ELECTRICAL SWITCHING EFFECTS
IN VW‘UM-BXIDE'COMPLEXES
* Thesis for the Degree of Ph. D.
MICHIGAN STATE UNIVERSITY
g 'RUEY JAM} YU
’ ' , 1972

 

 

 

 

 

LIBRARY

Michigan State
University

 

This is to certify that the
thesis entitled

ELECTRICAL SWITCHING EF‘VECTS
IV VA“IADIUM-OXTDE COMPLEXES

presented by
RUBY J ANG YU

has been accepted towards fulfillment
of the requirements for

Ph.D. . E.E.
degree in

///Z¢/ flw/

Major professor

Date 1/2/72

0-7639

 
  
 
     

      
    

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. “DAB 8 SONS. h
if! 90% BINDERY MW ;
[‘1 gram BINDERS
"30m. 9.19%.!!!

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ABSTRACT

ELECTRICAL SWITCHING EFFECTS
IN VANADIUM-OXIDE COMPLEXES

BY

Ruey Jang Yu

The electrical properties of oxidized vanadium
foils were investigated. One mil thick sheets of
vanadium, diced into approximately 1 mm by 1 mm squares,
oxidized in air at 475°C for up to one week, and then
mounted in a spring-loaded sample holder were found
to exhibit reversible, nondestructive, current-

controlled switching as well as inductance effects.

From the x-ray diffraction patterns and the
conductivity-versus-temperature data, it was concluded
that these oxidized samples were in the amorphous state:
furthermore, the oxidized layers were not V0, V02.
V205. etc. but rather appeared to be a combination of

oxide complexes.

The ratio of a typical sample's "off-state"
to "on-state" resistance was Observed to be approximately

100. The breakdown voltage ranged from 3 volts to 10

ii

volts and was found to decrease as the bath temperature
was increased. The delay time for the device to

switch was measured and found to be dependent upon

the ambient temperature, as well as the magnitude of

the applied voltage.

The inductance of a typical sample was found
to be strongly dependent upon the dc bias current but
independent of frequency, suggesting possible
applications as a varactor. For the case of zero dc
bias, inductances in excess of 100 mH were observed;

however, the Q of the structure was quite low.

An expression for the threshold voltage
required for the sample to switch from the low-
conductance state to high-conductance state was
obtained in terms of the sample's thermal conductivity
and rate of change of electrical conductivity with
temperature. The eXpression was found to be basically

in agreement with experiments.

The negative resistance of the unit was also
observed, and it was shown that the formation of a
current channel does not necessarily imply that the
material must exhibit negative-differential resistance.
Finally, a criteria for observing the negative-
differential resistance in current-controlled, bulk-

switching devices was established.

ELECTRICAL SWITCHING EFFECTS
IN VANADIUMrOXIDE COMPLEXES

BY

Ruey Jang Yu

A THESIS

Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of

DOCTOR OF PHILOSOPHY

Department of Electrical Engineering
and System Sciences

1972

(T\

To My Parents
and

to Suefen

ACKNOWLEDGEMENT

The author wishes to express his sincere
appreciation to his major professor Dr. P.D. Fisher,
for providing his thoughtful guidance, genuine
wisdom and encouragement during the course of this
investigation, and also for his painstaking review

of this manuscript.

Many thanks also to Dr. K. Subramanian for
his encouragement and help in the investigation of the
sample's structual properties. He also wishes to
acknowledge the other members of the guidance committee,
Dr. J. Asmussen, Dr. B. Ho and Dr. D. Montgomery for

their time and interest in this work.

Finally, the author wishes to thank his wife,
Sue Fen, for her patience, understanding and encouragement

in the entire period of his study.

LIST OF CONTENTS

List of Illustrations
List of Tables

Introduction

Experimental Procedure and Apparatus

NNN NNN
0.. .0.
(DUI-b WNH

Sample Preparation

X-Ray Diffraction Experiment
Temperature-Dependent, Electrical-Conductivity
Experiments

Switching-Delay-Time Measurement Procedures
Inductance Measurements

Experiments on Sample's Response to 600 Hz
Large-Scale Signals

Experimental Results

wwwwww
O. O.
O‘U'IDUNH

Structural Properties of the Unit
DC Conductance

Typical I-V Characteristic Curves
Switching-Delay-Time Constant
Response to AC Signals

Inductance Properties

Interpretation and Discussion

4.1
4.2

kbbbb
O O

wNNNN
hWNl-J

Introduction

Temperature Distribution and Switching
Threshold Voltage of the Device-~Sandwich
Configuration

Geometry

Boundary and Initial Conditions
Derivation and Results

Discussion

Temperature Distribution and Threshold
Voltage of the V02 Thin Film—-Planar
Configuration
Introduction
Geometry and Assumptions
Derivation and Results
Discussion

Negative Resistance

II

4.4.1 Introduction

4.4.2 Derivation, Results and Discussion

4.5 Negative Resistance--Isotherma1, Steady—
State Case

5. Conclusion

5.1 Summary of the Significant Results
5.2 Suggested Additional Research
Appendix A Radius of the Ring in the X—Ray Picture
of Pure Vanadium Foil
Appendix B Derivation of Equation 4.23
Appendix C Derivation of Equation 4.79

Bibliography

FIGURE

2.1

LIST OF ILLUSTRATIONS

Experimental Arrangement for Measuring Temperature
Dependence of the Sample's Electrical Conductivity.

Experimental Arrangement for Measuring the
Switching Delay Time for Various Bath Temperatures.

Bridge Circuit Employed in Measuring the Sample's
Impedance.

Experimental Arrangement for Displaying the
Sample's Response to Large-Scale Signals.

Back-Reflection, Pin-hole X—Ray Pattern for a
Pure Vanadium Foil.

Back-Reflection, Pin-hole X-Ray Pattern for an
Oxidized Vanadium Foil, Illustrating the Amorphous
Nature of the Oxide Layer.

Back-Reflection, Pin-hole X-Ray Pattern for the
Oxidized Vanadium Foil, Illustrating the Presence
of Both the Amorphous Oxide Layer and the Pure
Vanadium Foil Beneath the Layer.

Temperature Dependence of a Typical Oxidized
Vanadium Foil's Conductance Under Very Low—
Bias Conditions.

Temperature Dependence of a Typical Sample's

Threshold Current IC’ Threshold Voltage VC,

Threshold Power PC, Holding Voltage VH and
H.

Typical I-V Characteristic Curve for an
Oxidized Vanadium Layer.

Holding Current I

I-V Characteristics of a Typical, Oxidized
Layer for Bath Temperatures Varying from 23°C
to 60°C.

III

FIGURE

3.8

3.10

IV

Response of a Typical, Oxidized Vanadium Layer
to a Square-Wave Voltage, Illustrating the
Dependence of Delay Time on Biasing Level.

Response of a Typical, Oxidized Vanadium Layer
to a Square-Wave Voltage, Illustrating the
Dependence of Delay Time on Bath Temperature.

Response to a Square-Wave Voltage for a Bath

Temperature of 26°C with a Protection Resistor
Rp = lkohm, Illustrating that the Switching

Delay Time is Critically Dependent Upon the
Effective Source Impedance.

Response of a Typical Sample to a Large-Signal,
600 Hz Applied Voltage.

Frequency Dependence of Inductance for Various
DC Bias Currents.

Dependence of Inductance on DC Bias Current.

Frequency Dependence of the Small—Signal, AC,
Series Resistance.

Dependence of the Small-Signal, Series
Resistance on DC Bias Current.

Three Possible Thermal-Boundary Configurations.
The Sandwich Configuration Under Investigation.

Illustration of the Sample's Temperature at
x:o as a Function of Time for Various Bias voltages.

Critical Voltage Required to Actuate Switching
for Various Bath Temperatures.

Switching Delay Time as a Function of Applied
Voltage for a Typical, Oxidized Vanadium Foil.

Illustration of the Thin-Film, Planar
Configuration to be Analyzed.

Graphical Solution of the Time-Independent
Temperature Equation.

FIGURE

4.8

Graphical Solution of the Threshold Voltage
Required for the V02 Layer to Switch for

Various Bath Temperatures.

Voltage Required to Actuate Switching for
Various Bath Temperatures.

Illustration of the Section-Wise Linearization
of a Sample's Electrical Conductivity.

The Corresponding I-V Characteristic Curves
for the Various 0(T) Dependences Illustrated
in Figure 4.10.

Experimental Arrangement for Making Back-
Reflection, Pin-hole X-Ray Photographs.

LIST OF TABLES

TABLE

I. Delay-Time Data for Various Applied Voltages
as Depicted in Figure 3.8.

II. Delay-Time Data for Various Ambient Temperatures
as Depicted in Figure 3.9.

III. -%% [Tt(To)] of VO2 Thin Film at Various Ambient
Temperatures.

IV. Threshold voltage VC for Various Ambient

Temperatures.

VI

CHAPTER I

INTRODUCTION

During the past two decades, significant
progress has been made in terms of understanding the
properties of crystalline semiconductor materials.

,The mathematical treatment and subsequent understanding
of these materials has largely depended on the
simplification which can be introduced due to the
material's periodic nature(1). However, periodicty

is not vital in explaining the electrical properties

of all crystals, as can be seen by the fact that no

(2)

sharp drop in conductivity is observed at the
melting point of many materials. At this temperature,
long range order in the material disappears, yet

the electrical properties are only slightly altered.
The early investigation of these rather disordered
materials showed only that the electrical conductivity
was very low(2). The study was of interest
theoretically as an extension of the study of more

(2).

highly-ordered, crystalline materials

It was not until around 1960, that significant

(1.3.4)

research activity was undertaken in order to

deve10p an understanding of the electrical-conduction

properties of amorphous, chalcogenide and transition—

(7)

metal-oxide semiconductors . The interest in these

materials was largely due to the discovery that these

(3,4)

bulk materials possess switching and/Or memory

(7,8,10)

effects as well as negative-differential

resistances(ll'15'l6). Furthermore, some of the

materials were found to possess inductance prOperties(3).
Because of their unusual and potentially useful
properties(6'7), these materials could become the next

generation of widely used semiconductor materials.

Nevertheless, the promise of amorphous
semiconductor materials can not be realized until
their characteristics can be precisely defined and
their Operation controlled. The theory for amorphous
semiconductors is still not well defined(I3-15'18'22).
The nature of the change in the material during the
switching process has not been established;(1'15)
neither has the nature of the memory effect. Evidence
is fairly convincing that a structural change occurs
in the material at least in the memory switch(1),

but whether it is electrically thermally initiated

has not been settled. Boer(9'24), Berglund

(10)

(3,16)

and others have confirmed that the preswitching,
nonohmic behavior of amorphous semiconductors has
a thermal origin as a result of Joule heating. By
using irreversible thermolynamics and totally
neglecting effects due to lattice temperature

gradients, Ridley(11)

has shown that, in the steady
state, the current-controlled, negative-differential
resistance in a bulk device favors the formation

of thermal filaments. The negative-differential
resistance in an isothermal system was also discussed

by Burgess<26).

The effort of this thesis project has been
devoted to studying the electrical conduction,
switching and inductance properties of oxidized
vanadium foils. These foils(20) have been selected
for investigation for several reasons: First, the
approach used in sample preparation readily lends
itself to existing hybrid, integrated-circuit
technology. Secondly, the foils would cost less to
produce and would be easier to work with than the
reactively-sputtered thin films currently under
investigation by other workers(3'l7). unlike the

majority of the materials being studied by others,

vanadium oxide switches states for relatively low bias
voltages, when the heat sink is at room temperatures.
Finally, since potentially useful inductance effects
have been Observed in vanadium oxide, we wanted to

(3)

follow up the work of Berglund and Walden in order
to establish the feasibility of utilizing this effect

in fabricating useful integrated—circuit inductors.

(3)

The inductance effects Observed by Berglund
and walden possessed several undesirable properties:
The inductance was strongly dependent upon frequency
as well as strongly temperature dependent. In addition,
a series resistance of one thousand ohms was present
at the 6 mA "on-state", bias point. With the
intention of providing additional information concerning
the effects of bias current on inductance, as well
as uncovering possible methods of minimizing some of
the undesirable features of the structure investigated

by Berglund and Walden(3)

, we chose to investigate

the properties of an alternate sample configuration.
Furthermore, we wanted to extend the inductance-
versus-bias-current investigation into the preswitching

region in order to provide additional insight into

the inductance prOperties of oxidized vanadium.

(9'24) solved the governing

BOer and Ovshinsky
nonlinear, time-dependent, thermal-transport equation
through a Fourier expansion of the sample's
temperature T(x,t) into T(x,t) = §D§3eixert.
Although they arrived at a closed-form expression
for the threshold voltage, their analysis has many
draWbacks. Mbst notable is the fact that their
analysis is based on the assumption that
0(T) a exp(BT) over the entire temperature range
of interest, where B is a constant dependent only
upon the material chosen. In reality, 0(T) need
not (and does not) follow this exponential behavior
over the entire temperature range of interest. By
Taylor expansion of the temperature-dependent,
electrical conductivity and with the aid of Laplace
transform techniques, we solved the thermal-
transport equation, obtaining an analytical expression
for the threshold voltage. Our approach, involving
a piecewise linearization of 0(T), can be readily
adapted to a computer analysis, yielding whatever
accuracy is required in the determination of the

threshold voltage.

The relationship between the observed
negative-differential resistance and the formation

of current filaments in a current—controlled switching

bulk device is not well defined; therefore, an
attempt was made to determine if the observed
negative-differential resistance is a necessary
condition for the formation of a current channel

in a bulk device. The criteria established is developed

in Chapter 4.

CHAPTER II

EXPERIMENTAL PROCEDURE AND APPARATUS

2.1 Sample Preparation

 

Under favorable thermal boundary conditions,
many bulk materials, whose electrical conductivity
increases significantly with temperature and then
becomes saturated above some temperature, will

exhibited switching effects when subjected to a
(3.6)

2 "V2°3'

(4. 15)

certain critical bias voltage. V0

(17)
v205

examples of materials which exhibit these properties.

and chalcogenide glasses are

For vanadium dioxide, the abrupt change in
electrical conductivity at 68°C was found(18'19)
to be associated with a phase change from a

(17) Observed

monoclinic to a rutile structure. Futaki
switching in sintered pellets of V02 and Berglund
and walden(3) observed a similar phenomena in

reactively-sputtered thin films of V0 with their

2.
sample configurations in mind, as well as their
sample preparation procedures, we elected an

alternate approach to prepare our oxidized vanadium

samples, a sample preparation techniques and thermal

boundary configuration which proved to offer several

advantages over the others.

A pure vanadium foil of (20) 1 mil thickness
‘was diced into 3 mm.by 3 mm squares. After cleaning
the (TCE), these samples were oxidized in an air
atmosphere at temperatures ranging from 300°C to
600°C for periods ranging from a couple hours to one
week. The oxidized samples were then taken out

of the furnace and abruptly cooled in air.

After many observations, it was concluded that,
if the furnace temperature was maintained above 600°C,
the resultant samples were very brittle and difficult
to handle without fracturing. 0n the other hand, if
the furnace temperature was below 300°C, no matter
how long the foil was heated, no significant oxidation
occured. Strong evidence indicated that the best
furnace temperature for oxidation was around 450°C.

At this temperature, it took one week to oxidize
foils in order to obtain specimens possessing the
desired switching effects. Even after oxidization
for one week, chemical polishing showed that only
the outer section of the vanadium foil was oxidized,
indicating that the oxide layer possibly inhibited

oxidation from proceeding past a certain point.

Unless otherwise stated, all data presented
in this thesis are for oxidized layers prepared at

450°C for one week.

Once the oxidized vanadium layers were obtained,
many of their important prOperties were evaluated.
The rest of this Chapter is devoted to describing
the experimental methods used in determining the
following prOperties of the layer: (1) crystalline
structure of the layer, (2) dependence of the layer's
electrical conductivity on the layer's temperature,
(3) the switching delay time under various bias and
temperature conditions, (4) the inductance prOperties
of the material, and (5) the effects of large-scale

ac signals on the layer.

2.2 X—Ray Diffraction Experiment
The electrical resistivity of the oxidized

layer was estimated to be about 105 ohm-cm at room

temperature. This value is four orders of magnitude
higher than that of vanadium dioxide, and is much
higher than that of V 0 or V 0 (17'19)

2 3 2 5 '
this fact, it is unlikely that the oxidized layer

Based on

was a simple vanadium-oxygen compound. This should
not be surprising since the vanadium foil was heated
in the air without any control of the surrounding

environment.

10

If the layer is crystalline, conventional
band theory should be able to explain most of the
observed electrical effects. If, however, the layer
is amorphous, the prOblem of relating the experimental
results to a theoretical model becomes quite complex.
In order to determine the nature of the oxide layer
a back-reflection, pin-hole, x-ray study was used.
This technique is the same as the powder technique,
except for the type of the specimen. In the
pin-hole technique, sheet specimens are used instead
of powder specimen. The x-radiation used is

monochromatic.

According to Bragg's law, constructive
interference of x-ray will take place when
l = 2d sin 9,
where l is the wavelength of x-rays, d is the

interplanar distance, and 8 is the Bragg angle.

This would provide a sharp diffraction maxima
(lines) for crystalline materials at the Bragg angle.
If the material is polycrystalline, due to the
possible random orientations, these diffraction beams
will emerge as cones. A film intersecting these
cones will show rings, as the pin-hole diffraction

pattern. However, the amorphous materials, no such

ll

sharp peaks will exist due to the lack of long range
periodicy in the structure. These peaks may be

diffused.

The exact experimental configuration used
is described in details in Appendix A. It is worth
noting here, however, that the majority of the
measurements were done on a Strahlenschutzzulassung
PTB-526 x-ray machine, and that the x-ray tube
was operated at 30kV and lSmA with a typical sample
exposure time of 8 hours. The target was copper
and a nickel filter was employed, yielding a
wavelength of l.54°A. Results of the x-ray
diffraction experiment will be described in the next

Chapter.

2.3 Temperature Dependence on Electrical Conductivity
Experiments

The electrical conductivity of most metals
decreases as the temperature is raised. This is due
to the fact that carrier mObility decreases as the
result of increasing collisions with phonons, while
the concentration of carriers is almost unchanged.
0n the other hand, for crystalline semiconductors,
electrical conductivity always increases as temperature

is raised. The increase of electrical conductivity is

12

due to the increase in the density of free carriers
generated from either the impurity bands or valence
band by thermal agitation. For most of the amorphous
semiconductors, electrical conductivity is more or
less temperature, frequency and electrical field
dependent as is the case for ordinary crystalline
semiconductors; however, the manner in which the
conductivity changes is in general different. For
example, measurements of the conductivity of
amorphous semiconductors as a function of the applied
field's frequency shows an increase of conductivity
with increasing frequency, in contrast to band
conduction which exhibits a slow decrease(31).
Measurements of the conductivity of amorphous
semiconductors as function of the applied electric
field show that the log of the electrical conductivity
varies as the square root of the applied electrical
field<31). Evidence also shows that for temperatures
above room temperature, the conductivity of most
amorphous semiconductors has the same temperature
dependence as that of crystalline semiconductors;
however, for sufficiently low temperatures, a variation
in conductivity with temperature of most amorphous
semiconductors is exp(AT - %) instead of exp(BT-l),

where A and B are constants(31).

13

All the above information is critical to
the determination of the conduction mechanisms and
model for amorphous semiconductors. The so-called
CFO model is able to explain many of the above
mentioned Observations(31). Understanding the
temperature dependence of the electrical conductivity
of amorphous semiconductors has another significance:
the switching effects observed in amorphous
semiconductors are critically dependent upon the
ambient temperature of the amorphous layer: furthermore,
it has been confirmed that the heat generated in
the material due to JOule heating plays an important
role in the switching process. Ovshinsky(9) observed
that the critical voltage required to actuate
switching is inversely proportional to the rate of
increase of electrical conductivity with temperature.
Since the main concern of this research effort is
to investigate the switching effects of the oxidized
foils at temperatures in the vicinity of room
temperature, it is absolutely necessary to understand
how the electrical conductivity of the oxidized

layer varies with temperature.

The arrangement for measuring the thermal

dependent electrical conductivity is indicated in

14

Figure 2.1. The layer was held by a spring-loaded
holder, and the holder was placed inside of a
thermos. A thermometer was used to read the layer's
ambient temperature. In order to make sure that

the temperature distribution inside of the thermos
was uniform, as well as to reduce any thermal transient
effects, the temperature was not allowed to vary
faster than one degree per minute. Also, in order
to minimize the possibility of any spurious field
effects, the bias voltage across the unit was never
higher than 0.1V. This value was far below the

threshold switching voltage.

The protection resistor R.p was always chosen
to have a comparable resistance to that of the
switching device after breakdown. Two multimeters,
Fairchild model 7050, were employed to read the
potential drOp VS across the unit and the voltage
Vp across the protection resistor. The resistance

RS of the unit will be

and the conductivity of the layer will be

L L v
C(T) = =———P-——. 2.2
ARS ARp Vs

where L is the thickness of the oxidized layer, and

 

A is the contact area between the electrode and

the layer.

 

+l

 

 

a + To Voltmeter #1
(Fairchild 7050)

 

Trygon HR40 20m

‘VVVV\N--—

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Power Supply Sample
’VVV : To Voltmeter #2
:- Rp b . .
=100kn (Falrchlld 7050)
(a)
Thermometer
E/
: b
.3 % f Trygon HR4O
"Jf Power Supply
3% b———Thermos Bottle
Heating Element
(18 a-30w Resister)
(b)
Figure 2. l. (a) Schematic diagram of the electrical circuit used to

measure the resistance of the sample for various bath
temperatures. (b) The oven used to heat the samples.

16

2.4 Switching‘Delay Time Measurement Procedures

There are several important properties which

determine the usefulness of two-terminal, passive

'switching elements:

1.

"Off-state" resistance--The ideal off-state
resistance of a two-terminal switch is
infinite.

"On-state" resistance--The ideal on—state
resistance is zero for a two-terminal,
resistive-type switch.

Switching delay time--The switching delay
time is the time delay required for the
unit to switch after a command voltage
greater than the threshold voltage has
been applied. In an ideal switching
device, this delay time is zero.
Transient time--The transient time of

the switching device is the interval of
time it takes for the device resistance
to change from 0.1(ARS) to 0.9(ARS),
where ARS is the difference between

the device "off-state" and "on-state"
resistance. For an ideal switching

device, this time is zero.

17

5. Threshold voltage or current--The
threshold voltage is the voltage required
for the device to change states. The
desired magnitude of this voltage would
depend upon the specific application.

6. Hysteresis effects--Hysteresis effects
determine the voltage and current required
to maintain the switch in the "on—state"
once switching has occured. In many
applications, it is desirable that no
"holding" voltage or current be required
in order to maintain the switch in the

on-state.

As to the time duration of the switching process,
this may be the time required for the molecules of the
material investigated within the thermal filament to
reorder their structure. This time also may be
affected by the external load resistance and the
equivalent circuit capacitances and inductances. The
true mechanism underlying this switching action is still

uncertain.

The time delay in amorphous-semiconductor,

(9) related to the

switching devices is apparently
time required for the material to reach some critical

temperature. Once this critical temperature has

18

been reached, an unknown thermoelectric process is
activated: the state of the material changes abruptly
and a current channel forms inside the layer. Since
certain applications may arise which require hundreds
of switching elements to perform a single function,
if the switching delay time and the transient time

of each composite switching element is too long, the
total switching time for the system would be

prohibitively long.

From many observations of the investigated
layer, it was concluded that the transient time
is much-much shorter than the delay time required
for the switching action to occur. Hence, the
total switching time is limited by the delay time.
This is consistent with many reports from other
workers who have observed(4'15) the transient time
in various amorphous switching elements to be around
10-9 second. Since one of the main concerns of
this research effort is to investigate the preswitching
behavior of the oxidized vanadium foils and to
determine the limitations on the various switching
properties, the time delay for the unit to initiate
switching after the application of the command signal
must be thoroughly investigated in order to determine
exactly what practical limitations are placed on this

delay time.

19

The experimental arrangement for measuring
the delay time as a function of both ambient
temperature and applied voltage is depicted in
Figure 2.2. The layer was held by a spring-loaded
holder and was placed inside of a thermos so that the
ambient temperature of the device could be closely
controlled. A repetitive voltage step was applied
to the device. The switching delay time was measured
by comparing on a Tektronix Type 851A, dual-trace
oscilloscope the input signal with the system
response. Using this arrangement, the dependence of
delay time on ambient temperature and applied voltage
was determined. Results of this investigation are

presented in the next Chapter.

 

2.5 Inductance Measurements
Inductance effects have recently been observed

(3)

by Berglund and Walden in sputtered, vanadium-
dioxide, thin films. All of their inductance
measurements were made well above the switching
threshold. Although they observed inductances on
the order of lOmH for a 6mA bias, their thin-
film inductors possessed several undersirable
prOperties: The inductance was strongly dependent

upon frequency as well as strongly temperature

dependent. In addition, a series resistance of over

20

+

 

4T0 External Trigger
(Tektronix 581A Oscillosc0pe)

 

+

 

 

 

 

Tektronix 114
Pulse Generator

 

 

+ .
flTo Channel A

Rp =IOk-Qa +
To Channel B

Sample

 

 

Figure 2. 2. Schematic diagram of the electrical circuit used to measure
and display on an oxcillosc0pe the switching delay time for
various sample temperatures. The oven arrangement is the
same as that depicted on Figure 2. l.

21

one thousand ohms was present at the 6mA bias point.
All of their results were successfully explained in
terms of the thermal—transport properties of the

material.

Inductance is an important electrical property
of an electrical element. The inductance may be
detrimental to a switching device because it is a
parasitic parameter which may severely degrade the
switching time. On the other hand, in steady-state,
ac applications, high Q inductors are always a welcome
cirucit element, particularly if the inductive element
is compatible with integrated circuit technology. The
inductance properties exhibited by the layer
investigated thus have far reaching significance in

the modern integrated-circuit applications.

With the intention of providing additional
information concerning the effect of the bias current
on inductance, as well as uncovering possible methods
of minimizing some of the undesirable features of the

(3),

structure investigated by Berglund and Walden an
experimental investigation of these inductance

properties was undertaken.

22-

 

 

 

 

Detector
DC
Ammeter
Sample
‘I
I! R A
CB=82pf AC Signal
1.5V
9v -
m
CT
CB: 82 pf

 

///////

Figure 2. 3. Modified GR1650A bridge used in measuring the impedance
of the samples.

23

Two oxidized foils, with one side of each
chemically polished and silver painted, were sealed
to electrodes of the sample holder. The electrodes
were much more massive than the layer: consequently,
they served as infinite heat sinks. Since the
inductance of the sample was affected by both the
dc current flowing through the layer and ac signal
frequency, a systematic method of measuring the
layer's inductance under various dc current and
ac signal frequency was employed. By using a modified
General Radio Type 1650—A Impedance Bridge, the dc
bias current I was varied from 0 to 15 mA. The

B
ac signal current I was held constant at

S
approximately lOOu.A, while the frequency of the
ac signal was varied from 100 Hz to 25 kHz, the
approximate normal Operating range of the bridge.
The circuit employed is depicted in Figure 2.3. All
inductance measurements were made at room temperature.

Results of the investigation are described in the

next Chapter.

2.6 Experiments 0f Response To 600-Hz Lapge-Scale Signals
An electronic switching device would be a very

limited use if its prOperties change permanently as

a result of the switching action. An experiment was

designed, therefore, to evaluate the effects of many

24

switching cycles on the properties of these oxidized

layers.

The experimental arrangement is indicated in
Figure 2.4. The peak value of the input potential
was 9 volts, which is a little higher than the layer's
threshold voltage at room temperature. A sinusoidal
voltage was applied continuously to the unit over
the period of several days without noticable changes
taking place in the device's electrical characteristics.
It is concluded from this experiment that the switching

effect exhibited by the layer is non—destructive.

Detailed results of this aspect of the

investigation are presented in the next Chapter.

25'

 

L, To Channel A

 

 

+ Rp=10k .0. of Oscilloscope
HP 20‘.) CD +4, To Channel B

Audio of Oscilloscope
Oscillator

 

 

 

Sample

 

 

 

l;—

Figure 2. 4. Schematic diagram of the electrical circuit used in evaluating
the useful lifetime of devices when the devices are under
the stress of continuous, large-scale, AC signals.

 

CHAPTER III

EXPERIMENTAL RESULTS

3.1 Structure Properties of the Unit

Figure 3.1 shows the back-reflection, pin-
hole x-ray diffractiOn photograph of a pure vanadium
foil. A broken ring of radius 22 mm shows up,
indicating that the vanadium foil is polycrystalline.
The magnitude of the radius of this ring is what
we expect for polycrystalline vanadium. Figure 3.2
shows the x-ray picture of the oxidized vanadium
foil. In this photograph, no such ring is present;
on the contrary, the whole central zone of the film
appears to be exposed. This strongly suggests that
no sharp, coherent reflections of the incident
x-ray beam took place, indicating that the oxidized
layer was amorphous. Figure 3.3 shows another
x-ray picture of another oxidized vanadium foil;
a ring with a radius of 30 mm is present, as well
as the hazy background evident in Figure 3.2. This
photograph was taken using a different x-ray machine,
different oxidized vanadium foil, and different
distance between the film and the object, but other-
wise under identical conditions. This ring was

confirmed to be the result of the sharp coherent

26

 

 

foil.

ium

ray pattern for a pure vanad

x

-ref1ection,

Back

1

3

Figure

 

Back-reflection, x-ray pattern for an oxidized vanadium

llustrat

Figure 3. Z.

the amorphous nature of the oxide layer.

mg

i

11,

£0

28

 

Figure 3. 3. - Back-reflection, x-ray pattern for the oxidized
vanadium foil, illustrating the presence of both the
amorphous oxide layer and the pure-vanadium foil

beneath the layer.

29

diffraction of the incident x-ray by the pure
vanadium, indicating that the foil was not completely
oxidized. This is not surprising since the oxidized
layer was estimated to be around 1 micron thick.

The incident x-rays penetrated through this thin
layer and got reflected by the inner unoxidized
material. Besides this ring, which can be directly
attributed to pure vanadium, no other rings are
Observed in the photograph. This suggests that the

oxidized vanadium layer was in the amorphous state.

The oxidized layer will be primarily an
assortment of vanadium—oxygen complexes. There
will be a negligible amount of nitrogen-vanadium
and hydrogen-vanadium complexes present in the layer
because the solubility of hydrogen and nitrogen in

vanadium at 475°C is very low(32-34).

3.2 D.C. Conductance

 

Typical electrical conductance of the layer
in the temperature range from 25°C to 100°C is shown
in Figure 3.4. As indicated, the electrical
conductance is strictly an increasing function of
ambient temperature in the above mentioned temperature
range. This behavior can be approximated by the expression

0(T) = Go exp B(T-To) 3.1

‘3 O

 

 

 

 

l)

20 -

10 -

b be 8 _

a) 6 -
U
C.‘

m _
‘9

-c: 4 ‘
a.
O

O _
O)
.2
‘23

'23 2 ~
D:

1 1 l 1 l
20 4O 6O 80 100

Bath Temperature ( °C )

Figure 3. 4. Temperature dependence of a typical oxidized vanadium

' ' < < ,
£011 5 conductance. VBIAS Vc

31

for T near To' where T0 is the ambient

temperature, B = —A%-, and AB is the introduced

kT
energy-gap-like quantity. AB is strongly temperature
dependent and is estimated to be around 0.3 ev at

room temperature for the layer investigated.

Since pressure exerted on the oxidized layer
by the spring-loaded electrodes will alter the
layer's energy gap and thus its conductivity, all
the conductivity measurements were made under the

same pressure of 12 pounds per square inch.

Another approximation made in all the
measurements was that the junction resistance
between the layer and electrodes was negligible
and independent of temperature. This is because
contacts were ohmic and independent of the contact
materials. This ohmic behavior of the contacts can
be explained by tunneling through electrode barriers

(7)

of small screening length in highly disorder materials .

As mentioned earlier, conductivity is affected
by the strength of the applied electric field. If
the bias voltage is far below the threshold value,

the field effects will only be minor.

32

In Figure 3.5, the temperature dependence

of threshold voltage VC' threshold current IC

and threshold power PC required for the unit to

actuate switching in the temperature range from 25°C
to 100°C is summarized. Also shown in the temperature
dependence of holding voltage VH and holding

current IH for the unit to sustain its "on" state.
From this Figure, it is obvious that the threshold
current IC is almost temperature independent,

while the threshold voltage VC drOps down almost
linearly with increasing temperature. The explanation
of this is that the higher the ambient temperature,

the smaller the incremental temperature increase
required to bring the layer to the critical temperature

T where T the temperature required for the

c' c

layer to switch states: also, the higher the ambient
temperature, the higher the temperature rate of the
increasing electrical conductance of the layer:
consequently, as the ambient temperature increases,
PC and VC decrease, the amount of decrease being
strongly dependent upon the thermal boundary
configuration. An analytical expression relating
threshold voltage VC and the bath temperature will

be develOped and discussed in the next Chapter.

33

 

 

 

 

l4

 

l2

 

 

 

 

IO
m
.t.’
C:
D 3
>5
#4
S
.4.) 6
"-0
.0
H
< 4_ \x \\ A
\x‘ \‘ \
\“‘ \\~A
\‘\\ \
2". \\\\
1 1 1 l l 1 1 ,,
30 40 50 60 7O 80 90

Bath Temperature 0C.

Figure 3. 5. Temperature dependence of a typical sample's threshold
current Ic, threshold voltage Vc, threshold power Pc, holding
voltage VH and holding current 1H.

34

3.3 Typical I—V Characteristic Curves

 

A typical characteristic I-V curve is shown
in Figure 3.6. The main features of the switching
phenomena shown are the following: (1) The I-V
characteristic is symmetrical with respect to
the reversal of the applied voltage and current.

(2) The same symmetrical switching characteristic

is sustained when the electrodes are of different
contacting area or of different materials. However,
as the contact area is changed, the critical voltage
and current and holding current and voltage may
change. (3) In the highly resistive "off" state,
the material is ohmic, as the bias voltage increases
from zero up to some critical value. The typical
resistance of the unit in this region is around

10 kohm. (4) When the applied voltage exceeds the

threshold voltage V the unit switches along

C’
the laod line to the conducting state. During
this transitional state, negative resistance
sometimes is evident. (5) In the conducting "on"
state, the current can be increased or decreased
without significantly affecting the voltage drOp,
termed the holding voltage VH. In this "on"

state, the dynamic resistance is quite low (around

100 ohms). Typically the on-off resistance ratio

35’

 

Figure 3. 6. Typical I-V characteristic curve of an oxidized vanadium
layer. Horizontal scale is lV/div, and the vertical scale
is lmA/div. Bath temperature is 24" C.

36

is around 100. (6) As the current is reduced below

a characteristic value termed the holding current IH'
the unit rapidly switches to the original high
resistive state along the laod line. (7) The switching
process is repeatable. A series of pictures showing
the ambient temperature effects on the I-V character-
istics are given in Figure 3.7. The protecting
resistance in taking these I-V characteristic curves
was 200 ohm. The pictures were taken at ambient
temperatures 25°C, 35°C, 45°C, 55°C, and 60°C
respectively. From this experimental data, it is
Obvious that the threshold voltage is strongly
temperature dependent while the critical current is
essentially temperature independent. Reasons for
this behavior will be extensively discussed in the

next Chapter.

3.4 Switching Delay Time Constant

The response of a typical oxidized vanadium
layer to a squaredwave voltage of various amplitudes
is shown in Figure 3.8. All the experiments were
performed at 23°C: the protecting series resistance
was 10 kohm in each case. It was found that the
higher the bias voltage, the shorter the delay time

required for the layer to switch. Typical data

37

 

Figure 3. 7. I-V characteristics for a typical sample at (a) 23° C,
(b) 35° C, (c) 45° C, (d) 50° C and (e) 60° C. Horizontal
scale is lV/div, and the vertical scale is lrnA/div.

39

Table I

Delay Time Data for Various Applied Voltages
as Depicted from Figure 3.8.

Voltage Applied (Volts) Delay Time (Microsecond)
4.7 ----------------- 10
4.8 ................. 8
5.1 .................. 6
5.6 ................. 1,
Table II

Delay Time Data for Various Ambient Temperatures
as Depicted from Figure 3.9.

Ambient Temperature (°C) Delay Time (Microsecond)
26 --------------------- 7.0
30 -------------------- 4.5

60 -------------------- 4.0

39

depicted from the series of pictures are tabluated

in Table I

It was also anticipated that the delay time
should be strongly affected by the ambient temperature.
Figure 3.9 confirms this fact. Here once again the
protecting resistance was 10 kohm. From the
oscillosc0pe traces shown, one observes that the
delay time decreases with increasing ambient

temperature. Results are tabulated in Table II.

Figure 3.10 shows another oscillosc0pe
trace of the response of the sample to the same square
wave excitation except that a protecting resistance
of l kohm is used instead of 10 kohm. Comparing
Figure 3.8 with Figure 3.10, one notices that the
delay time of the investigated oxide layer in the
former case is much longer than that in the latter
case. This can be explained as follows: If RS
is the equivalent series resistance of the oxide
layer, Rp the protecting series resistance and
e(t) the applied square wave potential across both
RS and RP, then the instantaneous power p(t)

dissipated in the oxide layer will be

 

2
p(t) = (ii-ELF R = e(tl l 3.2
Rs+Rp S Rs (1+Rp/RS)2

40

 

   

I
rlii ._
a
,\
T

n a

 

fiwmcnm w. m. Womoovmo OH m gowns? 93%qu 485%ch H323. no u. mocwnolfiwfiw <oHnwmo. .25 wgowwncmm
Om are Home mwmbww we 3; e. .u <. Gov e. m <. on m. H < mom av m. o <. moauobeww mambo wm
no I moo\&<.

‘H

 

Figure 3. 9. Responses of a typical, oxidized vanadium layer to a
square-wave voltage for both temperatures (a) 26° C,
(b) 30° C and (c) 60° C. Horizontal scale is 10p. sec/div,

and the vertical scale is 2. 5 V/div.

42

Equation 3.2 implies that for a fixed value
of RS. the instantaneous power dissipated in the
oxide layer will decrease as Rp in increased. For
a typical oxide layer investigated, RS was around
10 kohm, while Rp employed in obtaining Figure 3.8
and 3.10 was 10 kohm and 1 kohm respectively. It
is obvious that there will be more power dissipated
in the oxide layer for the case depicted in Figure
3.10; consequently, the delay time for the oxide
layer to reach the critical temperature in order to
actuate switching will be shorter for the case of
Figure 3.10. This fact lends direct support to
the proposed theory that the observed preswitching

behavior has a thermal origin.

3.5 Regponse to A.C. Signals

 

The input signal is of 600 Hz with peak
amplitude of 9 volts as shown in Figure 3.lla. The
response of the unit to this signal at room
temperature is shown in Figure 3.11b. It was found
that the unit investigated can stand millions of
switching cycles without noticable changes in its

electrical characteristics.

43 __i_

    

101.;
sec

Figure 3. 10. Response to a square-wave voltage for a bath temperature
of 26° C with a protection resistor R1=lkfl instead of the

normal 10kt).

_l
7

 

_l
7

Response of a typical sample to a large-signal, 600 Hz

Figure 3. 11.
applied voltage. (a) Input-voltage waveform, and (b)

corresponding voltage drop across the sample.

44

3.6 Inductance PrOperties
The inductance effects observed by Berglund

and Walden(3)

in sputtered, vanadium dioxide thin
films were strongly dependent upon frequency as
well as temperature. All their inductance measurements
were made well above the switching threshold, and
were on the order of 10 mH for a 6 mA bias. Since
the switching and the inductance properties of
amorphous semiconductors are a direct consequence
of the material's thermal properties, and since
the temperature of any volume element is strongly
dependent upon the element's thermal boundary
conditions, the thermal boundary of the material
as a whole play an important role in ultimately

establishing the gross electrical properties of

the material.

The ac electrical properties of oxidized
vanadijm layers were measured as a function of both
frequency and dc bias current using a modified
General Radio Type l650—A Impedance Bridge. The ac
signal current IS was held constant at 100
microampeves, while the frequency of the test signal
was varied from 100 Hz to 25 kHz, the approximate
normal operating range of the bridge. Inductance

versus frequency data for various dc bias levels

45

is given in Figure 3.12. In Figure 3.13, the

effect of dc bias current on inductance is illustrated.
The ac series resistance RS versus frequency data
and bias current data are illustrated in Figure 3.14
and Figure 3.15 respectively. It was found that

the ac series resistance Rs, like the inductance,
was strongly dependent upon the bias point but
independent of frequency. It proved to be very
difficult to make any inductance or ac series
resistance measurements for bias currents between

0.3 mA and about 4 mA because when the load line
was adjusted so that the bias current would be

stable between 0.3 mA and 4 mA, the Q of the circuit
was too low to make meaningful measurements. Berglund

and walden(3)

proved that if the thermal hysteresis
effects were absent, one would expect the inductance
to be independent of frequency. For the sample

investigated, no noticable thermal hysteresis effects

were independent of frequency.

Inductance ( mH )

4-6

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

If 0 mA
v 4? <> o—-C !o———-o——-c 0 £1
102:
6 1: If .1 mA
t ‘J‘v— ’17 0+ —o——o-—o—-G +
4? W <9—
.. If .3 mA
2 _
10':
6 C
2 r-
L 43% ’H 0 CT 0 O O 1821-0 mA
100 1 I 1 111111 IWA—L 1H1?11?Ia.=1,5mA
2 3 4
10 .10 10

Frequency ( Hz)

Figure 3. 12. Frequency dependence of inductance for various dc bias

currents.

47

 

 

 

2 ..
102 f =10 kHz
6 Is: 100 HA
T°= 295°K
E 1
a 2 3.
v 1
o 1
g 10' '
«1 r1
8 -\
4 _ \
\
\
2 - \ \ \
100 1 1 1 1
4 8 12 16

Bias Current (mA)

Figure 3. l3. Dependence of inductance on dc bias current.

49

 

 

 

m

x

M .. acumen. W

C

n

a

.s.

.m N-

e

R

C

A l
_ c _ _ _ _ .c __ . _ . .i
. N u resume no 8.88

3.2393 Sammy

mdmcno w. 3. Tennessee. dependence 0». «so mngumwmsmr >0. mmnwmm nomwmnwsno.

(A)

Resistance

AC Series

49

 

 

 

1
f= 10 kHz
15:100 HA
T°= 295°K
1
— 1
I
1
2 h 1
I
102 i
Z 1
_ 1
6 Z ‘, °
” 1
f 1
- 1 1°
1 1’
2 — 1|,
1
I1
10] l 1 "4% 1 j l l I = .
.1 .2 .3 4 8 12 16 .18

Bias Current (mA)

Figure 3. 15. Dependence of the small-signal, series resistance on dc
bias current.

CHAPTER IV

INTERPRETATION AND DISCUSSION

4.1 Introduction

Ovshinsky(4) and Walsh(8) have shown that small
samples of certain chalcogenide glasses may be made
to switch reversibly from a low-conductance state
to high-conductance state through the application of
a voltage above a certain critical value. Berglund(3)
also found that vanadium dioxide films undergo a
phase change at 68°C and can be made switch if the
bias voltage is above a certain critical value. Baer
and Ovshinsky(9) have indicated that the preswitching
nonohmic behavior may arise from Joule heating within
the material and that the switching is a current—
controlled thermal breakdown. Joule heating causes
the temperature in a semiconductor to rise. Under
usual circumstances, the temperature increase will
be stable when the heat generated is balanced by the
heat dissipated. Since the temperature profile is
determined by the geometry of the semiconductor and
its surroundings, the threshold voltage required for
the device to generate enough heat and therefore
bring about thermal breakdown will be strongly

configurational dependendent.

SO

51

Generally there are three principal types
of thermal boundary configurations currently employed
by most Observers; these are depicted in Figure 4.1.
Futaki used the Pellet Configuration in investigating
the conductivity versus temperature properties of
sintered pellets of various vanadium oxide complexes(l7).
Materials investigated using this geometry would be
expected to evidence large thermal transient effects
due to the relatively poor contact made by the sample
and the heat sink. The Planar Configuration was
employed by Berglund and walden(3). For this
geometry, moderately good thermal contace can be
made with the heat sink provided the thickness of
the sample and the electrical insulating layer

between the sample and the heat sink are made

sufficiently small.

Ovshinsky, et al. have investigated the
electrical properties of various amorphous chalcogenides
using the Sandwich Configuration(9). Because of
the direct contact made by the sample with the metal
electrodes, this configuration affords the best
possible thermal boundary conditions in terms of
guaranteeing minimum thermal transient effects. The

presence of the high—conductance, low-field "on state"

is not entirely due to thermal effects: however, the

52

PELLET CONFIGURATION

Active Material \

  

 

 

Metal
7 Electrodes

 

 

PLANAR CONFIGURATION

% Metal Electrodes

_ Active Material
Electrical Insulator

/////7H=11n ,','/////////

SANDWICH CONFIGURATION

   

 

 

/IIIIIllll/lll/Il/I/I/I/I/l/I/

Active Material

MillII7IIIIIIIIIIIIIIIII[[1777

 

IIIIIIIIIIIIIIIIIIIIIII

Figure 4. 1. Three possible thermal-boundary configurations.

53

rapid and reversible transition between the highly
resistive and conductive states can have a thermal

origin from Joule heating of a current channel.

In the next section (Section 4.2). the
thermal transport equation governing the temperature
distribution within the oxidized vanadium layer
will be solved subject to boundary conditions
apprOpriate to the sandwich configuration. An
expression for the critical field required to
actuate switching from the low-conductance state
to high-conductance state at various bath temperatures
is Obtained. This expression for the critical

field is then compared with experimental results.

In Section 4.3, the technique and approximation
made in the analysis of the Sandwich Configuration
is extended to the planar configuration. The
threshold electric field requires for this
configuration to actuate switching is also obtained

and discussed.

In the last section (Section 4.4), by totally
neglecting the effects due to transverse temperature
gradient as did by most observers(ll'26), a criteria

for Observing current-controlled, negative-differential

resistance in a bulk, thin-film devices employing

54
the planar configuration is established and discussed.

4.2 Temperature Distribution and Switching Threshold

Voltage of the Device--Sandwich Configuration

4.2.1 Geometry

The first structure of interest (Figure 4.2)
is a thin, semiconducting sandwich of comparatively
infinite extension in the sandwich plane to the
thickness of the sandwich layer, having the temperature
dependence of conductivity given by Equation 3.1.
This configuration has gained significant pOpularity
recently since the results can be applied to the
D07-0vonic threshold switch (OTS)(9). The electrical
conductivity, specific thermal conductivity and
specific heat capacity of the semiconducting layer

and electrodes are 0 , k , c and o k

s s s d' d' cd'

respectively. The temperature distribution in the
semiconducting layer of thickness LS and that in
the two highly conductive electrodes are determined
by solving the thermal transport equation

C %%-- div(k grad T) = 3:E 4.1

where 3-E is the heat production rate and, J

and E are current density and electric field within

the sample respectively. This source term was

55
balanced by two others, C%%, the heat accumulation
rate and, div(kgradT), the rate of heat energy
conducted away to the bath through the sandwich and
highly conducting electrodes. No heat radiation
is concerned here. Since what is of interest here
is the preswitching behavior of the oxidized
layer before any thermal filament is formed, and
since the dimensions of the sandwich plane in both
y and 2 directions are much larger than the
thickness LS of the sample, it is reasonable to
neglect all thermal inhomogeneties within the
sandwich plane and consequently only the relevant
x-direction (normal to the contact area) in
Equation 4.1 need be considered. The origin of
this coordinate (x=0) shall be at the center
plane of the sandwich (Figure 4.2). For simplicity,
all bulk property parameters ki’ Ci (i=s,d) will
be assumed spatially constant and field, temperature
independent. The electrical conductivity 0d of
the electrodes will be so high that no significant
JOule heat is generated inside. Also the electric
field E inside of the sandwich layer will be
assumed space-independent, since space charge
effects will be neglected. Thus, the thermal

balance equation will be

56

2

 

 

B T (x,t) 8T (x,t)

ks $2 - CS -—€§:————-+ EZOS(TS) = 0

Ex
or
2
a TS(x,t) _.l; BTS(x,t) + E2 08(TS) = 0 4 2
2 K at K i

6x 5 s

for le _<_-éS—.
and

2
a Td(x,t) l 8Td(x,t)

-———a—E-——,=O 4.3

 

L
for |x| 235—.

A Ks Kd
where K = ——- and K -—— .
5 CS d Cd

TS and T are the temperatures of the semiconducting

d
layer and its electrodes respectively above that of

the heat sink substrate.

4.2.2 Boundapy and Initial Conditions

Equations 4.2 and 4.3 are time-dependent,
second-order partial, differential equations. It
takes two initial conditions and four boundary
conditions to determine the six integral constants.
The first two initial conditions will be Ts(x) = 0
and Td(x) = 0 at t=0; that is, before applying
a bias voltage, the temperature is everywhere the
same as that of the bath. The four boundary

8T
conditions required will be: (1) —S§-= 0 at x=0

57

for all time t. The highest temperature will be
assumed at the central plane where x=0. (2) The
second boundary condition imposed is that Ts(x,t)
be continuous for all time t across the contact
plane between the layer and electrodes. This is
because if Ts(x,t) is not continuous, then there
would be an infinite heat sink existing in the
contact plane. This is impossible. (3) The third

3T (x,t)
boundary condition is that ———-————- must be

Bx
continuous for all time t across the contact
plane. This is because if Ts(x,t) was not
continuous, there would be an infinite energy source
existing on the contact plane as in condition (2):
this is impossible. (4) The final condition
imposed is that the system's temperature drOps to
the bath temperature T0 at le = LS/2 + E where
6 is a small positive quantity which can be as
small as one wishes. That is, the temperature of

the system goes down to To in the electrodes

just over the junction plane.

Equation 4.2 and 4.3 are nonlinear, time-
dependent, second-order, partial differential
equations because of the nonlinear dependence of
electrical conductivity on temperature. There will

be no exact analytical solutions for these equations(23)

58

Semiconductor

Electrode

. , :p'
. . 1 (p.
‘ , .. ‘1'. - .
1' A

 

OU’kd’Cd/

Y

 

O'kC

S’S’S

/

Electrode

//

 

//. /

Z 5

Figure 4. 2. The sandwich configuration under investigation.

59

If instead, one assumes that 0(T) is field independent
and can be expanded into Taylor series around ambient
temperature To’ to the first approximation, one
will have

0(T) = 0(To) + 0T(TO)(T-TO). 4.4
where

_ £2
°T(To) ‘ BT|T=TO°

This kind of linearization of the electrical
conductivity 0(T) is justified by the fact that
the electrical conductivity of the semiconducting
layer investigated is relatively weakly temperature
dependent around room temperature. In addition,
one is only interested in the thermal behavior of the
system near ambient temperature To' the above
treatment will give us sufficient information. If
higher order terms are needed, than a computer

analysis would be required in order to solve the

nonlinear, differential equations.

Now after substituting Equation 4.4 into
Equations 4.2 and 4.3. a set of two ordinary second
order time dependent partial differential equations,

as shown below, is ready to be solved:

52TS(x,t) 1 aTS(x,t)
2 " 'K_' at + BsTs(X't) = -As 4'5
Bx s

 

60

 

 

L
for le g,;f .
521' (x t) an (x t)
and d ' __I_ d ' =0 46
8X2 Kd at
L
for IX] 2'—2§'o
2
E o (T ) 2
where Bs --——€%—Jl— "As E 0(T0) 4.6
5 KS

Note that this method of approaching the problem
is different from that by Bder and DOhler‘24). Instead
of a series expansion of electrical conductivity 0(T)
into temperature T, they expanded temperature
function Ts(x,t) into Ts(x,t) = gigie-ixxeat,
and solved the eigen equations. Their method proves
to be tedious and lengthy, and the assumptions they
make are buried in the analysis in such a manner that

it is difficult at times to determine the conditions

under which the analysis is valid.

4.2.3 Derivation and Results

 

By taking the Laplace transform of Equation

4.5 and 4.6, and by inserting the initial conditions,

one obtains for Ix] 3.7?

azvs(x.p) p A
2 - (E— - BS)VS(X'p) = - _ 4.7
8x 5

 

61

L
and for [XI 2.2?

 

a Vd(X.p) P
2 (“K“)Vd(X.P) = o. 4.8
ex d
where
v (X,p) = I” T (x,t)e-ptdt 4.9
S O S
and
A m _
Vd(X,p) = j Td(x,t)e ptdt; 4.10
O

p is some positive number such that the above

integrals converge.

From Equations 4.7 and 4.8,

L
for IX] 3-7?

 

qu -qu As
Vs(x,p) = o(p)e + E(p)e + 2 . 4.11
pqs
Ls
and for IXIVZ if L L
s s
‘qd(x‘ 2) qd(x’ 2)
Vd(x,p) = D(p)e + E(p)e , 4.12
P-K B
2 2 P
where q = -——§—§-, q ='—- and 0(p). 5(P). D(p).
s Ks d Kd

E(p) are all constants to be determined by

invoking the boundary conditions.

From boundary condition (1), one has

 

01(1)) = D(p) 4.13
From condition (2) and Equation 4.13, one has
LS A
2 e(p) cosh qS 3? + 82 = D(p) + E(p) 4.14

pqs

62

From condition (3), one has

L
2q8a1p1 sinh qs 3°- = -qS[D(p) - 31pm 4.15

From condition (4), one has

 

D(p)e + E(p)e = 0 4.16
Now, from Equation 4.16,
-2qd6
E(p) = -D(p)e .
and from Equation 4.15,
Ls
-2q a(p) sinh q -—-
D(p) = S 2 § 2 4.17
- qde
qd(1 + e )

Substituting D(P). E(P) into 4.14, one obtains

 

 

_ -As qd
2 2[ cOSh E§.+ sinh E§(l:§——-E—ii 4 18
qu qd qs 2 qs qs 2 -2q t °
l+e d
Putting Equation 4.14 into 4.11 yields
As L3 L3 l-e-2qde
V (x,p) = [l - q cosh q ——-+ q sinh q -—G——:f———J]

s pq: d s 2 s s 2 l+e 2qd€

4.19

The corresponding time dependent temperature distribution

function Ts(x,t) will be

TS (x,t) = m E-im VS (x,p)e dp 4.20

where a should be chosen so that no poles of Vs(x,p)

63

be to the right half plane. By taking the integral
along a closed path with radius R going to infinity,
and with the aid of complex variable integration

technique, it is easy to show that

T (x,t) = Z limt[(p—p )eptv (X,p)] 4.21
S n n S

pepn

640

where pn is the pole of Vs(x,p).

Due to the difficulties in finding the poles
of VS in Equation 4.19, it will be wise to take the
limit first in Equation 4.19 as 6 4 0 before the
inverse transform of Vs(p,x) is taken. Doing this,

one Obtains

s qd cosh qsx

 

 

limt V (x.p) = limt [1 - -
640 s 640 pg: h Ls . h Ls(l-e quE
q cos q -—+q Sln q -—- _
d s 2 s s 2 l+e 2qd€
A cosh q x
= __3. [1 _ ____.%_] 4.22
pqs coshqu?

After taking inverse Laplace transform of the
above equation (see Appendix B), one obtains the desired
time dependent, temperature distribution function

Ts(x,t) inside of the semiconducting layer as

 

 

 

 

 

 

 

( ) cT(To) [E l
T xt —
s ' o (T 5 _ 2 2
T o n—l LS rE OT(TO) - (2n-1)2n2 ]
4w L K L2
K S 5
_§. 2 ( ) 22
C ' E G T
[(_1)n l(2n-l)e s ( E 0 (2n-%) n )t
s L
s
_ (2n-l)wx
- (-1)n 1(Zn-1)] cos
L
s
K
_..§. 2
_ 4 E3 (-l)n ecs [E OT(TO) _ (Zn-l 2"2]t
N n=1 2n-l K L2
5 s
cos—QEE-l-Ei-ll 4.23
3

4.2.4 Discussion

 

The time dependent temperature function at
x=0 for various bias voltages is sketched in Figure 4.3.
It is Obvious that the temperature inside of the
oxidized layer TS. will increase without bound with

time if

 

2
E o (T ) 2 2
——T-—9- — 52”“1) 7' > o, 4.24
k 2 -
3 LS

where n is a positive integer.

If one takes n=l, the predominate term, one

sees that if the electrical field E is higher than

65

 

 

 

 

 

C
II
>4
‘3 V V
4> 3
s ’ Vazvc
5
.1.)
CU
s...
8.
E.
Q)
[—1
.2
g V2<VZ2
c2 JA<V2
Time

Figure 4. 3. Illustration of the sample's temperature at x : 0 as a function
of time for various bias voltages.

66

T)" .
Ls OT(To) , the whole system Will have a thermal

instability such that electric switching will occu

  

0n the other hand, if E is lower than 1L-

temperature Ts in the layer will be stable and
bounded: no switching effects will be expected to
be Observed. The desired threshold voltage required
for the device to switch at ambient temperature

To will thus be
1/Ks
c s c oT(To)

The threshold voltage is related quite
simple to the thermal conductivity kS and the
rate of change of electrical conductivity with
temperature. The higher the thermal conductivity
kS is, the higher the threshold voltage VC
required. This is because the heat generated inside
of the layer will be conducted away more quickly
if k8 is higher. Therefore, for the unit to
have thermal instability and in order for switching
to be initiated, the bias voltage should be high
enough to generate sufficient Joule heat to overcome

those conducted away to the bath.

67

From the expression for Vc given in
Equation 4.25, one observes that the threshold
voltage V6 ‘will be affected by the bath
temperature TO since OT(TO) is dependent upon
the bath temperature. For the amorphous semi-
conductors of interest, UT(TO) is a strictly
increasing function of T0 in the vicinity of
room temperature: therefore, Vc will decrease
as ambient temperature T0 is raised. This is

consistent with.what other workers have Observed(3-4).

The temperature dependence of electrical
conductivity of the samples investigated was shown
in Figure 3.1. Based upon this electrical
conductivity data and Equation 4.25, it is natural
to expect that the required threshold voltage VC
for the sample decreases with increasing bath
temperatures. Figures 3.7 shows measured
thermal dependence of the threshold voltages at
various bath temperatures. It is obvious that

they are qualitatively in agreement with the theory.

From Equation 4.25, VC at To', To", will

be

, _ k

68

z k
and V (T ") = w ’ u 4.27
c o oT(To

respectively.

Also from Equation 3.1, it is easy to show

 

 

that
oT(To') -4(To' - To")
———TO’T (Ton = exp. 1m 4028
Thus
VC(TO") OT(TO') -2(To"-To')
W — W — exp. 100 . 4.29

Figure 4.4 depicts the calculated thermal dependence
of the threshold voltage VC from Equation 4.29,

in the temperature range from 300°K to 440°K

for the case To. = 300°K. For comparison, the
normalized threshold voltage Observed in experiments

was also shown. One concludes that basically the

results are consistent with the theory.

The rise time of the temperature inside of

the oxidized layer will be, from Equation 4.23,

 

E§(E_EI£391._ E39
c k 2
s s L

s

for n=1. This is not exactly the switching delay

time but is in proportional to it.

Critical Switching Voltage (Normalized)

69

   
   

 

 

ll
1.0 ~
,_ - - - - Theoretical
.8 _ Experimental
.6 —
_ o\
.4 — ‘°
.2 L
23 30 4o 50 60 70 T

Bath Temperature (°C )

Figure 4. 4. Critical voltage required to actuate switching for various
bath temperatures. (At 23° C, the switching voltage was 4. 2V. )

70

From this eXpression, it is evident that

the switching delay time T will be affected by

d
both the applied bias voltage and the bath temperature
in the following manner: An incremental rise in

the applied voltage will cause an incremental

decrease in the time required for thermal run-

away to be initiated. This is because the rise

in applied voltage causes an incremental rise

in the instantaneous power being dissipated in

the sample; consequently, the temperature of the

sample will rise more quickly and thermal run-

away will be initiated earlier.

An incremental rise in OT(TO) will also
result in an incremental decrease in the time
required for thermal runaway to be initiated. This
is due to the fact that less Joule heating is
required to raise the sample's conductivity an
incremental amount: consequently, if the bias
voltage is held constant, the time required for

switching to be initiated is decreased.

Figure 3.8 shows the Observed time delay
of the unit for various bias voltages. From the

oscillosc0pe, traces it is Obvious that as the

71

bias voltage increases, Td decreases. Figure
3.9 illustrates how the time delay of the unit
is affected by the bath temperature. Once again

it is Obvious that all experimental data supports

the above theory.

More specifically, one can express

Equation 4.30 in the following manner:

 

 

 

2
_ LSCS/OT(TO)
V - Vc
2 vzks .
where Vc = E;TT;T. is the threshold voltage
required at bath temperature To. The delay time
T1. T2 corresponding to bias voltage V1, V2 at
the same bath temperature Tb will then be
L2G /o (T )
_ s s T 0
V1 - V
c
ch /o (T)
_ s s T o
V - V
2 c
respectively. Thus
T V2 - V2
l _ 2 c
if‘— 2 2 4.34
c

72

Equation 4.34 exPresses the normalized delay

time for various bias voltages at ambient temperature To'

For the case observed, To is 300°K, VC
is approximately 4 volts. Equation 4.34 is depicted
in Figure 4.5. Also shown is the normalized delay
time vs. bias voltage data Observed in the series

of pictures in Figure 3.8. From these two curves

in Figure 4.5, it is Obvious that basic agreements

exists between theory and experiments.

While considering a completely different
material and using a completely different analytic

(9)

approach BOer and Ovshinsky obtained a very
similar expression for the threshold voltage Vc'
They expanded the sample's temperature Ts(x,t)

into Fourier series summations which were in terms
of x and t, and then substituted TS into the
governing thermal transport equation. After solving
the Obtained eigen equation from thermal equation
and taking approximation, the desired expression

for threshold voltage VC was then derived yielding(9)

1

as T)
21':-

at temperature To, and o, the electrical conductivity

where kS is thermal conductivity, 0006 is

of the materials they investigated was

73

 

 

 

l
1.2 _ l?
‘\ - - - - Theoretical
1.0 — —— Experimental

2'6

41

.5

'5 .8 ~—

E

o
.2,

0 .6 —

E

i:

6‘

T1 .4 —

0

1

l4” .2 —- 0

Room Temperature, 23 C
0 1 1 1 I r
4 4.5 5 5.5 6

Bios Voltage (Volts)

Figure 4. 5. Switching delay time as a function of applied voltage for a typical
oxidized vanadium foil. (For a 4. 7V bias, the measured delay
time was 10 1.1 sec.)

74

o(T) = 000 exp B(T-TO) 4.36

Their approach proves to be much more
lengthy and tedious. Also, assumptions were buried
in the middle of their analysis, making it difficult
to follow and to understand implications of these
simplifications. All of our assumptions were
made at the very beginning, making it easy to follow
and gain physical insight of the meaning and
implications of the assumptions. Furthermore, our
approach lends itself much more readily to computer

analysis.

The discrepancy between theory and
experiments in Figure 4.4 is due to the fact that
higher order terms in the Taylor expansion of
o(T) were neglected. The radical derivation for
temperature above 55°C is because, above this
temperature, the assumption stated in Equation 4.4

is not valid.

All data and theoretical curves presented
here have been normalized because of the uncertainty
at the present time in the absolute values of ks,

Us and CS. This uncertainty exists because the

75

samples investigated were not composed of a single,
well defined vanadium-oxygen complex but rather

of an assortment of complexes.

Despite this uncertainty in the absolute
value of kS and Cs' the experimental evidence
presented suggests that preswitching behavior of
the oxidized vanadium foil is strongly dependent

upon the thermal prOperties of the materials.

4.3 Temperature Distribution and Threshold Voltagg

 

of the V02 Film--Planar Configuration

4.3.1 Introduction

 

The second sample geometry of interest was
first employed by Berglund and Walden(3) in their
study of thin-film inductance effects (See Figure 4.6).
Our motives for studying this configuration are
three fold: (1) To show how easily the techniques
and approximations made in the analysis of the
first configuration can be extended to explain
the important features of this new structure.

(2) To derive an expression for the threshold voltage
for this second configuration and compare it with

(3)

the experimental Observations of Berglund and Walden

Conducting film of
thickness 82 thermal
conductivity K and
heat capacity C

76

Metal electrodes
carrying current I

at voltage V

  

 

 

 

 

 

/

 

 

High thermal
conducfivuy
substrate at

/

Electrically
insulating film

of thickness SI

and thermal conduc-

ambient temperature tiviiy Ki

Figure 4. 6. Illustration of the thin-film, planax configuration to be

analyzed.

77

(3) To compare these two thermal boundary configurations
in order to determine which possesses the most

desirable switching properties.

4.3.2 Geometry and Assumptions

The structure of interest is a conducting
vanadium dioxide (V02) thin film of thickness 82.
thermal conductivity k and volume specific heat
capacitance C, placed on a thin electrically

insulating film of thickness S and thermal

1
conductivity ki. This insulator film in turn
resides on a substrate of high thermal conductivity
which acts as a thermal heat sink. Two parallel
electrodes of spacing £1 and length £2 are
deposited onto the electrically conducting film

as shown in Figure 4.6. Vanadium dioxide under
consideration has a phase transition at 68°C
accomplished by a discontinuous change in
resistivity of a factor of the order of 104 from

the high-resistance to low-resistance to low-

resistance state.

Since the switching and inductance

prOperties of V0 are a direct consequence of

2

the material's thermal properties, the nonohmic

behavior observed in the preswitching, low-conductance

78

state could be predicted by solving the thermal
transport equation, subjected to the proper

boundary conditions.

For simplicity, the following assumptions
are made. (1). There is neglible heat flow through
the electrical contacts.

(2). There is no temperature gradient
in the plane perpendicular to the film surface.

(3). There is no lateral heat flow in
the insulating film.

(4). The thermal conductivity of the
substrate is much higher than any others, so that
the insulator-substrate interface can be assumed
to always remain at the ambient temperature T6.

(5). Only conduction of heat is
concerned, heat convection or radiation will not
be considered.

(6). The thin film under consideration
is assumed to be homogeneous and isotopic; therefore.
the electric field in the thin-film layer is uniform.

(7). All the parameters ki' k, C are
all assumed to be temperature, electric field and

spatially independent.

79

4.3.3 Derivation and Results

The rate that energy is delivered to a
unit volume of the thin film active layer by
the conduction electrons will be E20(T), where
E is the strength of the externally applied
electric field and 0(T) is electrical conductivity
of the thin-film active layer. The rate that energy

is stored in a unit volume of the active layer is

a:
Bt'

through the SiO2 to the heat—sink substrate will be

C The rate at which energy is conducted away

k.T
1

$182
laterally out of a unit volume of the active layer
2

will be k é—gu Thus, from the conservation of

6x
energy, the heat flow equation will be

 

and finally the rate that energy diffuses

2 k.T(x,t) 2
a T(x,t) _ 1 _ g BTgxltl + E agrz = O 4.37
ax2 kSIS2 k at k

where T(x,t) is the temperature of the conducting

film above that of the heat sink substrate.

Due to the nonlinearity of the thermal
dependence of electrical conductivity, no exact
solution can be obtained for this nonlinear partial

differential equation.

80

By utilizing the same approach developed in
Section 4.2.2, one can simplify Equation 4.37 and
solve it subject to the appropriate initial conditions
and thermal boundary conditions, thereby obtaining
an explict expression for T(x,t) in terms of the
bulk properties of the material, the dimensions
of the structure and the applied electric field
strength. The principal assumptions made in the
course of the analysis are that 8(T) = 60 + (%%)T
and that the electric field is applied as a step
function at time t=o. Equation 4.37 can then be

rewritten as follows:

 

B T 5T
1 1 1 _
2'E—TE+31T1“A
6x 1
12

for lxl g_jr , and

 

for Ix. 2_7§3 where

 

 

6‘ 2 a"
1 2
260
B—(Efi k1).
1 k k8182
-ki
B:
2 ksls2
E20

 

and A = k

4.38

4.39

81

In order to determine the constants of
integration, the initial conditions and boundary
conditions must be specified. Since the external
electric field is assumed to be a step function
applied at time t=0, the initial temperature of
the thin film will be equal to bath temperature

and furthermore T1(x,0) = T2(x,0) = O.

The four boundary conditions specified

are as follows: (1). T is continuous across
‘2 31*
the interface Ix] = if; otherwise B;- would be

unbounded, and an infinite heat sink should reside
at the boundary.
8T . .
(2). S—- 18 continuous across
z x

the interface |x| = 7%; otherwise an infinite

heat source would exist at the boundary.

(3). Due to symmetry, '3; = O
at x=O for all time t.
(4). T remains bounded as x

approaches infinity.

By taking the Laplace transform, and with
the aid of the initial conditions, Equations 4.38

and 4.39 now become

82

 

 

2

O. V (Xop)
l P A

t—- B )v — - — 4.44
£2
for lX' _<_ —2—' and
2 - (I—(— - B2)V2 = O 4.45
ax 2

for Ix] 2_-;-, where

2
vl(p.x) = I6 T1(X.t)exp(—pt)dt, 4.46
V2(p,x) = $6 T2(x,t)exp(-pt)dt, 4.47

and p is some positive number such that the above

integrals converge.
Letting q1 = -—————-, q2 = —————— and then

solving Equations 4.44 and 4.45, one obtains

v (p,x) = G (p)eq'x + G (p)e-q'x + 1‘— . 4.48
l l 2 2
qu
and
2 L
qztx - 7%» qztx + 7%)
v (p.X) = G (p)e + G (p)e 4.49
2 3 4
where 61' 62' G3, G4 are constants of integration.

From boundary condition (1), G1=G2. From

boundary condition (4), G4=O. From boundary conditions

(2) and (3),

L £2

qu 1
5 2

Pq 2 .
l q2cosh ql 7T'+ q151nh q1 2

G1(P) =

83

Thus one obtains

 

q cosh q x w
v (p,x) = Ji— [1 - 2 1 J
l qu £2 . 22
l q2cosh q1 7?-+ q131nh q1 7?
4.50
For convenience, three cases of the above
equation will be discussed separately.
CASE A ----- q1 = q2 = q
This means that B1 = B2 = B. Equation 4.50
then reduces to
2 22
q( x) q(- +x)
v (p,x) = J}— [1 _ 1e 2 -l 2 ], 4.51
l 2 2 2
Pq
£2
Where le g —2- .
Letting (p-KB) = p', §FD= q'. Vl(p,x)
becomes
L 2
A 1 1 -q.(_22. X) 1 *1“? “0
V1(P'X) " 13 {ETD ‘ fie " 5.9 l
2 z
1 1 1 q"2 ’0 1"!“7’“)
' [(P'+KB)(1 fie " 22' :i}
4.52
With the assistance of an inverse Laplace
transform table, one can show that the corresponding
(27)

temperature function T1(X,t) in time domain will be

84

 

 

£2 22
A A (——--x) -—-—x
T1(x,t) = - 31‘- + -§L eKBt[1 - %erfc 2 - ierfc 2 :t
t bit
‘2
1 A1 { “(jf'-Xk/-B (2 :0
+ I "B— e erfc 2 __ -\/-KBt]
VAKt
‘2 ‘2
-(-§— +x)./—B (-2— +x)
+ e erfc[ -¢¢KBt]
2 t
2
(‘2— ‘X’v-B (‘2‘ ‘X’
+ e erfc -+./-KBt
t
‘2 ‘2
(T +X)\/-B ('3' Hi) .-.__._._
+ e erfc .__ +m¢LKBtJ
a/Kt
E
2
for IXI _<_ 3— .
From Equation 4.53, Tl(x,t) will be unstable
k1
only if B = (- -————0 > 0. As a consequence of the
kSlS2

assumption q1 = q2 = q, E2 §%%EL or E will be zero

k1
and B = B = -

l 2 kSlS2

This is the trivial case since no electrical field is

 

will be smaller than zero.

applied to the thin film. Also from Equation 4.53,

it is obvious that the final temperature T for

f
B < 0 ‘will be
Tf = limt T1(x,t) = o. 4.54
B<O
t-ooo

This is what one expects. No thermal instability

occurs in this case.

85

CASE B ----- q1 > q2.

Physically this has no meaning since if

 

q >qo
1 2 250
E "a? 1) k1
P—K(-—--——-_‘—_ >P+K I
l k ksls2 i l kslszi
2.3.9. . 9.9.
or E 8T < O. For the substance discussed, ET

is always larger than zero in the temperature range
interested. Thus, this case will not.be considered

further.

CASE C ----- q1 < q2.
80

For this case, E2 §T|> O, which is the
situation of most interest to us. With the aid of
the convolution theorem, one will be able to find
the inverse Laplace transform of Vl(p,x) in
Equation 4.50. Since the primary concern of this
section is to figure out the threshold voltage for
‘which the system will possess a thermal instability,
it is convenient to discuss the prOblem in frequency
(or p) domain instead of the time domain. This
is because the corresponding time-dependent,
temperature-distribution function Obtained from
Equation 4.50 will be too involved to be discussed
here. In order to be better prepared to discuss

the results of this analysis, the following

definitions and theorem are introduced:

86

Definitions
(1). If f(t) is classified as piecewise regular
if in every interval of the form 0 g_t1 g_t g_t2,
f(t) is bounded and has at most a finite number of
discontinuities.
(2). f(t) is of exponential order if there exists
constants a, M and T such that

e-at|f(t)l < M for every t > T,

where the greatest lower bound do of the set {a}

is called the abscissa of convergence of f(t).

Theorem

If f(t) and f'(t) are both piecewise
regular and of exponential order, and if the abscissa
of convergence of f'(t) is negative, then

lim p L[f(t)] = lim f(t),
p40 t40

provided that these limts exist<27).

In realistic applications of this theorem,
L[f(t)] will be known. Hence it is desirable that
conditions be expressed in terms of L[f(t)] rather
than f(t). This can be done, since it is possible
to show that the above theorem can be applied if
there is any value of p with nonnegative real part
for which p L[f(t)] is unbounded but can be applied

if no such value exists.

87

Now, from Equation 4.50,

 

./P-KB2 cosh./P—KB1 x
W (p,x) =
1 P-KB:L £2 . £2
x/P-KB2 COSh‘/P-KBl —2'— +‘/P-KB1 Slnh ‘/P-KB1 7

4.55

The poles of pV1(p,x) will be P =KB1 and those

1
for p for which

 

 

 

 

P--KB2
tanh \/P-KBl = - P-KBl
2 .62 _ k1
E 5T kSls2 ki
(Recall that B1 = k ‘-, and B2 = - ksls2).

If the electric field E is smaller than

k.
V/:—-—¥L- , it is easy to show that all the poles
S S 50
12-85

will lie in the left half plane. According to the

above theorem, the device temperature Tf as time

goes to infinity will be

T

f 11m T1(x,t)

t-ooo

= lim p V1(p,x)

 

p40
A ./IKB2| cosh./|KB1I x

z z
./|K32|[cosh ./'KBll—§2— + ./IKBllsinh Jim-1'32-

4.56

88

If T falls below the phase transition temperature,

f
where the electrical conductivity increases abruptly,
there will be no thermal instability; this is

because the temperature of the system will be always

bounded and finite for all time.

On the contrary, if the electric field E
k.
is so high that E2 >-—-4LSE ,
515236'

then the system will

be sure to have thermal instability and, therefore,

electric switching will be initiated.

Due to the nonlinearity of the thermal
dependence of electrical conductivity 0(T), the

final temperature T of the system, under the

f
/ k.
application of an electric field E, where E < —--$33-
315235'

will be higher than the value for Tf predicted by
Equation 4.56. This is because of the assumption
made neglecting higher order terms in the Taylor

series expansion of 0(T). Thus, the critical electric

 

 

k.
. _ 1 . .
field EC —.//r 30(To) is not exactly the de51red
S152 8T
threshold field, it is a little higher than the

desired value.

89

If the applied voltage falls below some
threshold value, the system investigated will be

eventually come to the state of virtual equilibrium;

2
that is, §1-= 0,-QJE = O. The equilibruim
at ax2

temperature Teq will be, from Equation 4.38,

2
T = E T . 4.57
31 52 eq 5( )

 

The graphic solution of the above equation,
as shown in Figure 4.7, gives two equilibrium
temperatures of the system, To and TB. If there
is a thermal disturbance T1(x,t), which is due
to thermal fluctuation, and where T1(x,t)<< Ta' TB'
then only Ta ‘will be the temperature at which

the system will remain stable. The reason is follows:

Since Tl(x,t) can always be expressed as

the superposition of a series summation, T1(x,t) =
Z} elxert, from the small-signal theory, it is
l,v

also true that 0(T) can be approximated by
0(T) = 0(Teq) + 0T(Teq)T1(x,t), 4.58

where [Tll << Ta. T Substituting 0(T) and

B.

T1(x,t) into the basic governing Equation 4.38, one

will have

90

 

 

 

 

l
l
O’lT) -—-1
A I sKiE2
E-t
“' E<EC ‘82
V I:
II
:E’ / i
> /
‘3 / I Ki 2
a / g as.
8 x l
O l
,’ I
1” I’ l
|
i l l L’
- T
Ta “5 B

Temperature T Above Ambient

Figure 4. 7. Graphical solution of the time-independent temperature
equation. T and T correspond to the equilibrium temperatures.

a F5

 

 

 

 

k k kS S

- l 2

C
or

v - i (E20 (T ) - ki ) - E x2 4 59
— C T eq 8182 C ’ '

l2 is always positive since A is real. Thus if

2 1‘1

v will be positive, and the temperature perturbation
T1(x,t) will grow up as time goes on. At the

equilibrium temperature Tfi' Gm(Teq) always larger

k.
than —-—l—§-. On the other hand, when the system
S S E
l 2 k
is at equilibrium temperature T . where o (T ) < --$-u
a T a 2
SlSZE

the negative value of v will cause the thermal
perturbation Tl(x,t) to decay. Thus the system
will be restored to remain stable at equilibrium

temperature To instead of TB.

From the above discussion, it is concluded
that in order for the system to reach the unstable
equilibrium temperature TB such that a phase
transition may occur, the applied bias voltage VC
should be such that To coincides with TB: that

is, as can be seen in Figure 4.7, the bias field

should be such that the heat dissipation line

92

k.
-——l—§-T is tangent to the electrical conductivity
S S E

l 2

curve 0(T) at Ta = T . The physical meaning of

B
this is that the Joule heat E20(Ta) generated
k.

1
$18

 

exactly equals the heat dissipated T. For

2

lower electric fields, the temperature of the system
will never go beyond Ta since the heat dissipated
will exceed the heat supplied for the temperature

T > Ta.

It is now clear that the desired threshold
voltage for this configuration to actuate the
thermal instability and consequently initiate electric

switching will be

 

VC = £1 - EC’ 4.60
k1
where EC is such that functions -—-——§-T and 0(T)
S S E
l 2
have only one point of intersection.
More specifically,
k1
V = E E = fl \/ , 4.61
c 1 c 1 Sison‘Tt)

where OT(Tt) is the rate of change of electrical

conductivity 0 of the thin film investigated at Tt'

where Tt is the tangent temperature of the

93

k.
dissipation line —-—-$§-T drawn from ambient
SlSZE
temperature To to 0(T). Through oT(Tt(To)),

V is affected by the bath temperature To as has

C
been Observed in most of the cases. (3'4)

4.3.4 Discussion

Due to the manner in which the electrical

conductivity O(T) 0f V02 depends upon temperature,(3)

one would expect that the higher oT(Tt(To)) would

be, the lower the threshold voltage VC would be.

This can be easily verified by the Observing data

of Berglund and Walden(3).

The temperature dependence of electrical

conductivity of V0 film can be approximated by

2
[1.52(4 - L39) - 1.74]
0(T) = 10 , 4.62

This o-T relationship is sketched in Figure 4.8.
ao[Tt(TO)]

'ET at various ambient

From this Figure,

temperatures T0 were estimated and are presented

in Table III. From this data, the required threshold
voltage was calculated using Equation 4.61 at various
ambient temperatures To and the results are

summarized in Table IV.

Electrical Conductivity

9+

 

 

 

.4 hem)" ,’

i /

/
/

0.6 /

T /

/
_ /
/
0.4-
/
/
/
L. /
/
/
/
//

0.2- //

/’ A Ki Ki Ki

_ 3: SzEa 31 32E?) 818133:

a
b
C

 

30 35 40 45 50 55 so
0
Temperature C

Figure 4. 8. Graphical solution of the threshold voltage required for the
V0 layer to switch for bath temperature of (a) 25° C,

(b) 238° c and (c) 50° c.

TABLE III

%%(Tt(To)) of V0 Thin Film At Various Ambient Temperatures

2
Ambient Temperature T (°C) é2-(T (T ))(O )-l
0 8T t 0 cm
25 ----------------------- .0167
30 ----------------------- .02
3S ----------------------- .024
38 ----------------------- .027
45 ----------------------- .04
so ----------------------- .06
55 ----------------------- .10
TABLE IV

Threshold Voltage Vc at Various Ambient Temperatures

Ambient Temperature TO(°C) Vc(Volts)
25 ----------------------- 93.5
30 ----------------------- 67
35 ----------------------- 61.3
38 ----------------------- 57.5
45 ----------------------- 47.4
50 ----------------------- 38.6
55 ----------------------- 28

95

96

In the above calculations the following
values for sample geometry and bulk parameters
were used:

k

0.06 W/cm°K, ki = 0.010 W/cm°K, 2 = 3mm

1
10-4cm., S = lo-Scm., C = 3.3J/cm3°K.

S 2

1

The threshold voltages measured by Berglund

and Walden(3)

at bath temperature 25°C, 38°C, and

50°C were 57 volts, 52.5 volts and 35 volts respectively.
Figure 4.9 shows the threshold voltages required for

the V02 thin film to accuate electric switching

at various ambient temperatures both by Equation 4.61
derived and experimental data measured. It is

obvious that good agreement exists between the experi-

ments and theory.

The discrepancy between theory and experiments
comes from the insufficient information on the

temperature dependence of electrical conductivity

(3)

of V0 film employed by Berglund and Walden . The

2
estimated threshold voltage from the derived Equation
4.61 may thus be a little higher than the experimental
data. Nevertheless, the theory confirms once again
that the switching effect in V0 thin films has a

2
thermal origin as predicted by other observers(3'l6_l7).

Critical Switch Voltage (Volts)

Figure 4. 9.

80

20

97

“—-Theoretical

Experimental

 

 

 

 

I l l l l

30 35 40 45 A 50
Bath Temperature (°C)

Voltage required to actuate switching for various bath
temperatures.

98

4.4 Negative Resistance

4.4.1 Introduction

 

Various negative-d1fferential-resistance
effects in solids have been observed. Most of these
effects are associated with junction or contact
phenomena(ll'25-26). The most notable examples
are the p-n—p-n junction devices and the tunnel
diode. Other negative-differential-resistance
effects may be classified as true, bulk properties
of solids. For example, the Gunn effect is a
voltage-controlled, switching phenomena which is
due to an inherent instability associated with
the nonlinear carrier mObility in certain solids.
Current-controlled negative resistance, as found
in many amorphous semiconductors and transition
metal oxides, is associated with electrical breakdown

which originates from a thermal initiated local

instability.

By totally neglecting effects due to lattice
(ll)

temperature gradients, Ridley has shown, using
irreversible thermodynamics, that the current-
controlled, negative-differential resistance in a

bulk device favors the formation of thermal filaments,

99

providing a current fluctuation occurs. The

negative differential resistance observed in an

(26)

isothermal system has also discussed by Burgess .

Presented here is an alternative approach
which assumes that a current filament is formed
inside of a bulk material and then by neglecting the
effects due to the transverse temperature gradient,

a critena for Observing current-controlled, negative-
differential resistance in a bulk device is

established and discussed. This method of approach

(11)

is opposite to that done by Ridley and is

(26)

more general than the case discussed by Burgess .

11)

In short, Ridley( discussed the adiabatic case,

(26) discussed the isothermal non—adiabatic

Burgess
case, while what will be discussed here is the case

of non—adiabatic non-isothermal conditions.

4.4.2 Derivation; Results and Discussion

 

The structure of interest is shown in Figure
4.6. It is a thin-film semiconductor of thickness
52' thermal conductivity k and electrical conductivity
0(T). 0(T) is eXpressed by Equation 3.1. The layer

is placed on an electrically insulating film of

100

thickness Sl which possesses a thermal conductivity
ki. This insulator film in turn resides on a
substrate of high thermal conductivity which acts

as a thermal heat sink. By totally neglecting
effects due to lattice temperature gradients, the

energy balance equation (Equation 4.37) reduces to

E20(T) =

 

S132
in the steady state, where E is the applied electric

field in the thin film.

Now take one unit volume element of the thin
film for consideration. Suppose that there is a
temperature perturbation due to thermal fluctuations
such that part of the thin film with area a has a
higher temperature Ta and the remaining part b

which has temperature Tb.

Let 0a, 0 , 0 be the corresponding

ao b' Gbo

electric conductivity of the thin film at
temperature Ta' Tao' Tb' and Tbo respectively, and

define

and

II
H
H'
3
*3
F]
p
0‘
U1

101

 

 

Then
0a = an + Ma(Ta-Tao) = 00a + M'aTa
and
016 = 0166 + ”bub-Tho) = 0616 + MbT '
where
A
0' ' = O - M T
0a 0a a oa
. .4
0ob - Oob - MbTob'
From Equation 4.63, one has
k1
0a = Goa. + MaTa = 2 Ta
SISZE
k1
Ob=oob +MbTb= 2Tb
S S E
l 2
Thus
-00 '
T = .
a
M (1- ----—-)
182 E Ma
and
T _ -00b
b — k. °
Mbil - ---—--->
S 182 E2 Mb

From the current density expression

J = 30 = E(aaa+bab) = Ea(00a +MaTa) + Eb(0

J8. + Jbo

4.72

4.73

ob+MbTb)

4.74

it is Obvious that J

102

is function of both E

and T: therefore,
ngEltz = dJa + de
dE dE dE
bJal anl BJa)| dTa (BJb)' dT.b
= + —— + (— —— + —— —— ,
0E Ta BE Tb a'ra E dE 6Tb E dE
4.75
where
aJ| — b - ' b ' b
BET — aoa + 0b — aooa + Gob + aMaTa + MbT ,
aJa
*BTa E = aEMa:
and
BJ
b _
8Tb E ’ bEMb'

these relationships can

 

and
dT
__§. _
dE %(1
and
dTb 2 Gob. (l -
dE Mb

can be Obtained from Equations 4.72 and 4.73.

the above Equations into Equation 4.75,

41

expre581on for dE

be obtained from Equation 4.74,

-3
s Ma kMEz) s 128 Ma E 4'76
SI 2
k. —2 k.
1 ‘ 1 -3
-—-——- 4.77
2) s s E
SZMbE 1 2M6
Substituting

one obtains an

103

dJ

dE'= aaoa, + bOOb' + aMaTa + beTb
1 k1 )‘2 R1 -2
+ 2ao°a'( - SlSZEZM S152M2 E
k. -2 k.
+ 2b00b,(1 --———£————) -———£—- 3‘2 4.78

2 S 8
$1828 Mb 1 2Mb

Equation 4.78 can further be simplified to

yield (Appendix C)

 

 

2M T 2M:Tb
[1 + Ca a] + b0 b[1 + S] 4.79
a 0a

For negative resistance to be observed, one requires

 

 

 

 

 

dJ
dE < O,
MaT
G 1 + 2-——41
that is (%)(329( O; )
b 1 Mb
+ 2 0
ob'
MaTa
O 2_ Ooa' -1
or (E)(—-—)( 21:)pr ) / 1. 4.80
+1
Oob'

For convenience, the electrical conductivity

0(T) is divided into two parts by the temperature T

t,
where Tt is the tangent temperature of the
k.
dissipation line -——$—§-T drawn from the ambient
SlSZE

104

temperature to the 0(T) vs. T curve. These two

parts may be represented as follows:

k.
M(T) < ———1—7 for T < rt 4.81
S S E
1 2
and
k1
M(T) >>-———-1f for T > Tt 4.82
SISZE

where M(T) = §%%£L| .
T

The following conclusions may then be drawn from

Equations 4.79 and 4.80.

(1) If the thermal fluctuation occurs when

temperature of the unit is below T then from 4.79,

tl

M and

since 0 o
oa" ob" a

aJ

aE > 0 for all T < Tt'
Mb are all positive. Therefore, no negative-differential
resistance will be observed. Physically, this is

because, for sample temperatures below T the rate

to
that the electrical conductivity increases with
temperature is so low that in order to meet and

overcome the heat conducted away to the bath, it is
required that a higher electric field be applied. Since
the only method of increasing the temperature of the

unit is to raise the electric field, no thermal

runaway condition exists.

105

(2) Consider the case where the thermal
fluctuation occurs at the time the unit temperature

just stabilizes at T For this case

t.
ma = mb'
and
an' = Oob' = 0'
From Equation 4.78,
dJ = a m Ta + b m Tb > 0. 4.83

This implies that the negative-differential resistance
will not be observed even if the temperature of the

unit reaches the characteristic temperature T For

t'
vanadium dioxide, this characteristic temperature
is the phase transition temperature. Thus it is
concluded that the switching action Observed in VO2
thin films does not start at the time when the unit
temperature uniformly reaches Tt'

(3) Consider the case where the temperature
of the device is uniform; i.e. b 4.0 and a 4 1.

For this case, no thermal fluctuation exists within

the material: consequently, from Equation 4.79,

dJ 2M T

EE=a0a L1+—-5§——":‘—J 4.84
oa

106

In order for the negative differential resistance to
be Observed, one requires that

9111/

dE\O°
Thus (MaT0a - Goa) > 0.
and hence
Ooa
M > .
a T0a

Therefore, for this isothermal case, the
switching action starts at the moment the temperature
of the unit reaches Tt'

(4) From the least entropy production rate
principle, the existing negative-differential
resistance favors the formation of a current channel.
From conclusion (3), it is obvious that even if no
current channel forms, the negative-differential
resistance may be Observed. Thus one concludes that
negative-differential resistance is not a necessary
condition for formation of a current channel, and
vice versa. From Equation 4.80, one observes that
the greater the rate of change of electrical
conductivity with temperature in the region of thermal
filament a, the higher the possibility of Observing
the negative-differential resistance. Usually area
a is much smaller than area b and that in order

for negative-differential resistance to be observed,

107

the temperature in channel a must be high enough so

0 M
a

T T
. b
that the ratios -3- and (T—-3—T)/(T———-T) are much
Ob Ooa' OOb'

larger than one. Therefore, even after the formation
of a thermal filament, if the channel is too small,
the negative-differential resistance might still

not be observable.

4.5 Negative Resistance for the Isothermal, Steady-
State Case.

 

After sectionwise linearization of any
temperature dependence of electrical conductivity
curve 0(T), (see Figure 4.10) one defines the

following regions Ri (i = 1,2,3,...,n) as

T1.g T.§ T as R

2 1
T2 _<_ T _\_ T3 as R2
Ti A-T-S 1+1 as Ri
Tn g T g Tn+1 as Rn.

Then the electrical conductivity at temperature T

within Region Ri can be approximated by

 

0(T) = Ooi + mi(T-Ti) 4.85
where
i-l
Ooi = 061 + .E; (Ti+1‘Ti)m1' 4'86
1—1
mi = limt C(Ti+l) O‘Ti) 4.87
Ti+17Ti Ti+1 - T1

108

and 001 is the electrical conductivity at temperature
Tl' Ooi is that at Ti' For convenience, T1, the
bath temperature which all the device temperatures
referred to, is set to zero.

For the steady-state, isothermal situation,

the thermal transport equation (Equation 4.37) reduces

 

 

 

to
2 kiT
E 0(T) = S S 4.88
l 2
Thus, the voltage V(T) across the device is given
simply by
v2 (T) = 822‘: = gig-gm 4.89
or alternately by
V2 mg.
(T) = SlSZMi(T—Ti)+ooi 4'90
for region Ri‘
Applying Ohm's law in region Ri' one obtains
the following expression for the current I(T) in
the film:
‘232
I(T) = V(T)0(T) L1 , 4.91

where (£282) is the cross-sectional area of the unit

investigated. One has for region Ri'

109

 

 

 

 

 

2
S L k..
2 ’ 2 2 1
I (T) 31 ) T [Mi(T-Ti_l) + aoi]. 4.92
From Equation 4.92,
T . IT .| 2 1/2
T=_2.£+._9£_(1+4I__1fl) :] 4.93
2 - 2 I 2
oi
where
M.T. - O .
= 1 1 01\
Toi - ( Mi ) 4°94
2 2
2 _ 32L2k1(M1T1 ' 001)
i 1
Note here that Toi may be positive or negative
depending upon where it is in the region. In addition,
since the temperature of interest is always above
that of the bath temperature, the negative sign in
Equation 4.93 should be discarded.
Substituting Equation 4.93 into Equation 4.89,
one obtains
|T .l 2 1/2
01 I (T)1
Toi/z+ 2 [1 + 4 I 2 J
2 2 oi
‘V (T) = V . 4.96
01 T . |T .l Ile) 1/2
___01+__<1=_L_[1+4
2 2 1 2
oi
where __.
2
lei
V . E -——-——- 4.97
01 SlSZMi

110

Several cases must be discussed:

Case 1 (Toi < 0)

This means that if the device temperature T
reaches region Ri and the temperature dependent

electrical conductivity 0(T) is as indicated by
a .
branch 1, then Mi < 521 ,

1

that is, the slope of

the straight line from origin to point (Ti’Ooi)°

For this case, Equation 4.96 becomes

 

 

 

 

 

 

 

 

12(T)
2 2 ‘ /2
k.L -1 + 1 + 4 I
V2(T) = 131M * g;
S . 12
1 2 1 1 + [1 + 4 I_i§1] /
Ioi
12 1/2 2
kiLi [<1+4 1012) ' 11
= s s M 2 , 4.98
1 2 i I
4 2
Ioi
Therefore,
2
I .
\1/2 -
k.L2 [(1+4 I .2) - 1]
_~ 1 l 01
V(I) — s s M I 4.99
1 2 i 2 | |
I .
01
and
-II -I
I(V) = °1 V 4.100

 

111

 

 

4.101

Thus
.91 = ('IOiI) V2 — 1)-2 V2 + 1) > 0
av voi 2 2
oi oi
while
2 |1 .| 2 -3 2
L%=-2(—V9_1—)(—Y—§_1) ‘71—(51—+3)>0 4.102
3V oi Voi oi oi

since from Equation 4.96, for T . < O.

V < voi 01

Case 2 (Toi > O)

 

This means that if the device temperature

reaches region Ri and the temperature dependent

electrical conductivity 0(T)

0 .
017 that is,
T1

electrical conductivity mi

is as indicated by

 

branch 2, then Mi >

of the straight line from origin to the point

(Ti'ooi)‘

For the case, from Equation 4.96, V > V

 

and one observes that I—V relation is
I: .| 2 —1
I(V)=—‘79-J-'—V(V2-l).
oi V

oi

the slope of

oi

T

is greater than the slope

4.103

112

 

 

 

 

Thus
I.| 2
p;_ '01) v __ )-—2 V2 )
av“ v. (V2 1 (Vz+1 <0, 4.104
01 . .
01 01
and
2 |1| 2 _ 2
-§—-I-—2( °1 )V -1)3 V V +3)>0. 4.105
8V2 V01 2 V 2 2
oi oi 01
Ooi
Case 3 (m1 = Ei—q

 

This is the case the slope of the electrical
conductivity mi is equal to that of the straight

line drawn from origin to the point (Ti,0 .).

01
- _ .61 ..
From Equation 4.96, V — Voi’ thus 8V 4 as can

be seen from Equation 4.103.

Case 4 (mi = O)

 

This is the case when the device electrical
conductivity is the temperature independent. From
Equation 4.88, one sees the linear relationship between

voltage V and current I.

Case 5 (mi 4 co)

 

In this case an abrupt change in the
electrical conductivity is observed as in some
metals or oxides at their melting points or phase

transition temperature. From Equation 4.103, one has

113

oiI

 

 

 

 

_ . 2 2 2 -1
I(V) — 11mt V V VOi (V - Voi )
miam
2 2
fl k. L k. -1
. 1 1 2 l 1
= limt S M.T.-0 .IV- —-————) V --——————)
miam 2 1 1 01 $182Mi ( $152Mi
2
0 . L k.
. 1 1\, V
= l1mt |T. - 01] ,—
mi” 1 Mi (81 (2 Liki
v - __.__)
2 SlszMi
- T zlki ; 4 106
_ i 81 V O 0
Thus
2
L k.
BI 1 1 -l
-—'= T. '—- < O. 4.107
6V 1 S1 V2

The following conclusions are drawn:
1. For the negative differential resistance to be observed,

the given 0(T) vs. T curve should be intersected

kiT

2
SISZE

from the origin. More specifically, positive

at least twice by the straight line drawn

differential resistance can only be observed if the

temperature of the unit is below Tt'

2. In any section of the I-V characteristic curve, the

curvature is always nonnegative, that is, the I—V curve

. d2:
18 always concave up, or, ——§g2 0.
dV

3. Any abrupt increase in electrical conductivity with
temperature can result in the appearance of negative

differential resistance.

114

Figure 4.10 illustrates a typical electrical
conductivity versus temperature relationship which
is exhibited by V02 and other semiconductors which
undergo phase transitions. The corresponding I-V
characteristic curve predicted in the steady—state
isothermal case for such a material is also depicted

in Figure 4.11.

(l5

 

 

 

 

 

GT
5
2
slope)? 3+0 .,-3
I
.I'
.l
>. I
I A
: slapem 5’77} - / s . ‘th
E a-OI .. ---------------------------- . 4
c : sloperlI-O
o - u
U u
./ :
0'03 _-_--__-__-__-_._ |
SIOpe=113/ I
01 """ 1 ' ,
0‘ Win]: “ape-712 E :
a . I
| I I l
i E/ slope- 1, "’1' :
l - '
I
: / i '
I . I I
1/ : i
/I I '
. ' 1 i
i i I
L 3 l ’
T‘ T

Figure 4. 10.

.4

N
.4

0

Temperature

Illustration of the section-wise linearization of a typical
sample's electrical conductivity.

Hb

Current

 

 

 

Voltage

The corresponding I-V characteristic curves for the

Figure 4. ll.
«(1‘) dependencee shown in Figure 4. 10.

CHAPTER V

CONCLUSIONS

5.1 Summary of the Significant Results
Nondestructive, reversible, current-controlled
switching was observed in oxidized vanadium layers
prepared using an extremely simple, sample-
preparation technique. Inductance effects and
current-controlled, negative differential resistance
were Observed in these oxidized vanadium layers.
The inductances were found to be independent of
frequency. This lends direct support to the theory
developed by Berglund and Walden(3); they predicted
that, if a material could be prepared which possessed
no thermal hysteresis effects, then the inductance
of the material would be independent of frequency.
Our material exhibited no thermal hysteresis effects.
Inductances for dc bias currents above and below
threshold were measured. For the case of zero dc
bias, inductances in excess of 100 mH were observed;
however, the Q of the structure was quite low.

Also, the observed inductances were found to be

strongly dependent upon the dc bias, suggesting

117

118

possible applications as inductors and varactors in
integrated circuits. Before this phenomena can be
used effectively, however, the series resistance

prOblem must be overcome.

The switching-delay—time constants were
measured and found to be on the order of 10-6 seconds.
They were also found to be dependent upon the bias
voltage as well as the bath temperature. The threshold
voltages required to actuate switching were also
measured and found to range from 4 volts to 10
volts, depending upon contact conditions, the dc

bias voltage and the ambient temperature.

The thermal-transport equation was solved
and an analytical expression for the switching
threshold voltage was obtained for the two most
important sample configurations. It was found that
the threshold voltage for both configurations employed
should be prOportional to the thermal conductivity
but inversely prOportional to the rate of change of
the electrical conductivity with temperature. In
addition, the threshold voltage for the sandwich-
type configuration was found to be independent of

the layer's thickness while that for the planar-type

119

configuration was found to be strongly dependent

on the film thickness as well as the thickness of

the insulating layer. This suggests that the
sandwich—type configuration will have a more constant
and stable threshold voltage. The other advantage

of the sandwich-type configuration over the planar-
type configuration is that the inductance effect is
independent of frequency over a much wider frequency
range. The fact that the threshold voltage is bath
temperature dependent suggests possible applications
as an thermoswitch. All of the fundamental conclusions
of the theoretical develOpment were verified in the
laboratory, thereby reinforcing the basic postulate
on which the entire analysis was performed: Current-
controlled switching in these devices is thermally

initiated.

5.2 Suggested Additional Research

Although these oxidized vanadium foils show
promise as being compact inductors, varactors and
thermoswitches, serious deficiencies still exist
with them: namely their large, series resistances,
their relatively poor stability and poor reproducibility.

These deficiencies are characteristic of virtually

120

all of the amorphous semiconductors, not just the

material selected to investigate here.

Work should be continued in h0pes of
substantially lowering the series resistance, improving
the stability and reproducibility, as well as
increasing magnitude of the inductance effects.

The following experiments are suggested:

(1) Improvements can be made in the overall

switching properties of the devices if, during
oxidation, the surrounding environment is more closely
controlled, and if after oxidation, the sample is
annealed in an inert atmosphere. This will

guarantee a more uniform, contamination-free

oxidized vanadium layer.

(2) The electrical contacts were relatively noisy,
due principally to the crude method in which contact
was made between the sample and the electrodes.
Evaporated gold layer or alloyed lead—tin contacts
should substantially improve the overall characteristics
of the devices.

(3) The memory effect observed in amorphous semi-
conductors has been confirmed to be associated with

(1).

microsc0pic structural changes within the material

121

This change in local order is initiated by the

thermal instability resulting in thermal filament
formation within the material(1). In order to provide
more information so that one is in a better position
to understand the observed switching effects in this
vanadium—oxide complex, micr0probe techniques should
be utilized in order to see if there is thermal-
filament formation within the oxidized vanadium layer.
(4) Also, in order to gain a more complete understanding
of the Observed switching phenomena, determination of
the oxidized layer's chemical composition through
chemical analysis is suggested.

(5) All the data presented in the discussion section
of the previous Chapter have been normalized; this is
due to a lack of knowledge of the magnitude of some
bulk parameters such as the layer's thermal conductivity,
heat capacity, band gap, and so forth. Measurements
of the above mentioned parameters are therefore
suggested in order to further refine the theory and,
consequently, obtain a better understanding of the
conduction and switching processes.

(6) The electrical conduction mechanisms in amorphous
materials is not well known at this time. Before the
switching effects investigated here can ever be

completely understood, a better understanding of the

122

conduction process in amorphous materials is required.
A detailed computer analysis, based on the models
developed in this thesis, would represent a positive
step toward develOping a more complete understanding

of the switching processes.

APPENDIX

APPENDIX A
RADIUS OF THE RING IN THE X-RAY
PICTURES OF PURE VANADIUM FOIL
The characteristic patterns made on
photographic film as a result of X-rays inpinging
upon solids may be understood in terms of Bragg's

law. The Bragg reflection law is
2d(h,k,l) sin 9 = nl, A.l

where l is the wavelength of the incident radiation,
9 is the reflection angle (see Figure A.l), d(h,k,l)
is the distance between the two nearest reflection
planes in the solid, n is a positive integer,

and h,k and 1 are the reciprocal lattice constants.
For this experiment, the K01 line from a c0pper

(29)

target was selected; consequently

l = 1.54 A.

In order to determine d(h,k,l), one has to
first determine which plane reflects the incident X-ray.
Vanadium is in b.c.c. lattice structure<29), and the

reflection intensity is prOportional to the absolute

value of the structure factor S(h,k,l). For a b.c.c.

124

I25

X- ray ‘

Source —-

 

 

 

 

 

 

Negative
Film

Figure A. 1. Experimental arrangement for making pin-hole back-
reflection, x-ray photographs.

126

lattice. S(h,k,l) is

S(h,k,l) = f[l + exp.[-iw(h+k+1)]}.
When h+k+l = odd integer, S will be zero, this
means no reflection is expected, if h+k+l = even

integer, S will be equal to 2f: this corresponds

to the maximum reflection condition.

Since d = ——°———— = 3— A.2
./1+1+0 \/2
where a = 3.03A is the vanadium lattice constant
at room temperature.
N0w, if one let n=l, one has, from
Equation A.l,
2 i492- sin e = 1.54
2
Therefore,
sin 6 = 0.359
9 = sin-10.359 = 21°
29 = 42°. A.3

The distance between the film and the object is W,

thus the radius of the expected ring will be

R = W 0 tan (w-Ze) = W - tan 42° = 0.9004W A.4

APPENDIX B

DERIVATION OF EQUATION 4.23

From Equation 4.22,

 

 

 

AS « cosh qsx
VS‘X'P) = ;;2'11 - Ls
S COSh qS '3-
_______ .E.
A K . cosh JP-KSBS fl?—
5 s [l _ s] B 1
P(P-K B ) L ° '

 

If one lets

and

one will have

 

 

A A AK
3 _ _ s s
2 ‘ .2 " P'(P'+K B)
pqs qu s
_§(1 1 ) B4
_ B P' P'+K B ° '
S S

127

128

 

 

 

 

 

 

 

 

 

 

Thus
VS(X.p) - VS(X.p )
= f§-[cl;)(l _ cosh qS x )
BS P' Ls
cosh qS ET
_ -( l )(1 _ cosh qs x )1
P'-l-KSBS Ls
COSh qS T
or cosh./P' 31-
v (x p') = f-S-D—L) (1 - S
S B P cosh./P' L5
2 af-
5
cosh \/P' —X—
_ l _ V 8)]
(P'+K B ) (1 L ’ 3'5
S S cosh./P' S
2/i—
s
The first term in the above equation is then transformed
to be (30)
AS KSBSt KSBSt 4 w I_11n —(2n-l)2w2KSt (2n71)W x
”fie "e (1+ 1? Z 2n-l e 2 C05 L )1
5 n=1 LS s
12n-1)2w2 }
-A m _ n [K [B - 2 ]t
= BS 3 Z} énii e s 5 LS cos (2n;l)wx 3.6

S =1 5

129

The second term is transformed to be

2:- (1 - I: F(U)dU>, B-7

where

x
cosh./P--KSBS -—-

-l[ s
L ”.1

cosh . /P—K B S
s s 2 r—-
KS

F(U)

 

 

4vK m S S L .
= 23[ 23 (-1)“’1(2n—1)e S .

L n=1
3

 

 

(2n-l)wx
]

L
3

Therefore,

>

(1 - ft F(U)dU)
0

kph}, kyhd

 

{1'2'412 1 22
n=l L (Zn-1) w

3 BS - 2

L

S

_ K _ 2 2 (2n—1)w./K x
[(-1)n l(2n-1)e s(BS - Q“ :2” 7T )1: cos L S
S

 

 

 

L
s

m (2n-1)wx
+ 23 (-1)n‘1(2n-1)cos 3.9
n=l LS J}

 

130

Thus, the temperature Ts(x,t) of the sample is

 

 

 

 

_ -l
Ts(x,t) — L [VS(x,p)]
_ 0(TO) {;m l
— 0 (T ) _ 2 2
T O —1 E§{E 0T(To) - 12n'1)2W2J
4w k 2
s L
s
k 1320 (T) 22
_§,( T o _ (Zn-l) w > t
c k 2
n-l s 3 L8
[(-1) (2n-1)e
(2n-l)wx
n-l
- (—l) (Zn-1)] cos L ~—
3
k E20 (T) 2
_§_ T o _ (Zn-12w ‘ t
m l n Cs ks L2 J
‘1‘“: 2 2:191 e 3
n=1
2n-1)wx
cos L 1}, B10

which is Equation 4.23.

APPENDIX C

DERIVATION OF EQUATION 4.79

From Equation 4.78,

 

 

 

 

dJ _
dE — aooa. + boob, + aMaTa + beTb
-2
+ 2 o (1 ki \-2 kiE
a ——————-—-—-—-— -———-—
oa' 2 S S M
SlsszE l 2 a
-2
k. _2 k.E
+ 2b0 (l --———¥L———> -—4L———
ob' 2 S S
slszsbs 1 2M5
= aooa, + boob, + aMaTa + beTb
k.M k.
+Za§§¥—T§E'2+Zb§-§%‘—T§E'2
l 2 oa' 1 2 0b'
If one lets
k.
a .—( 1 -M)T
oa S S E2 a a
1 2
and
k.
_ 1 _
Oob'-(SSEz Mb)Tb
l 2
one obtains
0 S S
-2 0a 1 2
E _ ( T + Ma) k.
a 1
(Gob'+ )S1sz
Tb Mb k.
1

131

132

Substituting E.2 into the above expression for

 

 

%%- (Equation C.4), one obtains
‘13— b MT bT
dE'— a0oa' + ob' + a a a + Mb b
MaTa
+ 2a . (Goa' + MaTa)
oa
+ 2b Mbmb (0 + T )
0 . 0b' Mb b
ob

Combining Equations (4.70) and (4.71) with C.5, one

obtains

 

 

 

M T T
%%-= aOa + bob + 2a a a 0 + 2b :b‘b Ob
oa' a ob'
2M T 2 T
=a0(1+—-§—9—)+b0(l+Mbb)
a 0 , b 0 ,
oa 0b

which is Equation 4.79.

BIBLIOGRAPHY

BIBLIOGRAPHY

S.N. Mott, "International Conference on Amorphous
and Liquid Semiconductors," 24-27 Sept. (1969),
Cambridge University.

A.F. Loffe, A.R. Regel, "Non-crystalline Amorphous,
and Liquid Electronic Semiconductors," A.F. Gibson,
ed., Progress in Semiconductors, Vol. 4, London:
Heywood and Co., 1960.

C.N. Berglund, R.H. Walden, "A Thin-film Inductance
Using Thermal Filaments," IEEE Trans. on Electron

S.R. Ovshinsky, "Reversible Electrical Switching
Phenomena in Disordered Structures," Phy. Rev.
Letters, 21:20 (1968), pp.l450-14S3.

A.H. Clark, "A Refiew of Band Structure and Transport
Mechanisms in Elemental Amorphous Semiconductors,"
J. of Non-crys. Solids, 2 (1970), 52-65.

J.G. Simmons, "Applications of Amorphous Semiconductors
in Electron Devices," Contemp. Phys., (1970),
V01. 11' n0. 1' 21-410

David Adler, "Theory Gives Shape to Amorphous Materials,"
Electronics, Sept. (1970), 61-76.

P.J. Walsh, R. Vogel, and E. Evans, "Conduction and
Electrical Switching in Amorphous Chalcogenide
Semiconductor Film," Phys. Rev., (1969) 178,
1274-1279.

K.W. Boer, S.N. Ovshinsky, "Electrothermal Initiation
of an Electronic Switching Mechanism in Semiconducting
Glagges," J. of App. Phys., vol. 41, no. 6, May
(1970), 2675-2681.

134

10.

ll.

12.

13.

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

- 135 -

A.C. Warren, "Switching Mechanism In Chalcogenide
Glasses," Electronics Letter (1969) 5, 461-462.

R.K. Ridley, "Specific Negative Resistance in Solids,"
Proc. Phys. Soc., (1963) vol. 82, no. 954-966.

M.H. Cohen, "Electronic Structure and Transport in
Covalent Amorphous Semiconducting Alloys," J.
of Non-Crys. Solids, 2 (1970), 437-443.

E.N. Economou, M.H. Cohen, "Energy Bands in One-
Dimensional Aperiodic Potentials," Phy. Rev.
Letter, vol. 24, no. 5, (1970), 218-219.

N.F. Mott, "Conduction in Non-crystalline Systems--
Part 1, Localized Electronic States in Disordered
Systems," Phil. Mag., 17:150, June (1968), 1259-1268.

H. Fritzsche, "Physics of Instabilities in Amorphous
Semiconductors," I.B.M. J. Res. Deve10p. 13
(1969), 515-521.

C.N. Berglund, "Thermal Filaments in Vanadium Dioxide,"
IEEE Trans. on Electron Devices, vol. ED-16 (1969),

Hisao Futaki, "A New Type Semiconductor (Critical
Temperature Resistor)," Japanese J. of App.
Phys., V01. 4' n0. 1' Jan, 1965’ 28-41-

F.J. MOrin, "Oxides Which Show a Metal-to-Insulator
Transition at the Neel Temperature," Phy. Rev.
Letter, vol. 13 #11 (1959), 34-36.

David Adler, Harvey Brooks, "Transitional Metals Oxide,"
Phy. Rev., 155 (1967), 851-860.

Alpha Inorganics z¥00394 Vanadium Foil.

C. Kittel, Introduction to Solid State Physics, 3rd
Ed., John Wiley and Sons, Inc. (1968), Chapter 2.

K.W. Boer, R. Haislip, "Semiconductivity of Glasses,"
Phy. Rev. Letters, vol. 24, no. 5, (1970), 230-231.

H.S. Carslow, H.C. Jaeger, Heat Conduction in Solids,
2nd Ed., Oxford Clarendon Press, 1959.

 

24.

25.

26.

27.

28.

29.

30.

31.

32.

33.

34.

- 136 -

K.W. Boer, G. Dohler, "Temperature Distribution and
Its Kinetics in a Semiconducting Sandwich,"
Phys. Stat. Sol. 36 (1969) 679.

B.W. Knight, G.A. Peterson, "Theory of the Gunn
Effect," Phys. Rev. vol. 155, no. 2, (1967) 393-404.

R.E. Burgess, "Negative Resistance in Semiconducting
Device," Can. J. Phys. vol. 38, (1960) 369-375.

C.R. Wylie, Jr., Advanced Engineering Mathematics, 2nd
Ed. 1960, New YOrk, McGraw-Hill.

S.R. Ovshinsky, E.J. Evans, D.L. Nelson, and H.
Fritzsche, "Radiation Hardness of Ovonic Devices,"
IEEE Trans. on Nucl. Sci., NS-lS (1968) 311.

B.N. Cullity, Elements of X—Ray Diffraction, (1956)
Addison-Wesley Co.

M.R. Spiegel, Mathematical Handbook, McGraw-Hill, (1968).
E.A. Davis and R.F. Shaw, "Characteristic Phenomena in
Amorphous Semiconductors," J. Non-crys. Solids
2 (1970) 406-431.

Hansen, Constitution of Binary Alloys, 2nd Ed.
1958, McGraw-Hill, Page 806—807, and Page 992-993.

A. Sieverts and A. Gotta, Z. anorg. Chem., 172,
1928, 1-31.

H. Hahn, Z. anorg. Chem., 258, 1949, 58-68.

   

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