Conduction Through Thin Titanium Dioxide

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Conduction Through Thin Titanium Dioxide Films
Citation
Maserjian, Joseph
(1966)
Conduction Through Thin Titanium Dioxide Films.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/MNGD-M461.
Abstract
Conduction through TiO
films of thickness 100 to 450 Å
have been investigated. The samples were prepared by either
anodization of Ti evaporation of TiO
, with Au or Al evaporated
for contacts. The anodized samples exhibited considerable hysteresis due to electrical forming, however it was
possible to avoid this problem with the evaporated samples
from which complete sets of experimental results were obtained
and used in the analysis. Electrical measurements
included: the dependence of current and capacitance on dc
voltage and temperature; the dependence of capacitance and
conductance on frequency and temperature; and transient
measurements of current and capacitance. A thick (3000 Å)
evaporated TiO
film was used for measuring the dielectric
constant (27.5) and the optical dispersion, the latter being
similar to that for rutile. An electron transmission diffraction
pattern of a evaporated film indicated an essentially
amorphous structure with a short range order that could be
related to rutile. Photoresponse measurements indicated the
same band gap of about 3 ev for anodized and evaporated
films and reduced rutile crystals and gave the barrier energies
at the contacts.
The results are interpreted in a self consistent manner
by considering the effect of a large impurity concentration in
the films and a correspondingly large ionic space charge.
The resulting potential profile in the oxide film leads to a
thermally assisted tunneling process between the contacts and
the interior of the oxide. A general relation is derived for
the steady state current through structures of this kind. This
in turn is expressed quantitatively for each of two possible
limiting types of impurity distributions, where one type gives
barriers of an exponential shape and leads to quantitative predictions
in c lose agreement with the experimental results.
For films somewhat greater than 100 Å, the theory is formulated
essentially in terms of only the independently measured
barrier energies and a characteristic parameter of the oxide
that depends primarily on the maximum impurity concentration
at the contacts. A single value of this parameter gives consistent
agreement with the experimentally observed dependence
of both current and capacitance on dc voltage and temperature,
with the maximum impurity concentration found to be approximately
the saturation concentration quoted for rutile. This explains
the relative insensitivity of the electrical properties of
the films on the exact conditions of formation.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
(Materials Science)
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Materials Science
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Mead, Carver
Thesis Committee:
Unknown, Unknown
Defense Date:
4 April 1966
Record Number:
CaltechTHESIS:07222014-092322045
Persistent URL:
DOI:
10.7907/MNGD-M461
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No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
8580
Collection:
CaltechTHESIS
Deposited By:
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Deposited On:
22 Jul 2014 16:46
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27 Aug 2024 22:27
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CONDUCTION THROUGH THIN
TITANIUM DIOXIDE FILMS

Thesis by

Joseph Maserjian

In Partial Fu lfillm ent of th e

Requirements

For the De g ree of
Doctor of Philosophy

California Ins titut e
Pa sa dena,

of Technology

California

1966
( S ubmitted April 4,

1966 )

ACKNOWLEDGMENTS

The author is indeb ted to his advisor,

Dr.

C. A.

Mead

for his inspi ration and encouragement and for making this
research possible.

Sincere appreciation is offered to the

California Institute of Technolog y
for their financial support.

Jet Propulsion Laboratory

In particular,

thank s

are offered

to the author's supervisors at Jet Propulsion Labora tor y
have been extremely generous and patient durin g
of this work.

who

the course

Special acknowledgment is due to the author's

wife Patricia for her many s acrifices during this endeavor and
for typing the final

manuscript of this thesis.

iii

ABSTRACT
Conduction

throu g h

have been inves tigated.
anodization of Ti

Ti0

The samp l es vvere prepared by either

or evaporation of TiOz, vvith Au or AI evap-

orated for contac t s .

The anodized samp l es exhibited consid-

erable hyst eresis due to e l ec tri ca l forming,
possible to avo id this problem vvi th th e
from vvhich complete se t s

hovveve r

it vvas

evaporated samples

of experimen ta l result s

tain ed and used in the analysis.
in c luded:

vvere ob-

Electrica l measurements

the dependence of current and capacitance on de

vo lta ge and temperature;

th e

dependence of capacitance and

conductance on frequency and temperature;

and transient

measurements of curren t and capac itance .

evapora t ed Ti0 2
cons tan t

fil ms of thickne ss 100 to 450

( 2 7 . 5)

and th e

op ti ca l di spers i on,

same ban d

the latt er being

evaporated film indicated an essen tially
s hort r ange order th a t cou ld be

Photoresponse measurements indicated the

gap of about 3

ev for anod i zed and

films and reduced rutile crys t a l s
gies a t th e

An e l ect ron transm i ss ion diffrac-

amorphous s tructur e with a
related to rutile.

( 3000

film was used for measuri ng the diel ectric

sim il ar to that for rutile .
tion pattern of a

thick

cont acts.

evaporated

and gave the barr i er ener-

iv
The results are interpreted in a
by considering the effect of a
the films and a

self consistent manner

large impurity concentration in

correspondingly large ionic space charge.

The resulting potential profile in the oxide film leads to a
thermally assisted tunneling process between the contacts and
the interior of th e

oxide.

gene ral

relation is derived for

the steady s tate current through s tructures of this kind.

This

in turn is expressed quantitatively for each of two possible
limitin g

types of impurity

distributions,

where one type gives

barriers of an exponential shape and leads to quantitative predictions in c lose agreement with the experimental re su lt s .
For films somewha t greate r

than 100

A, the theory is formu-

lated essent ially

in terms of only the independent l y

barrier energ ies

and a

character i s tic parameter of th e

that depends primarily on th e
a t th e

con tact s.

measured

sing l e

sisten t agreement with th e

maximum impurity

oxide

concentration

value of this parameter gives conexperimenta ll y

observed dependence

of both curren t a nd capacitance on de vo lta ge and temperature,
with the maximum impurity concen tration found to be approximately th e

saturation concentration quoted for rutile.

This ex- -

plains the relative insensitivity of the electrical properties of
the films on the exact conditions of formation.

· "

TABLE OF CONTENTS

lntroducti on • • • . . . . . . . . . . • • . . • . . . . . • • • . • . . . • • • . • . • •

1I.

Experiments on Anodized Ti0 2 Films • • • . . . • • . . . • • • •
A. Preparation of Specimens. . • • • • • • . . . • . . . • • • •
B . Measurements of Anodized Films • • •• •••• •..••
C. Discussion of Results on Anodized Samples ••.

10
29

ill.

Experiments on Evaporated Ti02 Films • . . . . . • . • • • • .
A. Preparation of Evapora t ed Specimens .•.•••••
B . Measurements of Evaporated Ti0 2 Films. . . . •
I. Thickness Measurements. . • . • • • . . . . . • . • • •
2. Optical Measurements . . . . . • • • • • • • . • .. •..•
3 . Photoelectric Measurements... • . . • • • • . • • •
4. E lectrica l Measurements. • • • • • • • • • • . • . . • •
C. Discussion of Results on Evaporated Films. . •

41
41
49
49
53
58
E2
80

rsz

Th eory . . . . . • . . • . • . . . • • . . . . • . . . . . . . • . . • • • • . . . . . . . . 9 5
A. The Barrier Potential . . . . . . . • . . • • • • • • • • • • • . 95
I. Impurity-Space-Charge Models . . . . . • • . • • . . 95
2. Image Force Corrections • . • . . • . • • • . . . . . . • I 02
3. Effective Barrier Approximations • • • . . . . . • • 103
4. Effect of Statisti ca l Fluctuations .• • . . .••.•• 104
B. Transport Theory . . . . . • . . • . . • • • • • . . . • . . • • • . . I 07
I. General
Equat i ons . . • • . • . . .• .. .. .. . •••••• 107
2. Current for the Int ermedia t e Case . •.. • . • . . 110
3. The Two Barrier Problem .•• •..•••••••.• 112
4. Formulation in Terms of Proposed

Barriers .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
C.

::SZ

Application of the Theory. . . • • • • • . . • • • • . • • • • • • 126
I. General Considerations •• . .•••••••.•••• • •• 126
2. Current Dependence • . • • . . . • • • • . . • • . . • . . • 128
3 . Capacitance Dependence. . • . • • . • • • • . • • • . • 139

Conclusions . . . . • • • . • . • . • . . . . . . . . . . • • • . • . • • • • . • • • . . 146

APPENDIX A. Properties of Rutile . . • • • • . • • • • • • • . . . . . . • •151
APPENDIX B. Experiments on Rutile ••••••.•.•••••••••• 157
REFERENCES • • . • • • . • . . . . . • • . • • • • . . . . • . . • . • . • • • • . • . • 163

INTRODUCTION

The subject of conduction through thin insulating films
has received much attention since the earliest observations of

Schuster ( )

The under-

in 1874 on tarnished copper wire.

lying theoretical principles have s ince

been well established

so that the primary cha ll ange remaining during the last few
decades has been to apply these principles to a
or mechanism which satisfactori l y
trical properties.

physical model

explains the observed e l ec-

Among the many specia li zed treatments

available in the literature,

none has

been found directly

applicable to the results considered here.

However,

theoretical description of the results has been derived in this
work which is c l ose l y

related to some of the earliest ideas.

Two important mechanisms based on relatively simple
physical models were first introduced in the

"thirties''

and

remain to this day often the starting point in attempting to explain many experimental observations of thin insulating films.
There is the well-known mechanism of quantum-mechanical
tunneling of electrons through a
.rnsu I atrng
regron,

or .
rg.rna II y

narrow potential barrier or

.rntro d uce db y

and developed further by Wilson ( 3 ) ,
Frenkel and Joffe(S),

in 1932.

F ren k e 1( 2 )

Nordheim (

) ,

1930
1n

and

Opposite this is the mechanism

involving th erma l excitation of electrons over. t he barrier,
developed by Schottky and his collaborators ( 6 )
years .

in the following

Each of th ese mechanisms proposed to explain the

properties of the ear ly metal-semiconductor rectifiers,

bu t

conf lict ed in their predictions of the direction of rectification
and o th er device properties.

controversy existed

for

severa l years over which of the se two processes applied to
the typical

rectifying devices of that era.

The thermionic

process eventually gained widespread acceptance ( 7 )

primarily

because it appeared to conform with the observed direction of
rectification,

gave the correct order of magnitude for current

based on reasonable estimates of barrier thickne ss ,
dieted a

temperature dependence i n qualitative agreement with

experiment.
littl e

and pre-

In fact,

this acceptance became so strong that

further consideration was g i ven to th e

tunneling mechanism

until recent years when attent ion focused on semiconductor
tunnel diodes,

the study of ve r y

thin insulating films in

connect i on with co ld emiss i on o f hot electrons (
ing between superconduct ing
The original

films (

tunneling theor y

(10)
extended by Srmmons

and tunnel-

has most recently been

and Stra tton

(II)

to describe more

generally the effects of image forces and other departures
from a

simp l e

rectangular potential barrier.

Most attempts

at quantitative comparisons of experimental results with the
theory have had relatively poor success.

This is not too

surprising in view of the simplifying assumptions necessarily
involved.

least two recent results can be cited which do

give satisfactory quantitative agreement.
and Chivian (

The work of Hartman

on extremely thin aluminum oxide films has

provided good agreement but required some arbitrary fitting
of unknown properties of the oxide.
extremely thin films

McCall's results

( 13 )

of mica cleaved from single crystals

demonstrated that it was possible

to introduce independently

determined properties of mica into the theory and still
agreement at least over a
The application of a
about SO

obtain

limited range of conditions.
tunneling theory to films thicker than

A is generally excluded because of the unreasonably

small currents predicted.

However this is true only if the

electrons must tunnel through the entire film thickness.
fa ct the original theory wa s

used,

the mechanism of rectification,
only a

during the controversy over

to describe tunneling through

very narrow ionic space char g e

the metal contact on a

barrier created at

much thicker semiconducting layer.

Relative ly thick barriers of this kind at metal-semiconductor
interfaces have been extensively considered in connection
with the thermionic process

( "Schottky emission")

that had

been further developed by Mott (
common l y

and

Schottky (

through very thin
been genera ll y

Schott ky

i g nored.

on the basis of such a

the con cep t of tunn e ling

type ba rri e rs appears

It wi ll be show n

results on tit a nium dioxide

( Ti0

can be exp l aine d

However ,

origina l theory had not been carried s uffi cien tl y

th e

far to

sa ti s-

observed e l ectric a l properties and

temperature dependence.

TIL,

to have

that th e expe rim ent a l

thin film s

physica l model.

factorily accoun t for th e

and are

Since the

refe r red to as Scho tt ky barriers.

origina l ideas were first introduced,

The th eory is extended in Chapter

considering in some detail the e ff ect of the ionic space

charge and th e

problem of charge tran s port throu g h

re s ulting barr i ers,

and l eads to

the

relationships th at ag r ee

with the experimenta l resu lts.
Although there i s
ical propert i es of Ti0

littl e

s tudies made of rutile

information ava il ab l e

thin films,

( Ti0

Castro

( 18)

Grant

( 16)

lit era tur e

Freder ikse (

Expe rim enta l evidence will

dica t es that the

film s tructure,

phys-

there have been detailed

ceramics a nd s in g l e

with comprehens i ve surveys of th e
reported by

on th e

crys t a l s

having been

a nd Hollander and
be g i ve n which in-

a lt hough highly disordered,

retains properties th at are sim il ar to that of rutile.

Some

of th e more r e le vant properties of rutile are di scussed in

Appendix A

and some experimental results on rutile are

given in Appendix B.
Experimental results have been obtained from both
anodized and evaporated TI0
described in Chapter IT,

films.

The anodized films,

exhibited effects that are attributed

to ionic migration induced by the applied voltage,
difficult to obtain reproducible results.
11 forming"

making it

These electrical

tendencies were successively avoided with the

evaporated films.

Therefore most of the detailed results

are obtained from the evaporated films described in
Chapter ill.

IT.

EXPERIMENTS ON ANODIZED Ti0
A.

FILMS

Preparation of Specimens
The techniques used for fabricating the anodized

titanium specimens may be divided into three parts:
preparation of the titanium substrate,
procedure,

and

( 3)

( 2)

(I)

the

the anodization

the deposition of metal contact .

Two methods were used in preparing titanium
surfaces for anodization.

The first method started with

20-mil-thick sheets of commercial-grade titanium

( 99% pure).

The sheets were cut into rectangular strips 7/8 11

x 3 11 and

heat treated under high vacuum for several hours.

This was

accomplished by passing approximately 100 amps along the
length of th e

strips in order to heat them to slightly below

their melting point

( 1800 oc) ,

while maintaining a

at the contacts to prevent buckling.
performed in a

The heat treatment was

Veeco 400 high-vacuum system equ ipp ed with

liquid nitrogen trap.

I ess t h an 10 -

slight tension

It was possible to maintain vacuums

torr dur1ng
th e

h eat treatment because o f t h e

additional

gettering action of titanium sublimating from the heat-

ed strip .

The heat treatment se rved to purify the titanium by

distilling out most of it s

impurities,

as well as crystallize the

metal with crystallites measuring the order of one centimeter

across.

Many

slippage lines appeared in the

The strips vvere subsequent l y

crystallites .

rectangles 3/16 11

111 and lapped

The central 3/411 of the

paper .

tropolished to a

cut int o

on one si d e

sma ll er

vvith #600 SiC

l apped sur fa ces vvere elec-

mirror finish using a

vvith the recommended so l u t ion.

i nd i vidua l

Bueh l er e l ectropo lisher

The samp l es vvere rinsed

repeatedly and boiled in disti ll ed vvater before anodizing.
The second method o f preparing the titanium
surface vvas accomplished by evapo r a tin g
t i tanium
• 030 11
11

{ 99. 999% pure)
1/2"

111

onto a

zone - refined

glazed a lumina substrate

{commercially obtained under the name

Smoothstrate 11 ) .

This substra te

smoother than common l y

vvas chosen because it is

used g l ass s lides or cover glass.

The evaporation vvas carried out in a
The titanium vvas evaporated from a

Veeco

4 00 system .

vva t er -cool ed copper

crucible by means of e l ectron bombardmen t.

The substrate

located 10 em above the titanium source vvas heated to 380°C
several m i nutes prior to and during the evaporation of the
titanium.

sublimati ng titan ium strip loca t ed in th e

upper

part of the be ll jar provided additiona l get t ering action as
mentioned above,

and insured vacuums of

during evaporation .
period of 5

l ess than 10

-6 torr

The evapora ti on vvas performed during a

to 10 seconds vvhile exposing the substrate

to the

evaporating source of titanium by means of a
shutter.

Films the order of 1000

observing a

movable

A thick were insured by

glass monitor slide located adjacent to the sub-

strate and closing the shutter at the point the slide just became
opaque.

The rate and thickness of evaporation were so

chosen to provide very smooth titanium surfaces.

The sub-

strates were placed directly into the anodizing solution after
this evaporation process.
The anodization process used an electrolyte
of either concentrated ammonium citrate or ammonium borate
solution.

The choice of either electrolyte had no noticeable

effect on the results.
and agitated with a

The solution was at room temperature

magnetic stirring rod during anodization.

The substrate was submerged to about three fourths of its
length at one side of the vessel with electrical contact made
to the exposed titanium.

platinum cathode was inserted

at the opposite side of the vessel.

The anodization was

carried out either at constant current or constant voltage.
The constant-voltage method which is normally used (

was

accomplished by applying the desired voltage across the cell
until the anodization current dropped to some small constant
va lue

(< 0 .I ma)

during a

period of several minutes.

The

constant-current method was used to gain some comparative

information on the effect of the anodization process.
of 0. 9,

2. 5,

and 8

ma were applied until the desired voltage

was reached across the cell.

After anodizing,

were thoroughly rinsed in distilled water,
pure nitrogen gas,

Currents

the specimens

dried with a

jet of

and immediately placed in the vacuum

system for the next operation of depositing metal contacts.
Gold was deposited as contacts on all anodized
specimens .

The gold was evaporated from a

through stainless-steel masks.

tungsten filament

In some cases,

single mask

was used having uniformly spaced holes ten mils in diameter.
In other cases,

two masks were used,

rectangular contacts 20
substrate,

one providing thirteen

250 mils spaced uniformly along the

the other with 6-mil

holes providing 12 groups of 12

dots each located between the rectangular contacts.

The rec-

tangular contacts were designed for photoresponse measurements and required film thicknesses the order of 500
was accomplished with a

glass monitor slide in a

ilar to that already described.
ular contacts,

which provided registration,

and the

This

manner sim-

After evaporating the

the mask was changed using a

rectang-

mechanical jig

remainder of the gold on

the filament evaporated through the second mask providing

gold dots at least 1000 A
formed in a

thick.

The evaporations were per-

vacuum of less than 10

-6

torr with the substrate at

room

temperature after it had been under high vacuum for a

few hours.
B.

Measurements of Anodized Film s
E l ectrica l measurements of the anodized films

were influenced by
effect s .

l arge hysteresis and time-dependent

T his was a

genera l characteristic of a ll the anodized

fi lms tested regard l ess of the method of preparation.
sequent l y,
part of a

the results presented in this sect i on are in large
qua litative nature,

ideas to be app lied

but are usefu l in supporting the

l ater.

The e l ectrica l measuremen ts
tacting a
of a

go l d

dot with a

m i cromanipu lator.

tact gave no no t iceab l e

were made by con-

po i nted 3-mi l go l d

wire with the aid

The pressure app li e d

by the point con-

effect on the e l ectrica l characteristics.

The time dependence of current with a
ly

Con-

relative-

l arge constant positive vo ltage app li ed at the g old contact

is typified by

th e

result

g iven in Fig.

was made by app l ying successive l y

sweeps.

The measurement

l onger pulses at constant

vo ltage from a low-impedance source,
current response

II- I .

whi l e

record i n g

the

on an osc illo scope at correspondingly slower

char t recorder was s tarted simultaneously with

the s l owes t sweep on the o s c ill oscope

discontinu it ies resu l t from some hysteres i s

sec/em).
effect s

The

even though

(.)

:::>

0::
0::

10-z

1o-•

10-·

10-a

Figure II-1.

1o-t

TIME, sec

lo-•

. .

lot

AT Au CONTACT

+ 1.5v APPLIED

r--_,._.

298°K

/ :__

Current versus time at large positive bias

lo-s

10-4

~.·,
/.
v·

SAMPLE
HEAT TREATED Ti SUBSTRATE
ANODIZED AT IOv (CONSTANT VOLTAGE)

10.

12
minimum pulse durations were applied in preceding pulses.
Other samples exhibited a
positive bias,

similar time dependence at large

the onset and rate of current rise and the max-

imum value increasing with positive voltage.

The current

rise would first appear in the v icinity of I volt positive bias
at room temperature and at much larger voltages at low
t emperatures as will be discussed later in this sect ion.
samp le s

The

undergo large changes in electrical characteristics

after being subjected to de voltages of this magnitude.

These

characteristics would usually partially recover over periods
of hours.
voltages

For lower positive voltages and for negative
(below breakdown) ,

no current rise in the time

re-

sponse was observed and the current would continue to decrease more s lowly from the foot of the initial decay to some
asymptot ic value.

At sufficient ly large negative voltages an

abrupt breakdown

( Bd

crease with tim e

would occur before a

current in-

could be observed.

The capacitance of the samp l es was
at 100 kc on a

Boonton 74C capacitance bridge.

measured
The sample

used in Fig. IT-I measured 595 pf prior to the above measurements,

682 pf immediately afterwards,

658 pf after severa l minutes.

partially recoverin g

The external series resistance

used in the measurements for Fig.

IT-I

was 100 ohms,

13
therefore the external time constant of the RC circuit was
only

0. 06 f- Sec.

The oscilloscope limited the response to

The observed decay time was approximately

about I f-sec.

100 f'sec which corresponds to the RC value obtained if one
takes R

to be the maximum resistance of the oxide occurring

at the minimum in the current response.

This

leads one to

suspect that the observed decay may be associated with a
redistribution of trapped electrons in the oxide film (

The de 1-V characteristics were obtained by
adjusting the applied current to the desired va l ue and allowing
sufficient time
The

for the voltage to reach a

nearly constant value.

results given in Fig. II-2 were obta i ned from a

s ample from the same s ubstrate used for Fig.

II-I.

log plot is included for compar i son with other results.

typical

logThe

measurements were made in this case by a lt ernately applying
pos itive and negative currents in decreas i ng steps starting
from the maximum current of 0. 03 ma.

This sequence of

measurements permitted the maximum changes to occur in the
sample during the initial m e asurements.
to e xhibit a

strong rectification property

The sample is seen
in this case .

larg er currents the samples wou ld usually break down or
undergo an irreversible change to a
characteristic.

lower impedance

SAMPLE
HEAT TREATED Ti SUBSTRATE
ANODIZED AT IOv (CONSTANT VOLTAGE)
Au +

Au_ Au_

j ~ J.
i /. i

Au +

102~--~--~~~--- 298oK--~--~r-~~--+--+----~

V]v

10 3 ~--~_,~~-----+-----+--~~;---~~--~--~
C\1

/X/

~10 4 ~--~~--~-----+--~-+----~~---H~--~----~
X/

wa::: 105~--~~--~---+-+-----+-----*----~----~----~

LOG v

SCALE~~--,.

10 6 ~---H----~-----+-----+--~~~~~----~----~

I /;,

j/l

I0 r---~-r---;-----+-----+---1~~--~----~--~

lo 8 r--+~-----;-----+-----+--~~----~----~--~

10-

10 ~--~----~----~----~----~-----L----~~~--~

VOLTAGE, v
Figure II- 2.

Current versus de voltage

IS
The capacitance was observed to vary considerably as a

Measurements performed on

function of de voltage.

most samples were complicated by the hysteresis effects mentioned above.

The result given in Fig. Ir-3 measured at IOOkc

typifies the kind of dependence observed and was selected from

sample exhibiting a

minimum of hysteresis;

itance returned to its initial

that is,

va l ue at zero bias

positive voltages had been applied.

( Ci)

its capacafter the

variation of capacitance

with voltage is suggestive of Schottky type barriers.
of this kind exhibit a
and voltage,

linear relation between

Barriers

( 1/ capacitance)

and for this reason the data given in Fig.

were plotted in the form shown;

however,

does not indicate this dependence .

II-4.

IT-3

the result c l early

The relation arising from

exponential barriers as discussed in Chapter
plot given in Fig.

In this case a

rsz: suggests the

reasonable correspon-

dence with the assumed dependence is obtained.
Samples prepared by anodiz in g

heat-treated

titanium sheet at constant current gave characteristics
similar to those discussed above except the drift and
hysteresis effects were enhanced.

For this reason these

samples were used to examine the effects further.
impractical

It was

to obtain useful measurements at room temperature

because of the complications arising

from the tendency of the

1.4

FAMILIAR THEORY: (C;/C) 2 : ( 1-V/f/>)
C::)

1.2

~ "(,-__

1.0

ro-o.

298°K

(>' 0.8

SAMPLE
HEAT TREATED Ti SUBSTRATE
0.6 ~ANODIZED AT IOv (CONSTANT VOLTAGE)
AREA= 5. 5 x 10-4 cm 2
Cis 588pf AT 100 kc

0.4

0 .2

-2

-I

VOLTAGE, v
Figure II-3.

Dependence of capacitance on de voltage

298° K

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

SAMPLE
HEAT TREATED Ti SUBSTRATE
-3~----+-----~-----+------+
ANODIZED AT IOv (CONSTANT VOLTAGE)
AREA= 5.5 x 10- cm 2
C; = 588 pf AT 100 kc
-4~----+-----~-----+------+
<)..- V= 1.40v
4>1 = 1.42 ev

-~'--:-----=-~'----'::-----"='""__._l~1___..l~
-6

-4

-3

-2

-1

(ljC-IjC;) 10 , fFigure ll-4.

Linearized plot of capacitance versus de voltage

18
samp l es to g r adually recover towards their initial condition
following the changes induced dur i ng each de measurement.
Howeve r,

at 77°K th e

sa mple wou ld remain in a

s tab le con-

dition d ete rmin ed by the maximum de vo ltage previously
app lied as l ong as they were ma inta ine d at this tempe rature
and th e
Fi g .

previo us maximum vo ltage was not exceeded.

II-5 i s

Curve A

this way.
of a

typical of severa l samples that were measured in

previous l y

ing currents.
vo lta ge from V
exceeded,

r epresents the i nitial d e

characteristic

untested samp l e

measured s l owly at i ncreas-

Curve B

result when decreasing the

wou l d

and would r ema in s table un til VB was

whereupon a

would resu lt from th e

new s tab l e

characteristic,

new maximum vo lt age V c'

of this k ind have been reported for other fil ms (
have been attribut ed to a
with a

curve C
Changes

21 22

"forming" process th at is assoc iate d

redistribution of i ons in the oxide

film.

observed at room temperature were simi l a r

The e ff ec t s

except for th e

fact that the characteristics would not remain stab l e
forming.

In thi s

and

case it appea r s

th a t th e

ion s

afte r

have suffic i ent

thermal energy in the presence of the b uilt-in field t o
immed i a t e l y
however,

s t a rt diffusing back into thei r

or i g in a l distribution;

comp l ete recovery is not usua ll y

a tt a in e d.

It might

be specu l a ted th a t some precip it ation of impuriti es occurs

~·

10I

.~ 11'

-vn./

ldBLE .

// -v/·/
l/

/ 8
/STABLE

I .I

_#
•'

·~/

77 °K

I .vf

fv8

.lo'RIFT

•'

./DRIFT

SAMPLE
HEAT TREATED Ti SUBSTRATE
ANODIZED AT 2 .5 rna TO 5v
C (100 kc) = 229 pf AT 296 °K =169 pf AT 77 °K

;·
.I

VOLTAGE, v
Fi g ur e II-5.

Ele c trical forming characteristics

20
during each forming process in a
sample permanently in a

manner that leaves the

lower impedance condition.

An attempt was made to relate the temperature
dependence of the 1-V

characteristic to this formin g process.

With a constant positive current applied
example),

( 10-

amps for

the sample would appear bi-stable during gradual
The

heating and cooling cycles between 77°K and 296°K.

resulting voltage was observed to switch erratically be tween

lower value characteristic of the formed sample,

exhib iting

meanwhile

relatively sma ll variation with temperat ure ,

and

higher value characteristic of the unformed sample, at this

time exhibiting a
errati c

much larger variation with temperat u re.

nature of thi s

The

process made more d eta iled measure-

ments impra ctica l.
Samp l es prepared by anodizing evaporated titanium
fi lm s

at constant vo lta ge also gave characteristics s imil ar to

those already described.

However in this case,

larger

conductance and capac itance per unit area was measured from
samples prepared a t th e

same anodiza tion vo lta ge .

They a l so

exh i b ited more uniform characteristics than obtained from the
other methods.

For this reason,

measurements were made

on these samp l es designed to avoid electrical
For de currents l ess than about ±I o-

amps,

forming effects.
the voltages

remained reproducible,
levels.

However,

with forming appearing at higher de

at higher levels it was possible to use

voltage pulses of sufficient length to observe an asymptotic
limit for the current that preceded the onset of forming,
vided the voltage was not too large.
way,

pro-

When obtained in this

the current is believed to be representative of the

undisturbed oxide film and will subsequently be referred to
The

pulse measurements were made using a

Rutherford pulse generator and applying consecutive l y

positive

and negative voltage pulses at increasing values while observing the limiting value of current on an oscilloscope.
gives a

typical

Fig. II-6

result measured at room temperature.

Measurements made in this way were nearly reproducible
providing breakdown had not occurr-ed , and ther efore are
believed to give the desired results.

The

direction of

recti/ ication is observed to be opposite to that obtained from
the de measurements given in Fig.

IT-2.

Both polarities

approach the limiting ohmic depencence at low voltages.
As mentioned previously,

the current decay follow-

ing the normal RC decay is characteristic of some internal
resistance of the oxide films and is interpreted to be associated
with trapped electrons.

Of interest is the current which

immediately follows the much shorter external RC decay.

SAMPLE

EVAPORATED Ti FILMS
ANODIZED AT IOv {CONSTANT VOLTAGE)
C { 100 kc) = 502 pf AT 298 °K
AREA=4 .7xlo-4 cm2

I vBd

Au-/ • Au+

Ii !I

298 °K

~ 10-4

v-' !

/~/

~(//

10-7

w-·

10-9
0.001

0.01

0 .1

VOLTAGE,v
Figure II-6.

Current (1 } v e rsus voltage

23
This current subsequently referred to as 1

could then be

interpreted to represent the current due to trapped electrons
before they

readjust to the steady-state distribution.

Measure-

ments of this kind were made by balancing out the external
RC decay and recording the current and v oltage transients
on a

dual-beam oscilloscope.

Figure II-7 illustrates a

transient response observed.

typical

The maximum current in the

transient was taken as 1 and the voltage measured at the same
time the maximum occurred.
manner at 298 °K

for a

to gether with the 11- V

The results obtained in this

typical sample are plotted in Fig.

II-8

curve measured from the same sample.

The 10 dependence is approximately ohmic and independent of
polarity in the voltage

range shown.

At large r

positive volt-

ages the current rise appears at the beginning of the transient response.

At the onset of this effect,

must be

equ ivalent to 11 and correspond to the point where the curves
meet.

Breakdown u sua ll y

occurs slightly above this point un-

less extremely short pulses are used.
tained by balancing the RC decay

The capacitance o b-

was 285 pf as compared

with 462 pf measured on the Boonton bridge at 100 kc.
difference which wa s

This

t yp ical in all samples measured is con-

s istent with the idea involving a

redistribution of trapped elec-

trons and consequently an additional space-charge capacitance.

VOLTAGE,
0 .05 vjcm
CU RRENT,

IJ.Lojcm --.....,

TIME, 2J.Lsecjcm

Figure II- 7.

Typical transient response

lo2

.-.cf.t
__, 0 • I
I •
• I

... 6-<1

I i

I .
I .
. I
I .

~"'

. I

,.O±Io

/ I

Au-f

~·Au+

I /I

298 °K

./ I /
/;

;·
. I

/./
//

SAMPLE
EVAPORATED Ti FILM
ANODIZED AT IOv
(CONSTANT VOLTAGE)
C (100 kc) = 462 pf AT 298 °K
AREA=4 . 5xi0- 4 cm2

./
...
-"'
...

10- 8
0.01

0 .1

VOLTAGE, v
Figure II-8.

Current (I

and I ) versus voltag e

100

26
Measurements of lo at both room temperature and
at liquid-nitrogen temperature over a

wider voltage range ar e

given in Fig .

(C 0 )

IT-9.

The capacitance

ing the RC decay at 77 °K was 260 pf.

obtained by balanc-

As mentioned above,

the departures from ohmic behavior occur when the current
rise appears at the beginning of the pulse response .

The

additional current must then be due to the mechanism associated with the current rise,
ionic origin

(forming) •

which has been attributed to an

The onset of the additional current

occurs in this case at approximately one volt at room temperature and ten volts at 77°K.
temperature dependence,
erature independent.

Although this effect has a

large

the ohmic region is relatively temp-

Measurements attempted at larger neg-

ative voltages usually resulted in abrupt breakdowns of the
samples and thus

no further information could be obtained

for this case.
The short circuit photocurrent response

of a

sample has been measured in order to determine the barrier
heights at the contacts as well as the band gap of the oxide.
The measurements were made at room temperature with a
Gaertner model 1234 monochromator and a

Reeder vacuum

thermocouple

The monochromatic

for a

calibration reference.

light was focussed on a

sample prepared with a

gold contact

SAMPLE
EVAPORATED Ti FILM
ANODIZED AT IOv (CONSTANT

VOLTAGE) C0 ;260 pf AT 77° Ki

AREA= 4 .6 x 10-4 cm 2

y //
/ v
/:I /
/:/
298°K

!/
w/

a. 10-l

77°K

10-3

0 .01

0 .1

VOLTAGE, v
Figure II-9.

Initial current (1 0 ) versus voltage

100

28
approximate l y

500

A thick as d esc ri be d above, enab ling

sufficient light to penetrate into th e

acti ve r eg i on of the sample .

The response was measured with a
synchronized with a

narrow-band amp lifier

cps chopper at the

The spectra l response R,

li ght source.

corresponding to the r e lati ve

photo-

electric energy response per incident photon is commonly p lotted as

m vs. hv ( the photon energy ) as a consequence of

the wel l-known Fowler dependence

where

(23)

(E-¢l

¢ i s the meta l - t o-ox ide work function.

arises when the photoelectrons assume a

zero probability of

penetrating the barrier at energies less than
ability for energies greater than

¢.

ed the case when the probab ility

This dependence

Cow ley

¢ and unit prob-

(24)

has consider-

undergoes a more gradual

transition wh i ch may be approx i mated to the next order by

linear dependence of probab ility

not too different from

¢ .

In thi s

( E- ¢)

for energies E

case one obta ins

dependence

( E-¢) 3

1 3
suggesting a p lot of R /

vs .

h~.

similar argument may

be applied to photoelectrons excited across the ox i de energy
gap .

Gobezi and Allen (

have shown that the above de-

pendence best describes their photo-response measurements

29
of s urf ace ba rri e r s

on various sem iconduct ors and of fer some

a lte rn ate possible exp lana tion s .
su it s

Since our experimenta l re-

also give better agreeme nt with this d e pendence ,

l atter plot is preferred for making th e
The
are g iven in Fig.

results obtained from th e
II-10.

and B) •

interpreted as th e

The ex trapolated va lu e

barri e r

I. 0

anodized sample

for rutile

height a t th e

ev which i s

(see

of I. 4 ev is

go ld-oxide contact .

sma ll reverse response also indicate s

of approx imate l y

ex trap o lation s .

The transition e nergy extrapo l a ted to

3 . OS ev corresponds to the energy gap
Appendix A

d es ire d

barrier he i gh t

be li eved t o

be due to the

Reverse b i asing gave the

titanium-to-oxide back con t ac t.
same va l ue of band gap,

but excess i ve noi se at l ower

energies did not permit a

ba rri er-he i ght extrapo la ti on .

th e

Discuss i on of Resu lt s

on Anodi zed Samp les

The l arge hysteresis and tim e-dependent effects
observed in the a nodized films have a lready been inte rpreted
to arise from ionic mi gra tion
film by the app li ed field.

pure l y

app l y

No a lt erna t e

induced in the oxide
exp l ana ti on based upon

e l ectronic mechanism can be visua li zed that might

to thin film s

and th e

(forming)

and accou nt for th e magnitudes of change

tim es involved.

On th e

o th er hand,

process itself m u s t be predom inantl y

th e

e l ect roni c

conduc tio n

since c l ear l y

"&:

"''

-2

_., ~·_,·

h v, e v

.-. v·/i'

~.---·--

l?~J
,X

) / RIGHT SCALE

Y"BIAS

- 15v BIAS/ l

P hoto-r esponse of anodized specimen (Au & Ti contacts )

1.4

F igure II-1 0.

LEFT SCALE

0 BIAS

SPECIMEN
HEAT TREATED T1 SUBSTRATE
ANODIZED AT 8 mo TO 8v

- 4

- 12 "''-ct

- 20

- 28

(..)

31
ionic currents of the required magnitude could not be supported in the thin films for more than an instant without material
breakdown.

It must be assumed that a

limited migration or

redistribution of the ions is responsible for large changes in
the electronic properties.
is accounted for by the

The effect of the oxide lattice ions

low-frequency dielectric constant,

and

consequently these additional effects must arise from impurities
(or struc tural defects)

in the oxide.

Very large concentrat ion s

of impurities are

re-

quired to have an appreciable effect on the electronic
properties of thin films as can be deduced by considering the
change in potential introduced in the oxide film by an ionized
impurity space-charge

(see Section TIZ A) .

for large concentrations of impurities and a

The requirement
correspondingly

large space-charge does not seem unreasonable since we are
d ea ling with a

highly disordered or amorphous-like structure

in the oxide films .

Furthermore,

hi gh as 5 mole percent
Ti0 2 , ,

impurity concentrations as

( I. 6 ·10 21 em -3)

can be introduced into

and from m easu rements reported on rutile one con-

eludes that a

relatively rapid diffusion of impurities can occur

even at moderate temperatures

(see Appendix A) •

Both donor

and acceptor t ype impurities have been observed in rutile,
however a

natura l tendency exists for Ti0

to be reduced

32
throu gh th e

formation of oxygen vacanc i es which act as donor

impurities.

Therefore in the absence of lar ge concentrations

of acceptors,

one would normally expect to obtain an excess

of oxygen vacancies and thus an n-type oxide
case 1

it would not be difficult for a

type of impurity

to enter the

ibly after their formation.
defects
is

(of a ll kinds)

e ither

lar ge excess of either

thin oxide films

during and poss-

The existence o f l arge densities of

a l so implies many trappin g cen t ers and

consistent with th e
On th e

film.

e ffects a lready ascribed to trap s .
assumption that an n-type

film is formed

in the anod i zed samp l es and using the resu lts obtained fr om
the photoresponse measurements 1
what

more s pecific mode l of th e

of I. 4

and I. 0 ev 1

we may dedu ce a
barr i er.

some-

The barrier heights

interpre ted to occur a t the gold and titanium

contacts respectively 1

are consi s t e nt with the differences in

work functions tabu l a t ed in th e

lite rature.

i ng in t he oxide as measured f rom th e
appear as illustrated in Fig . II-II a .
tha t the oxide is suff i c i en tl y
po tentia l approaches a

thick,

very sma ll va lue

po te nti al occu r r-

Fermi l evel 1

It i s
such

The

may

assumed in this case

that the minimum
in

th e

inter i or as de-

t ermined by

th e

excess conce ntra tion of donor type i mpurities.

The posi tive

space - charge th en ari ses from th e

purities represented by empt y

ioni zed im-

impurity states lyin g above the

Ti02

1.0 ev

FERMI LEVEL

a..

a::

VALENCE BAND

(.)

(a)

(b)

(c)

Figure II-11.
Representation of proposed barrier: {a ) at equilibrium ,
(b) with positive applied voltage, and (c ) with negative applied voltage

34
From Poisson's equation in one dimension

Fermi leve l.

where U

is the electron potential, X. is the static dielectric

constant,

and

impurities,

is the positi ve space-charge density of ionized

one sees that a positive curvature in the potential

is required which must lead to the kind of barrier illustrated
independent of the precise spa tial
impurities.
will

distribution of ionized

Only the detailed shape of the potential barrier

depend on this distribution.

itance,

as a

The measurements of capac-

function of de voltage,

have already implied that

the barriers may p ossess an exponential shape as will be
discussed in detail in Chapter ill".
requires in the above case a
for holes,

The

energy gap of 3

much larger potential barrier

and therefore their effect can be neglected.
When dealing with barriers as represented in

Fig. IT-I Ia,
which

we may consider the

familiar conduction theories

can apply at metal n-type semiconductor contacts.

mentioned in Chapter I,

the two limiting theories invol ve:

electron tunneling through the

barrier ,

emission of electrons over the barrier.

and

( 2)

As
(I)

thermionic

In case

(I) ,

reel-

ification is predicted with the easy direction occurring when
the metal contact is at a
semiconductor

negative voltage with respect to the

(oxide interior).

This arises because of the

35
much larger supply of e l ectr ons in the metal having energies
belovv the Fermi level as compared vvith that in the semiconductor .

smal l temperature dependence is predicted
In case

in this case.

( 2) ,

rectification occurs in the opposite

direction because of the reduction in the barrier height as
seen from the semiconductor under forvvard bias.

In this

case the electron supply occurs above the Fermi level

and

assumes the same type of Boltzmann dependence at either
side of the barrier.

large temperature dependence is

characteristic of thi s

case.

The experimental results suggest that either
limiting case may be approached,

depending on the sample

history and method of measurement.

With a

believed to exist at the gold contac t,

the easy direction of

rectification should occur for

larger barrier

the thermionic case vvhen a

positive voltage is applied at the gold contact.

This vvas the

situation observed in the de measurements at room temperature
given in Fig. IT-2.
The thermionic process also predicts an
exp ( qV /kT)

dependence in the forvvard direction.

The

observed forvvard dependence can be accurately described
by exp ( 41. SV)

belovv I. 4

volts,

prediction for room temperature

in good agreement vvith the
( q/kT=-40) •

The reverse

36
direction i s

not norm a ll y

particularly in thi s
were present .

subj ec t to a

sim ple interpretation,

case where e l ectrica l formi ng effects

Because of the

formin g

insta b ilities,

temperature depende nce was not measured directly.
ever th e

l arge tempe r a ture

r easonab l e

How-

d ependence observed in th e

fo r ming expe rim ents can be r e late d
appears to be

the

to this case.

evide n ce that a

later

There

therm i onic type

of process occurs at the Au contact when electrical forming
is al lowe d

to take place under th e

conditions described.

On the basis of th e above discussion the curren t
ri se observed at l a r ge positive vo lta ges
IT-li b

(Fi g .

IT-I) can be

interpre ted with th e

help of Fig.

vo ltages t he ba rri e r

a t the go ld con t act is suppressed and

the curren t becomes li m ited by the
nat ing

a t the ti tanium contact.

the f i e ld towa rd s

th e

tit an iu m

th e

top of th e

i mage - force l owering

con tac t,

as th e

in creases.

Also,

increasin g l y

im po rt ant as th e

must satura te

At large

of e l ectrons or i gi-

ca ti ons drift w ith

increasing the width

width o f th e

Ti barr i e r.

Ti barri er wi ll increase due to

of th e

em i ss i on )

supp l y

Impurity

of th e Au barrier and reducing th e
Emission over th e

as fo ll ows.

ba rri er height

( Schottky

field due to the addi tiona l spa ce-charge
tunne li ng through the bar ri er will become

when ei th e r

th e

barrier n a rrow s .
s uppl y

of impurity

This process
cations near

37
the gold contact becomes depl3ted or impurity saturation
occurs in the barrier at the titanium contact,

providing com-

plete breakdown does not occur first.
For reverse po larity,
cannot app l y

the opposite process

unti l sti ll higher vo l tages are attained,

since

nearly the entire voltage drop appears across the larger
barrier at the go ld contact until the applied field reduces
its effectiveness to less
contact.

This case

large vo l tages are

than the barrier at the titanium

is i ll ustra ted by Fig .

II-IIc.

Re l atively

required before this condidion is reached

because of the s l ow change in the barrier with reverse
v olta ge .

s l ower variation of current with vo l tage is there-

fore also expected in thi s
su it g iven in Fi g .

II-2.

case,

and corresponds to the re-

At sufficient l y

l arge vo ltages

onset of ion migration must u l timately be reached .
as previous l y
be fore a

stated,

However,

comp l ete breakdown always occurred

current rise could be detected.

consequence of the

the

This may be a

l arge fields already existing in the gold-

oxide barrier when sti ll higher vo ltages
from the titanium side .

This,

in turn,

initiates

io n

migration

could give rise to an

ionic co ll ision-ionization mechanism of breakdown as proposed
by

Joffe et al

(26 )

supply of titanium

Al so,

there exists vi rtually an un l imited

(or oxygen vacanc ies )

at the tita n ium contact.

38
The electrical

forming experiments performed on

samples which were anodized at constant current provide
additiona l evidence of the ionic processes discussed above.
In thi s

case it i s

believed that the barrier at the titanium

contact accumulates sufficient positive space-charge during
forming to enable tunneling to become the dominant mechanism
This is consistent with the temperature

of electron emission.

insensitivity observed under this condition.

The i nstability

observed during tempera ture cycling is attributed to the competing tendency of the

ionized impurities to relax towards their

orig inal distr ibu tion and thus tending to widen the barrier and
causing the electron em i ssion to temporarily revert to more of

thermionic type of mechanism.

Apparently,

the constant

current anodizat ion process results in l arger mobilities for the
ionized impurities so that the forming process becomes more
evident in this case.
In addition to the electrica l forming effects already
cited for other thin films,
served with

rutile.

simi lar effects have also been ob-

Ob servat 1ons

27 )
K un1n
et a I (
, on b o th

reduced and unreduced rutile with Ag contacts, were interpreted in terms of an e l ectrical
of the crystal.

Van

Raalte

generation of donors in the bulk

(28)

has made similar observations

on unreduced rutile with Au and Ti contacts but offers an

39
alternate explanation:

the electrons are injected at the cathode

gradually filling traps and increasing the density of conduction
electrons;
current

the

resulting space-charge which would limit the

(sci)

is neutralized throughout most of the bulk by

holes except at the anode where a

negative space-charge is

required to support field emission of holes into the bulk.
results appear to support this explanation,

His

however it is

difficult to reconcile this model with thin films since unreasonable trapping times would be required in order for the high
conductance condition to be maintained for long periods after
removing the applied voltage.

Experiments were also per-

formed during this investigation on reduced rutile with Au
contacts and are described in Appendix B.

These experi-

ments also revealed similar forming characteristics.
The results obtained from
evaporated Ti

samples prepared from

films indicate larger impurity concentrations

as compared with samples prepared from heat-treated Ti
sheet using the same anodization procedure.
is required to explain the
lances.

Also the

larger conductances and capaci-

reverse direction of rectification suggests a

tunneling mechanism.
or forming

This conclusion

The reduced effect of ion migration

is interpreted to be a

impurity concentrations.

consequence of the larger

One might argue on the basis of

40
impurity saturation;
reached,

that

is,

as lar ger concentrations are

the imp urities find fewer sites available and there-

fore must overcome

l a r ger ba rriers

next available site.

The

in order to move to the

field-induced drift of ionized impur-

ities would be correspondingly reduced.

Complete saturation

occurs when a ll possible sites are occupied and,
previously,

this limit may be as high as I. 6
In order to attempt a

properties of the films,

-3

more detailed study of the

it is c l ear l y

stable properties so that a

as mentioned

desirable to obtain very

complete set of accurate measure-

ments can be performed on individual samp les.

The evap-

oration method described in Chapter ill provides such samples
as well as offering other advanta ges .

ill.

EXPERIMENTS ON EVAPORATED Ti0
A.

Preparation of Evapo rat e d

F ILM S

Specimens

The fabrication procedure consisted essentially
of a
ed

sequence of evaporations in vacuum,

to give the desired configuration .

standard configuration obtained,
of crossed metal str ip s
surfaces of th e

using masks design-

Fig.

ill- I illu st rat es the

being composed of an arra y

(Au or AI)

which contact opposite

interlying oxide films.

Seve ral samples are

obtained from each of the four oxide film thicknesses in this
manner,

with the extremities of the metal strips providing a

convenient place for making so ldered connections.
The following sequence of operations was used
in the standard procedure:

(I)

cleaning of the substrate and

the materials to be evaporated prior to loading into the vacuum
system,

( 2)

vacuum outgassing of th e

loaded system,

evaporation of the first array of metal contacts,
of the Ti0

( 4)

( 3)

deposition

films,

( 5)

evapora tion of the second array of

metal contacts , and

( 6)

evaporat ion of Al

(not shown in Fig. ill- I).

protective

films

All evaporat ions were performed

without opening the vacuum system in order to avoid any
atmospheric contamination at the oxide-metal interfaces.

The

mechanical movements required for the successive operations,

42
such as repositioning the masks,

changing the evaporation

source materials and operating a

shutter,

through two high vacuum movable seals.
( 11 smoothstrate 11 )

The substrate

and evaporation source materials were clean-

ed in chromic acid,
water,

were accomplished

thoroughly rinsed and boiled in distilled

and dried with a

jet of pure nitrogen gas.

Maximum

precautions were taken to avoid collecting dust particles on
the materials prior to and during loading.

The loaded system

was out-gassed in vacuum at about 200 °C and then allowed to
pump for several hours reaching a

10-

torr before preceding.

vacuum of less than

final outgassing of the

substrate was performed by heating the substrate to 380°C
and allowing it to cool to room temperature just prior to the
first metal evaporation.
All evaporations were performed by electron
beam bombardment by moving the appropriate source into the
target position of the electron beam with the other source
materials located out of the way.

It was necessary to evap-

orate the AI or Au contacts out of a
this method.
of a

tantulum crucible with

The contacts were deposited during a

few seconds,

using a

shutter and glass monitor slide to

obtain film thicknesses the order of 500 to 1000
vacuum was less than 10

-6

period

The

torr during these evaporations.

UPPER AI ELECTRODES
0.005 in. WIDE

BOTTOM AI ELECTRODES,
0.005 in. WIDE
Ti02 FILMS
OF DIFFERENT
THICKNESS

SUBSTRATE
GLASS-GLAZED CERAMIC,
I X I X 0 .030 in.
Figure III-1.

Configuration for Al-Ti0 -Al thi n f ilm samples

44
The oxide films were deposited by slowly evaporating rutile crystal

( Ti0

at a

10-

pressure of 4

in the presence of pure oxygen

torr.

The oxygen gas,

metered

into the system during this evaporation,

serves to replace the

oxygen which dissociates from the

at the temperatures

Ti0

(c:::.: 1700°C) .

required for evaporation

sufficiently slow deposition rate

By maintaining a

( <:>< • 2A/sec),

the oxygen gas

has time to react with the dissociated molecules being deposited
on the substrate and thus form a
position of Ti0

in the film.

nearly stoichiometric com-

The shutter was opened after

steady evaporation rate was maintained and the mask re-

positioned at few minute intervals during the evaporation to
give the four thicknesses.

The thickness and rate w e re

e s-

timated during evaporation by observing color fringes which
formed on metal surfaces at fixed locations near the source.
Pure titanium was also successfully used in place of rutile as

source.

This method was not adopted however,

difficulties encountered in maintaining a

because of

uniform evaporation

rate.
The protective film of Al 2 o
same areas as the Ti0 2

films to a

was deposited over the

thickness of at least

IOOOA~

This served to protect the thin film samples from atmospheric
moisture during storage and is believed to insure more

45
reproducible results.

Saphire crystal was found to be a

convenient evaporation source of Al
of Al

o3•

Good insulating films

were obtained either with or without the use of

oxygen during the evaporation,
dissociation of A1

indicating that no appreciable

o 3 occurs at the evaporation temperatures

(c::I900°C).
The masks were separated from the substrate
by approximately 1/1 6

in. ,

the evaporated pattern.

giving relatively diffuse edges in

This is believed necessary in order

to avoid appreciable variations in the thickness of the oxide
film at the edges of the metal contacts.
micrograph showing the area of a
by the overlapping metal strips.

Fig.

ill-2 is a

photo-

typical samp le as defined
The phase contract optics

clearly reveals the gradual topological contour at the diffuse
edge of the strip.

The small blemishes in the film,

ed by the phase contrast,

exaggerat-

were usually related to the original

surface of the smoothstrate.

Samples exhibiting a

such blemishes were se l ected for measurements.

minimum of
The active

areas of the samples were measured from such photomicrographs,

interpreting the effective boundry to lie near the

outer part of the diffuse edges.
The evaporation process described makes no
attempt at improving the structural orde r

in the Ti0

films.

Figure III- 2.
Phase contrast
photomicrograph of
sampl e 270X

47
Rather than introduce new uncertainties associated with an
epitaxial process or heat treatment,

the intention was to con-

centrate on the disordered structure one would normally expeel from the process described.
the process really gives a
like structure,

In order to confirm that

highly disordered or amorphous-

an electron transmission diffraction pattern was

obtained from an evaporated film.
first preparing a
surface.

This was accomplished by

formvar film replica of a

smoothstrate

Formvar films were stripped from the surface and

suspended across small washers that would later fit into the
diffraction stage of the electron microscope.
was evaporated on the

The Ti0

formvar films with essentia ll y

procedure used for making the regular samples,
film thickness in this case was approximately

500

film

the same

except the

The

formvar was subsequently dissolved away leaving bare films of
Ti0 2

suspended across the washers.

lion pattern obtained from such a
ill-3a.

transmission diffrac-

sample is shown in Fig.

The pattern is characteristic of an amorphous-like

structure.

For comparison a

transmission pattern was also

made of finely pulverized rutile suspended in a
and is shown in Fig. ill-3b.

formvar film

The diffuse rings are the

Debye

rings which arise from the composite effect of the random
orientations of the crystallites of the

rutile powder.

The

rings

Figure III -3.
Electron transmission
diff r action patte r ns: (a ) evapo r ated
TiOz film, (b ) ruti l e powder
suspended in formvar film

49
are very diffuse because of the small size of the crystallites.
The maxima of these

rings occur at the same radial positions

as inflections in the intensity of the pattern shown in Fig.
for evaporated films.

ill-3a

Although this is not evident in the figure,

this was carefully checked with microdensitometer traces taken
This relationship

directly from the photographic plates.

suggests that the short range order in the evaporated films
is similar to that of

the

rutile structure.

The energy gap

measured from anodized films had already suggested that
such a

correspondence exists.

Measurements of Evaporated Ti0
I.

Thickness Measurements.

Films
precise know-

ledge of the thickness of the oxide film is clearly desirable.
In contrast to the anodized films,

the evaporated films can be

measured directly by an optical interference method,
they are not too thin.

Unfortunately,

providing

the range of thicknesses

of greatest interest in this investigation falls at the lower limit
in sensitivity of the interference method.

One reason for con-

centrating on this range of thickness is due to the fact discussed later that the thicker films exhibited greater electrical
forming tendencies which we clearly wish to avoid.
important reason stems from the

Another

ideas discussed in Chapter .II

involving the influence of ionic space charge layers adjacent

so
to each contact.

It is of immediate interest to investigate

these ideas in detail,
fluences due to a

avoiding if possible any i mportant in-

relatively wide oxide interior and corre-

spondingly other possible limiting conduction mechanisms such
as described by Mead

(29)

In spite of this constraint it was

at least possible to obtain an estimate of the thickness of
thin films by the interference method.
imens were prepared by a

the

few special spec-

modification of the process des-

cribed in Section rnA in order to permit a

means of making

thickness measurements as well as photoelectric measurements to be described later in this section.
thickness was evaporated on a
case,

film .

single oxide

cover g lass substrate in this

and in addition to the samp l e

of metal

contacts,

wider strip

was evaporated over an entire edge of the oxide

No Al 2 o 3

overcoat was deposited on these specimens .

The thickness measurement was made by multiple beam interterence,

using a

thallium

li g ht source

( 53SOA)

and a

silvered microscope slide for the comparison plate.

halfBecause

of some scratching of the spec im en surface during this
measurement,
measurements.

it was performed after completing al l other
The resolution of this method was limited by

surface irregularities to about 100A.
The thickness of insulating films

commonly

51
calculated from measurements of capacitance and area,
ing a

dielectric constant K

form of the material.

assum-

equal to that of the bulk ceramic

One difficulty in this method arises in

assuming the bu lk value of K

to apply to thin films,

particularly

in this case vvhere the accepted va l ue for ceramic rutile
about 100 and there is no guarant ee
cifically vvith a

rutile structure

that vve are dealing spe-

(over short ran ge)

rather than

some combination vvith other structural modifications
or brookite)

vvhich have considerab l y

(anatase

lovver values of )'(.

The other difficulty involves the assumption of a

good dielec-

tric vvhich in gene r al is not valid vvhen high impurity concen!rations are present such as postulated in Chapter IT.
difficulty is evident from

This

the large variations of capacitance vvith

frequency and temperature observed in these films as decribed
later in this section,
quite arbitrary.
of this method,

vvhich c l early makes such a

Hovvever,

because of the obvious advantage

measurements vvere designed vvhich attempt

to overcome these difficulties.

The capacitance,

measured by balancing the initial

voltage step,

to a

ca l culation

should lead in the

vvhen

RC transcient follovving a
limit of very short times

value characteristic of an ideal plate capacitor.

vvould require times short compared vvith the
of the electronic charge

in the oxide film.

This

relaxation time

Hovvever,

52
evidence will be given which indicates that the
process is characterized by a

wide

extending also to very short times,

relaxation

range of time constants
in general conforming

If we can assume

with the ideas discussed in Chapter IT.

that our measurement precedes most of the

relaxation process,

it may still be acceptable for our purpose .

This measurement

is considerably improved at lower temperatures where the
relaxation times are longer and therefore all such measurements were made at 77 ° K.

The capacitance thus obtained

will be referred to as C 0 •

The test arrangement used for

these and other measurements is described later in this
section .
The dielectric constant of the film was still
required in order to calculate the thickness of samples from
the capacitance and area measurements.

This was obtained

by preparing a

thick oxide film specimen with several

area contacts.

The thickness was measured by the optica l

interference method to be 3020

± roo A.

large

From measure-

ments of capacitance as just discussed and the contact areas,
the mean value obtained was l< =
value for "><,

27.5

± 2. 0.

Using this

the thicknesses calculated from the special

samples are compared with those es t imated from the int erference method as listed by the following:

53
SPECIMEN

(CALCULATED)

( ES TIMAT E D)

± 100 A

204 A

160

100

<100

194

140

± 100

457

300

± 100

The above results indicate no se rio us in consist encies a lthou gh
there appea r s
due t o

to be a

e ither a

lowe r

systematic difference.

Thi s

va lue of X

films or a

systematic e rror in th e

interference m e thod.

uncertainties are too l arge in the
va lu e

of x

Therefore

determ ination s

thi s

as j udged by the

be tter

was used f or thickness

of all other spec imens .

with t he different film

In any case t he

thin films to justify a

va lue

va lu es calculated were found t o

va r y

In these cases,
in a

the

consisten t manner

thicknesses depos ited on the substra t e ,

re l ative evaporation tim es.

Opt i ca l Measurements.

used for obtaining
tive index n

for thinne r

cou ld be

was a l s o

The same specimen

used for measuring th e

refrac-

and t he spec tral transmission character i s ti cs of

the oxide film.

In order to measure n,

film was deposited under th e

s tri p

of aluminum

oxide film with its

width ex tend -

ing beyond an edge and another s trip deposited over the
film in a
e d ge .

perpendicular sense with its

oxide

len g th over l apping the

The overlying s trip of aluminum film permitted the

54
measurement of th e
measu rin g

oxide

fi lm

thickn ess in t he usua l manner,

the shift which occurs in th e

At the adjacent r egion of the edge

a t the oxide film edge.

without the overlying a luminum s trip,
ference

interference fringes

sh ift of the inter-

frin ges occurs because of t he phase difference be -

tween th e

light rays which pass throu gh the oxide film befo r e

reflection at the aluminum surface compa r e d with the ra ys
which a re

r e fl ec ted directl y

the edge of th e

oxide

film.

sh i fts are sufficient for a

from

th e

Measu rement s

calculat ion of th e

smal l correction for th e

of the two fringe

refractive index .

diffe r ences in phase shift at th e

a l uminum su r face was also include d
result obtained is n

a l uminum surface beyond

1. 95 ±

in th e

(at 5350

calcu l ation .

Th e

A).

The spec tral transm i ss ion c h aracteristics were
obta ined from an area of bare oxide film on the g l ass
substrate.

The measurements were made with the same

apparatus described in Sec ti on l lB
response.

At each w ave length t he transmission was

measu r ed thr ough the
a l so throu g h

oxide film and g l ass subs t rate and

clear portion of the g l ass substra t e,

ratio being taken as approximate l y
film .

for measuring the photo-

th e

This neglects the effect of th e

be twee n

th e

optical density

th e
of the

differences in dispe r sion

two media on the r e l a ti ve change i n

the

r efl ectio n

55
Hovvever this becomes important only near the

coefficient .

natural aborption frequencies of the media.

Since the ab-

sorption frequency of glass occurs in the ultraviolet,

the

result should be applicable up to the first absorption edge
of the oxide film.

The transmission curve obtained in this

vvay is plotted in Fig.
is indicated by a
introduced by

ill-4.

The

transmission above 4 ev

dashed line because of the uncertainty

the glass in this range.

The maxima and

minima occuring in the transmission curve belovv 4 ev arise
from multip l e

reflections vvithin the film.

This permits an

independent calcu l ation of n as vve ll as g i ving its dispersive
character.

The calcu l ations are made using the familiar

(minima) ,

vvhere m

A /2d

relations,

internal reflections,
ness .

Fig .

(maxima)

and n

= ( m + 1/2) A /2d

corresponds to the integral number of

A is the vvavelength and d,

the film

thick-

ill-S includes the results of these calculations

and the value obtained above at 5350 A.

The agreement is

vvell vvithin the experimental uncertainty.

In the same figure
( 30)

are plotted the dispersion relations reported by

Devore

for rutile.

for the oxide

film

Apart from the smaller values of n

the dispersive character is quite similar to that of rutile,

indicating approximately the same natural absorption frequency .
This frequency apparent l y

corresponds to a

higher energy

a::

:1:

!::::

20)

Figure III-4.

llv, ev

_)\

\~!

---

film

- - --

,..,..,•

,·""

Transmission through Ti0

r\ I1\\)!r
30~0A THICK

EVAPORATED
TiO 2 FILM

(J1

3.4

3.2

3.0

~UTILE

2.6

o PHASE SHIFT METHOD

1\ ~

2.8

C::·

0 TRANSMISSION METHOD

c -DIRECTION

RUTILE a -DIRECTION

2.4

2.2

2.0

1.8

0.2

0.4

Figure III-5.

----

0.6

Ti0 2 FILM
..()..

0.8

Refractive index of Ti0

1.0

film

1.2

58
tran s istion 1
bands 1

for example 1

o- 2 ( 2p) and Ti+ 3 ( 4p)

between the

rather than to the smaller energy gap of 3

(See

Appendix A) •
3.

Photoelectric Measurements.

The photo-

response was measured as described in Section IIB.

The

results obtained from specimen # 16

are

given in Fig.

ill-6.

Th e

taken on different days.
gap 3.15-3.2 ev i s

having Au contacts

two curves represen t measurements
The extrapolated value of the energy

somewhat lar ge r

than the value of 3. OS ev

measured for both the anodized film and rutile.

This difference

may be attributed partly to the greater uncertainty in the
extrapo lation but more probably to a

systematic error that can

be assoc iat ed with the l arger s lit openings in the monochrometer that were

required for th ese measurements.

l atter would tend to shift the points in th e

The

direction of the

rapidly increasing response and could account for the difference.

The background level can be ignored in the extrap-

o lation from large response l eve l s
negligible effect in the R
barri er height

since in this case it has a

dependences.

is indicated by

Th e

go ld contact

the I. 45 ev extrapo lations 1

good agreement with the value I. 4 ev obtained from the anodized samp le and rutile
Th e

(see Appendix B).

photoresponse was mea su red from a

sample

,..,
'Q::

Figure III- 6.

145

---~

--::#

II v, ev

Photo-response of specimen No . 16 (Au contacts)

/;
.iL

/.~

~~~;::;q-.; ::X--l!- '/

SPECIMEN 16 ( 194 A)
Au CONTACTS

(Jl

60
with AI contacts from which extensive electrical measurements were also obtained,
sample had a

as described later.

thick protective overcoat of Al

expected to influence the

results.

ill-7 were obtained at

and

to th e

outer contact) .

±. 5

Although this

o3,

this is not

The results given in Fig.
vo lt s

bias

(polarity refers

much larger photo-response is

obtained from the contacts in this case which nearly masks
the effect of the band gap,
in the

response near 3

Section illC,
be p

the oxide

indi cated by only sma ll indentations

ev.

For reasons to be discussed in

films with AI

type and thus the

contacts are believed to

response below 3

ev can be inter-

preted in terms of holes excited into the valence band .

The

much larger response from the contacts in this case is consistent with the genera l belief that a
band exists compared with a

relatively broad valence

narrow conduction band

(see

Appendix A).
The large differences with bias can be interpreted in terms of different barrier heights

two straight lines

(dashed lines)

corresponding to th e

the foot of the curve
value of R

1/3

at oppo-

The zero bias curve can be decomposed into

site contacts.

(for holes)

by subtracting the values of

straight line that extrapolates from

from th e

total

and replotting the

similar separation of the

new

±. 5 volt curves

"''

-4

Figure Ill-7.

AI CONTACTS

hv, ev

L/
~·

/"O BIAS

,..-~.
--...._

...---·-·-·

Photo-response of sample No. 21-C4 (Al contacts)

SAMPLE C·4{ll ).)

SPECIMEN 21

62
is accomp li s h ed by subtra c tin g

the va l ue at the

lines)

Th e

see n

from each t ota l curve.
to

give approximate l y

zero ~ i as curve

(dashed
is then

give approximately the same two extrapola ti ons as
The barri er heights so determined

ob tained for each polarity .

and I . 7

foot

I. 4

I. 8 ev for th e

I. 45 ev for th e

under l y ing con tact

outer con t act.

The response above

ev exhibits broad maxima th at occur a t abou t t he same

energ ie s

observed for rutile
4.

(see Appendix B) •

Electrical Measurements.

connections were made by so l de rin g
in dium to the exposed ends o f th e
Using minimum lengths of th ese
soldered t o

heavier l eads a

emf ' s

5 m il copper l eads with

evaporated metal strips .

l eads,

they in turn were

short distance from the sample.

The heavier leads were permanen tl y
inals of a

The electrical

shie lded enclosure.

connected to the term-

In orde r

to minimize thermal

during l ow temperature measurements,

the leads were

carefully se l ected as matched pairs and arranged in a
metrica l manner.
enclosure was a

sym-

In some o f the earli er measurements the
shielded dewar so that the samp l e

be immersed under l iqu i d

could

The temperature dependence

was obtained in this case by a ll owing th e

li quid N

to evap-

orate and observi ng the temperature increase by means of

thermocouple a tt ached to the specimen .

gentle flow

63
of dry N

gas was fed into the dewar during this time to

prevent condensation of moisture.

The latter effect was not

completely eliminated at all times and would then result in
erroneous leakage currents across the insulation.

more satisfactory arrangement,

later measurements,
system.
sink,

used in all

enclosed the specimen inside a

The specimen was clamped against a

vacuum

copper heat

using silicone grease to improve the thermal contact.

One thermocouple was soldered with indium at the surface
of the specimen and another monitor thermocouple attached
to the heat sink.
maintained at a

Th e

temperature of the specimen

was

constant temperature between 76°K and

296 °K by controlling a

flow of liquid nitrogen or its cold

v apor through the heat sink.

Connecting leads from six

samples with two common leads were fed through an octal
seal in the vacuum base plate to a

switching box located

just below, Connections to the test apparatus was accomplished
with matched pairs of coaxial

cables from the two output

terminals of the switching box.

The leads from the samples

were sufficiciently short to permit transient measurements to
less than I

f sec and care was taken to insure negligible

leakage in the connections

< 10- 14 amps).

The evaporated fi lms were typically much less

64
sensi ti ve to the e l ectri ca l fo rmin g effects observed with ano dized films.

These e ffects would appea r

app li ed fi e lds 1

becom in g

and with thicker films.

mo r e

only at l arge

import ant at higher temperatures

Both AI a nd Au contacts were t ested 1

however samp l es prepared wi th AI contact s
better s t ability
contacts.

exhibited much

and r ep roduci b ili ty th an th ose prepared wi th Au

For this reason 1

most of th e

following results are

l imi t ed t o AI contacts .
The de 1-V
(#IS)

At room temperature the samp les were ohmic with

breakdown

( Bd)

occurring cons i s tentl y

of a

few tenths of a

thin specimen

with Au contacts are typified by the results given in Fig.

ill-S.

charac ter i s ti cs of a

vo l t.

At 77°K

a t voltages the order

(immersed under liquid

higher ohmic resistance results 1

relatively high vo lt age whe r e
power function of vo lt age

(""v

it sudden l y
14

persisting up to a
assumes a

h i gh

The intersection of the

extrapolated ohmic and h i gh voltage dependences occurs at
I. 4

volts 1

t he va l ue measured for the ba rri er height .

initial current 1

(see Section ITB)

at approx ima t e l y

the point of breakdown.

intersect s

s l ow,

approxima t e l y

with temperature to about 170 °K 1

the de current

The temperature

dependence of the current measured from a
vo lt bias exhib it ed a

The

sample,

•5

parabolic change

then in creasing more

SPECIMEN 15 (100 A)
Au CONTACTS

·1

I 0 AT 77 °K /

Bd
~"

~· °K
SAM/

.l/

/77°K-

~/
v_;
v-

9 - r- 1.4v ~

/./
0.01

0.1

VOLTAGE, v
Figure III-8.

I-V characteristics of specimen No. 15

66
rapidly vvith temperature before breakdovvn occurred belovv
room temperature.
The de 1- V
( #17)

vvith AI contacts i s

ID-9.

Thick specime n s

characteristics of a
typified by the

thick spec imen

results given in Fig.

such as this exhibited electrical

forming tendencies at room temperature and therefore the
room temperature data in this case vvere obtained vvith voltage
pulses by measuring the current asmptote on an oscilloscope,
as described for measuring 11 in Section liB.
de measurement vvas used at 77°K

standard

(immersed under liquid

Breakdovvn at 77°K vvas again observed to occur
at approximately 10 •

The initial current I

could not be

accurately measured at room temperature because of the
shorter relaxation times involved.
volt

exhibited a

The de current at

•5

temperature dependence accurately repre-

sented by exp(-1 0/kT)

from 77°K to 296°K,

After subjecting a
volt 100 cps for a

sample to ac forming at

fevv minutes at room temperature,

1.0

the

1-V characteristics vvould sh ift to higher vo lta ges,

vvith cor-

respondingly higher resistance at lovver vol ta ges.

In order

to obtain a

measurable current at lovver temperatures, a

volt bias vvas required.

1.0

The temperature dependence ob-

tained in this case could be represented by exp (-. 11/kT)

.'Bd

SPECIMEN 17 (454 A)
AI CONTACTS

.1'

v/t

I 0 AT 77°K/

.!I
1296

II~ v/1

i7°K

°K

SAMPLE_10

/ 1sd

sf/
··6

,''/

if
10-8
0.001

0.01

0 .1

VOLTAGE, v
Figure III-9.

I-V characteristics of specimen No. 17

68
be low 140 °K and exp (-. 23/kT)

at higher temperatures.

All subsequen t r esu lt s

were obtained from the

standard spec im ens measured under vac u um .
measurements we r e
The 1-V

made o n a

Keithley 6 108

T he 1-V
e lectromete r.

characteristics a t 296°K and 78°K obta in ed from

samp l es with va ri ous oxide film thi ck nesses are plotted in
Figs.

ill-10,

II,

and 12.

The l e tt ers A, B, C

present the success i ve l y
substrate .
fi l m

(A)

the samp l e

re-

thicker films depos ited on a

No reliable data w

of specimen #21.

uniform ity

a nd 0

specimen

ob tain ed fr om the thi nnest

Figs. ill-10 and II

of different samp les from the

illustrate

the

same oxide strip,

number r eferring to th e ir sequen tial order as

indexed from one end.

few abno rm a l samples had been

measured and are be li eved t o be r e l a t ed to fl aws that were
visib l e

under the

thicknesses a r e

phase con trast microscope.

The measured

seen to increase consistently with the

sequence of fil ms and the genera l shift in the 1- V

curves .

The measurements were not in genera l extended to higher
currents in order to comp l e te l y

avoid any forming effects .

For reasons discussed previ ous l y,
in more detai l the th i nner samp les .
I S are plotted comp l e te
temperatures,

it is desirable to examine
In Figs. ill-1 3,

14,

fam ili es of 1-V cu r ves at different

taken from

r epresentati ve samp l es of

and

/.0

SPECIMEN 21 o
STRIP 8(105 A)
SAMPLE I, 4 AND 7

/;'

/;'
296°K

/sl<»t<

lo-8
0 .001

0.01

0.1

VOLTAGE, v
Fi g ure III-1 0.

Comparison of I-V characteristics of No. 21-B

SPECIMEN 21
10- 11 - - - - STRIP C ( 114A}
SAMPLES I, 4 AND 7
STRIP D ( 125A)
SAMPLES I AND 4

10- L-----~------L-----~------~----~------~----~----~

0 .001

0 .01

0 .1

VOLTAGE, v
Figure III-ll.

Comparison of I-V characteristics of No. 21- C and D

I0- 3 1------+----~(\J

~ lo- 4 r----+-~~~~---~-----+------4---+---~~~-+----~

w 10-51-------+------4-------~-----+------4+----~~~---+----~

a:::
a:::

::>

(...)

I0- ~----~------~------L-----~------~------L-----~----~
10
0 .001

0.01

0.1

VOLTAGE, v
Figure Ill-12.

Comparison of I-V characteristics of
No. 20-A, B, C, and D

Jor------.------.-------.------.------,-------~----~----~

SAMPLE 21-84 (105A)

Jo-'r------+------~------4-------+-----~~--~~------~----~

Jo-3~-----+------~------4-------+-~~~~-----+------~----~

~ Jo-4~-----+------~------~~~~~~--~~-----+------~----~
0::

10-aL-----~-------L------~------~----~~----~-------L------~
0 .001

0.01

0.1

VOLTAGE, v
Figure III-13.

F a milies of I-V curves vs temperature
of sample No. 21-B4

lo- 1 ~-----+------4-----~~-----+------~------~~---+----~

SAMPLE 21-C4 (114A)

E lo-3~-----+------4-------~-----+------~~~--~-----+------4

0..

~- lo-•~-----+------4-------~-----+--~-+~~----r------+------4

a::
a::

(.)

10-~~-----+------4-------~~---+~~~~------~-----+------4

I0- 9 O~.O-O-I--~------O~.O-I-------~-----0~.1------~-------~----~----~IO

VOLTAGE, v
Figure Ill-14.

Families of I-V curves vs temperature
of sample No. 21-C4

SAMPLE 21-04 CI25A)

~ Jo-3r-------~----------r---------~---------+----------~~~~---------~---------~

~ IO~~------~----------r---------~---------+----~~~~----4---------~------~

0:::
0:::

~ Jo-5r-------~----------r---------~~~---+-~~~~-------4---------~---------~

0 .01

0 .1

VOLTAGE, v
Figure III- 15.

Families of I-V curves vs temperature of sample
No. 21-D4.

SAMPLE 21-C4{114A)

v!
//

1/;//
1-V?

C\1

o..lo-4

~~/

296°K

0::
0::

v·~

w 10-!5

:::>

10- 6

v>t'

~~~s·K

~I
/~/

v-x/
10- 9
0.001

0.01

0.1

VOLTAGE, v
Figure III-16.

Rectification characteristics of sample No. 21- C4.

SAMPLE 21-04(125A)

;jr;/

///;

o. lo-4

a::
a::

+/; ~//

10-~

/.#

::::>

y>4""

(.)

V_, v?
+J

I0-8

~·

x:/

""
10-9
0.001

0.01

0.1

VOLTAGE, v
Figure III-17.

Rectification characteris tics of sample No. 21-D4.

77
specimen #21,

strips B,

and 0

respectively.

The above results were obtained with negative
polarity applied to the outer AI contact.
were AI,

Although both contacts

differences encountered at each contact during the

fabrication process result

in some rectification.

As this

became of greater interest during the subsequent analysis

few months after the above measurements were made,

some

of the measurements were repeated to include both polarities.
In Figs. ill-16 and 17,

both ·polarities are plotted at 78 °K and

296 °K for the same samples of C
No significant change from

and 0

(Specimen #21).

the original characteristics occurred

during the intervening time.
Measurements of the capitance as a

function of

frequency and temperature were obtained using a

Boonton 75C

capacitance bridge for frequencies between 5 Kc and 500 Kc
and a

specially designed bridge for lower frequencies.

The

results obtained from the same two samples used for the
previous data are given in Figs. ill-18 and 19.

The dashed

curve indicates the cut-off effect predicted by a

1000 ohm

resistance in series with the limiting capacitance C
from

the pulse measurement.

obtained

The 1000 ohm resistance

repre-

sents the approximate value measured for the AI thin film
This rather high resistance was a

consequence of the

leads.

800

·-.

296°K

700

-........

'-........ ..............

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

600

500

Ill

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

"""

Nr,
---·- "

·-......... 160 -·--

.... ___ '·

-·~

-----·- -:·\"."""' ~\

---- f----- -----

Co-

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

208 - · ..............

78 ·-.......

400

- -·-·--

300

..... ,

o-f~j

1--

~,,
'\
,.~

200

\\.,
SAMPLE 21 - C4
AREA= 1.7 · 10-4 cm2

100
10
FREQUENCY, cps
Figure III-18.

Capacitance vs frequency and temperature of sample
No. 2l-C4.

800

700

600

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

.........

296° K

""· .................

500

-....._ ....... -.

400

208 ...............

Ill ·......_

........

300

I-

<{
a..
<{

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

78 ...... ..........

Co-

200

""""' .........

160 .............

.........

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

-............ -.........._

.......

' ·,

-. \

........... ' ~\
..........

\\
_______
.,
---- ---~~
_,~_Z
Co

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

SAMPLE 21-04
AREA = 2 .0 · I0- 4 cm 2

\\.

100
10
FREQUENCY, cps
Figur e lll - 19 .

Capacitan ce vs fr e que n c y and t e mp e r a tur e of sample
No . 21-D4 .

80
anticipated photo-response measurements r equir in g AI
about 500

A thick.

films

The ac conductance was also obtained

during the above measurements and i s
samples in Fig. ill-20.

given for the same

The conductance measured at other

intermediate temperatures could not be conveniently included
in the figures,

but li e

smoothly between the plotted curves.

The results are again compared with the effect of C 0
series with 1000 ohm
assymptotical ly

(dashed curves).

reaches th e

Figs. ill-14 and IS.

The 296 °K curve

same de va lue as obtained from

The conductance at

78°K

could not be

measured a t lower frequencies but presumably also approaches
the de limit at a

much l ower value.

The capacitance was observed to also vary
with de vo ltage.

In these samples

th e

varia tio n was small-

er at low vo ltages than observed for anodized samples .
Since some e l ectr ica l forming and eventually breakdown
occurred at the highest voltages,

the measurements were

re s tricted to o th er than the se l ected samples.
gives the result for a

representative samp l e

# 21-C measured at SKc on the
C .

Fig. ill-21
from specimen

Boonton bridge at 78°K .

Discussion of Resu lts

on Evaporated Films

It would be difficult to reconcile the differences
between AI and Au contac ts simp ly on th e

basis of

SPECIMEN 21
SAMPLES:
C4 (AREA= 1.7 • 10-4cm2)
X 04 (AREA= 2.0 •lo-4 cm2)

.c.

10-6

::::>

10- 7

I;
10-8

-1~
Co

I;
I;

10-lo~~~------~------~~----~------~------~------~
10

Figure III-20.

10 2
103
FREQUENCY, cps

Conductance vs frequency and temperature for
No. 2l-C4 and D4.

600

500

)..

~""

78° K

400

(.)

I{)

.:M:.

"a..I'

300

SAMPLE 21-CI
AREA= 2 .5 x 10- cm2

(.)

~ 200

(.)

100
0.2

0.4

0 .6

0.8

1.0

VOLTAGE, de v
Figure III- 21.

Capacitance vs de voltage

1.2

1.4 1.6 1.8 2.0

83
corresponding differences in barrier heights.

Upon com-

paring the effect of the different contacts on two samples with
similar oxide film thickness

(Fig. ill-8 and 10),

one notes

major differences in the 1-V dependences although the room
temperature conductance is nearly identical.

Also there is

the very appreciable improvement in the stability of the samples
having AI contacts · compared with those with Au contacts.
The only difference other than the barrier heights that could
conceivably be important is the presence of the corresponding
Au or AI impurities in the oxide.
process,

During the evaporation

with its associated heat of condensation,

one can

easily imagine an appreciable diffusion of the metal

into the

thin oxide films.

It is believed that the presence of AI converts
the normally n-type oxide into p-type.
act as an acceptor in rutile ( 16 ) ,

Alumi num is known to

with the A1

ions occupying

.4+ .
T 1
10n s1tes, yet p-type rutile is rarely formed because of
the strong opposing tendency of reduction that occurs during
the

normal crystal growth or diffusion process.

is also commonly used as a
rutile.

Aluminum

stabilizing agent in insulating

There has been evidence for the existance of p-type

rutile where iron was the known impurity(Z?, 3 I),
ions act as acceptors in the same way as AI

where Fe 3 +
In our case

84
the evaporation process may favor the formation of an excess
of AI acceptors.

One can imagine the AI diffusing through

the oxide during the deposition process,

compensating for

oxygen vacancies as well as continually reacting at the surface
with the oxygen vapor and generating additional acceptors.
On the other hand,

the Au impurities are likely to introduce

deep trapping states without affecting the

large excess donor

concentration introduced by oxygen vacancies.
In both cases we may assume that a

large excess

concentration of either donors or acceptors results
compensation would be very unlikely) ,

(exact

and the space charge

model discussed in Section IIC would only be changed with
respect to the type of carrier

(electron or hole)

volved in the conduction process.
valence bands were equivalent

that is in-

If the conduction and

(equal effective masses) ,

then

clearly no significant differences would result that could not
be interpreted in terms of differences in barrier heights.
However,

it is g enerally accepted that Ti0 2

a nd single-crystal rutile)
due to a

has a

small overlap of the Ti

ingly possesses a
Appendix A).

(both ceramic

very narrow conduction band

+ 3d states and correspond-

large effective mass for electrons

The valence band representing the

states is expected to be quite broad,

(see

o2-

with an effective mass

85
for holes probably similar to the free electron mass.

On the

basis of these differences we can account for the observed
results.
In the case of the very thin samples with Au
and AI contacts

(Figs.

dependence suggests a

ill-8 and 10},

the smaller temperature .

tunneling mechanism of conduction

especially at low temperatures.

. .

However,

the current from the famll1ar theory

( II }

if one calculates

usmg the values

measured for the oxide thickness and barrier height,
result is many orders of magnitude too small.

the

Even if we

allow for conceivably gross overestimates of the thickness
(or barrier height) ,

the observed 1- V

dependence

(-vn)

i-s

quite different than the nearly exponential dependence predicted
by this theory.

The space charge warping of the potential

discussed in Section IIC is capable of accounting for these
results as will be demonstrated in Chapter TIZ.

The differ-

ences between the Au and AI contacts can be qualitatively
understood by the

following.

In the case of Au contacts and at low temperatures,

the electrons tunneling through the first barrier

(adjacent to the negative contact)

are trapped by the narrow

conduction band in the oxide interior and may lose all their
excess energy,

reaching thermal equilibrium with the oxide.

86
Curre nt continuity must be sa tisfied by electrons tu nnel in g

out of the oxide throu g h

eve r

s ince

th e

su pply

with the oxide at th e

the second barrier.

of electrons in thermal equilibrium
dominant tunn e ling

less than in the metal contact,
appears across the
suppresse d
appea r s

to th e

secon d

energies i s

much

most of the applied vo lta ge

barrier until it i s

fermi-l eve l.

across th e

How-

n ear l y

Additional vo lta ge th en

fir st ba rrier and th e

creases much more rapidly.

current in-

The tran s ition vo ltage

therefore occurs at approximately the ba rri er he i gh t as
actua ll y
to

observed expe rim ent a ll y.

the transition to th e

Fowler-Nordheim region ( 32 ) for

an idea l r ectangu lar ba rri er ,
pe ndence in thi s
vo ltages th e
and th e

case

This also corresponds

howeve r

different.

the curren t de-

At su ffi c i en tl y

effect of the ba rri er s hape

d ependence shou ld app ro ac h

by Fig.

Thi s

form

a l so

results although it is not s hown exp licitl y

m-8.
In th e

tacts

l ess important

an exp(- 8 /V)

as in the ideal Fowl er-Nordh e im th eo r y .
agrees with th e

high

(Figs.

case of th e

ill-10 and II)

tunneling throu gh th e
excess ene rg y

thinner film s

with AI con -

and at low t emperatures ,

fir s t bar ri e r

holes

m ay not lo se a ll th e ir

in the broad valence band.

If th e suppl y

87
of holes at the dominant tunneling energies at the second
barrier is much larger than for equilibrium,

the current

assumes a

different form than that required for electron

tunneling.

This process which will be treated in Chapter

n:z:

predicts the more gradual variation of current with voltage
observed.
In thicker films with AI contacts
ill-9 and 12),

(see Figs.

the room temperature currents are seen to

exhibit quite different values that cannot be correlated with
the thickness.
(Fig.

ill-12)

This can be understood if specimen #20

has smaller effective barrier heights at the

contacts which encourages a
the higher temperatures.

thermionic type process at

At 78°K and at higher voltages

the different samples exhibit a

more consistent relation to

thickness that can be understood in terms of a
process at the contact barriers,

but where a

tunneling
larger fraction

of the applied voltage is required across the oxide interior
with increasing film thickness.

On the basis of our model

the holes tunneling into the oxide valence band which lose
their energy through collision processes

(phonon or impurity)

and become trapped before they can tunnel out through the
second barrier require an additional voltage across the oxide
interior in order to be released at the same rate and

88
maintain a

voltage dependence
thickness

The shape of the high

steady state condition.
("'Vn)

remains nearly constant with

(compare also with the thinner samples of Fig. ill-11

and supports the view that the primary dependence is governed by the barriers at the contacts.
(Fig.

ill-10),

er slope .

In the thinnest samples

the 1-V dependence exhibits a

somewhat small-

This can be understood if the oxide is too thin for

the barriers to extend to their normal effective distance from
each contact.

Alternately,

the effect of impurities from

the

contacts may become excessive in the oxide interior below
some critical thickness.
Additional

information on the influence of the

interior of thicker films was obtained from the experiments
described for specimen #17

(Fig. ill-9).

The activation

energies indicated by the temperature dependence are believed
to be associated with trapped holes.

After ac forming,

the

larger resistance at low voltage and the increase in the temperature dependence are believed to result
of some of the excess AI acceptor ions.
as Al

from the removal
This may occur

+ ions are induced by the applied ac field to move out

of more stable sites in the oxide,

permitting them to diffuse

with the assistance of the built-in field to the contacts where
they precipitate.

The smaller activation energy E

aor

.1 ev

89
may be associated with the AI
either E

or 2E

acceptor levels E A,

depending on

the

equaling

degree of impurity compen( loc. cit. , pp I 56-160).

sation as discussed by Molt and Gurney

The former applies when the density of excited holes is much
less than the density of compensating donors,
when the reverse is true.

• 23

ev,

and the

latter

The larger activation energy

observed above 140 ° K

after forming may be

associated with deeper acceptor states.

The reduced

acceptor concentration would result in less ionic space charge
and consequently longer tails in the barriers extending from
the

contacts.

Therefore more of the shallow acceptor states

would be depleted of holes,

allowing the deeper states to

dominate ?I the higher temperature·s.

These energies are

.much too large to result from the usual hydrogenenic model
of acceptor levels

(c.!. 01 ev).

However they may be under-

stood in terms of the large polarization energies associated
with the

release of trapped holes from

self trapping) •

Freder1ckse

( 17)

the oxide

(polaron

This argument has been used by
to explain the

in rutile at higher temperatures

large donor energies observed
(typically

= . IS ev).

Without dwelling further on the effect of the interior
of the thicker oxide films,
effects of the contacts.

we wish to concentrate only on the

This can be conveniently examined

90
from the

results on the thin film samples of. specimen #21.

mentioned above,

the results from film B

of this specimen in-

dicates that the oxide is too thin to be characteristic of the
normal barriers at the contacts.

and 0,

The slightly thicker films,

appear to be ideal for our purpose.

These films,

represented by the family of 1-V curves given in Figs. ill-13
and 14 exhibit an appreciable temperature dependence at low
voltages that cannot be attributed to a
However,

pure tunneling process.

neither can this temperature dependence be de-

scribed in terms of distinct activation energies.

In Chapter

r:sz it will be shown how the effect of the ionic space charge
and its associated barrier shape encourages a

thermally

assisted tunneling process that predicts precisely this kind
of behavior.
The results obtained from measurements of
capacitance and conductance as a
temperature

(Figs.

function of frequenc y

ill-IS through 20)

on the basis of the proposed model.

and

can also be interpreted
The large variation of

capacitance with temperature would be very difficult to explain
on any other basis.

For example,

the dependence is much

too large to be accounted for by the small temperature
dependence of the dielectric constant.

The theory derived in

Chapter ISZ predicts an effective barrier width that varies with

91
temperature,

and the corresponding variation of capacitance

with temperature will be shown to agree with these results.
The sharp cut-off in the frequency dependence has already
been attri buted to the series resistance of the leads.

The

frequency dependence of capacitance below its cut-off,

can

be interpreted on the basis of a

redistribution of space charge

in the oxide due to trapped holes in the manner discussed

in Section IIC for trapped electrons in anodized films.
is not possible to use the usual simple model (
ized by a

character-

single relaxation time for the trapped holes,

accurately describe the observed dependence.

and

distribution

of relaxation times could be chosen that could provide such

description,

However,

but this would offer no useful advantage.

the implications resulting

from a

distribution of

relaxation times a re more consistent with our model when we
consider the variation of potential with position in the barriers.
Trapped holes located at different positions and thus at
different energies in the oxide will necessarily find it more
or less difficult to shift about under the varying field.
The dependence of conductance on frequency
and temperature serves to compliment the capacitance results.
The eventual

saturation at approximately 10-

mhos near I me

is again attributed to the effect of the series resistance.

92
lower frequencies 1

the increase cannot be completely accounted

for by the effect of the series resistance but can be interpreted on the basis of the corresponding ac conductance of trapped
In Fig.

holes.
and a

ill-20 an inflection appears near 2

kc at 78°K

corresponding inflection can be detected for the capac-

itance of the sample in Fig. ill-18.
in the other sample

(Fig.

This is not as evident
This can arise if a

ffi-19).

prominent relaxation time exists for the trapped holes.
the basis of the idealized model that assumes a

single relax-

ation time we can construct an equivalent circuit such as
shown in Fig.

ill-22.

The idealized trapping effect is given

by the circuit shown in solid lines within the dashed rectangle

(c. f.

Ref. 33).

Also indicated in phantom lines within the

rectangle is the possible extension of the idealized circuit
to include a

distributed RC network for more accurately

describing the sample.

>> R2

Below cut-off

(l.U« 1/RsCo) and for

the admittance of the circuit becomes

+ (wR 1cSJ../R,
Y- I/R:2I + (wR,CJ2..
where the

relaxation time is R

is tic frequency

(W= 1/R C ~ 2 tr· 2 kc)
1 1

increases from the de value of I/R
it saturates at 1/R

2 1

Below the characterthe effective

conductance

and above this frequency

(when neglecting Rs) •

Correspondingly

r- -

L -

--J<.,A./'.,- -

---Av"./'.,- -

....J

L---n---"'1

""'

A~
vv

I I
I I

L - - - - - - - - _j
,...

I I
I I

Figure III-22.

""'

Equivalent circuit of sample.

94
the effective capacitance decreases from C
this frequency.

to C 0

above

This is in qualitative agreement with the

observed dependences,

but clearly a

more complicated dis-

tributed RC network is required for an accurate representation.

The pulse measurements for C

sistent with the ac measurements,

are seen to be con-

with the comparison

offering some reassurance that the limiting value was nearly
reached.

ISZ".
A.

THEORY

The Barrier Potential
I.

Impurity-Space-Charge Models.

In Chapter

n we discussed the influence of a large excess of donors in
the oxide film

( n-type)

on the potential shape of the barrier.

An equivalent problem is presented in the case of an excess
acceptor concentration

( p-t y pe)

as discussed in Chapter

when the potential is correspondingly taken for holes.

matter of convenience,

the following treatment will refer

specifically to the p-type case.
In order to obtain the barrier potential U
holes),

(for

we start as usual from Poisson's equation in one

dimension,

(1)

where in this case

is defined as the net negative charge

density which in g eneral is s ome function of x,
problem there fore requires a

knowledg e

f (x).

The

(=> (x) which in

turn depends on the di s tribution of ionized impurities.
Sing le ac·ceptor level with uniform distribution
a long x,
theory

Consider the familiar case in semiconductor

which deals with a

single acceptor level close to the

96
valence band and the acceptors uniformly distributed in the
semiconductor with density N A.

Let the potential"'< be

measured from the Fermi level at the adjacent metal contact,
and let U

=--r; at the limit of the space-charge layer

(x = d).

Beyond this layer the positive charge of holes

balances the negative charge of acceptors

( ~ =0).

In the

space-charge layer the density of holes decreases as
exp [ - ( U+

"l) /kT] and becomes negligible when

· ( U+~) /kT ~ I (depletion of holes) •

Thus,

for U

one can use the approximation

( 2)

(constant)

The barrier potential may be evaluated for t he general case
in which there may be an applied field if we let 'Y/
the voltage drop V

include

across the space-charge layer;

that is,

( 3)
where

4 o represents the equilibrium value of

'1. .

will neglect any resistive effect of the oxide interior beyond
the space-charge layer and therefore consider the field in
in this region to be zero.

The bound a ry conditions

>:eft is convenient to use potential and energy interchangeably
with the understanding that electron volts ( ev) is the unit of
energy.
Thus U may also be understood to represent the
energy of holes at the valence band edge measured from the
Fermi energy at the contact.

97
are then

u (0) =
U(d)

= -"'(

(~~)X =d

( 4)

where W

is the metal-to-oxide work function at the

contact.

Equation

(I)

the above conditions,

is readily integrated subject to
yielding

U+1'( =

( 5)

and

(6 )

similar expression is correspondingly obtained for the

opposite contact.

Clearly the film thickness must exceed

2d in this case.

Barriers of this type have been treated

extensively ( 34 )

for more moderate impurity concentrations

than considered here,

particularly in connection with a

thermionic conduction process
Chapter

(Schottky emission,

c. f.

I).
Uniform energy distribution of impurity states.

It is unlikely that the above assumptions on
should apply to thin films.

For example,

we can consider

the effect of a

distribution of acceptor sta t es of density

N (E)

wide range of energies in the forbidden band.

over a

98
Then clearly

must increase with U

more states become ionized.
distribution of states,
N (E)

as correspondingly

Let us assume a

uniform

defined for convenience by

N A/W (constant) •

acceptor states at any x

The total density of ionized

in the barrier must then be

proportionate to the height of the valence band above the
quasi-Fermi level

( imref)

at this x.

The value of the

imref in turn depends on the steady-state transfer of
trapped holes in these deep lying states.

Since the trans-

fer of trapped holes becomes progressively easier as one
moves from the barrier maximum to the tail of the barrier,
the inref will correspondingly change more slowly 'with
increasing x,

asymptotically approaching its limiting value.

reasonable approximation is to assume the imref to be

constant throughout the space-charge layer,
introduced at small x

since the error

would only become appreciable at

very large applied fields.

The space-charge density is

then approximately

(7 )

Equation
conditions

(I)

may then be integrated subject to the bound a ry

U(O) = W

U(x-+co) -==-->(

u +-rz

( 8)

obtaining

-x/s
(9 )

where

(I 0)
Since a

similar expression also applies at the opposite

contact,

Eq ( 9)

might only be expected to apply when the

film thickness w
ness w

is several times S.

is accurately taken into account,

the contacts are assumed equivalent,

U+'Y{ Thus,

When the finite thick-

cosh ( ~ -

(W-r~)

for w/2s >I,

and for simplicity

one obtains

fi)

( II )

cosh(~)

Eq (II)

can be approximated by the

simpler expression of Eq ( 9)

applying at each contact,

with any small error at the center of the oxide film
absorbed in the value of

"/.o •

Single acceptor level with exponential
distribution along x.

If we assume a

single shallow

acceptor level with the density given by N A e -x/s,

measured from each contact independently,

we arrive at

essentially the same result given by Eq ( 9).

Such a

tribution of acceptors

(A I

ions)

can result from a

dis-

100
rate-determined diffusion process that occurs during the
formation of the films.

For example, consider Fick' s

diffusion law

dN
2J t

where 0

( 12)

is the diffusion coeffecient of the AI

If the rate of accum-

under the conditions of formation.
ulation of N

depends on some rate of reaction R

the AI and the available oxygen,
Eq ( 12)

in the oxide

we might write

between

aN=RN.

is then integrated to give

= NA e

where N ( 0)

-'YR/D x

is defined as N A

( 13)

to conform with the above.

This result is identical with the required form is we
simply let 1D!R =

The Pois so n-Boltzmann distribution.
Frenkel
Hucke! theor y

( 3 5)

has dtscussed how the Debye-

of electrolytes can also be applied to solids

when under thermal equilibrium with its surfaces
(or interfaces) •

The theory treats the influence of

coulombic interactions of ionized impurities under thermal
equilibrium which is expressed by the Poisson-Boltzmann
equation

(in one dimension),

101

sinh (U+!l)
= kT
kTe

( 14)

Debye length and T e

is the equilibrium

where L

is the

temperature.*

If we let L

d 2 U/dx 2 '::k(U+~)/s 2

larger values of U

the problem becomes

( U+ 'Y{) /kT

identical to the above when
becomes

[c.f.

(smaller x) ,

rapid decrease of U

with

< 1;

Eq(l)

that is,

and

(7)]. ·

Eq ( 14) predicts a

However,

Eq ( 14)
For

the boundary

conditions at the surfaces are different during the process
of formation and also the attainment of equilibrium is
doubtful.

In any case,

Eq ( 14)

serves to suggest still

another possible reason for anticipating barriers similar
to the exponential

form

given in Eq ( 9) •

For lack of more precise knowledge of ~ (x),
the subsequent analysis will compare the effect of the exponential barriers of Eq ( 9)
barrier of Eq ( 4).

with the usual space-charge

However,

the emphasis will be later

given to the exponential barriers,
leads to a

primarily because it

better correspondence between the theory and

experimental results.

*T e may refer to some high effective temperature during
formation of the film, with the resulting distribution
II frozen in 11 at lower temperatures.

102

The effect of

Image Force Correction.

image forces at the contacts should also be included in
describing the effective barrier potential.

The image force

correction is approximately

u. -

(I 5)

where )(0
( X:

= n

is the optical value of dielectric constant

2 )

and x

contac~.

is measured from

The effect from the

the corresponding

opposite contact can be

considered negligible for our case,

i.e.,

q/4x0 w<< W.]

The effective barrier is thus represented by the sum

u + u ..
The barrier height due to the image-force
lowering can be determined by evaluating
point x

where its derivative is zero.

+ U.) at the

The results

obtained for the assumed barriers are

( 16a)

and approximately

( 16b)
for the quadratic barrie~s of Eq ( 4) ;

and correspondingly,

( 17a)

103
and

w(t-z~)

( 17b)
The above approx-

for the exponential barriers of Eq ( 9) •
imations make use of the assumptions

x 0 /d, x 0 /s << 1 ,

which should be valid in most cases of interest.
The

Effective Barrier Approximations.

effective barrier potential
the form given above.

+ U.)
is awkward to use in

more useful approximate form

may be obtained as follows.

For

> x0

the barrier can

be nearly represented by the original expression for U
we

replace W

cp and translate the origin to x 0 •

< x0

contribution for

and represented by a
and U ( -x

may be added as a
parabolic potential

The

small correction

fitted at U ( 0)

¢>

The errors introduced by these two

approximations are in opposite directions and therefore tend
to cancel one another.

The effective potential

then becomes

for the · quadratic barrier of Eq ( 4),

¢ [I - (~Jz]

U-+7( -::::::: (¢+'Y()(I + ~l
where

and

,I..

't'

t:;:

yZ 'frq NA

><(¢+!£)

are given in Eq ( 16);

barrier of Eq ( 9) ,

for

x < 0

( 18a)

for

x ::> 0

( 18b)

( 18c)

and for the exponential

104

p [I - (~)2 J

'::::1

for

x< 0

( 19a)

for

X> 0

( 19b)

->t/s

U+"l_ ~

(¢+-rc) e

14;~NA

where

and

x 0 ,

( 19c)

are given in Eq ( 17) •
4.

Effect of Statistical Fluctuations.

One

might also consider the effect of fluctuations in the very
narrow barrier due to the discreet nature of the ionized
impurities.

This is ignored in the classical meaning of

E'(X) W~ich implies a continuous function of
given in Eqs(2)

and

(7).

Henisch

(34)

such as

has suggested

the possible limitations of this classical treatment in very
narrow barriers that are only a

few times thicker than

the average spacing of the ionized impurities.

continuous function of

However,

f(X) , and correspondingly U ( x) ,

clearly exists for describing the statistical average over
the entire macroscopic volume of the film.
properly take

If we were to

into account the effect of fluctuations from this

average at each increment of area,
ingly have to include the statistical
of encountering each fluctuation .
accounting is carried out,

we would

correspond-

frequency or probability
When this statistical

we find that the effect of

lOS

fluctuations in reducing the barrier tends to be offset by the
associated probabilities describing the frequency of their
occurrence.
One way of visualizing this is to consider the
limiting case where a

local

fluctuation results in the com-

plete eradication of the barrier.
example,

This limit can occur,

when acceptors are found aligned in a

chain at

each lattice site in the increment of area in question,
permitting a

lattice of normal sites of average spacing

the probability of not finding a

length

is given by Clausius'

We are fr e e

thus

metallic-like transfer of holes along the chain.

If we consider a
~,

for

:::

normal site in a

formula (

chain of

e - x/a

(20 )

to assign acceptor impurities to abnormal

sites of this chain if appropriate allowance is made in the
value defined for a.

Equation

( 20)

may then be interpreted

to represent the desired probability for finding an acceptor
chain.

It ma y

alternatel y

be interpreted simply as the prob-

ability of not encountering a
collisions)

normal lattice site

(no lattice

and in this sense is equivalent to the usual

probability predicted for quantum mechanical penetration of a
barrier.

The value of

1/a, which must take into account

the presence of the impurities

(e.g. ,

acceptors)

and the

106
contacts,

is then correspondingly equivalent to the usual

attenuation coefficient characterizing the. average barrier
in quantum theory.
The above argument serves to suggest a
similarity between the effect of the average barrier on its
penetrability and the effect of local fk1ctuations.

This sim-

ilarity is characteristic of the familiar "correspondence
principle 11

of quantum theory that states the equivalence

between a

quantum mechanical and classical description

of a

system represented in terms of macroscopic averages.

The difficulty in applying

this principle to our system

arises from the apparent failure of the macroscopic representation in one dimension although in reality we are still
dealing with and measuring a
dimensions) •

macroscopic system

(in three

The question th en arises on whether the

quantum mechanical effects predicted for our one-dimensional
microscopic model should then be valid for our macroscopic
system.

We can only imply that this is true from the

simple argument given above for lac k
On the other hand,
dence
view.

of a

rigorous proof.

we can turn to the experimental evi-

found in similar cases that provides support for this
For example,

the well-known theory of tunnel

diodes has very successfully used the classical description

107
of the barrier in predicting the experimental results,

even

though the barrier widths in this case may t ypically represent on the average

only two impurity separations.

therefore feel justified in using a

classical

representation

of the narrow space-charge barriers in our proposed
model for the Ti0
B.

films.

Transport Theory
General Equations.

1•

The current density

that crosses an arbitrary barrier along the x-direction may
be expressed in the general

::T

form :0:<

[ c. f.

Eq(l3)

of Ref. (II)J

-E/kT

P(C) In [ : ::-

( 21 )

-"?

where

(22)

is the familiar Richardson 1 s

coefficient,

P (E)

is the trans-

mission probability of the barrier at the energy E,
the applied voltage across the barrier,
imum energy allowed.

and - "'{ is the min-

The quantity A/kT times the log

term in the integrand is sometimes referred to as the supply
function and represents the difference in the flux of holes

>!thermal equilibrium at each side of the barrier, however the
effect of a non-equilibrium distribution of holes in the oxide
can be important and is treated in part 3 of thi s section.

108
(or electrons)

+ dE

along the x-direction in the

range E

incident from opposite sides of the barrier.
In considering the barrier at the first contact

in our model,

it is convenient to define E

level of the contact.

'1.

The energy -

the edge of the valence band in the
to the

limit of the space charge

at the Fermi

must represent

oxide,

corresponding

layer as defined in Section

An additional contribution to the
current may result from
band at lower energies.

impurity states in the forbidden
However this

11 impurity

conduction 11

may be considered to act in parallel with the above process
and should normally be negligible except at low voltages
The second barrier may be treated

and temperatures.

in an analogous manner by transforming the energy origin
to the Fermi l e vel of the second contact.
The transmission probability can be approximated by

P(E)

(23)

+ e G(E)

where

eCE)

j '(u(x)- dx ,

(24)

>'to

are the roots of U (x) -E=O,

and oC

is the factor

109

~ Sm* /11 assumed to be constant.

Equation

sents an extension of the familiar W. K. B.
for P(E)[i.e.,
and Good

(37)

P(E)=exp(-e)J

( 23)

repre-

approximation

and was used by Murphy

to describe the effect of energies near the

barrier maximum in the case of electron emission from a
metal into a

vacuum.

The expression is exact for an ideal

parabolic barrier and therefore represents a

good approx-

imation for energies in the vicinity or above the barrier
maximum .

At slightly lower energies

( E< ¢;) ,

P (E)

approaches the familiar form of exp ( -8).
One can obtain from Eq ( 21)

the expressions

which apply to both limiting cases discussed previously,
that is,

(I)

emission.
E~ 0;

pure tunnelling and
In case

(I)

( 2)

pure thermionic

the dominant energies occur near

thus 8(E)>> I and P(E)~exp(-8).

The currents

may then be eva luated following Stratton (II) by using the
expansion

+fE
When c 1 kT < 1 and f
can be neglected,
Eq ( 21)

2..

+-· ..

(25)

the integral of

converges and gives

- b, (

-c1

I- e

v;)

(26)

110
In case

( 2)

we may use the limits E/kT
P(E)

-::r

....-·o

>> 1, and

> ¢

< ¢

which leads to the familiar expression for the thermionic
current

-¢/kT(

A e

I - e

-V,/kT)

( 2 7)

In most barriers commonly considered,
above two limiting cases [ Eq ( 26)

( 2 7)

the

represent pract-

ical solutions and the intermediate case in which the dominant tunneling energies occur between 0

and

cp, would

only apply over very narrow ranges of voltage and temperature.
and

( 19)

However,
will

the space charge barriers of Eqs ( 18)

in general

require a

solution to the intermedi-

"pure 11

tunneling case of Eq ( 26).

ate case in addition to the
2.

Current for the Intermediate Case.

Since

this case applies when the dominant energies lie within
the range 0

< E < ¢ , we may assume
E/kT,

Equation

j =

e (E:) >> 1

( 21)

then reduces to

t'T

-V,/kT
-[8(£") -1- E/J.(t- e
)J"" e
dE •
-'1

roo

(28)

The integral may be evaluated by the familiar saddlepoint
approximation,

that is

by integrating about the energy

Ill
in which the bracketed function in the exponent is a
minimum.

The saddle-point energy E

is determined by

(29)

Expanding the function about E

where

8(E0 J + E'a/kT

fJ and

(lJ2_)

1- \d E

(30)

( 31)

one obtains the result

. _

A _,fjf

J - kTVV e

-(3 (

-\1, /kT)

/- e

The limiting condition for which Eq ( 32)
apply is E

c. f.

> 0.

Eqs ( 2 5)

(32)

can be expected to

This is equivalent to the condition
and

( 2 9)]

(33)
and therefore compliments the condition c

< 1 which

applies to the case of pure tunneling in Eq ( 26) •

It is useful to compare the results for the two
cases as c

kT-+ 1 •

From Eq ( 32)

- b, (

and from Eq(26),

I- e

one obtains

c,\It)

112

J ~ A [ ~;~(1rc,k T)

J e- (I - e-c, V,) .
b,

11 c kT /sin(rr c kT) from

We see that the divergent factor
Eq ( 26)

is replaced by the finite factor c

"/'fr /f 1 from Eq ( 32) •

This suggests the following extrapolation between the two
limiting cases for c

Since

kT ~ 1 :

--2

where

(c 1 kT)

-bI

( I- e

_ [ sin(1/c 1kT)

1Tc 1 kT

f / 'il c~

-c,V,)

1/~
1rc,

is normally small,

important in the vicinity of c

kT=1.

was confirmed by carrying out a

(34a)

]-/·

(34b)

its contribution is only
The validity of Eq ( 34)

numeral calculation using

the saddle-point method with the general expression for the
integrand in · Eq ( 21)

and assuming reasonable values of

Similarily Eq ( 32)
approximation for c 1 kT>1.

was found to represent a

good

( 32)

are

Equations

equal at their common limit c

and

kT=I.

The Two Barrier Problem.

proposed for the Ti0

effect of two barriers,
two metal contacts.

( 34)

The model

films requires that we consider the
one associated with each of the

In order to treat this problem we

must make certain assumptions on the steady s tate energy

113
distribution of holes
oxide.

(or electrons)

in the interior of the

We will describe the two limiting extremes one

might consider and then propose a

third intermediate pro-

cess which is in complete agreement with our results.
One extreme corresponds to the situation in
which the holes tunneling through the first barrier lose all
the energy acquired by the applied field to the oxide lattice.
The energy distribution of holes may then be described
as for thermal equilibrium except for the corresponding
change in the

(quasi)

fermi

level in the oxide.

The two

barriers then act as two independent impedances in
series and the applied voltage must divide in the appropriate manner between them.

In this case almost the

entire voltage will occur across the second barrier until
it is sufficiently suppressed by the applied voltage 1
the current is limited primarily by the
supply of holes in the oxide.

since

"quasi-equilibrium"

This was the situation pro-

posed in Chapter IT for electrons in the narrow conduction
band of n-type oxide films.

The problem must in general

deal with two simultaneous implicit equations representing
the equal currents through each barrier 1
of its portion of the applied voltage 1

each a

function

and the exact solution

must then be obtained by numerical or graphical means.

114
The other extreme corresponds to the situation
in which the holes tunneling through the first barrier retain
the energy acquired by the applied field during the time requi red to tunnel through the second barrier.

In this case

let us represent the supply functions of Eq ( 21)

ed functions Y
ive regions.

genera liz-

with subscripts added to denote the respectThe current through each barrier is then

written

where only y

and y

retain the usual thermal equilibrium

dependence at each contact,

that is,

In [ I -+ e -£/kT]

and

-(£+V)/kT]
y2. - AkT In [I -+ e
and Y

correspond s

to the ge n e raliz e d

supply function for

the oxide inte rior.

S ince the steady state condition re-

j [ Y, P, .- Y.,P.,_- Yo(P. +P,_)] dE
dO

-"2
In this case the integrand must vanish at all E

and we

115
then obtain

Y, P, + y2- R.

R + P2.

may then be eliminated from the current equations,

yielding

where V

e e, + e e,_
is the total applied voltage.

(35)

This result is directly

analogous to the familiar problem of quantum theory which
treats the transmission of a
barrier (

) •

particle through an ideal double

The result of Eq(35)

predicts a

rapid de-

crease in the contribution of the second barrier with
applied voltage.
We do not believe that either of the above
extremes applies in the case of holes entering the oxide.
However the holes are

likely to lose much of their energy

after entering the oxide without necessarily reaching thermal
In this case

equilibrium with the oxide.

let us assume that

the supply functions corresponding to the oxide interior in
the steady state can be represented l;>y

Yo = >.(E) kT In

-(E-t"Y()/kTJ

I -t- e

(36)

116
where the coefficient
function,

'A (E),

which modifies the equilibrium

varies relatively much more slowly with energy

over the range of interest,

kT ~~~

that is,

We may then evaluate the current through each barrier
about the respective saddle point energies as described
previously,

but obtained separately for the forward and

reverse components of current.

We then obtain

(37)

where we have stipulated the same

>. at each of the saddle

points corresponding to the reverse component at the

first

barrier and the forward component at the second barrier.
This is justified since these two saddle points will in general
differ only very slightly and

A (E)

sufficiently slowly in this range.

was assumed to vary

The continuity condition

j -i =0 then gives
1 2
(38)

117
We may then eliminate

).. from

jOI

and obtain

jo 1 j2.0

j,o jo2 -

Eq ( 37)

(39)

J02.

yt 0 ~ 0

In general if

the

intermediate case

( c 1 kT > 1) will always apply for both current components
and j

20 , as well as the reverse

component at the first barrier jO

and thus we must always

at the second barrier,

have

(V-V,)/kT
J2o

and either

AVJf«' e-(3,

for c

1 kT > 1

_A!;L
(c1 kT?·

-b,
for c

kT< 1 •

In most cases of practical interest we can use the
approximations*

'·' (3 1 and (32. are evaluated with respect to the energy zero
defined at the Fermi level of the first and second contact
respectively as a matter of convenience in applying general
equations derived for either contact.

118

(3,

(.32

where b

v-v,

(40)

b 10 - b11 {v, ~

are constants independent of voltage

and temperature,
perature and b

- (322fT1 +

b10

and b

6,0 - (3, {

,-,,

is a

and

l""":z.2.

are functions only of tem-

function only of voltage.

These

approximations are particularly good for the barriers
being considered as can be verified from the derivations
The current from Eq ( 39)

given later in this section.
then becomes for c

and for c

where G

= kTG e

kT>1

'-"/kT (

kT< 1

G (

-V/kT)

t- e

~"L1#-

( 41a)

-V/kT)

kTe

is the zero field conductance given by

and B

( 4lb)

(42)

is given by Eq ( 34b).

The limiting conductance

is seen to take the form one would expect when considering the two barriers in the zero field limit as two resistors in series.

The reverse component in Eq ( 4lb)

119
normally negligible whenever the equation is applicable
(i.e.,

kT<1).
The vo ltage across the first barrier V

be determined from the condition that the total

can

space change

in the oxide does not change from its equilibrium value
(i.e. ,

This condition is

no space charge is injected).

implied by the introduction of ).. in the above derivation;
that is,

the supply function at the second barrier depends

primarily on the non-equilibrium excess of higher energy
holes which has a

negligible influence on the total charge

density of the holes in the oxide.
written

jw( f- fJ dx

This condition may be

(43)

where

f' and fa correspond to the space charge density

of the assumed barriers under an applied field and at
equilibrium respectively
Equations

[i.e.,~ =(X/41l"}d 2 U/dx 2 ] .
( 41)

and

( 42)

can only apply if

the second barrier is not completely suppressed by the
voltage,

since only under this condition does

the entire derivation have any meaning.

>. and thus

Also when the

total applied voltage is greater than the barrier height,
direct emission of holes from the first contact over the
second barrier,

without being trapped in the interior,

120
becomes increasingly more probable and may become the
dominant process.

Therefore a

practical upper limit on

the applied voltage when using Eqs ( 41)

and

( 42) ,

may be

considered to be approximately the barrier height at the
second contact.
4.
Barriers.

Formulation in Terms of the Proposed

The relations required in the theory will be

subsequently evaluated for each of the proposed barrier
potentials given by Eqs ( 18)
defining

and

( 19).

e in Eq ( 24) may be evaluated exactly for both cases,

the initial portions of the two barriers
Eqs (I Sa)
to

Although the integral

and

( 19a)

(x < 0)

given in

normally represent small corrections

which we may approximate by a

constant ~ eval-

uated at V=E=O

[we will continue to refer to the main

contribution for X

> 0 as 8].

same form for both cases

The correction

(with appropriate

6.. has the

x 0 ) , . giving

(44)

The remaining relations will be treated separately for
each type of barrier with subscripts added to distinguish
between the first and second contact.

121
Quadratic Barriers.
Inserting Eq ( ISb)
first barrier

1 [(¢,

into Eq ( 24)

e, = oe

we have at the

1/2

+ "() ( 1 -

~ )"-- E] d x

where

Letting z

(cp1+ '7_) ( f - ~)2

E + "(

V(¢1+-'Y[)/(E+1) (I - ){/d)
1/(¢ +''1)7( E:+'"()
e, = oC
1/z2.-1 dz.

the integral reduces to

where the parameter T
the

oxid~

is a

characteristic temperature of

defined by

(45)

After integrating 1

one obtains the result

The other expressions required in the theory for the first
b arrier

[c. f.

Eqs(25) 1

(29) 1 (30)

readily evaluated using Eq ( 46)

[ N.B. ?'{ =V 1 -

and

(31)

may be

and are give n as follows

"'lo] :

~ [11 + r,

(47a)

122

c, - SiV\h-lyf /kTo .

(47b)

1/ ( LJ I< To 1 t{i 4-r,)

(47c)

¢, sech (~) -1 tanh (.~j

EOI

(47d)

¢, +~ tanh(~) - .!L. + .6..,
II

(:3, -

[ ~kTc;(¢,+1)

a', -

tanh(=¥-) sech ~)]

(47e)

-l
( 47f)

The transition between pure tunneling and the intermediate
case

(i.e.,

kT=I or E

=0)

is given by

1 - ¢,/sinhz_ (~1
where larger values of

'1_

(48)
correspond to pure tunneling.

The expression for

62 ,

with the energy

measured from the Fermi level of the second contact,
be obtained directly from Eq ( 46)

~- V

and

rp, with

by replacing

may

'1. with

that is,

+ E"+:Z-V - E+?[-V sech~f£+-r<-Vl
e =¢++]-V[/1
kTo [V
Pz.+;z-V f->2 +~-v
·y¢ +"(-Vj ·
2.

(49)

The other expressions required for the second barrier may
be obtained from Eq ( 49) ,

giving

(SOa)

123

P -lnz-V
tanh(To\ + ~ +6:z..
1<. lo
T"l
kT

(SOb)

[ 4kTa (c/>,_+"(-V) tahh (~) sech (~)]

-I

(SOc)

Exponential Barriers.
Inserting Eq (19b)

into Eq(24)

we have at the

first barrier

where

- ><, /s,

(¢,+~)e
Letting z

=E+-~

=~(¢,+'1.)/(E+1f) e

1/¢, (£'+')ll)

~To

-)(As1

'\f(¢,+-rz)/(E+'7._)
"V":z2 - I ,J z

where the parameter T 0

,the integral becomes

is a

characteristic temperature of

the oxide defined by

(51)

After integrating we obtain the result

r:n -~cos~]

e, = !&
k To l

't"'t

¢,

¢, -t 't

(52)

The following expressions are then readily obtained from

124

[ 1 -~ ctn ~ J +A,

,
kTc,

it dn-Yi; /z

[~ct~'~ +1}/skToyt

The saddle point E

(53a)

kTo

(53b)

(53c)

is determined by

[c. f.

Eq ( 29)]

(54)

Since the intermediate case applies only when
and thus for all reasonable barriers

'1 << c/>1 ,

(E 01 + 1 )/(¢1 +'l() << 1,

we may use the expansion

and obtain from Eq (54)

2..

rA [ 11"'/2,, L2.To/T + I

E"OI + ~

(55)

Then to the same approximation we obtain

(3,

kTo I -

(7-J-o + 1)jn z kTo r},

'!rYB

2.Tc,/T+ f

_:1_
kT

+b.,

(56a)

(56b)

The transition between pure tunneling and the intermediate
case

(i.e.,

kT=1)

is given ' by

125

(57)

- 11:2 ]2_

:!1.
¢,

~ 11; /ctr~- ~

- l2."/o I+ I

where lower temperatures corresponds to pure tunneling.

e2. , with the energy

The expression for

measured from the Fermi level of the second contact is
obtained from Eq (52)
changing subscripts,

replacing

with

'1 -V, and

that is,

(58)

The other expressions required for the second barrier
are obtained from Eq (58)

E02+ 1(- V

- ¢z

above,

[ 1r/2Z.'Ta/T+I

f32.

¥2

r/>2 [ 1 _ tr /B

ZT;,/T+- I

with the results

]2.

(59a)

J V-22

(2-fo + IJ3;42:z ~!;, ¢'2.

/<.T

+ .Llz.

(59b)

(59c)

126
C.

Application of the Theory
I.

General Considerations.

preliminary

comparison of the theory with the experimental

results

for

both the quadradic and the exponential barriers has shown
the latter to give much better agreement.

Therefore,

will use the theory for the exponential barriers exclusively
in the subsequent analysis.

Since the experimental

results

for the thin samples with AI contacts display similar characteristics,

we will

sentative sample

limit our analysis to a

( #21-C4) •

single repre-

The latter is one of the samples

which appears to be consistent with our simplified model;
that is,

in satisfying the assumption that the barriers assoc-

iated with each contact extend to their practical

limit and

that the interior separating the barriers is negligible.

will also provisionally make the additional assumption that
the equilibrium potential in the interior
zero and thus

'1. = V 1 •

'1_0

is approximately

This will be confirmed by the re-

suits of the subsequent analysis.
The condition imposed on V
quires that V

= V/2

by Eq ( 43)

if the intrinsic work function W

reis

equal for both contacts as should be the case when both
contacts are AI.
and

The differences in barrier heights

¢1

¢2. obtained from the photo-response measurements

127
must then arise from differences in impurity
adjacent to each contact
defining a

concentrations

{interface or surface states) •

single characteristic temperature T

[c. f.

In
Eq {51)]

we have already implied that the effective impurity distribulion

{"" N A)

However,

is essentially the same

for both barriers.

since this does not take into account differences

in concentrations in the immediate vicinity of the contacts,
no inconsistency is presented by the differences in

¢.

The

assumption of constant T 0

was made for simplicity and will

be shown to represent a

good approximation for the exper-

imental

results.

analysis.

We will thus use V

=V /2 in the subsequent

Also since the voltages obtained in the measure-

ments of sample #21-C4 do not exceed the barrier heights,
the theory represented by Eqs { 41) ,

{ 42) ,

and

{ 43)

should

apply in all cases.
The theory is then formulated in terms of two
important parameters,

which should relate to the photo-

electric measurements and T
reasonable

which is adjustable within

limits based on its defining equation .

wish to show that a

single value of T

diet the correct dependences
ita nee,

each as a

We then

can consistently pre-

of both current and capac-

function of both voltage and temperature.

128

of Eqs ( 41)

Current Dependence.

and

( 42)

The general relations

can be simplified further since we can

¢ 2 < ¢ 1 , from the photoelectric measurements.
The larger exponent involving ¢
in the expression for G

assume

will always dominate for either

polarity.

Thus after sub-

stituting the appropriate expressions from Eqs ( 40)

and

(59)

and defining the reduced temperature variable

1T/z

(60)

2 7;,/T + 1

we obtain for the zero field conductance

(in mho/cm 2 )

( 61 )

The intermediate case applies when*

[c. f.

Eq (57)

(62)

and becomes

= 2 kT G sinh ( ~T)

(63)

The pure tunneling case applies when

(64)

>: ¢ is used to denote either
, where
is to
1 or
be used when ne g ative voltage is applied at the outer contact
(negative polarity) and ¢
with positive polarity.

129
and becomes

(65)

where

1.14- • J0 Tc,

_:A_

iJ kJo

and B

is given by Eq ( 34b) •

imation

ctva-~ ~

limiting case at the

(66)

+ f:j. I

assume c

exp[-bro + ~(tLi,¢)]

We have used the approxIn the

in the coefficient of Eq ( 66).

higher voltages and at 77°K we may

< 1/2 and thus B ~ 1 as can be verified by

the results of the subsequent analysis.

We then have for

this limit

( 67)

The nearly V
experimental

results for j

dependence exhibited by the

in this range suggests that we

look for an approximation of this dependence from Eq ( 67) •
One finds that a

[ i . e . , ·n =

maximum exists in the theoretical slope

dlnj/dlnV]

occurring at V/2¢=.30,

The value of the maximum then becomes

n max
However a

+ .1775" kTo
more useful

relation which gives a

(68a)
more

130
effective average near the maximum of the gradually changing n may be given by

+ _L
6kTo

(68b)

The slopes measured from sample #21-C4

give n

6. 5 and 5. 4

respectively and thus
we have the

ratio

[c. f.

Fig ill-17]

for the positive and negative polarities

¢,/k~-:::- 33

and

¢2./k~-:::- 26. 6.

Thus

¢,/¢2 ~1. 24, which is the proper ratio

corresponding to the photo-response measurements from
which we may select

(within the experimental uncertainty)

cp2 =I. 40 ev.

the values ¢, =I. 7 5 ev and

We may compare the theory with the experimental data of Fig.

ill-17 in this

versus ¢V'v/z¢ct;'vv/z

range by plotting log

for both polarities.

( j/V)

The result

given in Fig :r::sr-1 exhibits the predicted linear dependence
with the slopes corresponding to 1/kT InfO.

more precise values of T
the two polarities

(i.e. ,

which is essentually equal

620 °K and 630 °K),

justifying our assumption of a
¢/kT

We thus obtain

estimated from n

constant T 0 •

for

therefore
The values of

are seen to compare reasonably

well with the more precise values obtained in this way.
The value of j/V obtained by extrapolating to the zero of
the abscissa must equal
of b

given by Eq ( 66) •

The value

is thus determined and gives very nearly the same

13 1

SAMPLE 21-C4
7S•K

-v
10-3

THEORY

.s:;

THEORY

4>2·1.40ev

cj> 1=1.75 ev

To =630°K

To=620°K

ho=25.9

-lt=32.8

b 10 =37.8

b'{) =37.7

6,=5.0

6,=4.9

........

10-4

10--~----~~----~~--~~----~~----~~----~~----~
0.4

0.~

0 .6

0 .7

0 .8

0 .9

1.0

4> ./W2cj> cot- ~
Figure IV - 1 .

Limiting c a s e of pure tunneling- comparison of the ory
with e xperim e nt.

1.1

132
result for each curve
b 10 = ¢

/1<.1; + 6 1,

(i.e. ,

=37 .8,

Since

37. 7).

the correction .6 is then obtained,

giving

5.0 and 4.9.
The experimental

ill-IS can be

results of Fig.

compared with the theory at low voltages given by Eq ( 63) ,
by plotting the data in the form

j/ 2kTsinh ( V /2kT) vs.

The result in Fig • .ISL-2 indicates the predicted constant
dependence on voltage except near the transition points
The deviation can be partly accounted for by

slight overestimate of the predicted intermediate tunneling

current at the transition point.
arise from a

A (E)

An additional effect may

slight shift in the saddle point energy due to

which was neglected in the theory [c. f.

Eq ( 36)]

This could be interpreted equivalently as an increase in the
effective hole temperature at the larger applied voltages
leading to deviations such as observed.
The constant asymptotes must according to
the theory equal

given by Eq ( 61).

We may then com-

pare the experimental results for G

obtained from Fig.

TIZ-2 with the theory by plotting log

versus t

using the value T
given in Fig.

=620°K.

TIZ-3.

The result so obtained is

The deviation of th e

experimental

points at low temperatures from the predicted linear

...

.,c

...

"'uE

•0.001

---

Figure IV- 2.

10-

10- r

10-

1o-•

I (II

0 .01

VOLTAGE, v

0.1

• 78 (I= 0 .093)

" 9 2 (1=0.114)

123 (/=0.142)

2 ~~

235 (/=0 . 250)

• 298° K (I= 0 .303)

192 {1=0. 210)

v~2ct>

T0 =620° K
4> 2 =1.4 ev

-/TRANSITION

Intermediate case at low voltages-comparison of theory with
experiment.

SAMPLE 21-C4

134

SAMPLE 21-C4

VI-THEORY

To =620° K

£1

----

1 •1.78 ev
I

kTr=33.4

b,o = 38.1

~-- - 7 - ~-IMPURITY
~,=4.7
CONDUCTION

0.2

0.1

0.3

Figure IV- 3.

Dependence of zero field conductance on temperaturecomparison of theory with experiment.

135
dependence is attributed to tunneling into impurity states
(impurity conduction)

which was neglected in the theory.

By adding an impurity contribution G

10-

mhos/cm

t- 3 / 2

equal to

to the indicated asymptote,

the solid curve representing a
points.

one obtains

good fit to the experimental

The theoretical asymptote gives a

value of

¢ 1 =1. 78 ev which is in good agreement with the predicted
value of 1.75 ev·.

Correspondingly,

The extrapolated value of Gt pression given by Eq ( 61)

¢ 1 /kT 0

becomes 33. 4.

at t=O must equal the ex-

and thus we obtain b

10

=38 .I and

.6 =4. 7 which is also in good agreement with the values

obtained above.

It is interesting to note that the small

differences are in the correct direction one would expect
from the effect of an additional image force lowering of the
barrier height with an applied field,

which was neglected

in the theory.
The temperature dependence predicted when
pure tunneling applies at the higher voltages is given by the
coefficient B

exp

[1r4 (¢,-p)
t]
kTo

appropriate barrier [i.e. ,
for negative polarity

J.

where B

is evaluated for the

¢ 1 for positive polarity and ¢2.

Inserting the appropriate express-

ions one obtains for -. 5 volts,

136

-:I:
and for

ex

+ • 5 volts,

where j

e ( 5.1 5 t)
~ sin ( r7~) + .194
'liT
1.194

1.142

~~ sin (r}()) + .[42

corresponds to the current at T=O ° K. These re-

lations are compared with the experimental results in
Fig. r::s:L-4.

Although the observed dependence is somewhat

steeper than predicted, a
by a

better fit could clearly be obtained

small adjustment of the parameters.

This would hardly

be justified in view of other second order effects which have
been neglected such as the temperature dependence of the
dielectric constant.
The correction 6
expression given by Eq ( 44)

may be compared with the

which may be written in the

form

11

(Xo) JA

S S 1 kTo

where
[c.f.

Eqs

(17a)&(51)]

Inserting the values obtained for zero applied voltage and
using

><0 =n 2

(thus

oe

3. 3

from Fig.

1.03(ev)-l/Z

m-S and assuming

A - l ) one obtains

m* /m=l

137

[/

/,

a,
C\.1

0..

f----- 1 - -

-.... - -

+ 0.5v..,... ...., ....,

_....

rot'-

./

...., ....,

...., /~

~--'"""'

~,

,""

_L_

v'

B 2e

7r(cp,-cp2) t
4k To

- 0 .5v

100

Figure IV -4.

300

Dependence of tunneling current on temperaturecomparison of theory with experiment.

138

and

b.,

3.0

Therefore the value predicted for .61 by our approximations
underestimates the measured values of 4. 7-5. 0.

This is

quite understandable if the point made earlier is valid,
is,

if a

that

different effective impurity concentration occurs

immediately adjacent to each of the contacts, . accounting for
the different barrier heights.

We would then expect a

corre-

spondingly larger .6 at the larger barrier than would be
characterized by the parameter T
both barriers.

Alternately,

which is common to

there may be a

small contrib-

ution from the oxide interior which was neglected and would
appear as an effective increase in ~

It is of interest to calculate the maxim u m acceptor concentration N A

from the definition of T

That is if we assume 'K =27.5,

m*/m=1

given by Eq (51).

and T

= 620°K,

we have
~~

c:::t.B·IO

or about 5. 5

mole percent.

-3
em

This corresponds roughly

to the saturation concentrations quoted in the

literature.

139
3.

Cap a citance Dependence.

The capaci-

tance may be expressed in terms of the sum of the effective
barrier widths in the oxide film,

that is

(69)

where C/A

is the capacitance per unit area,

and x

represent the effective widths of the first and second barrier
respectively and .6X

represents the

image force corrections

( ~ 2 X0

total width due to the

plus any additional contri-

bution of holes trapped in the oxide interior which depends
on frequency as discuss ed in Section illC.

We can express

x 1 <;1nd x 2 by the barrier widths at the saddle point energy
from Eq (55) ,

that is

U(x)-I

- (cA +~)e-x, /s,

¢, t2.

or

x,

s, [!Vl (I+ ~ )

2 lnt

(70a)

and similarl y

Xz.

s2 [ In (I + V~ ) - 2 In t

(70b)

wh e re

repres ent s

22

th e

polarity correspondin g
tact

(~ ) •

applied de voltage with positive

to positi v e

voltage at the first con-

We have used the intermediate case for the

s addle point energy

(cp tl

on the assumption that only small

ac voltages are used in measuring the capacitance.

The

140
de voltage will only a ffect the steady state barrier shape
as indicated by Eq ( 70) ,

with the de current acting inde-

pendent of the small ac fluctuations.
is nearly independent of de voltage when the voltage and the
difference in barrier heights are not too great.

If we write

and
and expand to second order terms of
we obtain

x1 + xz

V/2 ¢0

and

~¢/2 ¢o,

so[-41n t -t 2~o e~~~oV)]

( 71 )

s, +

(72)

where

so

52..

Therefore the capacitance changes negligibly with de
voltage when

vjz~o << 1

At de voltages larger than

the barrier height at the negative contact the voltage can
no longer be assumed to divide equally between the barriers
as assumed for Eq ( 70) ,

and a

greater share of the voltage

must then occur across the positively biased barrier resuiting in a

net increase in the total barrier width.

The

capacitance will then decrease until the barrier extends
across the width of the oxide.

This kind of dependence on

de voltage is in qualitative agreement .w ith the
in Fig m-21.

result given

One also observes that the decrease in

141
capacitance occurs sooner for positive polarity which is
consistent with the fact that the larger barrier
at the outer contact.

The onset of the

¢, occurs

rapid decrease in

capacitance is seen to occur in the vicinity of the barrier
height associated with the negatively biased contact.

The

limit where the barrier extends across the oxide film could
not be reached before breakdown occurred.
In the case of the anodized films with Au and
Ti contacts we can expect a
voltage.

larger variation with de

We have proposed that an electronic conduction

process occurs for these samples and that the voltage must
divic:Je between the two barriers as with two independent
series resistances.

Since the barrier at the Au contact

was seen to be much more effective than at the

Ti contact

and therefore dominates until it is nearly suppressed by
sufficiently large positive voltages

(at the Au contact) ,

we

can express the capacitance dependence on voltage in this
case approximately by"~

where

L:1Y.0

is the total effective barrier width at zero

,~The change

in sign of V corresponds to the electron
conduction process for this case.

142
applied voltage.

more convenient form is

(73a)

(73b)

where

c.I

and

(73c)

is the capacitance at zero applied voltage.

In Fig. TI-4 we

have plotted the data in the form corresponding to Eq ( 73)
using the average value obtained for the Au barrier height
(i.e. ,

¢. =I. 42 ev) •

The predicted linear dependence agrees

within the experimental uncertainties,

with the expected

deviation occurring when +V approaches
measured value of the slope,
1< =

27.5 and A = 5.5

Eq(73b)

10

Cs =

¢, .

5400pf,

From the
and using

-4 em 2 , we calculate from

10.7 A .

It is interesting to compare this

value with that obtained from t;)e above re su lts for sample
From the definition of T

#21-C4 ·.

given by Eq (51).

and assuming

s2

m~'/m=l , one calculates s 1 = 12 .I

A and

10.9 A.

The similarity in the values of s

for different

kinds of samples is required by our model in order to describe reasonable barriers,

consequently these

results are

143
also encouraging.
The temperature dependence of the capacitance
at zero de bias is also predicted by Eqs
the samples with AI

contacts.

( 69)

and

( 70)

for

This may be expressed in a

more convenient form by

where s 0
at 5 kc

A]

exp

is defined in Eq ( 72) •

(see Fig. ill-18)

log plot of the

(74)

- 16frs0 C

The experimental

is plotted in Fig . .ISZ-5.

data taken
The semi-

results exhibits the predicted linear dependence .

with the . slope giving

.328
Using the values of s

1 and s 2 g iven above for m>:'/m =

1,

we have s 0
reasonably

II. 5 A

and thus obtain 1< =

35.

good agreement with the value X =

since the uncertainty in the area

This is in
27.5

alone is easily

of course we have assumed m >:• =

m.

± 2. 0,

± 10% and

There is also the

possibility that the value 2 7. 5 may have been reduced by

lower dielectric constant in the interior of the thick oxide

film used in the measurement.
The value of Ax/4s

may be obtained from the

position of the curve and gives at 5 kc,

Ax/ 4s 0

• 97.

0 .5

144

0 .4

[\

SAMPLE 21- C4 _

0 .3

v- -K- =
l61rs0

7. 25 · 10- 2 t/m 2

0 .2

DECREASING
FREQUENCY

0 .I

5kc

0 .05

25

30

35

40

45

Ale, m2 /t
Figure IV- 5.

Dependence of capacitance on
temperature- comparison of
theory with experiment.

50

145
This value decreases with decreasing frequency as indicated
by the corresponding increase in capacitance
ID-18)

which we have

(see Fig.

related to the effect of trapped holes.

In the de limit we would predict that .D.x/4s 0

should approach

The results are quite consistent with this
prediction,

expecially since the values obtained depend on

the difference of two larger quantities and become very
sensitive to the error in the area measurement.

146

V.

CONCLUSIONS

The electrical properties of thin. Ti0
interpreted in terms of a

films have been

physical model that considers the

effect of large impurity concentrations on the shape of the
barrier potential.

If we consider only the general

features

of such barriers before making any specific assumptions on
the impurity distribution in the oxide,
at a

qualitative understanding of a

we are able to arrive

variety of results that

would defy explanation on the basis of the usual model of
insulating films.

For example the electrical

forming effects

observed in anodized films are readily understood in terms
of the change occurring in a

narrow space charge barrier

as the distribution of ionized impurities is disturbed by the
applied field.
1- V

Similarly one can qualitatively interpret the

characteristics and the capacitance variation on the

basis of the general model.

required in order to formulate a

more specific model was
quantitative description of

the properties.
In formulating a

quantitative theory we have treated

two different barrier shapes

(quadratic and exponential}

that arise on the basis of certain assumptions concerning

147
the distribution of ionized impurities.
barriers may in a
the

The two kinds of

sense be considered limiting cases for

impurity distribution so that the analysis of the experi-

mental results provides us with a

means of evaluating the

kind of distribution that actually occurs in the barrier.

We

have found that the theory formulated in terms of the exponential barriers is in good agreement with the experimental
results,

and therefore conclude that the actual barriers can

be very nearly represented by an exponential form.

In

Section IS7A we have offered three possible physical
reasons for predicting barriers of this kind which are
essentually indistinguishable on the basis of our results.
However the sensitivity of the barrier to the contact metal
(Au or AI)

suggests that the most important reason may

be associated with a
contacts.

rate-limited diffusion process from

On the other hand,

the possibility of a

the

this effect does not exclude

Poisson-Boltzmann type of distribution.

Futhermore the impurity states may also be distributed over
different energies in the forbidden band
reason)

in addition to either or both of the other possibi l-

ilies and still

(the third possible

obtain the' same kind of barrier.

serious difficulty discussed ( 7 ' 34 )

with the early interpretations of a

in connection

tunneling-type process

148
at metal-semiconductor contacts involves the extreme sensitivity of the barrier resistance. to its effective width.

This

is readily resolved in terms of our model since we have
considered the barrier only in terms of the interaction between the oxide and the metal contact and the maximum
impurity concentration that occurs in the oxide.
ysis of our experimental

The anal-

results has indicated that the im-

purity distribution is determined by the saturation concentration of impurities

(acceptors)

fore it does not depend in a

at the contact and there-

sensitive manner on the precise

conditions for its formation or the

total

thickness of the film.

The effect of film thickness has been interpreted in terms
of an additional effect in the oxide interior which we have
successfully avoided in our detailed treatment of the experimental

results by an appropriate choice of samples.

In order to obtain a

consistent description of our re-

sults we have concluded that the evaporated titanium oxide
films with AI

contact s

must be p-type.

This conclusion is

of particular significance in view of the fact that it is ordinarily difficult to obtain p-type rutile in bulk form.

For

samples prepared with Au contacts we have offered more
qualitative arguments

that

indicate that the oxide films in

this case are probably n-type.

149

key device introduced in the development of the

theory involves the treatment of the energy distribution of
holes in the p-type oxide films under applied voltage.
introducing an unknown multiplier

that only modifies the

equilibrium distribution at higher energies in a
manner and eliminating
state conditions,

{\

By

reasonable

from the equations under steady

we arrive at explicit relationships for the

current and capacitance as a

function of voltage and

temperature.

The success of the theory in describing the

experimental

results leaves little doubt as to the validity of

this approach.

We therefore suggest that this approach may

also be useful in other two barrier problems where intermediate transitions of an indeterminate nature are involved
in the steady state process.
Finally,

our Ti0 2

essentually amorphous,
related to rutile.

films which were shown to be
exhibited properties that could

be

We therefore are inclined to believe that

the short range structural order of the films is essent ially
that of rutile.
too far,

However,

this relationship cannot be carried

since the large degree of disorder in the film could

easily permit a

similar relationship to other structural modi-

fications of Ti0 2

(anatase and brookite) .

In this respect

the values measured for the dielectric constant and refractive

ISO

index of the

films are more consistent with the lower

values reported for the other crystalline forms of Ti0 2 •

151

APPENDIX A
'PROPERTIES OF RUTILE
Rutile is the most stable crystalline form of Ti0
Other forms

(anatase and brookite)

are irreversibly conThe

verted to rutile vvhen heated betvveen 700 and 920 °C.
crystal structure of rutile is in the tetragonal class vvith

lattice constants a

= 4. 5937 A

and c

= 2. 9581 A.

The

structure may be considered composed of slightly distorted
Ti0 6

octahedra vvith one pair of Ti-0 bonds being slightly

longer than the other tvvo pairs

(I. 94 and I. 99 A).

bonding forces often considered to be ionic
and

o2-

ions),

The

(based on Ti 4 +

have been shovvn to also possess an appre-

ciable covalent contribution (

16

) •

The energy band structure may be considered in
terms of the electronic configurations of the

o 2-

filled 2p levels of the

ions.

The

ion are broadened to form the

valence band.

The normal electronic configuration of the

titanium atom is

( 4s)

(3d)

outside its argon core,

and

the conduction band is generally believed to arise from
unfilled 3d levels of the Ti
ing a

the

+ ions, vvith the 4s levels form-

broad band at higher energies.

Morin ( 39 ) has also

152
indicated on the basis of orbital
also be split into narrow de.

theory that the 3d band may

and db.

bands .

for the observed low mobilities in rutile,

In accounting

it is generally be-

lieved that the 3d levels overlap slightly to form a

narro-w

3d conduction band and not extending to the much broader
4s band.

There still

remains some question about the

band structure although extensive experimental results
have been reported.
The reported measurements of the band gap vary
considerably and we can only attempt to give our interpretation.

Gronemeyer (

measurements,
eratures,

40

taken over a

obtains from conductivity
wide range at higher temp-

two well defined activation energies which are

interpreted in terms of energy gaps of 3. 05 and 3. 7
From optical

transmission measurements,

energy gap of 3.03 ev.
duced rutile,
2. 8

to 3. 0

erature.

he obtains an

From photoconductance of re-

he obtains energy gaps in the range of

ev,

ev.

about

the value increasing with decreasing temp-

The evidence suggests that the 3d band occurs

at about 3. 05 ev above the valance band,
energy possibly associated with the db
the 2. 8 to 3. 0

ev energies,

with a

with the 3. 7

or 4s band,

ev

and

donor impurity band.

153
Most of the

results reported by other investigators can be

related to one of the above values or some combination of
them.
3. 4 ev;

For example,

Rudolph(JI)

reports 3.12 ev;

and Sandier [see Ref ( 16)]

2. 8

ev.

Earle(41)

Although the

optical absorption edge in thick rutile crystals is consist e ntly
observed near 3
De Vore (

30

ev,

the optical dispersion reported by

and others indicates a

higher natural absorp-

tion energy above 4 ev that may relate to the influence of
the broad 4s band.
peaks at 4. 0

This is consistent with the reflectance

and 4. 13 ev reported by Nelson and Linz (

from oriented single-crystal
ular and parallel

rutile

for directions perpendic-

to the c-axis respectively .

The optical

absorption edge for thin polycrystalline films w as

also

found by the above authors to occur near 3. 9

with

similar results reported by Gronemeyer (
ed in this investigation

42

40

(see Section illB).

ev,

and also obtain-

It would appear

that the absorption due to the narrow 3d band which is
important in relatively thick crystals becomes less important
in thin films and one then observes the stronger absorption
due to the much broader 4s band.

Conductivity measure-

ments at high temperatures reported by Hauffe
indicate an energy gap of about 3. 9

[see Ref

ev that may similarly

( 16 ~

154
relate to the greater importance of the broad 4s band at
high temperatures.
in Appendix B

Our measurements on rutile described

are also consistent with the above int c rp-

retation of the band gaps occurring near 3

and 4 ev,

corresponding to the 3d and 4s conduction bands respectIn addition the photo-response measurements also

ively.

suggest an additional impurity band at about 2. 8 ev.
The conductivity of rutile is usually generated by a
reduction process,

which is commonly believed to form

oxygen vacancies,

each of which may act directly as a

T.4+ .
cause two a d Jacent
1ons

d onor center or a I ternate I y

to convert to T1.3+ d onor s1tes.

Grant (

16

) has summarized

the effects of various other impurities investigated which
can be classified as either donors
Ta,

Nb)

or as acceptors

(e.g. , W,

(e.g.,AI,

Fe,

P,

Ga,

Sb,

Y).

V,
The

latter have been characterized by their ability to compensate
for donors,

however p-type rutile also has been observed

under certain conditions (

27

31

) •

The impurity concentra-

tions generally tend to saturate at a
Ehrlich (

43

few mole per cent ,

and

has determined the maximum concentration of

oxygen vacancies before structural changes occur to be
mole per cent.

Although no detailed information has been

found in the literature on the diffusion constants,

the

ISS
conditions reported for obtaining uniformally
samples (

40 44
) indicate large values for the diffusion con-

stant of oxygen vacancies in rutile,
3'10- 8

reduced rutile

that is,

the order of

cm2/sec at 3S0°C and 10-S cm 2 /sec at 800°C,

and thus an activation energy the order of

•8

ev.

These

values are consistent with the large effects of ionic drift
observed in rutile under applied fields.
The dielectric constant of rutile is given as 173 for
the c-direction and 89 for the a-directions.

The average

of 117 compares with values the order of 100 measured for
ceramic rutile.

Similarly ceramic rutile exhibits electrical

properties comparable to single-crystal rutile,

with electron

mobilities for reduced specimens measuring the order of
• I to I. 0

cm

volt- 1 sec- 1•

The additional scattering due to

gra in boundries in ceramics appears to be masked by the
strong coupling of electrons with the polar modes of lattice
vibrations

( polarons) .

Correspondingly lar ge effective

masses are usually quoted for reduced rutile ranging from
I to 1000 times the free electron mass but more commonly
indicated to be the order of 2S.
narrow 3d conduction band,

This is consistent with a

the very large

representing the larger polaron mass.

values possibly

Although extremely

large donor concentrations are usually produced in reduced

156
rutile

20
in excess of I o
em - )

(e.g.

degenerate,
around

the material

is not

exhibiting distinct activation energies centering

.I-. 2

ev at higher temperatures and the order of

• 01 ev at very

low temperatures.

The large values of

activation energies cannot be interpreted in terms of the
usual hydrogenic model of donor states but may be interpreted by the polaron self-trapping energy as discussed by
Frederikse (

17

).

The low energies may be associated

with the true binding energy of the donors.

The polaron

effect is strongly related to the coupling constant ocp as

d .rscusse d b y

F ro
.. h Ire
. h ( 45 ) •

for the restrahlen wave
investigators,

obtarns

OCP~

and the

If one uses the values of 50 f'

length,

as measured by several

refractive index of about 2. 5,

4 ( m >'·< j m ) I I 2 •

If

oCP is greater than about 6,

strong coupling occurs accompanied by a
mass.

one

Thus one can easily expect a

very

large polaron

strong polaron effect

in reduced rutile since m* /m is believed to be greater
than 2.

157

APPENDIX

E>Single crystal rutile purchased from Linde Air
Products was sliced into specimens approximately
.I x

•2 x

lengths.

I. 0

em,

Th e

specimens were reduced at 650 °C in H

for ten minutes.

with the c-direction parallel to their

The resistivity

was approximately 2
published data,

ohm-em,

of the reduced specimens

and comparing this with

the carrier concentrations at room temper-

ature are estimated to be the order of 10
donor concentrations greater than 10

20

18

cm- 3

10

19

cm-

and

The spec-

imens were prepared with Au contacts by immediately

evaporating a

thin film of Au

(~ 500 A)

on surfaces freshly

exposed . after fracturing the specimen in a
cleavage could not be obtained,

but a

true

smooth surface was

exposed when the specimens fractured,
along a

vacuum .

lying approximately

plane perpendicular to the c-axis.

The areas of

the gold contacts were defined after the evaporation by
carefully scraping the Au film away from the boundry of
the desired area .

The opposite contacts were made by

soldering indium to the other surfaces of the specimen.
They introduced no detectable contact resistance at room

158

temperature.
The photo-response from the ·Au contact was measured with -I. 5 volts applied at the contact.
are plotted in Fig. B-1
IIB.

The results

in the manner described in Section

The large response extrapolates to 3. OS ev,

believed to represent the energy
to the 3d conduction band.

the value

gap from the valance band

second peak above 4

arise from the additional contribution

ev may

from the 4s band.

Direct observation of the energy gap to this band is partially
masked by the effect of the lower 3d band but appears to be
in the gene ral

vicinity of the suspected value of about 4

(see Appendix A) •

The tail in the

ev

response which extrap-

olates to 2. 8

ev is believed to represent transitions to a

donor band.

The small response at lower energies is

interpreted to arise from electrons excited from the Au contact into the conduction band of the underlying rutile crystal,
and thus the extrapolation to I. 4 ev gives the barrier height
at the Au contact.

This is in close agreement with the re-

suits obtained from both anodized and evaporated Ti0 2
with Au contacts

films

(see sections IIB and illB).

The capacitance at 100 kc was measured at room
temperature as a

function of de voltage.

The capacitance

IS9
exhibited considerable drift with time unless the sample was
first electrically formed by applying . a

sufficient positive de
The results

voltage at the Au contact for several minutes.
obtained from a
at

sample at room temperature after forming

+. S volts for IS minutes are plotted in Fig. B-2.

usual theory of a
dependence

The

metal-semiconductor barrier gives the

(A/c)

= Bl
donor concentration and V

where N

is the

is the built in potential at the

barrier equal to the barrier height less the equilibrium
potential at the limit of the space charge region,

(i.e. ,

Thus the result gives the predicted linear
dependence of
I. 3S ev,

(A/C)

vs.

with the intercept equaling
The

consistent with the measured barrier height.

'Y{ 0 ~.OS ev which is quite reason-

difference would predict
able,

although the experimental accuracy does not permit

this result to be taken too seriously.

The electrical

forming

is believed to cause positively ionized donors to drift out
of the space charge
correspondingly

region into the interior of the specimen,

reducing the concentration N

the space charge

region.

at the tail

of

From the slope of the dependence

the effective donor concentrations N 0
approximately 6 ·10 17 cm-

is estimated to be

which is considerably

less than

the average value for the crystal but is consistent with the

160
forming process proposed.
Optical

transmission measurements were obtained from

polished slice of unreduced rutile,

perpendicular to the c-axis.

sliced I mm thick

The result indicated a

sharp

absorption edge at 3. 06 ev in agreement with the results
obtained by other investigators and the value believed to
be the E)nergy gap

(to the 3d band) •

No effects of a

donor

band at lower energies could be detected in this case.
small residual

transmission barely detectable above the

noise persisted out to approximately 4
been related to the 4s band.

ev and may have

Q::

f'()

ol

21

41

6f

..J

11 v, ev

·'-·--·1

./

./

.,

Photo - r e sp o ns e of rut ile (Au contac t).

/Jt

;·-·-.........

fRIGHT SCALE

Il

I I

Iii

~/

Figure B-1.

/I \
L I

(INCREASED SLIT OPENING)

,_

II

LEFT SCALE

-r-

Au CONTACT ON REDUCED RUTILE
-1.5v BIAS

r --- ------

_./

Io

-18

Ls

-Q)

~24 ~

132

-440

48

16 2

C\J

".

<0

"""'

Au CONTACT ON REDUCED RUTILE
AFTER FORMING AT +0.5v FOR 15min

~~

298 °K

~ '-.. ........,

~~

.......

...........

-1.2 -1.0

-0.8

-0.6

-0.4

-0.2

0.2

0 .4

0 .6

0 .8

VOLTAGE, v

Figure B- 2..

Dependenc e of capacitance on de voltage.

1.0

1.2

1.4

163

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