

29
For values of say

30 we may approximate this

exoression by
A
<-/J\e. (
;

*■

)

The following table shows the results;
TABLE II
f

C.

^(A)

j

lop

20

0

770 x 109

100$

0

7

8.8 x 109

l.i

26

0.21 x 109

.027

Case of Large Charge Transfer from Insulator to Metal
An analysis for Figure 4 similar to that for Figure

3 shows that the volume charge density in the insulator is

(11 ( w ; -

* { rf ( , . ( ) to},

except that now <sAVt is positive and $ A V U

negative.

We solve for the potential energy variation X

~ —e^

by means of Poissonlls equation;
A*

I ~ ^j'/^JftTrXi->-eAK-f)4T]~F[--(Xx'feAK ' A t Jj
where
Letting

K T { £ - ( \ ■'C*V2 -J-) .
^

* 1

\ tc Al/t - J

W

? t = 2t/ A T -;

30
Integrating with the boundary conditions that as 4.
the field strength vanishes, we have

121

(£j -

.1

;

Introducing the relatlon[12]

j F ± ( r])d'i + —
3

=-

**

Ir

^

(fi),
V

and integrating from o to x we see
/
2AA'

^ f*
I «■
\t

h ri (Y+tJ)-jFi (y-'feJ-jfl fSf«)+M(-fJj
where
q _ e&Vu
t01
Tr
Here the positive root is extraneous, and we discard it.
If we restrict the upper limit to ^

- f0l -t £

where £ ? o

and is sufficiently large, we may retain only the first
term giving
(3)
h (r + fJ/J *
_

C.AVi

We now have an expression for the potential versus
distance in the insulator.
From Chapter III equation (5) we have the corres­
ponding equation for the metal
(J*)

^

= J, - A

We solve f o r e A l / and c

c o h e r e now e_AV, is positive

in equation (3) and (4) by the

condition that at the boundary the field strength must be

LlJ

i—

or
o

<

3
below Fermi Lev^l
of Filled Band of

<

■wsx

Schematic Energy-Band Diagram;
Fermi Level of Metal
Insulator, lop of Filled Bend of Insulator abov® Top

CONTACT

CO

L,

AFTER

of
'-'etal

cr
o

Figure

CONTACT

CO

BEFORE

o

LlJ

32
continuous and that at infinity
(5)

f

'ai

-f c A [/ r J

dud

I

1.

-i

= J ,

Evaluation of the Quantities e^i/, t e ^ ^ a n d 1 .

From equation (2)

(6 )

-

I

^rjJ

/? [ X <r't *** ~ e

At -0
and from (4)

(7)
Eliminating J

from (5) gives

e f A V ' = % r \ +^

.

Continuity of field strength requires that (6) and
(7) be equal:

Placing

^

5 (f

< W

* JF (^,-V^)/kT

gives

8

( )

-

j j FI (f)l

where A

=

4 + 57
B

= ? / ( z^

=

Equation (8) may be solved graphically for e A ^ ,
to be the normal mass of the electron,
T = 300° K, and

^

Ja/

6 7 3 x/o

= 2

—2
ok.

€t = ^ =

Taking
3,

we S et
/7 77 \

■*= 6 . l £ y / o

7 -/
^
Ptw
i

(^°)

33

Therefore

B = 2.38 and A*= 2.38 ,

For a specific example

v.,

*>X=

f
and

u0l_- -s.frZ.v,,

<f,ot T-£.L.7e.vy

=

A = - 6 9 .5 ©
x

We are to plot therefore
just as in Figure 3.
a t f = 20,
Now

tyVj ^

The curves are found to intersect at

y = 22.

^ KtjXzi - c A K j ^ r

therefore,

^

c a

= iO "}

V^— ~ Z c,v.t

and assuming <T = -3 e.i/.,
$

=

= ~

+

y

”

^*

^

Finally,
e ^1/ - J - <f>ot = .2.3
Knowing the shift of ootential and the position of
the Fermi level at equilibrium, we are ready to calculate
the surface and volume charge density in the insulator.
2.

Calculation of the Surface and Volume Charge

Density. As in part 3 of Section B we have for the surface
charge density

kT

W

s

(z f l V

f

z r

,

,1

I F r J f I r i UJp

‘

.

<* ,7h

The variable t has the same values here as in part 3 of
Section B since ^

, therefore, the results are identical

3^
to those in Table I on page 28 except the charge has the
opposite sign.
The volume charge density has the form

which is also identical with the result of part k of Sec­
tion B.

Table II, in the same section is applicable with

the charge of opposite sign.
D.

The Case of Small Charge Transfer from Metal to Insulator
In this example, Figure 5, electrons escape by thermal

agitation from the filled band in the metal to the empty band
of the insulator.

A small amount of charge is lost by the

metal in a small spatial region, and the same amount is gain­
ed by the insulator In a large spatial region.
Applying Fermi-DIrac statistics to the energy band
structure we again arrive at the expression
(1 )
for the volume charge density.

Letting X

for Poissonl|s equation

and f
For a specific case let us take
v

£ e /r

If =-//€.k

(pw '

Z-

- ~ * V we have

Figure

5.

BEFORE

CONTACT

or
o

<
i—

zx

CO

LU

I—

<

Schematic Fnergy-Fand Diagram:
Insulator, Top of Filled Band

AFTER

CONTACT

CO

<

1“
LlJ
Fermi Level of Metal above Fermi Level
of Metal below Empty Band of Insulator

h<
of

o

CC

o

O

36
We see from the figure that f and
with If*- f

are negative quantities

for all x,and so we may neglect the first term in

the expression for the charge density provided we are not too
far from the contact, giving for Poisson,ls equation
= - g
Since /

/! (!) .

is negative we may also use the series representa-

tion[l2] for /u ( 0

and obtain
n

ti
S- /

If f

S

Is sufficiently large and negative, we may approximate

the series by retaining only the first term.

The error

is about(.6% If f ~~k, and negligibly

small for f-

the greatest value of f

our example.

occurring in

-ZO »

We finally have
=

- 2

j

i-

e f

A

= -g n ( \ ) .

Solving in the usual manner subject to the condition that
at infinity the field strength vanishes, we find
(2)
where

Xu - ATJ^[(Cj-y# +2J
r

=

3^
-('AVrfyr
—
e.

j
j,
D, - C

JrT
(3)
where

For the metal we have

OOf - (jj0i — -

di 1/ C

^1

^

~^

Is negative.
We make use of the boundary conditions at the contact

as before in order to solve for the unknowns' eA\/ , a n d e A V ^ .

37
It can be shown that t/^Vl is negligible compared with eAl/
We can then take

J ^

^

and

e z>l/ -

~~

z

.

~ 2- 2-3e i/

The surface charge density is computed in the usual manner:
w

w

-

with*
U

x

i r - f z

= c

( o)

/

=

3

,
^ 'J

^ ^

roj - ^ _ c^,i/ - - f.
oi

S'Td7c out C*n toS /

«•

4,

/OWi .

The volume charge density is

(si p, w - - ^ ( z g a f r a ) /
-

“ «<r.

and
It can be seen that the charge density falls to 1/A
o
of its value at the contact in 750A.
In cases where
differ by say 0.5 to 1 e.v. this distance ranges from about
^
no
10
to 10
angstrom units, a value much greater than ob­
tained in the previous two cases.
E.

The Case of Small Charge Transfer from Insulator to Metal
Figure 6 represents an example where electrons escape

from the insulator only by thermal agitation from the filled
band in the insulator to the half-filled band In the metal.
A small charge Is lost by the insulator from a large spatial
region and an equal charge is gained by the metal within a
small spatial region.

38
Upon application of Fermi-Dirac statistics to the
energy band structure, we have for the volume charge density
again

(1) f>.(*) =

[ r { h - r ) - r i (r)j .

PoIssones equation becomes

n
where T£_--C l/ > f - - { X ^

and

&TT \ L(z^kT)
=

-j—

\— ^ — J

We solve In the same manner as

Section D,except

that we

neglect the second term in (l)

compared with the first,

obtaining

(2)

IT = - ft T

=

with

L

Q

+ (£i)%]

' kl

e

A

{s%z°)

and

= P(\)QX .

.

(&V-VAkT
C *

J

t

*'/$t
For the metal
where

we have O'- tuffi- - e

l/ c

(•*■ £ o)

is positive.
Making use of the boundary conditions as before, we

solve for cAl/

and e A

ible compared with e h l ^
example, take
then

^

.

As

in Section

, and so jf ^
c - Jc .v.

e AV£ - J - lg7 +_|T =

- / ev/"

^

is neglig­
. As a specific

s

*“

<

Figure

CONTACT

ZD

BEFORE

UJ

6.

AFTER

CONTACT

ZD

CO

<

UJ

I—

cr
o

CO

Schematic Fnergy-Band Diagram:
Fermi Level of Metal below Fermi Level of
Insulator, Top of Filled Band of Metal below Bottom of Empty Band of Insulator

cr
o

I—

o

bO
The surface and volume charge densities follow the
same form as those of Section D with the exception that the
charge is of opposite sign.

It is, therefore, not worth­

while to discuss this case further here.

CHAPTER V
INSULATOR-INSULATOR CONTACTS

A.

General Behavior
In this chapter we discuss two cases of lnsulator-

insulator contacts.

The energy band structure is represent­

ed in Figures 7 and 8.

Figure 7 illustrates conditions

where electrons can escape from the left-hand insulator to
the right-hand one only by thermal agitation from a filled
band into an empty band at higher energy.
charge

(perhaps 10"”

Hence, small

statcoulombs /cm2 ) is lost by the left-

hand insulator from a large spatial region (perhaps 10

fi

n

A )

and an equal charge gained by the right-hand insulator within
a large spatial region.

B’igure 8 is the case where electrons

can spill from the filled band of the left-hand material into
the empty band of the right-hand material.
Large charge, say
k
2
10
statcoulombs /cm , is lost by the left-hand insulator
from a small spatial region, say

10 A°, and equal and oppo­

site charge is gained by the right-hand material in a small
spatial region.

The following calculations give the quanti­

tative data of the previous arguments.
B.

Case of Small Charge Transfer
The insulator on the right has a band structure

identical with that of the insulator in Figure 5.

bz
We have, therefore, a volume charge density

(1)

^

= y,7 e

[

(1-r)

ff)J

- /p

The solution of Poisson®s equation is
(2 )
where

J)

and

B2.

=
=

r
a

1

J

r(k)

_ -3*
1

7 T e

*

The insulator on the left has
lar to

~feAV/*

a band

that of theinsulator in Figure

structure simi­

6.The volume

density is

(3)

f / V - //Te

f^JLL'j’'^/T r>,_

- P ffjj

^

and the solution of Poisson“s equation is
/.

(A)

= -ZkTJy,[-(C^
r,
2> * e

where
and

B

=

/

'■y

t

P,J >

r
J C' ~

(*£*)

3,
{cA^' s)/* T
7r &

A P (\)
/

'

After equilibrium has been reached we see from
Figure 7 that at infinity

(5)
(6)

and

+

- |T - J

,

charge

?.

CONTACT

C O N TA C T

CO

a>

cr

o

<
>'

a

cr

CO

ZD

cr

o
Fermi Level of Insulator 1 above Fermi Level
Insulator 1 below Lmpty Band of Insulator D

h-

Schematic Energy-Band Diagram:
of Insulator ?» Filled Band of

AFTER

<

Figure

BEFORE

cr

o

co

44
These equations along with the condition at the
boundary that the field strength must be continuous deter­
mine the unknowns

and eA\/ .

From (2) and (3) respectively

<AX I

.
=

L-O
and

^

2A 1 ( C l V
\1

= ikT / f \ j
2.

The continuity of field strength requires
c r

c y
=

a

X

'

Inserting values for the G 9s and D ”s, and assuming the effec­
tive mass of electrons in the two insulators to be the same,
we find that

(7)

f=

+X*i) .

From (6) and (?)
e AVf

-

M
(x
2m I OZ

-J r ),
Q\ /

and from (5) and (7)
cAl/

=

.

For a specific case let us take K,'~7j 3ro/ ^
This gives

I

i'e,!/ v e A \ Z - „ / e i/

-w/

and e AVJ_ r 2.c./,

The expressions for the surface and volume charge
densities in the right-hand insulator are identical with
those in Section D,Chapter IV.

We have, therefore, a charge

density and depth of penetration which behave

in exactly

^5
the same manner.

The left-hand Insulator has the same form

as the Insulator in Section E, Chapter IV, and, therefore,
the charge density and depth of penetration are similar to
those found In that section.
C.

Case of Large Charge Transfer
In Figure 8 the right-hand insulator has a band struc­

ture the same as that of the insulator in Figure 2.

There­

fore, as developed in part 1 of Section B of Chapter IV, the
charge density is
(1 )
and the solution of Polsson's equation is
(2 )

The insulator on the left has a band structure the
same as that of Figure k.

Therefore, as derived in Section

C of Chapter IV, the volume charge density is
(3)
and the solution of Poisson's equation is

After equilibrium has been reached it is seen from Figure 8
that an infinity

cr

Lu

I-

<

I—
O
<
1—
z
o
o

LU

cc
o
Ll

GQ
CO

2

tr
o
h<

Ll

CO

3

o

o/
OC
O
K

<

3

CO

cn
o

3

CO
Fermi Level of Insulator 1 above Fermi Level of
of Insulator 1 above Rot tom of Emoty Band of

3

J^n
Fnerg.y-Band Diagram;
?, Top of Filled Band

hz
o
o

Schematic
Insulator

<
<

6.

h"
O

Figure

ol
(T
O

LU

k?
(6)

-y^ = j

These equations, together with the requirement that
at the hounciary the field strengths must be equal, enable
us to determine

e

and e A

.

We

find, assuming the

effective mass of the electrons in the two substances to be
equal,

eAV<

=

S

=

and

For a specific case let us take XOj_ = - 7 e . v J ir
£^^ = ~ / o e u . ,
= - £ e.K, and X 6/ = -

.

Then
=
d

=

eAV/ =

— 6.5 c.i/.j
- /.5 *. K,
2.e. i/,.

This case is illustrated in Figure 8.
Equations (1) and (3) for the volume charge densities
for the two insulators are identical with those of Sections
B and C of Chapter IV, respectively.

The potential energy

variations are identical with those of part 1 of Section B
and Section 3 of Chapter IV.

Therefore, we conclude that

the volume and surface charge densities and depth of penetra­
tion of charge in the two insulators follow a form identical
with those of parts 3 and *4- of Section B in Chapter IV.

CHAPTER VI
APPLICATIONS
A.

Other Features of Contacts
Hersh and Montgomery[l]

describe the experimental

techniques and the results in a 3tudy on the static elec­
trification of filaments rubbed against one another, and
Harper[2] describes the charge transferred by spheres
touching one another.

In the previous chapters we have

presented a quantitative account of the charge transferred
when several types of contacts are made.

At the present

state of knowledge we cannot hope for a complete explanation
of static electrification.

Experimentally, the physical and

chemical natures of the contacting surfaces are not known,
nor are the temperature and the force at contact.

Theoret­

ically, we are not certain that the substances used possess
the necessary periodicity for the application of the band
structure of solids.

However, we believe that some progress

can be made, and we have approached the problem on the
simplified picture that charge is transferred between two
perfectly periodic structures and that part of the charge
remains when separation occurs.
Our calculations were carried out under the assumption
that the bodies in contact were in equilibrium.

For

i+9

metal-raetal contacts equilibrium should be attained instantan­
eously, with a large charge per unit area of contact.

In

metal-insulator contacts, where electrons can spill freely
from a filled band into an empty band, the transfer of
charge near the interface should take place in a very short
time.

We see from Figures 2, k, and 8 that owing to the

smaller gap between the filled and empty bands of the In­
sulator as compared with the other cases, an additional
charge is furnished by thermal agitation to the empty band
at large distances from the interface.

The charge Is small,

but the gap is large enough that an appreciable time may
elapse before equilibrium is attained.

The cases represented

by Figures 5, 6, and 7» where electrons are transferred from
one body to the other by thermal agitation alone, should
cause a long time to be

required for equilibrium, with only

small amounts of charge being transferred.

We could expect

appreciable charge In these cases only if the Fermi levels
are nearly at the same level in the isolated substances or
if the temperature is very high.
In our picture we assume that the charge which has
been transferred remains localized at the point of contact
in the case of insulators but not of course in metals.

Ex­

perimentally, we know that charge can be localized on
insulators; but theoretically an electron in the conduction
band is free to move throughout the crystal upon removal of
the attractive force due to the oppositely charged layer
of the other substance.

G-onsalves[l3] has proposed a theory

50

to explain localization based on surface states.

Seitz

[p.320-326] discusses localization from the point of view of
imperfections on the surface, and Mott and Gurney [p.86-88,
12*4-131 J put forth several models to trap electrons in the
volume of the crystal.

We merely say that some mechanism

exists whereby electrons are immobilized within the volume
of the insulators.
Harper[2] has made a study of the case of metal
spheres as to how much charge remains upon separation.

His

results show that the fraction of charge remaining is prac­
tically independent of the speed of separation, amounting
to perhaps one-half.

The other half is returned by tunnel­

ing through the gap during the early portion of the separ­
ation.

Harper does not consider in this respect any metal-

insulator or Insulator-insulator contacts, but it is clear
that more charge will be tunneled back in these cases be­
cause of the greater intensity of field strength per unit
area.

Let us assume that perhaps one-tenth of the charge

remains upon separation in the insulator-insulator cases.
We now turn to comparison of our theoretical results with
experiment.
B.

Comparison with Experiment
A qualitative prediction of the theory is that a

triboelectric series exists, that is, that substances can
be ordered such that any substance above another substance
in the series will become say positive when the two

51
substances are rubbed together.

This rule follows from the

circumstance that the electrons flow from a substance with
the higher to that with the lower Fermi level.

Because the

value of the Fermi level depends upon temperature, molecular
structure, and state of strain as well as chemical composi­
tion, we can expect a definitely ordered series only if the
state of each substance is carefully specified, or if we
restrict ourselves to substances possessing well-developed
Fermi levels.

Triboelectric series are given in many places

(see for instance

reference 1).

The existence of these

series is evidence for some of the qualitative features of
our picture.
Much more convincing evidence would be provided if
we could predict the position of a substance in the series.
Unfortunately we have almost no data on the positions of
the energy bands in the substances used in the various ex­
periments,

Hersh and Montgomery (nrivate communication)

have made a start on predicting the values of certain of
the energy parameters in the case of high polymers.

They

have been able to correlate the relative positions of sev­
eral synthetic polymers in the triboelectric
the properties of the monomersC 1^,153.

series with

It is too early to

say how meaningful this correlation is.
On the other hand, we may make use of the theory
outlined in the earlier chapters of the present work to test
our picture by a semi-quantitative procedure.

We examine

the sign and amount of charge transferred when a set of

52
metals and insulators is rubbed against one another.

Then

we try to assign a set of values to the various energy para­
meters, and see if a consistent set of predictions is found.
At the present stage of investigation the details of the
mechanism of charge return--in particular the dependence
upon resistivity— during separation are so little known
that we must restrict ourselves to only a semi-quantitative
treatment.
Figures 9a and 9b, reproduced from reference 1,
Figures 26 and 29, show the experimental results of rubbing
insulators on insulators and metals on insulators,respectively.

We may

correlate these results somewhat by the
f o r

energy level scheme shown in Figure 10^where*the photoelectric
work function has been taken as the value for the Fermi energy.
Agreement is obtained in every case with regard to the sign
of charge transferred and in most cases with order of magni­
tude.

The Fermi level for magnesium lies above the bottom

of the nearly-empty band of nylon, and so a large charge is
lost from magnesium to nylon.
Section B of Chapter IV.

This case is discussed in

Aluminum has its Fermi level below

the nearly-empty band and as explained in Chapter IV, Sec­
tion DJ a small charge is lost from aluminum to the nylon.
The Fermi level for iron lies below that for nylon but not
low enough so that electrons can spill from the filled band
in the nylon to the iron and so^ as studied in Section E,
Chapter IV^ a small number of electron

is gained by the metal.

For platinum we see that electrons may spill from the filled

NYLON

ACETATE

POLYETHYLENE

ACE

+70° /

CHARGE(fLfl c)

+400F

+

POLY

200-

0

-200

20

40

20

40

20

NORMAL
FORCE (g)

40
ACE

-

NY­
LON
NYLON

-400 Fiaure 9a

NYLON

PO L YE TH YL EN E

acetate

♦ 300 -

CHARGE {fifi C)

+ 200

♦

100

-

Fe

40
Fe

20
NORMAL
.

-300

40

FORCE (g)

-

Figure 9b
Charge Transferred as a Function of Normal Force between
Various Metals and Insulator Rubbed Together

10.0

CM
00
i
l

POLYETH.

o

ACETATE

o

m
U
CD

00

in

CsJ

ro

o

4-J
CD

in

E

qj
aJ

P-.
Pu

NYLON

>>
bO-—'
U CD
CL *->
Ci
—I
[£1 *
O

o

a

c

p~*
c:■»->
<£)

cp

METALS

rO

a- a

E <D

CM

<s>
I
I
I
I

a>

CD
Ll _

C

r-<

b: cd
CD C

CD-M
<c

CL

55
band of the nylon into the half-filled band of the metal
and, therefore, as explained in Section C of Chapter IV,
large charge is lost by the nylon to the platinum.

We have

one serious disagreement in our diagram, acetate on nylon.
In this case we would expect a small charge transfer, as in
Section B of Chapter V, but a large one is observed.
The data in reference 1 show that the maximum sur­
face charge density measured is from 10 to 100 statcoulombs
per square centimeter.

Under normal conditions of tempera­

ture and pressure the breakdown strength of air corresponds
to a surface charge density of 5 to 10 esu on a plane sur­
face, and perhaps ten times this value for cylinders of the
radius used in the experiments.

According to our calcula­

tions in part 3 of Section B and part 2 of Section C of
Chapter IV and Section C of Chapter V, we have densities of
transferred charge of the order of 10,000 esu.

Hence, we

have an amount of charge transferred which Is far more than
adequate to explain the amount measured.

The charge re­

maining is decreased by a factor of ten, say, through the
tunneling effect, and finally by a factor of 10 to 100
through atmosphere breakdown.

On the other hand, If the

mechanism is that described in Sections D and E, Chapter IV,
(electrons boiled up to an unfilled band), the maximum
charge transferred over an energy gap greater than 0.5 ev
is only 5.0 x 10” 2 statcoulombs per square centimeter.

The

chief point to be established here is that large amounts of
charge can be transferred even between insulators by the

56

mechanism of Chapter V (electrons soilllns' ir.tc an incom­
pletely full band) without calling upon local heating, or the
existence of surface states.

We cannot reach a more de­

finite conclusion without additional experimental evidence
on the position of the energy levels.
The same explanation would be valid for the results
of Harper[2] on the light contacts of insulating surfaces
with metals.

Harper found that relatively large amounts of

charge can be transferred when spheres of certain insulators
are touched against spheres of various metals.

These pheno­

mena can be explained on the basis of the mechanism just
discussed.

On the other hand, when the insulator is an

exceptionally good one, the charge measured is negligible.
We believe that this finding can be explained by the large
degree of reverse tunneling which must occur when the in­
sulator is very good.

For then the charge localized at the

point of contact cannot flow away, and the resulting high
field will produce a greatly increased tunneling.

Of

course, the energy band structure might be such that only
small charge would be transferred, but Harper studied enough
different Insu3.ators that it seems impossible for all of
those showing little electrification to'have the appropriate
energy band structure.

CHAPTER V I I

CONCLUSION
The theory put forth in the preceding chapters
appears to be able to explain semi-quantitatively at least
the basic phenomena in static electrification.

Our results

are adequate to explain the experimental results without
Invoking surface states.

It is, of course, entirely possi­

ble that they do exist, and may even be the major cause of
the charging process.

The so-called "hot spots" proposed

by Frenkel[l6] likewise may exist, and are the primary
source of charging in certain cases where a large charge
is observed when a small one would be predicted on the
basis of energy bands alone.

It Is evident that much more

experimental evidence must be acquired before the proper
choice can be made among the various theories.
The treatment could be refined by considering the
problem in two or three dimensions[l?], and by including
exchange and correlation energies in the energy term for an
electron in a solid.

It is doubtful whether it is worth

while to refine the treatment until additional experimental
data on static electrification are available or until there
is more information on the actual values of the energy
levels in the solids under consideration.

REFERENCES CITED
1.
2.

S. P. Hersh and D. J. Montgomery, Textile Research J.
25, 279 (1955).
W. R. Harper, Proc. Roy.
111 (1953).

Soc. A205, 83 (1951); A 218,
-----

3.

N. F. Mott and R. W. Gurney, Electronic Procease a in
Ionic Crystals, Chapt. 5, (Oxford, 1940, 1948)'.

4.

H. Y. Fan, Phys. Rev. 6l, 365 (1941); 62, 388 (1942).

5.

A. H. Wilson, The Theory of Metals, 2nd ed., Chapt.
1— 2, (Cambridge, 195377"

6.

A, Sommerfeld, Z. Phys. 47, 1 (1928).

7.

W.

V. Houston, Z. Physik. 47, 33 (1928); 48 449

.

A.

H. Wilson, Proc. Roy. Soc. A 133, 458 (1931).

8

9.
10

.

(1928).

H. Hellmann, Einfiihrung in die Quantenchemle, p. 102,
(Franz Deuticke, Vienna, 1937).
F.

Seitz, The Modern Theory of Solids, Chapt. IV,
(McGraw-Hill, New York, 19^077

11.

A. Lande, Quantum Mechanics, p. 229, (Pitman, London,
1951).

12.

J. McDougall and E. C. Stoner, Philos. Trans. Roy. Soc.
237, 6 7 (1938).

13.

V. E. Gonsalves, Textile Research J. 2J3, 711 (1953).

14.

R.

15.

J.

16.

J. Frenkel, J. Physics (U.S. S.R. ) J5, 25 (1941).

17.

S. M. Skinner, Bull. Am. Phy. Soc. 1, 292 (1956).

E. Honig, J. Chem. Physics 1 6 , 105 (1948).
D. Morrison and A. J. C. Nicholson, J. Chem.

Physics

20 , 1021 (1952).