ELECTRONIC STRUCTURE OF Al 203
VUV REFLECTIVITY MEASUREMENTS FROM ROOM TEMPERATURE TO 1100 OC
by
Roger Harquail French
B.S. Cornell University, Ithaca
(1979)
Submitted to the Department of
Materials Science and Engineering
in Partial Fulfillment of the
Requirements of the
Degree of
DOCTOR OF PHILOSOPHY
at the
MASSACHUSETTS
INSTITUTE OF TECHNOLOGY
February 1985
Massachusetts Institute of Technology 1985
Signature of Author
Department of Materials Science and Engineering
October 30, 1984
Certified by
R. L. Coble
Thesis Supervisor
Accepted by
B. J. Wuensch
Chairman, Departmental Committ e0n Graduate Students
ARCHIVES
MASSACHUSEPIS INS ITUTE
OF TFrHNOLOGY
MAR 2 5 1985
U39.~ 2~
3
- 1 -
ABSTRACT
Electronic Structure of Al203: VUV Reflectivity
Measurements from Room Temperature to 1100 *C
by
ROGER HARQUAIL FRENCH
Submitted to the Department of Materials Science and Engineering
on January 11, 1985 in partial fulfillment of the
requirements for the degree of Doctor of Philosophy.
the
of
The
0.1
A Vacuum Ultraviolet (VUV)
Spectrophotometer was built
to measure
near normal incidence reflectivity
and transmissivity from 6 to 15 eV
samples heated without contamination from room temperature to 1100 OC.
relative energy resolution is 0.01 eV while the absolute resolution is
eV.
The temperature dependence of the electronic structure of single
crystal Al203 was studied.
Room temperature measurements on oriented samples show three main
features: an 8.8 eV absorption edge corresponding to transitions from the
0 2p bands in the top of the valence band to the Al 3s bands in the bottom
of the conduction band,
a 9 eV peak which is attributed to exciton
formation, and a 13 eV peak that is due to transitions to the Al 3p bands
in the conduction band.
The 13 eV peak shows orientation dependent
features that have not been reported previously; for E
perpendicular to
the crystallographic c axis a shoulder at 12 eV and a peak at 13.1 eV are
observed, while for E parallel to the c axis a doublet peak is observed at
12.9 and 12.3 eV.
These features have been interpreted in terms of the
available room temperature electronic structure models.
Even though these
measurements did not identify the direct or indirect nature of the minimum
band gap of the material,
the 8.8 eV edge is assigned to the room
temperature optical direct band gap of the material.
With increasing temperature the three features in the reflectivity
broaden and shift
to lower energies; the shifts
were
linear over the
temperature range studied.
The linear temperature dependence should hold
up to 1500 OC where anharmonicity of the lattice
vibrations becomes
noticeable.
The temperature dependence of the band gap is given by E (T),
where P
is the linear temperature coefficient of the band gapE and
E
(T=OK) is the T = 0 K apparent optical band
gap of A1 2 03 '
E (T) = E "(T=0K) -
$3=
g
g
-3
-4
1.11x10
eV/K +1-li 10
- 2
PT
o
E "(T=OK)
g
=
9.1 eV +/-
0.25
The lattice stabilization energy (A)
represents the difference
between the optical and thermal band gap energies and arises W
to the
difference between the time scales of electronic fcitations (10
hz) and
the concomitant response of the lattice
(10
hz) to the excited
carriers.
A leads to a decrease in the carrier creation energy for
thermal excitation processes.
A was not determined in this work, and no
reasonable estimates exist in the literature
In previous attempts to analyze the high temperature intrinsic
electronic conductivity of Al 03, the conductivity was used with assumed
models for the carrier mobili'y to determine the band gap energy.
With
the knowledge of the high temperature electronic structure presented in
this work, a different and more fruitful approach can be pursued.
The
character of electrons and holes, whether they are large or small
polarons, determines their mobility.
Information on the character and
mobility of carriers in Al 2032is scant at best: electrons have mobilities
on the order of 1 to 100 cm /V sec and appear to form large polarons,
while the mobility of holes has yet to be observed.
Therefore, it seems
reasonable to use the results of conductivity measurements, with our
knowledge of
the high temperature
electronic
structure,
to derive
information about the electron and hole mobilities.
This analysis
suggests that the holes in Al203 form small polarons with a thermally
activated mobility.
Thesis Supervisor:
Title:
R. L. Coble
Professor of Ceramics
- 3 -
Table of Contents
Chapter 1 Introduction
14
Chapter 2 Literature Review
17
2.1 Physical Properties
2.2 High Temperature Electronic Properties
2.2.1
2.2.2
2.2.3
2.2.4
2.2.5
Conductivity Results
Mobility Measurements
Photoconductivity Measurements
An Electronic Defect Calculation
Summary
2.3 Electronic Structure and Optical Properties
2.3.1 Electronic Structure Models
2.3.1.1
2.3.1.2
2.3.1.3
2.3.1.4
Cluster Models
Band Models
Franck-Condon Effect
Polarons
2.3.2 Optical Properties at Room Temperature
2.3.2.1 Summary
17
20
20
23
25
25
27
29
29
30
32
35
36
39
42
2.3.3 Temperature Dependent Electronic Structure
Theories
2.3.4 Temperature Dependent Optical Properties
2.3.5 Summary
43
48
52
Chapter 3 Research Goals
64
Chapter 4 Experimental Apparatus and Procedures
65
4.1 Literature Review - Techniques of VUV Spectroscopy
65
4.1.1
4.1.2
4.1.3
4.1.4
4.1.5
Light Sources
Optical Elements
Monochromators
Detectors
Optical Spectrometers
66
67
68
68
70
4.1.5.1 Room Temperature Spectrometers
4.1.5.2 Temperature Dependent Spectrometers
- 4 -
71
72
4.1.6 Special Problems of High Temperature Spectroscopy
73
74
4.1.7 Heating Methods
4.2 High Temperature Vacuum Ultraviolet Spectrophotometer
75
4.2.1 CO2 Laser Heating Techniques
4.2.2 Vacuum Ultraviolet Spectrophotometer
4.2.3 Data Acquisition and Signal Processing
75
77
80
4.3 Resolution and Accuracy
82
4.3.1 Energy E
4.3.2 Reflectivity R
82
83
4.3.2.1
4.3.2.2
4.3.2.3
4.3.2.4
Systematic Errors
Random Errors
Polarization
Signal to Noise Ratio at High Temperature
84
85
86
87
4.3.3 Temperature T
87
4.4 Conclusions
89
Chapter 5 Experimental Results
95
5.1 Room Temperature Results
95
5.1.1 Reflectivity of Sapphire
5.1.2 Orientation Dependent Features
5.1.3 Transmissivity of Sapphire
5.1.4 Other Forms of Al 20 399
5.1.5 Other Materials
5.2 High Temperature Results
5.3 Calculated Results:The Temperature Coefficient S
5.4 Summary of Results
Chapter 6 Discussion
96
97
98
99
100
101
105
118
6.1 Introduction
6.2 Room Temperature Optical Spectra and Electronic
Structure
118
119
6.2.1 Comparison of Optical Spectra to the Literature
6.2.2 Room Temperature Electronic Structure
6.2.2.1 Summary
119
120
124
- 5 -
6.3 Temperature Dependent Optical Spectra and the High
Temperature Electronic Structure
124
6.3.1 Optical Results
6.3.2 Temperature Dependent Electronic Structure Models
124
127
6.3.3 High Temperature Electronic Structure
of a Al2 03
23
6.4 The Role of High Temperature Electronic Structure and
Lattice Properties in Determining
Electronic Carrier Character
and Creation Energy
6.4.1 Polaronic Nature of The Electronic Carriers
6.4.2 Thermal versus Optical Band Gaps
129
131
132
133
6.5 Intrinsic Electronic Conductivity in A1 2 03
135
6.5.1 Factors That Control Conductivity
135
6.5.1.1 Carrier Concentration
6.5.1.2 Carrier Mobility
6.5.2 Calculation of a mg
e
6.5.2.1 Evaluation of A
6.5.2.2 Derivation of ph
6.5.2.3 Summary
138
e
Chapter 7 Conclusions
7.1
7.2
7.3
7.4
135
137
139
140
142
145
Reflectivity Measurements at Room Temperature
Temperature Dependent Reflectivity Measurements
High Temperature Electronic Structure
Implications for Conductivity
- 6 -
145
146
147
148
List of Figures
Page
FIGURE
2.1
Coordination
clusters
Al2 0 .
for
Aluminum
53
coordination octahedra
and oxygen coordination
tetrahedra
are
shown.
The two sets of Al - 0 bond distances are distinguished by
Oxi, Ox1' and Ox2, Ox2'. From reference 12.
FIGURE 2.2 Plots of refined thermal B factors as a function
of
temperature.
Straight
lines represent harmonic
regions of
thermal vibrations for Al and 0. From reference 11.
53
FIGURE 2.3 Diagram of conduction mechanism in single crystal
From reference 13.
54
Al 20 3
FIGURE 2.4 Effect of 0
partial
pressure
single crystal Al 203 From ref 2erence 13.
on
conductivity
FIGURE 2.5 Absorption (1)
and photoconductivity
of Al203 at 295 K. From reference 21.
for
54
spectra
54
conduction
ionization
55
(2)
FIGURE 2.6 Optical energy levels in Al 20 3
the
band edge E , the optical band gap E9
and the optical
energy E.0. Cprom reference 22.
FIGURE 2.7 Molecular orbital energies
calculation of Brower. From reference 25.
of
the Al203
cluster
55
FIGURE 2.8a X-ray photoemission spectra (PES) of the valence
band of Al 20
From Balzarotti, reference 26.
56
FIGURE 2.8b X-ray photoemission spectra
band of Al 20
(PES) of the valence
56
From Kowalczyk, reference 27.
FIGURE 2.9 Molecular orbital energies for the AlO6
calculation of Tossell. From reference 28.
FIGURE 2.10 Band
particular directions
reference 31.
structure of Batra
in
the
rhombohedral
for a Al203
Brillouin
-9
cluster
56
along two
57
zone.
From
FIGURE 2.11 Batra's complete band structure of a Al203 along
various directions in the rhombohedral Brillouin zone (shown as an
inset).
From reference 31.
57
FIGURE 2.12 Batra's total and orbital densities of states for
57
a Al 20
The broken curve in the top panel shows the XPS results of
Balzarotti. From reference 31.
- 7 -
FIGURE 2.13 Ciraci's band structure of a Al203 along
directions in the Brillioun zone (shown as an inset).
reference 32.
FIGURE 2.14 Ciraci's total
for a Al2 0
From reference 32.
and orbital densities
of
three
From
58
states
58
FIGURE 2.15 Band structure and density of states of the ideal
a Al203 (0001). Sd and S
are the surface state bands.
From
reference 32.
58
FIGURE 2.16 Illustration of the Franck Condon Principle. The
energy of the ground and excited states are shown as a function of
the configuration coordinate x.
The -energy of the fast optical
process is E while the slow thermal process is shown as E .
59
FIGURE 2.17 Representation of the localized potential wells
induced
by
electron-lattice
interaction
for
(a) the
large
polaron-weak coupling limit
and
(b) the small polaron-strong
coupling limit. From reference 18.
59
FIGURE 2.18 Reflectivity of Al203 with E perpendicular to the
From Loh, reference 38.
60
c axis.
FIGURE 2.19 Transmittance
x) and reflectance at
angles of incidence of Al 20
From Arakawa, reference 40.
various
60
FIGURE 2.20 Dielectric constants of Al 20 . The dotted curves
show Arakawa's separation of the exciton from &ie absorption edge.
From reference 40.
60
FIGURE 2.21 Reflectance of Al 0
for E parallel
and
perpendicular to the c axis. From Rublof, 3 unpublished research.
61
FIGURE 2.22 Reflection (1)
and absorption (2) spectra of
Al 20 3
The continuous curves were taken at 295 K and the dashed
curves at 4.2 K. From Abramov, reference 42.
61
FIGURE 2.23 Temperature dependent transmissivities of A12 03
from room temperature to 900 OC. From Gilles, reference 55.
62
FIGURE 2.24 Variation of an energy on the absorption edge
with temperature using the results of Gilles as calculated by
Harrop, reference 56.
62
FIGURE 2.25 Transmittance curves (a) of A12 03 as a function
of temperature from 20 to 380 *C, and the temperature dependence (b)
of the T equals 0.05 point on the edge. From Yakovlev, reference 57.
r
FIGURE 2.26 Transmittance curves of Al 0 as a function of
temperature from 20 to 175 0 C. From Laufer, reerence 58.
62
- 8 -
63
FIGURE
2.27 Reflectivity of Al 0
as a function
of
temperature from 90 K to 400 K. From Rubloft, 3 unpublished research.
63
FIGURE
2.28 Reflectivity of Al 0
as
a function
temperature from 21 to 450 OC. From Blum, r 2eerence 62.
of
63
FIGURE
resolution.
nm
91
FIGURE 4.2 High Temperature Spectrophotometer. Monochromator
and sample chamber built by Acton Research.
91
4.1
H2
Lamp
spectrum.
FIGURE 4.3a Two file baseline.
100
gm
slits,
0.4
R = rb / 'b.
92
FIGURE 4.3b Baseline of a four file background corrected
reflectivity.
Taken in transmission position with no sample,
therefore T = 1. No offset is evident.
Four file reflectivity
calculated by R = (r/rb (ib /i).
92
FIGURE 4.4 Absolute
different slit widths.
for
93
temperature
93
FIGURE 4.5
calibration curve.
Laser
energy
power
resolution
versus
versus
thermocouple
FIGURE
4.6
Optical
pyrometer
thermocouple temperature calibration curve.
energy
'temperature*
versus
94
FIGURE 5.1a Reflectivity of laser float-zone grown sapphire:
natural basal face, E perpendicular to the c axis, T equals 21 OC.
107
FIGURE 5.1b Reflectivity of Union Carbide sapphire:
plane, E perpendicular to the c axis, T equals 21 OC.
c
in
107
plane,
E
108
FIGURE 5.2b Reflectivity of Al 20 3: c in plane, E parallel to
the c axis, T equals 21 OC.
108
FIGURE
5.2a
Reflectivity
of
Al 0
perpendicular to the c axis, T equals 21 CC.2 3
0C.
c
in
FIGURE 5.3 Transmissivity of Al 20 3: R plane, T equals 21 OC.
109
21
FIGURE 5.4 Reflectivity of gel-amorphous Al2 3: T equals
109
FIGURE 5.5 Reflectivity of Lucalox polycrystalline
equals 21 CC.
0C.
Al 0 :
23
T
110
FIGURE 5.6 Reflectivity of single
crystal MgO:
T equals
21
110
FIGURE 5.7 Reflectivity
crystal LiF:
T equals
21
111
of single
0C.
- 9 -
FIGURE 5.8 Temperature dependent reflectivities of Al 03: T
(from bottom to top of figure) equals 21, 300, 500, 700, 90 , and
1100 OC. Each division on R axis equals 0.01.
112
FIGURE 5.9 Reflectivity of Al203:
the c axis, T equals 300 OC.
E parallel to
113
FIGURE 5.10 Reflectivity of Al203: c in plane, E parallel to
the c axis, T equals 700 OC.
113
FIGURE 5.11 Reflectivity of Al203:
c in plane,
c in plane, E parallel to
114
c in plane, E parallel to
114
FIGURE 5.13 Demonstration of the method used to determine the
energy of peaks: peak energies are 12.9 and 9 eV, T equals 21 OC.
115
FIGURE 5.14 Demonstration of the method used to determine an
arbitrary energy on the absorption to determine the temperature
coefficient: at R equals 0.20, energy is 8.7 eV, T equals 21 OC.
115
FIGURE
temperature.
116
the c axis, T equals 900
*C.
FIGURE 5.12 Reflectivity of Al203:
the c axis, T equals 1100 OC.
5.15 Energy of
FIGURE 5.16
temperature.
Energy of
the
'13 eV' peak
as
a
the
'9 eV' peak
as
a function
function of
of
116
absorption
117
FIGURE 5.18 Energy of the '9 eV' peak as a function of
temperature.
This data is from a single closely controlled
run;
note the linearity of the results.
117
FIGUP 6.1 Relationship between optical band gap E 0, thermal
band gap E
and lattice stabilization A energies.
g
144
FIGURE 6.2 Electronic conductivity versus partial pressure of
showing the electronic and ionic compensation points.
From
Kroger, reference 104.
144
FIGURE 6.3 Compensation point conductivity plotted versus l/T
to give the activation energy for conductivity.
From Kroger,
reference 104.
144
FIGURE 5.17 Energy of the R = 0.20 point on the
edge as a function of temperature.
H2
-
10 -
List of Tables
Page
TABLE 2.1 Calculated A Values From Rubloff's Data
TABLE
Temperature
5.1
Linear
Regression
Fit
Of
Feature
51
Energy
To
TABLE 5.2 Standard Deviation And Standard Error Of p
TABLE 5.3
TABLE 6.1
TABLE
Lowering
6.2
104
p (Temperature Coefficient) Values For Al203
105
emg
e
Calculated
103
139
Mobility
-
11 -
Ratios
And
Fermi
Energy
141
Acknowledgements
I am
throughout
approach
indebted
this
to
to Prof.
research
R.
L.
project.
formulating
Coble
In
for his guidance
addition he
research problems.
The
taught
and
me
guidance
support
a creative
of
Dr.
Hans
Jenssen was instrumental in my becoming an experienced spectroscopist.
I would also like
to acknowledge Prof. R. Raj of
Cornell
for my
first exposure to research.
The aid of Pat Kearney
in teaching me
the way around the
lab was
essential and I thank him.
I would like to dedicate this thesis to my parents who have supplied
me
with
an
education
and
an outlook
that
made
this
document
possible.
Their support and assistance throughout my life is gratefully acknowledged
and appreciated.
The
students
in
the
department
were
a
major
asset
both
to
my
education and the quality of life during my tenure.
I would like to thank
Todd
their
Gattuso,
Art
Keigler
and
preparation of this document.
thanks
to Yet Min-Chiang,
Morris
for
assistance
In addition to these three people
Alexana
Dynys and many others for the
Patti
Roshko,
technical
- 12 -
Jeff Gambino,
and non-technical
in
I owe my
Brint Corb,
Joe
discussions we
had over the years, and the friendships we have built.
I gratefully acknowledge
through
contract
number
the support
DE-AC02-76-ER02390
possible.
- 13 -
of
the Department of Energy
for
making
this
research
Chapter 1
Introduction
Aluminum Oxide (Al203)
is an important ceramic material in both its
single crystalline and polycrystalline
applications.
Traditionally
used
abilities
or as a high temperature
acquiring
new
module,
importance
in
insulator)
devices.
polycrystalline
ceramic
ceramics research.
a
temperature
temperature
applications
as
structural
insulator,
a
it
is
chip
carrier
crystal substrate material
for SOI
circuits,
major
as
high
and as
applications
prototypical
an optical material
worldwide
ceramic
for
are
in
as
a
fundamental
The ceramics research has focused on understanding
high temperature defect
of Al 203;
Its
and
its
and room
a single
integrated
opto-electronic
for
electronic
a thin film insulator,
(silicon on
forms for structural and electronic
the
structure, diffusion, sintering and creep behavior
the defect formation energies, specific
defects, their charges
and mobilities are fundamental to all of these processes.
Al203
system.
per
is
a
complex
material;
it
is
not
an
ideal
isotropic
model
Anisotropic due to its hexagonal crystal structure, with 10 atoms
rhombohedral unit cell, Al203 defies simple modeling.
It is a highly
ionic, highly insulating material, characterized by a large electronic band
gap,
where
many
properties
are
controlled
- 14 -
by
trace
impurity
atoms.
has been focused
in the past 20 years
research on Al203
Considerable
The charged nature
elucidating mass and charge transport in the material.
of all atoms and defects controls
on
their transport properties,
hence many
studies have addressed the intrinsic and extrinsic defect structures.
Conductivity has been the major scientific tool used
the
defect
chemistry of Al 203
Various
diffusion
to understand
experiments,
and
mass
transport experiments such as grain boundary grooving, sintering and creep,
have
also
been
used
to
Al 20 3's
explore
fundamental
defect
chemistry.
Surprisingly, the fundamental intrinsic disorder type (Schottky or Frenkel)
is
still
have
controversial.
been
The data
inconsistent
nonspecific:
parameters
and
parameters
of
the
collected
inconclusive.
measured
sample,
and
on
from different
These
depend
on
both
ionic
measurements
intrinsic
Without fundamentally characterizing the samples
laboratories
and
it
and
are
extrinsic
electronic
defects.
has been impossible
develop a coherent picture of the basic nature of Al203
too
to
and the processes
that control its behavior.
Toward
this
goal,
necessary to separate
data.
Electronic
in
separated
a
I explore
and
ionic
contributions
conductivity
is
a
Al203
Concentrating
on the electronic
of
origins,
electronic
and
Determination
analysis.
which
mixed
their
electronic
Significant progress
experimentally
to
experiment
by
conductor
conductivity,
carriers,
measure
of the
fundamental properties
the superimposed processes
experiment.
number
the
identify
mobilities
structure
toward
identifies
the
- 15 -
Al 203
in conductivity
the
conductivity
a
transference
at
high
can
temperatures.
intrinsic
specific
of Al203
temperature
or
carrier
in
this
dependence
the
extrinsic
is required
such is presented
be
number
one wants to determine
their
in
observed
of
bands.
for such
thesis,
of
the
electronic structure of Al2 03 , from 25
- 16 -
OC
to 1100 OC.
Chapter 2
Literature Review
In
this
chapter
the
literature
will
be
reviewed
understanding of the electronic structure of Al203 at
the first
to
develop
high temperature.
In
section the physical properties of Al203 will be reviewed.
the experimental
electronic
information that gives insight
properties
will be
In
presented.
an
Then
into the high temperature
the
third
section
a
more
detailed
discussion
pursued.
First, the room temperature electronic structure is discussed by
considering
the
measurements.
presented,
of
electronic
electronic
structure
structure
models
and
optical
and
the
properties
available
is
optical
Next, the temperature dependence of electronic structure is
consisting
of
the
temperature
dependent
electronic
structure
theories and the available temperature dependent optical measurements.
2.1 Physical Properties
a Al203
exceptional
high
is
(9 on
the
high temperature mechanical
quality single
techniques
a hard
crystals of
such as Verneuil,
moh
clear
crystal,
and electrical properties.
a Al2 03
Czochralski 1 ,
- 17 -
scale),
can be
grown by
a
with
Large
variety of
Edge-Defined-Film-Growth
2
,
and
in a large variety of shapes and sizes.
the Heat-Exchanger-Method
are
three
stable
major phases
at
phase
all
unstable
intermediate
hydrated
aluminas,
of A1 2 03 :
temperatures,
cubic
and
the
the
phase
the
a phase
y
phase
temperature oxidation of aluminum metal.
Pauling ionicity of 56%.
transfer
between
ionicity.
is
low,
Al
phase
formed
amorphous
is
There
the
thermodynamically
a
thermodynamically
is
through
the
can be
of
dehydration
formed
by
the
low
Al2 03 is an ionic material with a
Effective charge calculations show a large charge
and
0
atoms
suggesting
an
even
larger
degree
of
The band gap is large and the lattice point defect concentration
making Al203
a 'good' insulator.
Its melting point
is 2051
OC,
while its Debye temperature has been variously reported to be 1050 K5, 935
K6 or 695 K
.
The
index
of refraction
and dielectric
constant
at
room
temperature are respectively 1.7604 and 9.3 perpendicular to the c axis and
1.7686 and 11.5 parallel to the c axis.
The a phase consists of a hexagonal close packed, AB oxygen stacking
sequence,
atoms.
with
two-thirds
of
the
octahedral
The presence of three aluminum layers -
voids
filled
sites parallel
atoms followed by a void.
are
occupied
allows
to the c axis
P A y B a A P B y.
there are
aluminum-aluminum
repulsion
to
is in distorted tetrahedral coordination with Al atoms.
can be
seen
two aluminum
The fact that only 2/3 of the octahedral sites
distort
octahedral (0 h point group) coordination of the aluminum.
clusters
aluminum
in sequence a, A, y - leads
to a unit cell with twelve layers in the sequence A a B
Along the octahedral
by
in figure 2.1.
the
ideal
Each oxygen ion
These coordination
The space group of Al203
is D63d
or
- 8
9
R3c. With two different Al-0 bond lengths of 0.186 nm and 0.197 nm. The
fundamental
atoms).
unit
cell
is rhombohedral,
containing
two
formula
units
(10
The hexagonal cell, containing 30 atoms has lattice parameters a =
- 18 -
0.475 nm and c = 1.299 nm.
The
structural changes associated with changes
thermal expansion of the lattice,
changes
are:
in temperature
in the distortion of the atomic
coordination of the atoms and increasing
atomic vibration as demonstrated
by
expansion
the
Debye
Waller
factors.
Thermal
temperature, and at 1100 OC is 8.84 x 10-6 /C
and
7.96
x
10-6 /C
parallel
to
the
c
of
varies
Al203
with
perpendicular to the c axis,
axis.10
There
have
been
two
diffraction studies performed to determine the structural changes occurring
with temperature.
showed
the
changes
concentrated
distance.
on
in
the
the
the
difference
between
distance.
high
temperature,
for
occurring
each
reported to change
the
two
Al-0 bond
and
interatomic
from 0.1971 and
to 0.2024 and 0.1880 nm at 1900 OC.
but the Al
temperature.
of
The
the
Al
The Debye
sub-lattice
At
1900
has
increased,
Debye-Waller
thermal vibrations, were 0.33
OC.
at
lengths
In other
increases
with
The oxygen sub-lattice moves toward a more perfect hexagonal
order with temperature,
coordination
lattice
changes
The Al-0 bond lengths were
temperature.
increasing
crystal
specific
0.1852 nm at room temperature
words
C1 1
High temperature x-ray diffraction results to 1900
OC
factors,
the
distortion
distortion
with
which
an
increases
of
the
increase
represent
in
the
with
octahedral
the
Al-Al
magnitude
of
nm2 at room temperature and 2.25 nm2 at 1900
Waller factor B =
87 u , where u equals
the
rms
thermal
displacement, so these correspond to atomic vibrations of 0.0064 nm at room
temperature and 0.0168 nm at 1900 Oc.
The structural
neutron diffraction.
2.2.
this
These
are
changes to 2000 OC were
The Debye-Waller factors measured are shown in figure
linear up to 1500
temperature
is
studied by Aldebert12 using
attributed
0
C. The departure
to
- 19 -
the
onset
from linearity above
of
anharmonic
atomic
vibrations.
This
implies
are
three
higher energy phonon modes are
temperature
energy increases
amplitude
the
vibrations
lattice
there
become
with the
associated
of the
anharmonic.
approach
to
regimes:
below
above 750
excited,
modes present, and
This
melting,
of
region
the
although
structure is maintained up to the melting point.
the
Debye
OC thermal
above 1500 OC
anharmonicity
corundum
is
crystal
Aldebert also observed an
increasing difference between the two Al-O bond lengths, and the increasing
distortion of the Al Octahedral coordination.
2.2 High Temperature Electronic Properties
In this section various results from the ceramics literature will be
that
cited
give
insight
into
the
electronic
properties
of
Al 20 3:
the
conductivity, the mobility of carriers, the energy for optical creation of
carriers,
and
the
difference
between
optically
and
thermally
created
carriers.
2.2.1 Conductivity Results
There
have been numerous
studies done on the conductivity of A1 0
2 3
both in nominally pure or intentionally doped,
samples.
Most conductivity results reported
were
to
due
sample.
surface
It became
and
obvious
gas
conduction,
that
higher
- 20 -
single
and polycrystalline
in the 60's
not
the
purity and
and
early 70's
conductivity
better
of
the
characterized
samples were needed to successfully model the defect chemistry.
studied
the
conductivity
and defect
many different dopants,
and has
characterized samples.
This has
has
shown the
dopant
activity
electronic
conductivity
Still
even
Kroger's
with
understanding
of
the
reported
served
of
lowest
chemistry of Al2 03
a
large
the
quantity
in Al203
and
compensation
extensive
conductivity
extensively, using
of data on well
to advance the modeling of Al 203
hydrogen
at
Kroger has
study
and
we
defect
has measured
point
lack
yet
a
structure
the
reported.
comprehensive
of
A1 2 0 3 ;
for
example, different experiments still suggest different fundamental disorder
mechanisms.
energy
The temperature dependence of the electronic carrier creation
E (T)
g
is
an
important
factor
A1 2 0 3 , but
conductivity behavior of
necesary
up
for
the
to now such
modeling
information
of
has
the
been
lacking.
A1 2 0 3
understood
2.3).
from
indentified.
constant
At
the
numbers
By
ionic-electronic
conduction
Using a technique
transference
curves
is a mixed
at
(see
plot
of
behavior
can be
Kitazawa 1(see
figure
spurious conduction,
experiments,
temperature,
corresponding
mechanism
to avoid
looking
conductor whose
the
three
regions
variation
figure
2.4)
to transitions
of
one
ionic
conductivity
were
of
conductivity
can
from n to
and performing
see
ionic
the
V
with
pO 2
shape
of
at
the
to p type conduction.
low temperatures (T<1400 OC) the observed conductivity was predominantly
ionic
and
independent
of
p0 2 .
At
higher
temperatures
and
low
pO2,
the
sample was an n type electronic conductor, while at high p0 2 the sample is
a p type (hole) conductor.
Kroger
has
observed
conductivities
Kitazawa
and with a different
contests
Kitazawa's
conduction
lower
transference
number
mechanism
results.
- 21 -
than
those
behavior,
Kroger15
reported
and
has
by
therfore
used
a
different
conductivity
technique
to
extract
a
band
gap
carrier creation energy from conductivity measurements.
or
electronic
The V shaped a vs
pO 2 curves taken under non-equilibrium conditions, where new atomic defects
are not created show the compensation point of the material at the base of
the
V, where
sample.
equal
numbers
Kroger used
an
of
electrons
acceptor
hydrogen, which acts as a donor.
and
protons,
and
the
analyzed,
incorporating
dominated
holes
sample
are
present
and
in
the
'titrated' with
The conductivities due to ions, electrons
transferences
identification of a due to e',
and
h',
V
were
measured,
''', and Al.'''.
allowing
the
The data was then
the mobilities of the various defects,
permitting
the extraction of the equilibrium constants K for the mass action relations
that
then
yield
the
carrier
creation
energies
and
defect
If the electronic carrier creation energy is known,
energies.
formation
and used in
the calculation, then the mobilities can be determined.
It
is
useful
non-equilibrium
between E
and
structure
and
to
further
analyze
titration experiments,
P
(the
models
the
theory
because they show
temperature coefficient
of
the
material.
of the
The
gap)
behind
these
the relationship
and the defect
electronic
and
hole
conductivities are given by:
a =N[e]q# e , and ah=N[h]qVha
Here N is the number of Al203 molecules per cm3, [e] and [h] are the
concentration of electrons and holes per molecule, q is the electron charge
and g is the appropriate mobility.
eh
2
0 ->
e'
At the compensation point,
ipeph
+ h'
K..
1
- 22 -
a h
creation of
the mass
one
gives
intrinsic
electronic
carriers.
for
equilibrium constant
relation
action
Assuming
large polarons this
has the form
K.=C T
1
3
-Eg/kT
e
= E 0(T=OK)-PT
g
3
P/k -E0 /kT
therefore K. = C
e
e g
Now
E
g
1
K. ~ C e
1
or
e (+.)/kT
The linear model of E (T) is confirmed for Al203 in this work.
analysis
leads to values of 10 +/- 2.1 eV and 11 +/- 1 eV for E opt(T=OK)
g
and a value of 0.65 +/- 2.1 eV for A,
and
thermal
gaps.
the difference between the optical
The
large
calculations
raises doubt
about
For example,
the assumption that both carriers
well
established
analysis
This
but
is instructive
strongly
number
the
of
assumptions
validity of
affects
the
these
are
used
in
specific
numbers.
large polarons
interpretation..
these
is not
But,
the
in showing the role of the temperature dependence
of the electronic structure in defect chemistry.
To
modelling
summarize Kroger's reported values of the band gap of Al203
conductivity data,
the thermal gap energy, Eg,
g
the optical band gap,
E 0,
g
was
for
10.35 eV and
was 9.7 eV Both of these energies correspond
to a temperature of 0 K.
2.2.2 Mobility Measurements
The mobilities of electronic carriers in Al2 03 have been measured in
- 23 -
Hall
effect
mobility
photoconductivity
related to the
is
In
mass.
and
a
large
carrier type,
band
curvature
of
material,
gap
whether
i.e.
experiments.
the
the
the carrier
In
'band'
semiconductors,
through
mobility
affected by
also
or
a band
The mobility serves as a good indicator of carrier type.
effective
the
is
is a polaron
the
electron.
Additionally, the
mobility is required to estimate the conductivity.
Hughes16
sapphire,
measured
inducing
transient
photoconductivity
2
sec,
assigned to coduction band electrons.
was
independent
temperatures,
mobility,
and is not
large polaron without
appropriate
indicative
mobility
of
from
of temperature
to photoconductivity
of the
intrinsic
trapping would exhibit
sapphire
100 to 300 K.
and
He notes that this may be
500
to
at
these
low
carrier mobility.
900
Dasgupta17 measured the
K
and
observed
electron
mobilities, increasing with increasing temperat ure, of 0.1 to 5 cm /V
Dasgupta
results
but
states.
At
these
the
than
temperatures
or
impurity
phonons
for
majority
the
mobility
scattered
carriers
observed
a
cm2/V
mobility;
hole
of
mobility
100
was
not
sec
were
should
Green18
with
observed.
was
At
from
the
up
His
impurity
to
more
the
Hall
to 1000 K,
assigned
a
being
studied
high
to
correspond
traps
19
sec.
electrons.
also
at temperatures
which
contributed
originated
mobility,
scattering.
of optically created
also
carriers
and may have
mobility
mobility
may have
mobility
not optically created
modulated
efficient
hole
that
concluded
carriers were
trap
that
considered
A
mobilities on the order of 15
to 100 cm 2/V sec due to optical phonon scattering.
Hall
Carbide
He reported that the hole mobilities
were not observed and therefore were much lower.
trap-modulated
Union
free carriers with low energy x-rays from
The mobility measured was 3 cm /V
a
in
to
and
electron
temperatures
he
concludes that phonon scattering is dominant and that the results represent
- 24 -
the intrinsic carrier mobility.
2.2.3 Photoconductivity Measurements
is
Photoconductivity
a
good
to
technique
identify
the
energy
required to optically create mobile electronic carriers that contribute to
the observed conductivity.
It has usually been used to study the carriers
given off by color centers and other mid-gap states, but,
higher energy optical
which
true
there
has
excitation,
valence band
been some
transitions
to
it can serve to identify the
to conduction band
ambiguity
features
excitation
in assigning
in
if performed with
the
the
optical
occurs.
energy
spectra,
the exciton peak observed in the optical
differentiating
absorption edge.
energy at
In Al 203
of band
to
specifically
band
in
spectra from the
Kuznetsov20 21 studied the photoconductivity of sapphire
from 7.4 to 9.9 eV at room temperature
(see
figure 2.5).
Here one can see
the onset of photoconductivity between 8 and 9 eV going to a peak at 9.25
eV.
At 9.4 eV the photoconductivity increases very rapidly due to band to
band transitions
edge.
If one neglects the photoconductivity due to thermal dissociation of
excitons,
observe
and the large joint density of states at the absorption
and
its
projects
onset,
the
one
absorption
finds
that
edge -photoconductivity
back
the
to
onset
of
band
to
band
photoconductivity occurs at 9 eV.
2.2.4 An Electronic Defect Calculation
The atomistic
techniques used to calculate point defect
- 25 -
properties
in ceramics
and
are
generally
electrostatic
crude
repulsion,
atomic size defects.
calculations
their
aim
where
a
instructive
electronic
large
to
is
an
to calculate
the
point
charges
properties
of
But in the case of a material such as
electron-lattice
review
'defects'.
techniques
consider
They do not operate on the correct scale to calculate
electronic energy levels and states.
Al2 03
is
that
the
The
results
from
calculation
interesting
interaction
an
It
present,
atomistic
of electronic
hybrid.
is
is
calculation
states with
serves
it
to
for
atomistic
highlight
many
characteristics that might be ignored if the problem were considered to be
one of electronic states existing outside the context of the dynamic atomic
structure
(as exists upon invoking the Born Oppenheimer approximation of a
static
lattice).
lattice
calculation to determine
relaxation,
polarons.
Colbourn
and calculated
and
the
Mackrodt22
have
the differences
relative
performed
the
necessary
introduced by the lattice
stability
of
large and
small hole
They found the large hole polaron state to be 0.4 eV more stable
than the small hole polaron state.
They calculated
the energy of electronic
'defects'
by starting
with
experimentally determined electronic energies in their calculation and then
determining
electronic
the
lattice
energies
contribution
in
the
calculation
to
the
are
the
defect
energy.
optical
The
band
gap,
two
E 0,
g
corresponding
to
conduction band
the
edge,
valence
E ,
band
that
is,
to
the
conduction
band
energy between
conduction band and the vacuum level ( see figure 2.6).
optical ionization energy, E.
1
E.0 = E
0
decomposed
the
and
the
of
the
bottom
They then define an
that is the sum of these:
+ 1E 1.
The optical
be
,
energy
into
ionization energy
two
components,
is experimentally determined.
an
- 26 -
optical
lattice
term,
EL,
It can
that
Madelung
includes
and
range
short
the
electronic
shell relaxation
due to electron
energy of the lattice
polarization
and
contributions
(the
thermal version of this term, ELt would also contain the relaxation energy
Lo
due to ion core motion) , and an effective ionization potential a eff
corrpponding to the reaction
0
(lattice) -> 0 + ev.
vac
Due
to
small
potential
differences
also
has
a
in
the
thermal
final
state,
counterpart
the
effective
that
8e
differs
ionization
from
the
These energies are related to the optical
optical one by 0.5 eV for Al 20
ionization energy by
E=E
0
E
The
L
lattice
eff
contribution
potential is determined.
is
calculated,
and
the
effective
Then the thermal band gap E
ionization
can be calculated:
g
E t= E.t -E
= E t+8e
- E.
g
i
c
L
eff
c
Using 9.5 eV for the optical band gap,
band edge, EL
equal to 18.4 eV,
found to be -7.9
eV and Et
and
1 eV for the conduction
and ELt equal to 16.1 eV,
is 8.4 eV.
then e effis
This leads to a thermal band gap
of 6.7 eV, 2.8 eV less than the optical gap energy.
These lattice calculations are done
therefore,
if
the thermal
will
also
g
at 0 K only,
the optical band gap energy changes with temperature,
gap energy,
shift
at
the
temperature optical gap.
gap, E 0,
for a lattice
is 9.5
eV,
using
this non-temperature-dependent
same
rate,
and
be
2.8
eV
less
and
then
calculation,
than
the
high
To summarize, Colbourn, assuming the optical band
calculates
a thermal
band gap,
E9,
g
of
6.7 eV which
corresponds to the activation energy for carrier creation E crea at 0 K.
2.2.5 Summary
- 27 -
The
conductivity
results
reveal
creation and carrier mobility, but
From
effects.
the
mobility
it
the
combined
of
effects
is difficult to separate
measurements
we
see
that
the
carrier
these
mobility
two
of
electrons has been studied, but the observation of hole mobility has been
obscured
by
indication
of
the
the
electrons.
The
photoconductivty
room temperature
mobile
carrier
measurements
creation
give
energy.
an
The
calculation of polaron defect state energies demonstrates the importance of
the lattice to the electronic structure at high temperature.
- 28 -
2.3 Electronic Structure and Optical Properties
Optical
structure,
energy
measuring
levels
mobility
Still,
that
property
the
a
the
while
that
is
model
energy
avoiding
and
of
the
of
is
effects
of
presented.
temperature
involved
electronic
the room temperature
in
of
is
any
the
followed
electronic
material's
electronic
allowed
factor
of
electronic
the
transport
optical
structure
then
a
between
optical properties
This
on
probe
additional
interpretation
review of
Al203
can
difference
the
necessarily
analysis
basic
measurements
electron
measurement.
measurements
is
available.
and electronic
by
structure
a
Here
a
structure
discussion
and
require
the
of
the
available
temperature dependent optical measurements.
2.3.1 Electronic Structure Models
There
are
two
structure of solids:
and
the
atomic
distinct
the band
cluster
chemistry.
All
vibrations
allowed
approaches
used
in modeling
molecular
the
electronic
structure calculation of solid state physics
orbital
calculation
of these models are done for T = 0 K.
in
the
models.
The
bond
of
solid
state
There are no atomic
lengths
used
in
the
calculations are those appropriate to the material at room temperature.
The band structure models can give the most complete description of
the
transport
perspective
properties
highlights
of
bonding
a
solid,
and real
- 29 -
while
space
the
molecular
structure.
Band
orbital
structure
bands.
as
carriers
electronic
cluster
A
bond
between
that
occur
is
most
bands
are
flat
Neither
of
atoms.
in a situation where
tightly
bound
reside
in a
specific
can
the
encompass
changes
interaction
strong electron-lattice
leads
The cluster approach could model the before
to relaxation of the lattice.
and after
electrons
models
these
ionic
an
to
appropriate
and the
free-electron-like
curved,
by highly
demonstrated
model
the
material where
that exhibit delocalized
to covalent materials
models are most appropriate
transition state calculation,
state well through a
but
this has
yet to be done for
Al 203
2.3.1.1 Cluster Models
The
cluster,
atomic
molecular
same
cluster
structure
for
coordination,
by
represented
As
separation.
orbitals
The
without
by
the
the
interact
atoms:
calculation
is
atomic
separation
orbitals
between
with
the
infinite
solid,
the
set
of
the
real
atoms
The
are
popular
octahedral
space
using
a
electrons
at
decreases,
Exclusion
bonding, nonbonding and antibonding orbitals.
the
in
in
periodicity.
atoms
Pauli
The most
surrounded,
when
the
bulk.
done
long range
calculating
small cluster of atoms of
the
Al
invoking
in accord
as
an
seven
O's.
six
elements
symmetry
uses
Al203
local perspective
are
and
for
method
the solid by a
electronic structure23 represents
the
orbital
the
Principle,
infinite
atomic
forming
In approaching the limit of
molecular
orbitals
would
broaden
continuously into the bands of band structure theory.
The
ionic
localized
character
of
or molecular
Al203
is
nature. of
demonstrated
- 30 -
the
by
electronic
the
fact
that
structure
the
and
cluster
models can
compared
reproduce the major features
to the band
of the electronic
structure models.
calculations on Al203 was
the
structure when
The method used for both cluster
Self Consistent Field,
X a Approximation,
Scattered Wave Cluster method of Slater and Johnson. 2 4
Brower's25 results for an Al203
in figure 2.7. In the model r
like.
cluster with D3 symmetry is shown
is s like, r2 is pz like and
r3
is p'
p
The atomic character of the various groups of levels is shown.
top of the valence band is of eu 0 2p character.
The
The next available level
is an Al 3s level at 8.75 eV followed by an Al 3p level at 9.4 eV.
this one would deduce
The
cluster
bulk;
only
for
this
study
lacks
the
true
3s levels.
coordination
neither the Al or the 0 atoms have their full complement
nearest
bands
chosen
a band gap of 8.75 eV from 0 2p to Al
are
neighbors. Therefore
sparse
and
the
levels
the bandwidths
are
of
the
valence
too small.
and
From
of
the
of first
conduction
The valence band is
1 eV wide and does not show the dual peak structure that is observed
in the photoemission spectra (PES) of the valence band (see figure 2.8).26
27
Tossell
cluster
with
28
0h
has
calculated
symmetry
(see
distortion (D3d) from octahedral
the calculation.
4
eV wide valence
levels:
the
molecular
figure
2.9).
orbitals
He
choose
for
to
an
A10 6
ignore
9-
the
symmetry present in sapphire to simplify
The low lying 0 2s levels are separated by 14 eV from a
band.
The valence
band
consists
of two groupings
of
the top of the valence band is 0 2p like while the bottom of the
valence band is a mixture of 0 states with Al 3s and Al 3p states.
This
agrees with the PES results that show a 17 eV separation and a 6 eV wide
valence band.
to
the
large
The successful characterization of the valence band
amount
of
0
-
0
interaction
-
31
-
in
the
cluster
is due
chosen.
In
there
contrast,
is no Al - Al interaction in this cluster,
There is an Al 3s level at
expect a narrow conduction band in the model.
3
7.6 eV and Al
levels at
p
11 eV,
leading one to
agreement with other
in qualitative
This cluster model would suggest a 7.6 eV band gap between 0 2p
models.
and Al 3s levels.
2.3.1.2 Band Models
Band structure models of the electronic
insight
By
into the details of the electronic
considering
the
flatness
ionicity,
covalency
or
carriers.
The density of states
structure can supply
or
of
the
invaluable
structure than cluster models.
the bands
localized
(DOS)
structure can give greater
one
or
can gauge
delocalized
the
degree
nature
that can be derived
of
of
the
from the band
information on the detailed
structure
of
the bands and permit the calculation of the joint density of states (IDOS)
that
should
between
closely map the energy and intensity of possible
the valence band and conduction band.
transitions
There have been four band
structure calculations performed on Al203 in the past 15 years.
performed
got
the
results
first calculation
that
calculation led
bands.
were
unreasonable,
but
the
ad
hoc
technique
nature
of
that did
used
not
a
semi-empirical
surpass
method
the knowledge
that
to
calculate
is derived
a
the
1982 Batra31 published
the
results
of his
Ciraci and Batra32 published
- 32 -
first
principles
band
from the
cluster models and in addition led to a forbidden gap of about 36 eV.
and in 1983
and
to excessive band widths of 16 and 17 eV for the valence
Evarestov30
structure
not
in 1970 using a simplified
Douglas 2 9
In
calculation,
the results of a semi-empirical
calculation that also discussed much of the available
on Al2 0
These
calculations
the
art
two
that,
in
calculations
while still
our
are
both
high
spectroscopic
quality,
rudimentary in comparison
understanding
of
SiO 2 ,
signal
a
complementary
to the
new
data
state of
stage
in
our
understanding of Al 203
Batra
performed
a
first
principles
LCAO
(linear
combination
of
atomic orbitals)
extended tight binding method calculation on a Al203 with
an
in
understanding
Al
with reference
interest
oxidation of
consistent
or
even though
ionicity
SiO 0
to
iterated to determine
it is a first
and
the
principles
bonding
as
relate
This calculation was not
stability of
calculation,
the answers.
the
exchange
a is an input variable that can change the values of E
the bandwidths.
they
to
self
Also,
parameter
the band gap and B
The symmetries of the bands were not identified, so it is
not possible to discuss optical transitions from individual bands.
The band diagrams for selected directions (see figure 2.10) and all
directions
in
calculated
value of
Caution
of
a
the
is
0.75,
but
appropriate,
calculation.
The
2.12.
E
is 8
Zone
eV,
(see
figure
while
the
2.11)
valence
is called for with respect to these numbers.
decreases to 5 eV.
more
Brillouin
The
with a
1, the
gap
Batra states a value of
compensating
for
is
in
the
6
The
eV wide.
The accepted value
to
11
ae= 0.95, and E
problems
of
top frame
of
the
states
from
this
shows the total DOS,
DOS
on
to
the
calculation
eV while
B
= 9.5 may be
techniques
of
the
is
shown
in
figure
while the lower frames show the
constituent
lowest lying bands are 0 2s in character.
character,
increases
band
shown.
The band gap is found to be direct, at the zone center.
density
decomposition
=
are
atomic
orbitals.
The
The valence band shows a 2 peak
the bottom 4 eV of the valence band is made up of hybridized Al
- 33 -
while the top 2 eV of the valence band is comprised of
3s and 0 2p bands,
PES results
(see
figure
2.8).
while it
is a large density of states
There
in the
The bottom of the conduction band consists of Al
top of the valence band.
3s states,
is in good agreement with
The structure of the valence bands
0 2p bands.
major peak in the conduction
is stated that the first
band appears 4 eV above the conduction band edge, and would correspond to
Al 3p states.
The other recent band structure calculation was done by Ciraci and
Batra.
This was a semi-empirical
photoelectric
threshold
energy,
This
and broad
Only broad
from PES experiments.
used.
calculation that used the band gap,
study was performed
features,
features,
no
to interpret
such
specific
the
as
the
bandwidths,
structure,
available
were
experimental
spectral results and to supply more insight into surfaces of Al203 and the
The effects of effeptive charge on the atoms
process of oxidation.
covalency)
and
widths
positions
and
variables,
the
the
inter-orbital
of
the
band structure
matrix
bands
were
model
was
elements
on
the
investigated.
then
fitted
valence
Using
to
(i.e.
the
band
these
two
previously
mentioned experimental data.
The band structure calculated
the
energy
indirect,
gap
between
valence
with a value of 8.7 eV,
value of 8.9 eV.
band
and
0 2p
which
0-Al bands by Batra,
assigned
in Al203
band
is found to have 3 different regions.
followed with hybridized
present
conduction
the direct gap would occur at
of the valence band is due to hybridization
were
Ciraci found
to
be
r with
a
The lowest lying bands are associated with 0 2s states.
The valence band itself
part
the
is shown in figure 2.13.
as
and Al 3p states.
and the bands are non-flat.
- 34 -
The lowest
of 0 and Al 3s
states,
Both of these regions,
demonstrate
the
covalency
The top of the valence band
is
of
predominantly
0
2p
character,
In figure 2.14
total and orbital DOS are
the
band width calculated is 12 eV,
energy
(see
Upon comparing
figure
of 3
shoulder
excessive
larger band
of hybridization 0 contributes
interpretation
are
very
2.8a)
regions
observed
width and
flat,
width,
in
The valence
A larger degree
can be
seen by the
the conduction
band
in
the valence band to the PES results of
one
sees
in the
in PES.
shoulder
shown.
larger than Batra found.
in accord with the
this calculation.
Balzarotti
bands
followed at higher energy by bands of Al 3p character.
Al 3s character,
greater degree
the
The bottom of the conduction band is of
demonstrating their ionic nature.
of covalency,
and
that
the widths
valence
But
in
band
recent
in Balzoratti's
agree,
would
work
spectra
while
explain
of
the
the
low
Tasker,33
the
are
attributed
to
experimental error.
To study the defect
introduced by
structure
see
one
edge.
the dangling
including
surface
mid-gap band
Ciraci suggests
normally attributed
and
states
in A12 0 3 , Ciraci calculated
bonds
formed on a
states
2 bands
that the
to exciton
is shown
that
the
(0001)
surface.
in figure
2.15.
introduce
states
The band
Here we can
states below
the band
features observed near the band edge and
formation may
actually
arise
from
these
surface state bands.
2.3.1.3 Franck-Condon Effect
The electronic
a static
are
lattice,
outside
their
structure models discussed previously only consider
the effects of lattice relaxation and atomic vibrations
scope.
When
the
lattice
- 35 -
changes
configuration,
the
position of the electronic
energy levels
will
shift.
A qualitative model
for this is given by the Franck Condon Principle.
The
large
Franck
Condon
differences
between
observed
for F centers
the
of
case
Principle
alkali
emission occurs at
(
was
first
absorption
energies
)
anion vacancies
halides,
1 eV.
F center
developed
and
to
explain
emission
the
energies
in alkali halide crystals.
absorption
may occur
The difference between these
at
In
2
eV while
energies,
1 eV, is
called the Stokes shift and represents the effect of lattice relaxation on
the electronic energy levels.
breathing
mode displacements
by a configuration
Now
referring
excited
total
lattice,
figure
x
2.16
example,
that
which
then
these
represents
shows
the
can be
the radial
energy
parameterized
displacements.
of
the
ground
states as a function of the configuration coordinate x,
state
such
The
for
coordinate
energy minimum
excited
E .
to
If we consider the relaxation to consist of
it
occurs
as
lattice
for
an
the
ground
at x .
optical
state
occurring
A transition that
transition,
will
we see the
while
for
the
is fast relative to the
occur
will then relax with the emission
at x 0
and
vertically
requiring
of phonons
to
its
new
equilibrium position at x,, where the emission energy will be given by Ee.
The
difference
between
these
energies
is
the
3
centers in Al203 it is on the order of 3.1 eV.
Stokes
shift,
and
for
F
4
2.3.1.4 Polarons
In ionic materials, with locally defined electronic structures, the
interaction
between the
The
carriers,
charge
electrons
electrons
and the
or
holes,
- 36 -
lattice
can
becomes very
be
dressed
by
important.
phonons,
or
and
a
polaron
created.
lattice
whereby the
created.35
is
Both
the carrier
energy of
and
electron
hole
is reduced
can
polarons
be
Large polarons are formed where there is a weak electron lattice
interaction
In
fields,
relaxation
lattice
and the relaxation
(see
figure 2.17),
this case
the
field
leading
energy of
is
to small displacements
the polaron
relative to the nearest lattice atoms.
of strong
electron-lattice
self-trapped
spread over a large
coupling,
in its own lattice
is
independent
of
of
in the
the atoms.
its
position
A small polaron occurs in the case
and leads
field.
area
to a particle
The electron's
its position, so it tends to reside near an atom.
effectively
energy depends on
Large polarons can move
**
in an applied field, but their effective mass m
is larger than the band
*
effective mass m
A small polaron can move only by a thermally activated
.
hopping process similar to that of an atomic defect.
The strength of the electron-lattice interaction determines whether
polarons
should be considered in the strong
interaction limit
for small polarons, or in the weak limit for large polarons.
polaron
coupling
constant
a,
determines
these
two
limits.
appropriate
The Frohlich
36
It
is
a
dimensionless measure of the lattice deformation or stabilization energy.
a/2 = (deformation energy x 2n)/(h W)
e
2m w 2n1/2
27
=--------------------
-
4hw
h
-1
x(e~,
For a <
are
-1
-co0
1 we are in the large polaron limit, while for a > 10 we
in the small polaron limit.
a for Al203 was calculated by Hughes to
- 37 -
be
2.7, while
Kroger
calculated
a
value of
3.1. These
were
calculated
without regard
to the differences between the optical phonon frequency W
most effective
for lattice deformation around holes or
and
can
electrons
respective
have
character.
distinct
values
of
a
electrons.
that
Holes
determine
The measured carrier mobility is a good
their
indicator
of polaron type; mobilities below 0.3 cm2/V sec are associated with small
polarons.
The polaron rest mass is given by
**
m
or for a>1
*
= m (1 + a/6)
= m
a < 1
(1-.0008a2)/(1-(1/6)a+.0034a2
**
This leads to a value of m
Due
to
the
sparsity
electrons and holes
polaron
optical
is
temperature
also
the
for Al 0 .
2 3
is useful
for conductivity
strongly controlled
is
S
experimental
it
phonon branch that most
frequency
to
of
in A12 0 3,
mobility models
polarons
of 1.75 m
one
that
by
the
data
on
the
of
to refer to the theoretical
calculations.
vibrational
strongly
mobility
couples
should be used
The mobility of
frequency W0
to the
to
of
carrier.
determine
the
This
the Debye
BD. Reported values of the Debye temperature vary from 695 K
1000 K depending
on
the
reference.
The
'best'
value
may be
695
K
because it was calculated from an experimental phonon spectra for Al 20337
But this value
is much lower than the other values quoted,
that
For
for
MgO.
large
polarons,
the
fundamental equation given below.
- 38 -
mobility
is
and less than
determined
by
the
4A
plp
O /T
1/2
3 ()
**
m
4
q
1/2
ip
3 (7)
From
a.0
this
one
calculated
m
can
/
[T0
WI
--
aw
see
mobilities
that
by
uncertainty
an
order
of
in
0
or
magnitude.
w
can
The
change
small
the
polaron
mobility is not determined by a fundamental equation and is of the form:
SP
=
3/ 2
Ap
-Eh/kT
where Eh is the activation energy for hopping of the polaron.
2.3.2 Optical Properties at Room Temperature
Over
crystal
the
years
there
have
been
sapphire (A12 03 ) performed
many
optical
studies
to better understand
as an optical window or a geological material.
Most researchers studied
the complexity of VUV measurements
higher
Only
analysis.
in
the
past
single
its applications
only the absorption edge,
energy
of
15
become available over a considerable energy range.
years
have
restricting
measurements
The measurement of the
reflectivity or absorptivity of a solid is among the most direct means of
measuring
the
Therefore
the
structure
of
energy difference
optical
the
spectra
valence
and
between allowed
allow
one
conduction
- 39 -
electron energy states.
to
profile
bands
with
the
ease.
electronic
The
room
temperature
spectra
optical
in
energy
the
range
to
appropriate
an
intrinsic electronic structure study will be reviewed.
studied
Loh38
of
surfaces
natural
the reflectance,
of Al203
The
sample
spectra
corresponds
spectra
(see
was
to
figure
to
larger
absent
peak
is
at
the
features
and
shallow
slope
7
13.2
eV.
in R below
results.
lower
in other work,
such
as
surface
Kramers Kronig
In
basal
the
plane
c
9.1 eV,
sample
the
the
sharp
9
incidence,
measurements
axis.
an edge
was
Cr
surface,
so
the
Apparent
in
his
starting
doped,
absorption
eV peak
addition,
at
yet
10 eV
conspicuously
the
are
edge
not
in
intensity of
show prominent
at
10
eV and
agreement
the
with
exciton
of
optics
data,
or
but
He
sample.
interpreted
did
most
is
problems
not
his data
the
peak
and may have been due to experimental
of his
were
taken point by
transfer peak.39 His spectra
contamination
analysis
of
and a smaller peak at 12 eV followed by
The
but
literature
than
eV,
angle
The
(ruby).
a
to
is a peak at
eV Cr charge
structure,
present
E perpendicular
followed by a shoulder at 10.6
a
45 degree
6 to 14 eV and were
covering
cut
2.18)
a
doped with Cr
performed at room temperature,
point.
at
to
perform
show an
exciton peak at 9.1 eV, and a 10 eV band gap.
The
most
comprehensive
done
work was
Williams 4 0
and
by Arakawa
studying anodized films on aluminum and single crystal Al2 03 oriented with
the
c
axis
lying
polarization
orientation
and
in
the
their
plane
of
the
spectrometer
the machine.
but
surface.
do
They measured
not
oblique
They
do
mention
note
the
incidence
their
data
refraction
functions.
n
Their
using
+ik,
the
reflectivities
Kramers Kronig
dielectric
for
analysis,
constant
corundum
- 40 -
are
a
reflectance
calculating
and
the
shown
in
the
sample
transmission of corundum at room temperature from 8 to 28.5 eV.
analyzed
of
of
in
the
energy
figure
They
index
loss
2.19,
where one
peaks at
a peak at 9.5 eV,
can see an edge,
13,
16.5 and 20.5
shows a shoulder at 9 eV.
of band
correspond
to band
to peaks
identified the plasma
transitions
in
the
joint
occurring
at 9.9
The
eV.
density of
higher
states.
In
energy for collective oscillations
below the
energy peaks
addition
of
they
the valence
Beyond these three assignments of features
spectra they were unable
to
identify any other peaks or attribute
optical features to the electronic structure.
found,
the dielectric constant
or bound electron pair,
electrons as lying at 25.8 eV.
in the
In figure 2.20
They choose to assign this to a separate peak
to an exciton,
corresponding
onset
eV.
a shoulder at 11.6 eV and
One interesting result they
which demonstrates the relationship between physical and electronic
structure, was that in the anodized disordered films, the exciton appeared
1.1
eV
lower
transitions
in
they
energy
at
considered
8.4
to
eV,
and
the
occur
0.5
eV
onset
lower
of
at
band
9.4
to
eV.
band
Other
features in the spectra did not shift from film to crystal.
Rubloff
in unpublished
from 8.3 to 27.5
and perpendicular
energy so it
measured
the
to the c axis.
The temperature
dependent
They show (see figure 2.21)
and an absorption edge at 10 eV.
in
results will
an exciton peak
The spectra do not go to low
is not possible to determine if the absorption edge starts at
10 eV or if the exciton is superimposed on an edge.
problem
reflectivity of Al 203
eV from 90 K to 400 K and for light polarized parallel
be discussed in section 2.4.4.
at 9.2 eV,
work
deconvoluting
the
exciton
and
the
Arakawa had a similar
edge.
dependent features of their spectra are pronounced.
there is a broad peak centered
at 12.8 eV,
The
orientation
For E parallel to c,
while for E perpendicular
to c
the peak is at 13.2 eV with a slight shoulder at 12.3 eV.
Abramov42 in a paper on electronic structure and optical properties
- 41 -
measured transmission and reflectance
from 4.2 to 20 eV from 4.2 to 295 K
on a sample cut to present a basal plane.
2.22)
shows an exciton peak at 9 eV,
Their reflectivity (see figure
a small shoulder at 10.4 eV,
From the steepness of
small peak at 11.7 eV, and a major peak at 12.8 eV.
the
absorption
They
absorption
edge.
exciton
at 9.25
is
less than 9.5
determine
their exciton peak
edge
say
an
seems
absorption
eV at room temperature,
eV.
They measured
to be
superimposed
starts
edge
and the
It was more intense
Rubloff's
spectra.
8.5
optical
The exciton peak
the edge with a halfwidth of 0.6 eV and a low binding
0.1 eV).
at
photoconductivity, and
an energy gap of 9.25 eV.
another
on
the
eV.
The
band gap
from this
is
they
is superimposed
on
energy (less than
for E perpendicular to c which agrees with
They also reported that
the highest joint density of
states occurs at 13 eV.
2.3.2.1 Summary
Even in the most recent work the identification of the 9 eV peak as
arising from excitons is not definitively reported.
observed arises from an exciton peak below
as to whether the
reflectance
the onset of band
to band transitions,
superimposed
on
Investigators
agree
relative
absorption
upon
energy
shape of the reported
electronic
structure
reflectivities
observed
the
from
to
8.8
models,
different
to
9.2
eV.
Disagreement persists
edge
it
is an exciton peak
to
band
or whether
of
positions
band
of
major
transitions.
features,
but
the
reflectivities varies.
Due to the lack of
the
not
investigators
transitions.
Two groups
- 42 -
The
did
exciton
observed
a
attribute
peak
small
has
the
been
shoulder
at
-10.5 eV.
12.3
eV,
For E perpendicular
and a major peak
to c,
a shoulder is seen between 11.6 and
at 12.8 to 13.2
eV.
For E parallel to c, a
broad peak is observed at 12.8 eV.
2.3.3 Temperature Dependent Electronic Structure Theories
In the past 30 years there
have been many attempts to
electron-phonon problem so as to understand,
view,
the
cause of
energy levels
temperature
in solids.
This
and pressure
is
a very
point can address the shifts observed
new
material
of
technological
from a fundamental point
induced
shifts
of
the
theory
has
of
electron
complex problem which
in most semiconductors,
interest
treat the
at this
but for each
needed
to
be
reformulated.
Beyond the complexity of the calculations, it does not seem
that
necessary
all
the
understanding
terms
have
of the electron-phonon
even
b'een
determined
problem and the
to
allow
temperature
an
induced
shifts in solids.
All
of the theories separate
the
temperature
induced
shift in the
band gap into two terms.
dE
dT
The
that
dE
P
dT
dE
V
dT
ther. exp.
temperature induced shift is separated
is due to the electron-phonon
into a term at constant volume
interaction,
and a second term due to
the effect of lattice dilatation that accompanies thermal expansion.
- 43 -
Fundamental Models
In
the
electronic
fundamental
structure
approach considers
the
there
of
the
temperature
have been two approaches
the effect
chemical-thermodynamic
of phonons on the
approach considers
dependence
taken:
the
electronic
physical
states,
only
first
work
problem as
work
in
are
states that
gap.
the
physical
approach was
done by Radkowsky 4 4
specifically
at
Muto
from the uncertainty
interacting
with
leads to energy
and
Fan
phonons,
Self-Energy
theory
the
of phonons on the
is added
principle:
undergoing
developed
effect
He
AE/At
viewed
= h/27r.
transitions
the
The
between
in the band
model served
limited
independently
the
electron's
self-energy.
This
by
the number of phonons
to reasonably
, while above 0
and leads
shift of the
D
the shift
is proportional to 2nkT/hw.
explain results observed in Si and Ge.
success with other materials
because
it
and
is an effect
Their model predicts an exponential
D
Fan45 and
emission
independent band structure
band gap below the Debye temperature 0
linear because
of
electron's
on to the temperature
to energy level shifts.
has
materials.
level broadening and a decrease
Oyama46 considers
absorption
is
ionic
This was
This theory has not met with success and is no longer used.
The
that
aimed
resulting
electrons
while
the effect of the electrons
with the aim of understanding the shifts observed in insulators.
the
of
The review here is after Cohen. 4 3
on the phonons.
The
models
requires
This
Yet it
a detailed
knowledge of the electron-phonon coupling constants which are usually not
known.
The
.47
Antoncik
Brooks-Yu
Debye
Waller
theory
developed. independently
by
and Brooks and S. C. Yu in unpublished work 48 takes a different
- 44 -
point of view on the problem.
Here they calculate
temperature dependent
band structures by modifying the crystal potential with structure
or Debye Waller terms that represent
with
temperature.
As
the
factors
the vibration and shift of the atoms
temperature
increases
the
atomic
potentials
become shallower and more spread out, leading to spreading and shifting of
In this theory,
bands.
temperatures
far
quadratically
temperature,
far
above
below
with
the
the
there are three distinct temperature regimes.
the
Debye
temperature.
shift
Debye
is
temperature,
For
the
temperatures
band
gap
above
temperature.
Because
the
shifts
the
linear, becoming exponential for
For
Debye
temperatures
Brooks-Yu
theory
only
requires modification of the potentials used in a standard band structure
calculation,
it
semiconductors.
has
seen much application
in Group
IV,
III-V,
and II-VI
This theory can supply results for some systems that are
accurate to within +/- 30%.
Even
though
calculating
two
the Brooks-Yu
theory
temperature dependent
theories
discuss two
shifts,
different
ignored,
the
leads
increasing
to
incorrect
self-energy
succeeded
the
shift,
but
terms
greatest
is the belief
act
use
that
in
these
in opposition.
The
theory even when the self energy
in the atomic potentials
in certain cases,
results.
The
band gap with temperature
decreasing band gap.
the
and
These uncertainties are sufficient to cover up
self-energy contributions,
terms
that
lies in the uncertainties
Debye Waller factors used.
seen
there
terms
reason for the success of the Brooks-Yu
terms are
has
HgTe
but the
and
ignoring self-energy
CdTe
alloys
show
an
Brooks-Yu theory predicts
a
A study by Guenzer 4 9 showed that the neglect of Fan
might
explain
the
HgTe
in uniting the two previous models
demonstrating
the
uniqueness
- 45 -
and
case.
Allen
and
Heine 5 0
into one coherent model for
complementarity of
the
two
theories.
This
temperature
theory
dependent
may
be
the
correct
shifts
of
the
electronic
semiconductors,
but
electron-phonon
coupling
because
of
the
constants
unified
structure
in
difficulty
it
has
theory
not
seen
for
the
observed
in
determining
much
the
application.
Allen and Cardona performed the unified calculation for Si and Ge in 1981,
and corrected
it
in
1983.
They
found that
the
self-energy terms
could
reduce the temperature coefficients by a factor of two.
The chemical
considers
work
to this
area was done by Thurmond51
electronic
thermodynamic
potential
approach
problem
is
one
the effect of electrons on the phonons of the solid.
in this
Various
thermodynamic
of
structure
variables,
the
features
the
carriers,
followed
are
energy
the band
gap
in
corresponds
corresponds
Energy for creation of electron-hole pairs,
and
The first
recently by Heine.
identified
Fermi
that
terms
to
to
of
the
the
therefore,
52
their
chemical
Gibbs
the
Free
enthalpy
and entropy due to carrier creation can be calculated and considered as a
function of temperature.
considers
the
effects
Waller
factors
them.
The
because
bonds
thermal
simultaneously
due
to
excitation
without
the
corresponds
as
the
anomalous
anti-bonding
conduction
observed
it
enters
a
bonding
to
is
and
explicitly
the
Debye
consider
softening
of
the
This arises
to breaking
one
from a bonding
orbital.
it
or
more
valence band
The
thermodynamic
and should also be able to explain
in HgTe,
increases due to mode stiffening.
band
band
is that
self-energy
electron hole pairs.
electron goes
success in Si and Ge,
results
need
of
in the material,
approach has met
the
and
creation
electron
an
of this method
that must be determined
an
to
advantage
expansion,
of
orbital
the
of
only thing
phonon modes
The major
because
in
that
case
the
gap
When an electron enters the conduction
orbital,
- 46 -
in
distinction
to
the
other
semiconductors.
determines
Still,
This
the
sign
point
of
may
the
due to the difficulty
be
the
fundamental
temperature
coefficient
of determining
the
consideration
in
any
electron's
that
material.
role
in mode
softening, this method has not met wide acceptance.
Empirical Models
Due
to
the
computational
they are not readily applicable.
that
are
basically
fundamental models.
to E
curve
complexity of
the
fundamental
theories,
Two empirical models have been advanced
fitting
models
and
The first by Varshni53 gives
do
not
contradict
a quadratic
the
dependence
below the Debye temperature, and a linear dependence above the Debye
temperature.
It requires two parameters a and
from experimental data.
#,
which are not derivable
His equation is
E =E -aT 2 /(T+A).
g o
Ravindra54
simplified
Varshni's
equation removing
the
parameters
and replacing them with 0D'
Eg=E0- ((2.25xlo-5D)-4.275x10-3 )T2 /(5(T+5sD-1135))
From
the functional
they only consider
dependency of
the Debye Waller term.
these models,
one can
see
that
They have been applied by the
authors to Si, Ge, SiC, GaAs, InP and InAs with reasonable results.
Linear Models
Most of the results presented in the literature use a simple linear
- 47 -
model
for
the
assumed model
temperature
dependence
of
or the data itself may be
the band
gap.
truly linear,
This may be
an
a distinction that
is. important.
E (T) = E9
g
-
T
g
Here the
linear temperature coefficient
range
10
of
to
10
eV/K
and
A
is usually found to lie in the
a positive
decreasing band gap with temperature.
value corresponds
to a net
The linear dependence should apply
from theory for temperatures on the order of the Debye temperature.
2.3.4 Temperature Dependent Optical Properties
Temperature
dependent
optical measurements
have been performed
on
Al 0 over the years due to its use as an optical window material, and to
2 3
develop an understanding of the processes that lead to temperature induced
shifts
of
absorption
edges.
Most of
these
studies
have been performed
below the exciton peak and absorption edge due to the difficulties of high
energy
spectroscopy.
The transmittance
was measured.
T
r
Generally the
falls below some arbitrary
transmittance
limit of the material
limit is defined as the wavelength where
value,
usually between
0.1 and 5 percent.
Because of the limited energy range covered, the specific character of the
edge
studied
may not have been determined.
what
appears
to be
result
fundamental
arbitrary
coefficient
and
can
a chromium peak was
to sapphire.
affect
the
The
In some cases the
studied
choice
calculated
shift of
and then reported as a
of a cutoff value
value
of
the
for T
r
is
temperature
5, while any experimental uncertainty or drift in the absolute
magnitude of the temperature T between runs also introduces error in
- 48 -
P.
The
first reported study on the temperature
of sapphire was done by Gilles 5 5
transparency
absorption coefficient of single crystal
5.9
to
8
energy
eV
(see
figure
to determine the
2.23)
These
origins of
induced
in 1952.
She measured the
sapphire from 20 to 900 OC from
spectra
the
do
not go
absorption
to
studied,
same time a peak at 6.9 eV would seem to be attributable
These
spectra were
He
results.
second
is
analyzed by Harrop56
them
and third
to
order
a
power
terms,
plotted
nonlinearity.
figure
2.24
while
and
at the
to Cr impurity.
conductivity
found
to use
noticeable
a linear model
The data he extracted from Gilles
versus
energy
and
shows
noticeable
One of the complicating factors in using these measurements
P is the low lying peak at 6.9 eV that is discrete at low
to determine
temperature
in
expression
then proceeded
eV/K.
high enough
to explain observed
law
but
P value of 1.5 x 10
yielding a
data
fit
shift of the
while
it
is
incorporated
in
the
absorption
edge
at
high
temperatures.
Yakovlev
in 196157 measured transmission up to 8 eV
samples heated from room temperature
are
shown
in
figure
2.25
along
to 380 OC.
with
the
nm)
(140
on
The transmittance curves
temperature
coefficient
they
calculated for Tr = 5%. From these one can see that the transmittance edge
has
not
retained
temperature
its
coefficient
shape
with
increasing
temperature.
for their transmittance
cutoff
The
measured
and others
chosen
by this author give an idea of the scatter in the data.
# (T =5%) = 1.5 x 10 -3 eV/ K
r
# (T =20%)= 1.4 x 10 -3 eV/ K for 85
r
# (T =20%)= 2.0 x 10
No
attempt
was made
eV/ K for 20
to
explain
C<T<380 OC
0
C<T<380 *C
the data
letter on an experimental problem of interest.
- 49 -
since
this
was
a
short
Another
material
figure
study,
for
high
2.26)
the
motivated
by
temperature
transmittance
an
interest
in
photochemistry,
edge
by
of sapphire,
a
suitable
Laufer58
up to 8.75
window
shows
(see
eV (140
nm)
moving to lower energy for temperatures from 20 OC to 175 OC. The value of
0
calculated
experiment
by this
covered
author
for T
=
r
10 %
a small temperature
is
range.
1.9
x
10
trends
is made
in
ion
established.
In
size
from peak shift.
and
exciton
transitions,
but
is
not
rigorously
Al2 03 is said to be different, and no explanation is given.
1969
sapphire
Reilly60
specifically
to explain their results on alkali halides in terms of
Hunter59
studied
the
temperature
transmittance limit of VUV window materials
and
eV/ K. The
The authors
mention the importance of distinguishing peak broadening
An attempt
-3
from
10 K
to
373
K.
explaining the observed
A
of
such as the flourides,
companion
shifts
dependence
in
paper
terms
was
the
quartz
published
by
of photoexcitation of
excitons of reduced energy at thermally distorted regions of the lattice.
Hunter
refers to the fact that
chooses
the
materials
they
temperatures
i.e.
cut
off
wavelength, X
found
a
linear
approaching
the value of
the value
10 K,
to be
co
range
the
of
for
slope
P is not well defined,
at
0.1<T <0.5. For all the
X
above
r
changed
of
shift
in this
case
results
for
co
was
attributed
sapphire,
while
zero,
LaF3 was the
induced shift.
large
at
The
impurity content
.
sapphire were
taken for
temperature, corresponding to 8.75 eV.
below the exciton peak.
to a
eC,
approached
P decreased to zero at low temperatures.
controlling the value of X
The
20
and
only material studied that did not show a temperature
lack
and
a X
co
of 142 nm at room
This is on the absorption edge and
They do not show the actual Tr edge measured for
but from the high temperature portion of the temperature
- 50 -
shift
curve,
a A value of 1.75 x 10-3 eV/
K from 250 K to 373 K can be deduced.
This value is in agreement with those measured previously,
and the change
of 0 with temperature is a new finding.
Rubloff61 studied the reflectance of single crystal Al 203 from 8.3
eV to 27.7 eV using
synchrotron radiation and continuous scanning.
detailed spectra were obtained
and
perpendicular
to
temperature dependent
and 400 K.
center
the
showing reflectivities
c axis,
curves were
exciton peak,
each of
taken
for
'fused'
for E both parallel
Al 20
for E parallel
In
addition,
to c at 90,
300,
From this small amount of data this author has calculated peak
temperature coefficients
calculated
and
Very
the
13
eV Al
by measuring
the temperature
figure 2.27).
for 3 features
3p
peak and
geometric
changes
peak
in the
spectra,
the
18
eV peak.
centers
at
the
the 9 eV
These
half maximum
from 90 to 300 K and 300
were
for
to 400 K (see
Due to the lack of multiple runs over which to average,
the
results are only accurate to within a factor of two.
TABLE 2.1
-1.3 x 10 1.3 x 10-3
1.5 x 10-3
-6 x 10 -4
6 x 10
6 x 10
9eV peak
13eV peak
18eV peak
Notice
Calculated A Values from Rubloff's Data
eV/K
90 - 300 K
300 - 400 K
that A
is larger
with Hunter's findings.
above
room temperature
It is very unusual
to shift to higher energies with temperature,
small
scans.
a
size
than below,
in agreement
that the exciton peak appears
this may be due to the very
of the exciton peak and its position at the beginning
of the
Blum calculated the A value of Rubloff's exciton peak and reported
positive
value
of
5 x
10-1
eV/K
in contradiction
analysis of the data.
- 51 -
to
this
author's
Blum62 measured the reflectivity
0
25
C, 275 OC,
10-4
eV/K
and 450
for
attributes
it
the
0
of sapphire
C (see figure 2.28).
rate
of
shift
of
his
from 9 to 12.5
eV at
He reports a value of 1.4 x
spectra with
to a decrease in the band gap.
temperature
and
He also analyzed a series of
peaks on the high energy of the exciton peak, at 9.5, 9.9, and 10.1 eV and
attributes
stated
them
to
the
that his data
is
excited
states
of
a
Wannier
type
exciton.
He
consistent with a room temperature band gap of
10.3 eV for Al
203
2.3.5 Summary
From
the
literature
results we
can
see that
the
room temperature
optical properties of Al 0 are generally agreed upon, but the position of
2 3
the fundamental absorption edge is not well established.
Some researchers
have reported that the edge lies at 9 eV while others have reported a 10
eV
This
edge.
studied.
disagreement
The
assignment
may be
of
the
due
to
features
the
in
limited
the
energy
optical
ranges
spectra
to
electronic structure transitions has been retarded due to the lack of high
quality electronic
structure models.
This has changed recently with the
band structure calculations of Batra and Ciraci. The valence band is found
to consist of two major sub-bands,
-
0 character
in
the bottom of
consists of two major sub-bands,
of 0 2p character
the valence band.
from
the
0
2p
The
conduction
band
of Al 3s charcter in the bottom and Al 3p
character at the top of the conduction band.
band gap of Al 0
23
at the top and of Al
The reported T = 0 K optical
lies between 7.5 eV and 9.5 eV and is due to transitions
sub-band
in the
top
of
- 52 -
the
valence
band
to the
Al
3s
sub-band in the bottom of the conduction band.
In
terms
electronic
of
structure,
understanding
the
field
the
is
temperature
still
young.
dependence
The
theories
temperature dependence are very complex and theorists are still
with the simpler cases of the covalent semiconductors.
results
we
temperature.
know
that
This
the band
gap
has been modeled
of
as
Al203
reported from 1.4 x 10~4 to 2 x 10-3 eV/K.
the
of
the
struggling
From experimental
decreases
a linear
of
with
increasing
process with P values
None of the data has been of
good enough quality, or over a large enough temperature range to determine
if the true functional form of E (T) is linear.
g
The role of the lattice in the electronic
seriously developed.
Al203 where
structure
has not been
This is a very important topic for materials such as
strong electron-phonon
interaction gives rise
localized electron-lattice interactions.
- 53 -
to polarons and
C
4
Ox
Ox2
FIGURE 2.1 Coordination clusters for Al 0
Aluminum coordination
octahedra and oxygen coordination tetrahedra are2 s own. The two setsof Al
- 0 bond distances are distinguished by Ox1, Ox1' and Ox2, Ox2'. From
reference 12.
X102
1B(,n,)
AL
-AL
.0
.5
-
I
0
0
i'oa
.T(60
2000
FIGURE 2.2 Plots of refined thermal B factors as a
temperature.
Straight
lines
represent
harmonic
regions
vibrations for Al and 0. From reference 11.
- 53.-
function of.
of thermal
ELECTRONIC
L)
FIGURE
2.3
Diagram
of
conduction
mechanism
in
single
crystal Al 203 From reference 13.
CONDUCTION
1600
w
n-TYPE
p-TYPE
w 1400
23
aw
IONIC CONDUCTION
1200-
-14
-12
-I0
-8
-:
LOG P.2
0
-2
-4
at
-4
SINGLE-CRYSTAL A1,O,
1650 *C
-5
-0.2
FIGURE 2.4 Effect of 0
partial pressure on conductivity oi
single
crystal
Al 20 3
From
reference 13.
0.2
1500*0
0
1-
3-r0
~
'
A
-61-
-O'MATC
'''
-0.15
ESIMT
0.15
0
0.2
1400 -C
-7t-
a
1200 C
"0
0
an
-14
-12
-10
-8
-6
-4
-2
LOG P
02
a-
FIGURE 2.5 Absorption
(1)
and photoconductivity (2) spectra
of A1 2 03 at 295 K. From reference 21.
tu
*
q
*
,
-
S .
a...,
w
20
A,
- 54 -
cv
0
VACUUM
E
Ei
CONDUC TION
SAND
Eg
Ei
0
SAND
OL2p, VALENCE
FIGURE 2.6 Optical energy levels in Al 20 : the conduction band edge
E , the optical band gap E 9 and the optical ionization energy E.. From
reference 22.
0 ATM
AlA CLUSTER
r
r,
At
ATOP
r
45j-
Xs
2P
Al3s
-Lo 1-
-1.307
:A00326
-*.'395
FIGURE 2.7 Molecular orbital
energies
for the A1 2 03
cluster
calculation
of
Brower.
From
reference 25.
..
-1.295
*
/00-00
L8
p-L3$
Z
-LS
0
2p
h
-2.0 I-
*2LdIO a 2s
'000
-2.5
-LAN
0440
C
W
-.0
30
4-LI
41612
4.612
-5.612
.0.0,
Al
Al 2p
u
4.0
21
-. 135
-L.35
Al2
- 55
-
In
E
65f-Z 7
I
is
-
.02s
.
0
U
(b)
A12 0 3
%0-
0
VB
C
S
WS
N75
1X
7485
Et4m -
24
1VB
C
'-
0
-Y
Binding
FIGURE 2.8a X-ray
spectra
photoemission
valence
(PES) of the
From
band
of
Al2 3.
Balzarotti, reference 26.
energy
(eV)
FIGURE
2.8b
X-ray
photoemission spectra (PES) of the
valence
band
of
From
Al 0
Kowalczyk, reference 27.
Oh point group
orbital
sy mmetry type ---
eg tg tu
+
161
Al 3p{ 12F
FIGURE 2.9 Molecu1Dr orbital
energies for the A10
cluster
6
calculation
of
Tossell.
From
reference 28.
c.b.
Al 3s
4-
a)
0 2p f
v.b. Al-C
{
r
0OW
-*-
-4
-12 -
o 2s -16
-20
- 56 -
p.
-
*0~
14
12
FIGURE 2.10 Band structure
of Batra for a Al 0
along two
23
particular
directions
in
the
rhombohedral Brillouin zone.
From
reference 31.
10
I
8
0
S-2
a)
-4
-6 -14
Ar0
~
12
-16
E, =8
.-
-18
4
-20
Z
r
x
0
PD
FIGURE 2.11 Batra's complete
band structure of a Al 0 3 along
various
directions
in
the
rhombohedral Brillouin zone (shown
as an inset).
From reference 31.
4
-8
-
'.9
16
KpDS
16
r
i~
t
X A
MOPU-F Z
A
r
a
kv
I
Y
Bond structure for A1203
0
I
Oxygen 12p)
a,
a
C
FIGURE 2.12 Batra's total
and orbital densities of states for
a Al 2 03 * The broken curve in the
top panel shows the IPS results of
Balzarotti. From reference 31.
C
a
Oxygen (2s)
a
Aluminium
(3s)
Aluminium 13p)
-20 -16 -12 -8 -4
0 4
Energy (eV)
- 57 -
8
12
4-
CCB
0.
FIGURE 2.13 Ciraci's bind
structure of a Al 2 0j? along three
directions in the
illioun
zone
(shown
as
an
inset).
From
reference 32.
-4 '>>g8.7eV
,VBM'
-8
>-12
>- -16
j -20
UVB
-24LYB
UVBm
LVBM
-28
TDOS
(a)
-32
-36
UVB
LVB
000S :Oxygen
Za
j
z A
r
XVBm
__
xr
i
_Y
A
Y
CB
2p
(b)
CM,
U.'
FIGURE 2.14 Ciraci's total
and orbital densities of states for
a A2 0. From reference 32.
-
LA
Oxygen 2s
-32
1c)
Aluminum
3p
Id)
Aluminum
3s
(e)
-24
-16
-8
ENERGY (eV)
5
0
CB
0
\Sp
'-- 5
FIGURE 2 .15 Band structure
and density of s tates of the ideal
a A12 0 3 (0001). Sd and S are the
surface
bands?
From
state
reference 32.
ASd
._10
UVB
LV-15
u .20
=-25
-30,
LVB
-35,
M
r
r
- 58 -
M
K
K
r
DOS
Excited electronic state
aIJ
U
I)
E.E
0
Lowest electronic state
X1
x
FIGURE 2.16 Illustration of the Franck Condon Principle. The energy
of the ground and excited states are shown as a function of the
configuration coordinate z. The energy
of the fast optical process is Ea
while the slow thermal process is shown as E
ve
Lattim.
sit"
-
a.
%
Latt
pet atLal
~ea
UmakLeaplia
Yr
b.
Stteng Cap11ag
FIGURE 2.17 Representation of the localized potential
wells induced
by electron-lattice interaction for (a) the
large polaron-weak coupling
limit and (b) the small polaron-strong coupling limit. From
reference1 8 .
- 59 -
24
020
-i
-
Is
2.18
FIGURE
Reflectivity of Al 03 with E
perpendicular to tie c axis.
From Loh, reference 38.
--
4I
-
-03
~6
S14
12
Sto
I
-
I
I
I
-
a
6
5
-
14
15
A1
U
U
I
2
bID
7 8 9 10 11 12 13
Photon Energy (eV)
-,.oum
0.6
4
9C.,
II'
-l
Ii.
6
I-I
FIGURE
*J0#\.
0.4
2.19
(x)
and
Transmittance
reflectance at various angles
of
incidence
of
A1 0 3
40.
From Arakawa, reference 2
ha
U
z
4
0.2
a,
00
DO
PHOTON
DER.Y
WA
6'
CORUNDUM
z
z
0
C-)
0
10
15
20
25
30
PHOTON ENERGY (eV)
FIGURE 2.20 Dielectric constants of Al 0
Arakawa's separation of the exciton from
e
reference 40.
- 60 -
The dotted curves show
absorption edge.
From
0.6
M
1 2 03
used
0.5
0.4
0.3 4E-.
0.2
0.1
0
10
15
20
25
30
Photon Energy(eV)
35
FIGURE 2.21 Reflectance of Al 03 for E parallel and perpendicular to
the c axis. From Rubloff, unpublished research.
FIGURE 2.22 Reflection (1)
of
and
absorption
(2) spectra
Al 0
The continuous curves were
taien at 295 K and the dashed
curves at 4.2 K. From Abramov,
reference 42.
I
20%
2
#a
-
.
U 10
.
4-
0
06
10
14
Energy (eV)
-
61 -
0
2.0 1
U
18
a
.2
FIGURE
2.23
Temperature
dependent transmissivities of Al 0
from room temperature to 900
From Gilles, reference 55.
-
-
In
.
7.75
6.89
6.20
Energy (eV)
p~11~11~
I
1000
FIGURE 2.24 Variation of an
energy on the absorption edge with
temperature using the results of
Gilles as calculated by Harrop,
reference 56.
I
I
I
I
I
I
I
0
T (K)
S
.600
S
200I'
6I
I
7.0
I
I
I
I
8.0
7.5
E(eV)
0
I
S..
0.
Ill'
'U
a..
E-e
.-
liso
U
V
V%
10
1451)
I..
El
ad
4,
106u
/0o
a
/fee
1JIM
Wavelength, A
FIGURE
2.25
IS
S..
I..
1
-
"
-
100
Transmittance
curves (a) of Al2 0 3 as a function
of temperature from 20 to 380 *C,
and the temperature dependence (b)
of the Tr equals 0.05 point on the
edge. From Yakovlev, reference 57.
- 62 -
--
1OO
jig
405
Temperature, 'C
'
50
40
FIGURE
2.26 Transmittance
curves of Al203 as a function of
temperature from 20 to 175 SC. From
Laufer, reference 58.
z20
I-
t0
A
1400
0.20
I
I
1450
1500
I~I
*
A12 U3 Lii|
RS= 0.01
0.15/
Ve
FIGURE
2.27
Reflectivity of Al 0
c 0.100.05 -
05
10
15
In
20
25
30
Photon Energy (eV).
13
I
I
I
I
Sample I
12
-
o45t),&C
CO
FIGURE 2.28 Reflectivity of
Al203 as a function of temperature
from
21
reference
as
a
function
of
temperature from 90 K to
400 K.
From
Rubloff,
unpublished research.
to
450
*C. From
Blum,
S
10
-V
62.
9
-I
8
9
- 63 -
10
I1
Photon energy (eV)
12
Chapter 3
Research Goals
In an attempt
and
as
an
aid
to improve our
in the
analysis
fundamental understanding
of conductivity
results, the
structure as a function of temperature
was studied.
the
band
temperature
coefficient
of
determination of the electronic
The
major
thrust
of
temperature
vacuum
ultraviolet
incidence
the
reflectivity
the
carrier
work
from
gap,
involved
to
electronic
This work determines
thereby
allowing
the
concentration at any temperature.
the
spectrophotometer
7
of Al 203
15
eV
on
development
of
to measure
near
samples
heated
a
high
normal
from
room
temperature to temperatures above 1000 OC.
Reflectivity measurements
confirm
the
spectral
features
Experimental
electronic
cannot
be
electronic
available
electronic
assigned
to
measurements
structure
structure.
of
models,
electronic
temperature
supply
on
this
temperature
structure
the
derived,
From
room
fundamental
features,
theoretically
at
a
electronic structure can be developed,
of
considering
and
the
structure
which
temperature
picture
performed
dependent
information,
the
were
the
observed
features.
shifts
at
this
dependence
high
of
of
time
the
temperature
the role of the lattice
on the creation energy of the electronic carriers and their mobilities.
- 64 -
to
Chapter 4
Experimental Apparatus and Procedures
The
incidence
aim
of
this
reflectivity
wavelength range
research
from 6
has
been
to 15 eV,
from 210 nm to 82 nm,
heated from room temperature
to
in the
measure
and
construction
spectrophotometer.
signal processing
work.
First
optical
New
on oriented single crystal
to temperatures greater
and heating
techniques
optical
VUV
capabilities,
will be
vacuum
spectrophotometer
will
Then
be
the
samples
ultraviolet
geometries,
have been developed
reviewed.
(VUV)
They involved the
the literature on VUV spectroscopy techniques
spectrometers
temperature
elements,
normal
than 1000 *C. These
of a custom high temperature
optical
near
Vacuum Ultraviolet
are. measurements that have never before been performed.
design
the
design
discussed,
advanced
during
this
and previous
of
followed
the
by
high
its
resolution and accuracy.
4.1 Literature Review - Techniques of VUV Spectroscopy
Due to the increasing need to understand the bonding and electronic
properties
of
solids,
liquids,
gases
the advent of high intensity tunable
and molecules,
in conjunction
synchrotron sources,
- 65 -
with
research in the
VUV
has
been
spectroscopy
growing
in
techniques
recent
years.63
VUV 6 4
for the
65
There
are
various
65 67 which cover
texts
the
on
range of
samples studied in the VUV. The most comprehensive work is that of Samson.
4.1.1 Light Sources
The two major sources
scanning,
source
or
continuous,
for continuum VUV radiation
reflectivity measurements
and the glow discharge
are
lamp used with flowing
appropriate
the
to
synchrotron
hydrogen gas.
The
synchrotron source generates continuum high intensity radiation from 0.01
nm
to
200
nm
(100,000
eV to
6
eV) using
electrons
accelerated
in
a
circular orbit at speeds greater than 0.99 times the speed of light.68 The
phenomena of UV and X ray radiation from a cyclotron was first
the
1940's69
synchrotron
National
70
and
light
has
sources
Laboratory,
Standards,
now
for
Stanford
and DESY in Hamburg,
lead
to
the
materials
science
University,
Germany.
construction
The
the
light
work
noticed
of
is
dedicated
at
National
in
Brookhaven
Bureau
of
truly a continuum
with no
superimposed line spectra and only one broad hump in the spectrum
related
to
the
average
energy
of
the
electrons
used
to
produce
the
lamp.
For
radiation.
The
other
standard VUV source
continuum radiation
The gas
choice.
the
order of
applied.
ground
in the range of
the
glow discharge
interest here hydrogen is the gas of
flows through an insulating capillary,
1 torr,
where
a high field of
The gas molecules are
state
is
atoms
lead
at a pressure
on
approximately 400 V/cm
is
ionized and collisions
to emission of
- 66 -
radiation.
among
The
ionized
atoms
are
and
then
reaccelerated
by the
field until
radiation-producing collision.
glow
discharge
continuum
are
their
is
sufficient
for
a new
This leads to a dark-light banding of the
down the
length
of
a
number
of
large
energy
approximately 0.01 nm intrinsic
the
capillary.
intense
halfwidth,
Superimposed
atomic
leading
hydrogen
to rapid
on
the
lines
of
variation
of
intensity with wavelength (see figure 4.1). This dense structure is one of
the
major
difficulties
encountered
in
radiation
below
applying
this
lamp
to
continuum
spectroscopy.
In
addition,
all
180 nm
is
strongly
absorbed
air, so all optics must be in vacuum of less than 5 x 10-5 torr.
at
wavelengths
transparent
below
window
'windowless'.
nm
105
(11.8
materials,
This means the
system at 10-5 torr.
so
lamp
to
attain
the
at
the
LiF
cutoff,
measurements
the
be
are
no
performed
1 torr must look directly into the
is required,
necessary
To work
there
must
Differential pumping of a special
between the lamp and spectrometer
required
eV),
by
high
aperture mounted
and high pumping speeds are
vacuum with
such
a
substantial
'leak'.
4.1.2 Optical Elements
There are no transmissive
in the VUV below 105
optimized
for
Coatings of
function
reflective
this
nm.
optics
Reflection optics
region
have
115
nm.
coatings for
Gold,
and
gratings and mirrors.
- 67 -
on
the
order
of
higher reflectivities but
platinum
for use
such as mirrors and gratings
reflectivities
MgF 2 on aluminum can have
below
such as lenses or prisms
osmium
are
Optical
30%.
do not
the
preferred
design
therefore
minimizes
the
Monochromators
number
of
reflective
use concave
to
surfaces
focusing gratings that
retain
intensity.
simultaneously disperse
the light by wavelength and collect and focus it on the exit slit.
4.1.3 Monochromators
The
with
most
resolution
Seya-Namioka
simple;
the
popular
the
of
a
optical
monochromator
few
tenths
layout
design
of
developed
entrance and exit slits
grating
and
gratings used
only
the
grating
a
a
nanometer
in
1952.71
small
is
72
monochromator
based
The
on
design
are fixed on the Rowland circle
is
rotated
to
in these monochromators are concave
combine the collimating optics,
scan
wavelength.
focusing
the
is
of
The
gratings,
to
the plane grating and the focusing optics
all into one reflection from the grating.
gratings
for
The major disadvantage of these
is that they are highly astigmatic.
This is partially corrected
by using holographic, aberration corrected gratings.
4.1.4 Detectors
There are various
be
separated
events
of
into
types of detectors
photon
counting
and current measuring
incident
light.
detectors
detectors
Only detectors
for use in the VUV.
to
measure
for measuring
suited
the
They can
single
gross
to current measuring
photon
intensity
will be
discussed here.
The most popular detector uses
a
sodium
salicylate
phosphor,
to
translate the VUV radiation into visible light which is then detected by a
- 68 -
photomultiplier
tube
mounted
system,
in the
The blue
pipe
(PMT)
to visible
and emits blue
to
the
PMT.
This
it
is
The
phosphor
is
struck by VUV photons.
system through a window or light
is
Absorbed
sensitive.
response dramatically.
Still,
detector
light.
light when
light is then sent out of the
environmentally
OC.
sensitive
simple
water
to
and
make,
aging
but
will
it
is
change
its
In addition it decomposes at temperatures above 80
the
detector
of
choice
since
the
active
electronic
detector, the PMT, is sealed and the phosphor may be redeposited at will.
Another sealed detector utilizes a photomultiplier tube with a LiF
In this case the photocathode
window.
to
VUV
photons.
exposed
to the
This
is
a
very
system vaccum.
of the detector responds
stable
But,
detector
because
it
because of the window,
directly
is
never
it does not
function below 105 nm.
Nude
actual
vacuum photodiodes are
detector
signal
is created
an
They are very simple,
with a voltage
applied.
sensitive,
material
their
with
detector where
not
on
the
them,
whereby
photons
sensitivity
is
they
incident
low
the
in a sealed
consisting of a cathode and anode
They can be used with a phosphor
evaporated
or
of a
in the system vacuum,
detector tube.
salicylate)
example
at
are
more
on metal.
low
energy
(e.g.
water
With
but
no
they
sodium
and
age
phosphor
are
more
environmentally durable.
Electron Multipliers can also be mounted directly in system vacuum,
and are basically a photodiode with a 16 stage dynode
chain to yield 106
higher gain than a diode.
Electron multipliers are used predominantly as
ion or electron detectors,
but they will respond to high energy photons.
They
are
response
solar
at
blind
if
used
energies below
without
the work
a phosphor,
function of
- 69 -
i.e.
they have
the dynode
no
materials.
Typically the dynode material is BeO on copper, and has no response beyond
350 nm.
Anyone
who has used optical detectors
such as PMT's will
realize
with horror the impact of having the detector mounted in the vacuum system
and exposed
therefore
to air regularly.
A PMT would be ruined
the detector of choice
For hostile
applications
if brought to air,
is usually sodium salicylate with a PMT.
electron
multipliers
with no
phosphor
have
the
advantages of high durability and high gain.
4.1.5 Optical Spectrometers
The
design
of
VUV
reflectometers
require
that
each
feature
be
chosen to best compensate for the difficulties of working in the VUV.
basic
problems
optics
optical
to overcome
in a reflectometer
and detection techniques.
elements
must
be
lie
in the
light
Due to low reflectivity,
kept
to
a
minimum.
The
The
source,
the number
superimposed
spectrum of a H2 lamp, leads to a very dynamic lamp signal.
of
line
Small errors
in determining X will lead to large errors in the calculated division R =
r/i
of
the
reflectivity
reflected
r
R.
the
Since
and
incident
lamp
i
operates
signals
on
a
to
determine
flowing
gas
the
design,
instabilities in gas pressure or flow lead to random fluctuation in light
intensity.
Optical
nude detectors
their response.
due
to the
radiation.
age with
time
and
sample
and
of
system
the
and
hard
surfaces,
vacuum cycling
Alignment of optics,
vacuum
All
surfaces
sample
- 70 -
to
be
or
leading
in
to
and detectors
to ascertain
spectrometers
detector phosphors
due
is
changes
difficult
to lack of visible
discussed
perform
normal
incidence
reflectivity,
and some also measure
transmissivity and variable
incidence angle reflectivity.
4.1.5.1 Room Temperature Spectrometers
The most successful reflectometer design was developed by Rubloff 7 4
75 for near UV spectroscopy and then extended for use in the VUV to 36 eV
with
synchrotron
detector,
radiation.
It
is
a
single
beam
apparatus,
used is a PMT with a sodium salicylate coated light pipe.
the
about
an axis perpendicular
incident and reflected
rotation
removes
on-off
nature
of
of i
the
to the
intensity
temporal
simultaneous measurement
the
one
and thereby avoiding the problems of matching or calibrating the
optics and detector response of two dissimilar beam paths.
rotates
using
70
light,
The light pipe
alternately
times a sec.
instabilities,
lamp
The detector
The high speed of
and
approaches
and r of a dual beam system.
detector
signal
collecting
can be used
for
In addition
AC detection
techniques.
They quote an accuracy in the absolute value of R of +/-
and relative
accuracy much better
than that.
They have
data and reported no problems with their system.
the
3%,
taken very good
The single beam rotating
light pipe design has become very popular for VUV spectroscopy.
A
different
approach,
taken
by
single beam single detector apparatus,
ellipsoidal
or toroidal
two
separate
the
sample
is
focusing mirror.
incident
on
the
is
a
This mirror rotates, directing
detector.
visible PMTs were used to detect the VUV.
- 71 -
77
where the light is incident on an
the light alternately from the detector to the sample.
from
groups,76
The reflected beam
Sodium
salicylate
Reported absolute
and
accuracy was
+/-
3 %, and relative accuracy of +/- 0.2 %. Because of the existence of
another
reflecting
surface,
signal
strengths
are
lower
and
system
stability is reduced.
C.
F.
Dickinson78 used a pulsed H2 lamp source to achieve greater
intensity and,
to compensate
for the temporal
they used a dual beam reflectometer.
measuring
fluorescence of
sample where
The incident signal was acquired by
sodium salicylate
that obstructed part of the beam.
deposited
on
a nickel
grid
The rest of the light traveled to the
the reflected beam hit
fixed light pipe.
instability of the source,
sodium salicylate
on the
The signal was measured by a second PMT.
front of a
The r detector
could be placed in a no-sample position and the two detector systems cross
They
calibrated.
mention
problems
of
aging
and
contamination
of
the
optics and sample.
4.1.5.2 Temperature Dependent Spectrometers
Various
of the
measurements
'transparency
have been made on the
limit'
of Al203
temperature
dependence
and other UV window materials
over
limited temperatures ranges.79 80 81 These were all performed with single
beam
spectrometers
highest
using
temperatures
one
reached
PMT with
sodium
were
OC,
110
salicylate
180
OC
and
phosphor.
300
OC,
The
and
no
specific problems with the measurement were mentioned.
Blum82 83 measured the reflectivity of Al203 from 9 to 12.5 ev from
room
temperature
monochromator,
salicylate
a
to
450
single
rotating
light
OC.
beam
He
used
single
a windowed
detector
pipe detector.
- 72 -
He
H2
lamp,
system
used
Seya-Namioka
using
a
a molybdenum
sodium
heating
element
in
alumina
insulation
as
a
mentions contamination of the sample
In
furnace.
addition
windowed lamp.
the
vacuum
furnace
to
heat
the
sample.
He
surface, which may originate from the
system
contaminated
the
optics
and
the
The maximum temperature of 450 OC was limited by the black
body radiation from the hot sample saturating the visible PMT.
4.1.6 Special Problems of High Temperature Spectroscopy
In
high
additional
These
temperature
problems
problems
radiation
system;
on
all
are
the
are
small *furnace
to
the
occur,
heating
sample.
There
colder
sample
addition
to
will
to the
in
in
the
be
unintentional
area,
0
C.
with no furnace,
Therefore
detectors
damage.
sample
anything
and
optics
Assuming
still
the
the
system.
For
heat
be
doping
at
a
from
in
of
the
the
the hot
the
sample.
The
a
a
is
the
sample
to
the
other
Heating
If
there
elements
of
the
then
temperature
reflectance
body
on
sample.
flow to
heating
sample.
black
effects
sample,
are
spectroscopy.
of
aging
of
higher
the
VUV
effects
severe
heat
must
with
there
than
the
onto
the
sample
problems
in
are
sample of 2 cm2
requires on the order of 30 watts to get above 1000
exposed
in
a clean
radiates
used,
to
changes
caused by the heat load from
soley
and the
deposition
to
measurements,
high temperature
vacuum
element
leading
method
acquisition,
used
outgassing
reflectivity
associated
heating
related
substantial
the
those
signal
is
VUV
all
the
to
VUV
sample
of
the
its
are
sample
must be
very
sensitive
that does
input heat
very durable,
not vaporize
into
the
to
yet
environmental
impurities,
spectrometer.
the
This
rules out the use of sodium salicylate or other phosphors because they are
- 73 -
unstable,
surface
decomposing
exposed
to the
with a good heat
Effects
that
be taken
at
low
temperatures.
sample's
sink so
black
that the
The
body
The
radiation
coatings
must
be
on
any
in contact
coating does not heat up and de-adhere.
occur more slowly than these
into consideration.
optical
catastrophic
failures must
also
of the optics will vary
characteristics
slowly with time, and must be regularly calibrated.
The
effect
of
another concern.
greater
become
from
the
effects
visible
than
in
the
this
background.
on
the
sample's
radiation
The black body radiation
intensity
essential
the
light
from
situation
The
reflected
to
light
signal.
signals
the
If
Lock-in
incident
radiation
can
the
have
this
problem
after the sample.
is
to use
If the
a
is not solar blind),
monochromator
to
are
techniques
light
two
detector
signal
separate
responds
to
then the black body
radiation will be detected as an incident light signal.
avoid
detected
sample can be of 105
source.
recover
body
it
the
from the
a H2
black
and IR radiation(i.e.
on
The normal way to
discriminate
the
light
incident radiation heats'the detector, then
its
dark current will
increase dramatically due to the greater ease of thermal
electron
from the
emission
cathode.
For
successful high
temperature
VUV
spectroscopy all of these problems must be addressed.
4.1.7 Heating Methods
The
various
measurements
above
heating
room
indirect
methods
radiation
from a furnace,
where
methods
temperature
the
sample
and direct
that
can
is
have
be
been
separated
heated
by
used
into
thermal
heating methods using
- 74 -
for
two
optical
types:
contact
light
and
incident
upon
the
sample
to
heat
it.
Resistance
furnaces
using
tungsten,
molybdenum, or graphite elements, and alumina insulation have been used84
but
the
large
outgassing
thermal
and
mass
of
achieve
the
same
system.
The
optical
radiation
the
furnace
temperature,
and lasers
the 1950's
deposition
on
the
requires
thereby
sample
are
severe.
Also
that greater power be used
increasing
the
heat
load
on
or direct heating methods have used both Xe arc
of various
sorts.
to grow high purity float
the
Xe arc lamps have been used
to
the
lamp
since
zone or skull melted crystals.85
The
major difficulty is collecting the light from the arc and concentrating it
on
the
sample.
optical
Being
heat
CO 2
source
coherent
gas
lasers
that has
operating
been used
and nondiverging
the
at
10.6
microns
for high purity
laser
radiation
is
are
crystal
another
growth.
relatively
86
simple
to concentrate on the sample.
4.2 High Temperature Vacuum Ultraviolet Spectrophotometer
4.2.1 CO2 Laser Heating Techniques
A
CO 2
temperature
microns
Al 20
microns,
gas
laser
reflectivity
was
chosen
as
measurements.
the
heat
The output
source
of the
for
laser
the
high
is at 10.6
in the infrared and coincides well with the natural absorption of
The
it
IR absorption
is decreasing
of
sapphire
with
n
the
starts
index
- 75 -
at 5.6
of
microns
refraction
-
and,
1.
at 10.6
Therefore
the
incident
reflected
laser
radiation
beam being
is
totally
absorbed,
formed in the reflectometer.
without
a
stray
The laser is a Photon
Sources, 400 watt, axial flow, continuous beam laser belonging to Dr. John
Haggerty
of
non-tunable
from
125
the
Ceramics
industrial
watts
to
Processing
welding
450
Research
laser.
watts,
The
but
by
Lab
operating
without
reflective
a
operated
power
rear mirror of the laser.
0.5 watts.
a
to operate
nitrogen,
the
meter
placed
Fluctuations
at
the
were
partially
in the meter were +/power fluctuations were
The laser beam was of high quality TEM00 mode above about 14
but below this power the beam had a dumbbell power distribution.
The beam
line to deliver the laser radiation to
water cooled silver overcoated Si wafer mirrors.
water cooled
optics
copper aperture
before
from being destroyed due
into the
system were
beam
thereby
intersected
minimizing
oversplash
entering
the
sample used
The beam went through a
the
system,
to protect
to accidental misalignment.
ZnSe high power
with indium wire welds.
CO2
is
Power measurements
The laser stability was much higher;
negligible.
watts,
battery
It
Using a rudimentary pulsing
ability the minimum stable power was 1.5 watts.
with
M.I.T.
laser was designed
continuous power could be lowered to 9 watts.
done
at
IR laser windows
The
the
widows
sealed for vacuum
The laser beam geometry was designed so that the
the
danger
plane
to
of the beam on the
of
the
the VUV
delicate
sample exited
optics
only
detectors
the
at
and
the
sample,
optics.
Any
system through a window
into a water cooled aluminum beam dump.
To minimize the heat load in the spectrometer,
heated,
the
sample
stand
having
minimal
only the sample was
contact
with
the
sample.
Therefore the sample was mounted horizontally resting on three points of a
sapphire
tripod.
Sapphire was chosen to
- 76 -
minimize
contamination of
the
sample
at
surface
high
of the sapphire
The
incident.
The
temperatures.
beam
sample,
diameter
laser
the
was
same
the
beam was
incident
surface where
same
as
the
on
the
top
the VUV beam was
wafer
diameter.
A
nichrome heat shield surrounded the sample, with holes to allow access for
the incident,
sample
reflected and transmitted VUV beams,
tripod
stand.
A
shutter was
used
optics from the black body radiation
It was opened for
temperature.
the laser beam and the
to obscure
of the sample
data taking.
the detectors and
during heating up to
In addition the detectors
had heatshields in front of them to reduce detector heating.
4.2.2 Vacuum Ultraviolet Spectrophotometer
The
physical
design
of
the
The signal acquisition
discussed.
next section.
spectrophotometer
and processing
will
now
will be covered
be
in the
The design of the reflectometer VUV optics, the chamber and
the monochromator was developed here at M.I.T. 8 7 in conjunction with Acton
Research
Corporation
constructed
figure
The
diffusion pump,
and a foreline
the
Acton
the reflectometer
4.2).
system.
of
vacuum
MA.88
chamber,
system
did
optics
consisted
with a mechanical
trap.
ARC
of
the
detailed
design
and the monochromator
a
6
inch hydrocarbon
and
(see
oil
fore pump using low vapor pressure oil
A liquid nitrogen trap was between the pumps and the
Precautions were taken to minimize the back streaming of oil from
pumps
into
the
system.
Only
hydrocarbon
oils
were
used;
the
silicone oils form SiO2 upon combustion by the VUV photons, destroying the
optics.
aperature
A
flowing-gas glow-discharge
and
separate
mechanical
source
pump
- 77 -
with a differential
allowed
windowless
pumping
operation.
These were made by VUV Associates.89 High purity hydrogen gas was used to
create usable continuum radiation from 82.5
nm to 225 nm.
Due to the horizontal sample geometry of the sapphire sample stand,
and the demands of the VUV optics, the monochromator was mounted with the
monochromator
of
the
VUV
incident
slit
horizontal,
radiation
on the top
is
or in other words the plane containing all
vertical.
In
of the
sample.
surface
this
manner
the
radiation
The monochromator
is
is an ARC
model VM-502 0.2 meter Seya-Namioka type monochromator, with a holographic
aberration
corrected
grating
with
1200
grooves
per
millimeter.
The
inherent resolution of this grating is 4 nanometers per millimeter of slit
width.
The
grating
was
overcoated
reflectivity below 105 nm.
motor drive were also used.
with
Bilateral
osmium
due
adjustable
slits
A movable MgF 2 filter,
to
its
high
(30%)
and a synchronous
with a cutoff at 115
nm (10.7 eV) was installed in the monochromator so that during low energy
scans, the higher order radiation could be blocked.
is
7.5 to
15 eV while use of
the filter allows
The high energy range
the low energy scan to
cover 6 to 9.5 eV
The reflectometer
application
dual
beam
of lock-in techniques,
optical
installed on the
400 hz.
itself is designed
setup.
exit slit
The
to use chopped
and uses
subminiature
a beam
tuning
of the monochromator,
light for the
splitter
fork
modulating
to create
chopper
the
The light is then incident on an osmium coated focusing
mirror beam splitter.
a
is
light at
spherical
The mirror is used so that its spherical aberration
takes the image of the slit
and focuses
it
to a square spot.
One half of
the beamsplitter takes one end of the slit image and directs it upward to
the incident intensity detector,
focusing to a spot on the detector.
The
other half of the slit image is directed down to the sample, where it is a
- 78 -
square
spot approximately 4 mm on a side.
to the
reflected detector,
through
the
sample
stand.
The
optical
reflected light.
to
is
or alternatively the light can travel straight
a transmitted
path
is then reflected upward
lengths
signal
are
detector
identical
for
located below
the
both
and
incident
The beamsplitter is movable along the length of the slit
allow balancing
beamsplitter
to
This
the
signal
removable
levels
without
or
loss
for
of
calibration purposes.
alignment
for
The
cleaning
or
replacement.
The
standard
sample is
a 1.12 cm diameter disk, polished on one
side and typically 500 pm thick.
sample
tripod
stand.
The
It rests on the 3 points of the sapphire
sample
stand
is
kinematically mounted
in the
reflectometer chamber, with micrometer adjustment from outside the system,
to
permit
installed,
adjusting
the
the
sample
is
sample's
tilted
height
with
and
the
lamp
angle
of
on until
tilt.
the
Once
light
is
incident on the reflection detector.
After extensive measurements
our high temperature measurements,
chosen.91 These
have 16
stages of dynode
regenerated
Their
response
nude metal
have copper-beryllium
3000 volts applied.
be
on the various possible detectors
by
oxide
electron multipliers
dynodes,
amplification yielding
are
for
were
solar blind
and
a gain of 2 x 106 with
They are durable under incident IR radiation, and can
heating
in
is not matched,
oxygen
at 400
OC to
remove
so a calibration procedure
contaminants.
was
required,
which is discussed in the section 4.2.3.
Additional features of the reflectometer allow remote monitoring of
the sample.
A sapphire viewport window allows sighting on the edge of the
sample
for
optical and IR pyrometry.
allows
for
thermocouple measurement
An 11 pin electrical feed-through
and other electrical
- 79 -
measurements
on
the sample.
4.2.3 Data Acquisition and Signal Processing
Lock-in
signals
detection
techniques
were
from the black body radiation
used
of the
to
discriminate
sample,
the
VUV
and to improve
the
signal to noise ratio.
Scans of 100 nm were taken from 80 nm to 180 nm
for the higher energies
and from 130
in the spectral range.
computer using
supply
shaft
The data was acquired on a Hewlett Packard92 87-XM
two computer controlled
lock-in amplifier,
pulse
nm to 230 nm for the lower energies
corresponding to the
encoder was
trigger
pulses
simultaneously.
digital
installed
to
the
voltmeters,
one
for
each
incident and reflected signals.
on
digital
A
the monochromator
drive
shaft
voltmeters,
were
triggered
To be able to accurately reproduce
which
to
H
the hig'hly dynamic
lamp spectrum, so that the division of these rapidly varying signals would
be
flat,
scan.
data points were
Since
typical
slit
taken
every
width
of
1/2 angstrom,
the
2000
monochromator
was
points
100
to
or
a
500
microns leading to 4 to 20 angstrom resolution, a minimum of 8 points were
used on each peak.
To
reduce
the thermal
noise of the detectors
time constant was used in the lock-ins.
and system,
a 1 sec
Scans were taken at 4 nanometers
per minute, so this time constant, or integration time, did not remove any
meaningful information from the signals.
Due
unmatched
method
of
to
the
detectors
fact
that
calibrating
that
are
the
this
not
is
very
system
a
stable
without
- 80 -
dual
a
beam
in
spectrometer
their
sample
response,
present
and
with
some
to
compensate for the detector behavior was needed.
taking
a
detector
spectra
'background'
in
the
run
with
transmission
files were acquired
reflected
path background,
no
sample
position
and
ib,
the
and
(below
in this manner,
This was accomplished by
the
the
'reflected'
sample
stand).
one corresponding
incident
light
Two
to rb,
path background
the
signal.
The reflected detector would then be put in the reflected position (above
the sample), a sample mounted and a sample run taken.
The reflectivity R,
referred
reflectivity,
to here as
a four file background
corrected
would
then be calculated by
R = (r/rb ) x (ib /
and
Here each term, (r/rb)
its characteristic
(ib/i),
response.
compensates the appropriate path for
A comparison of
the
uncorrected
two
file
background and a corrected four file background of the spectrophotmeter is
given in figure 4.3. In addition, a background run can also be taken after
a sample run and the
two background runs,
used
four
to
calculate
corresponds
These
are
to
used
the
a
file
background
spectrometer baseline,
between
every
sample
run
the
i.e.
corrected
since
to
no
check:
four
files,
can be
reflectivity
sample
is
the
state
that
present.
of
the
spectrophotometer, that no damage has occurred to the optics or detectors,
or that
present.
the system gives a value
The baseline
of unity for the result with no sample
value. is one because
transmission position with no sample,
has been a very powerful method
the baseline
hence no absorption
is taken in the
is
seen.
to compensate for many of the
problems of VUV spectroscopy.
- 81 -
This
inherent
4.3 Resolution and Accuracy
Any consideration of experimental work must consider the resolution
and accuracy of the results.
VUV reflectivity of Al203,
For this study of the temperature dependent
the parameters were:
E at which a spectral
feature is observed,
at
and
a certain energy,
the
the wavelength X or energy
the value of the reflectivity R
temperature T of
the
sample under
study.
Each of these will be discussed separately.
4.3.1 Energy E
The resolution in energy can be determined in two ways: by the slit
width of
the monochromator
system.
The grating
or by
the
spectral
eV)
were
overall
in the monochromator has a inherent resolution of 4
nanometers per millimeter of slit width.
15
resolution of the
taken with 100 pm or 200
The higher energy scans (7.5 to
gm slits,
while
the
lower energy
scans (from 6.5 to 9.5 eV) were taken with 500 pm slit width.
In figure
4.4 the variation of energy resolution with photon energy is shown for the
3 slit
widths used.
scan the
resolution
At all energies except the high energy limit of the
is
always
below
0.1
eV,
and
at
all
energies
it
is
below 0.14 eV.
The overall
spectral
resolution is worse than the
due to noise in the reflectivity
The hydrogen
slit
resolution
remaining even after signal processing.
lamp produces a very dynamic
- 82 -
( 103
) signal with very sharp
features.
Upon
division
reflectivity curves.
Therefore
0.1 eV.
these
features
The size of these
it
is
hard
to
can
appear
lamp features
attribute
arising from the sample or the system.
a
in
the
calculated
is on the order of
feature
of
For this work,
this
size
only features on
the order of 0.3 eV or larger were considered to be representative
sample.
the
of the
The origin of a specific feature can be checked by comparison to
lamp
feature;
as
spectra,
sample
and by
observing
features shift, while
position of a feature,
once it
to within an
accuracy
order of
absolute
The relative
resolution.
+/-
0.01
reproducibility
the
lamp
dependence
features do not.
0.1
relative
is known
eV as determined by the
uncertainty
the
The energy
has been attributed to the sample,
of +/-
of
accuracy in energy is higher than this,
This
eV.
temperature
slit
on the
is determined
by
the
of the grating position, which leads to a reproducibility
of the monochromatized
light wavelength of+/-
0.05 nm.
It is the relative
accuracy in energy that determines how accurately one can determine energy
shifts between spectra,
as one does
in measuring
the
temperature
induced
shift of a feature.
4.3.2 Reflectivity R
The
resolution
in
the
certain energy can be separated
random.
measurement
of
the
reflectivity
into two types of errors:
R
at
systematic
a
and
Systematic errors lead to an offset of R from the true value for
the sample.
Random errors are
unpredictable
data.
- 83 -
and
lead
to scatter
in the
4.3.2.1 Systematic Errors
For
optical
a
study
spectra
reflectivity,
Ultraviolet
the
four
number
of
the
temperature
only
the
relative
is used in analysis.
Spectrophotometer,
induced
shifts
reflectivity,
Still,
absolute
not
present,
system
variables.
The
reflectivity
magnitude
could be determined by measuring
standard.
to
Due
the
difficulties
features
the
of
a
in
absolute
in the High Temperature
Vacuum
is calculated,
file background corrected method, so as to
of
of
using
constrain a larger
spectral
offset,
if
the value of R for a reference
of
establishing
VUV
reference
standards, and the fact that such knowledge was not needed for this study,
this was not done.
compare
the
values
Instead, to determine if an offset is present, one can
measured
in
this
study
Most measured reflectivities
literature.
to
those
for sapphire,
reported
in
the
at the top of the
13 eV peak, are less than 0.2. Arakawa93 measured reflectivities up to 0.3
in a very carefully controlled study.
Some spectra taken in this work had
reflectivities below 0.25, but most reflectivities
0.28 to 0.32 for the highest peak,
other
in agreement with Arakawa's work.
studies probably had low light collection
offset
sample.
of
the
measured
R values
were in the range from
from
the
efficiency
higher
leading
true values
The
to an
for
the
The reflectivities measured in this work are of the correct order
of magnitude
so as to suggest that no fixed offset errors are present
in
the measurements.
Another systematic
error is due to the changes in the spectrometer
during the actual spectra acquisition.
by environmental
exposure of the
This is aging of the system caused
optics and detectors, and aging induced
- 84 -
by
the
heat
identical
load
state
features
on the
system.
from scan
to scan,
appearing
Another
in the baseline
source of baseline errors,
is scan to scan instabilities
vacuum
system
detectors
leaks.
In
they age
leading
due to
exposed
the
time
As
in
baseline,
relative
affected.
The baseline
order of +/- 5%.
not
in
in the baseline
the adsorbed
the
shift
high
to the
of the
as the run proceeds.
then
is
gaseous
temperature
radiation of
run,
causing
an
and
species.
and consequently reflectivity
baseline
the
system
to tilts
addition, during
during
the
errors,
in things such as gas flow in the lamp,
and beamsplitter are
therefore,
Therefore
scans,
or
the
the
sample,
shifts
in the
long as these cause only smooth shifts
observed
reflectivity
in these
is
not
measurements
adversely
was on the
A more serious problem was. observed in some runs where a
notch would appear at
10.2 eV
(124
This was not
nm).
sample dependent,
appeared in some scans and not in others, and at times led to a depression
or offset
in R at energies below the notch.
The notch would appear in the
measured reflectivity as a feature approximately 0.4 eV wide and therefore
precluded analysis of features
it was present.
Much work was done to determine the origin of the notch,
but no explanation was found.
aberration
was
in the area of the notch in spectra where
definitively
The fact that the notch was a spectroscopy
established
where all spectral features would shift,
energy.
In
addition
the
notch
in
4.3.2.2 Random Errors
- 85 -
dependent
runs
but the notch stayed at the same
appeared
materials.
temperature
in
spectra
taken
on
other
Random errors in R lead to fluctuations in the values of R measured
over
a set
zero
of
errors
of reflectivities.
the
lock-in
These
amplifiers
in calculating R.
Using
can be due to
leading
to
the change
small drifts
division
in the
and
multiplication
in R at 15 eV,
this variation
was found to be +/- 0.03 in R or +/- 12%.
4.3.2.3 Polarization
By varying the crystallographic orientation of the sample face, and
relying
on the inherent
spectra were taken.
as arising
axis.
polarization of the
light,
This allowed the differentiation of spectral features
light,
after being
incident
on the
predominantly polarized with its electric vector
transitions
with
c axis
that occur parallel
perpendicular
to
the grating
permitted
perpendicular
to
the
broadens the features in the reflectivity,
to determine
grooves,
a
sample
c
it
The same sample
or
a basal
as
face
will show the transitions
axis.
Because
temperature
was not considered essential
the exact polarization efficiency of the optical system.
this had been needed it
of
is
grating grooves will
to the c axis.
sample with the c axis as a sample face normal,
are
grating,
E parallel to the grating
of the sample and parallel with the
show the
that
monochromator
Then samples mounted on the sample stand with the c axis in the
surface plane
the
dependent
from transitions parallel and perpendicular to the crystal's c
The
grooves.
orientation
If
could have been done by measuring the reflectivity
a function
of
the
angle
orientations.
- 86 -
of
incidence
for
two
sample
4.3.2.4 Signal to Noise Ratio at High Temperature
At very high temperatures,
calculated
good
Lock-in
reflectivities.
signal to noise ratio for the
to noise
improvement
that
can be
the DC case is 105, i.e.
still
random noise was
be discriminated.
amplifiers
were
also present
used
spectrometer.
Still,
achieved using
lock-in
to establish
techniques
Operating near
this condition,
the
appear as
the
a band of noise
calculated
smoothing
R
+/-
20%
spectra
function
represent the
slit
on both sides
of
R.
To
would
be
smoothed
that
over
the noise can be 105 larger than the signal and
in the output.
approach
a
the best signal
subject to overload and random noise appears
could
in the
encompassed
determined
of
remove
the
or
by
are
This would
average value of R and
average
computer
same
lock-ins
this
with
number of
noise,
a
flat
data points
resolution of the monochromator.
the
box
as
Therefore
no information from the sample was lost.
4.3.3 Temperature T
The temperature of the sample is the third parameter measured.
sample was heated with a
this
laser beam with a
coupled with the radiation heat
gaussian power distribution,
lead
to
a radial
temperature
The VUV light reflected from the sample was only incident on a
gradient.
4 mm by 4 mm
gradient
loss
The
square
in the center of the 11 mm diameter
sample,
in the part of the sample actually used for measurement
to be less
than the uncertainty
in the
temperature.
so the
is taken
The temperature of
the sample was determined indirectly by performing calibration runs with a
- 87 -
thermocouple
to
determine
the
curve and the temperature vs.
of
a
sample
furnace,
of
this
and made
temperature
geometry, a
of a material
thin
that
disc,
is
calibration
The temperature
contained
transparent
in
in the
a cold wall
visible,
is
Optical methods are not reliable because the
sample is not a black body ( emissivity e =
the
power
optical pyrometer reading.
very difficult to determine.
instead
laser
vs.
emissivity varies
with
1 ) or a grey body ( z < 1 ),
temperature.
The
problem
is
not
a
normal radiation heat loss problem, because a volume emissivity as opposed
to a surface
emissivity must be used.
thermal mass
is very sensitive
In addition a sample of such low
to, heat conduction losses,
and may not be
able to reach equilibrium with a thermocouple bead.
The
temperature of the center
by installing
(.010
inch)
a 75 pm (.003
inch)
region of the sample
was calibrated
Pt/Pt 10% Rh thermocouple
in a 250 gm
groove in the back side of the sample and firing it
with alumina cement.
The thermocouple reading
is accurate
in place
to +/- 0.5 OC,
and in this configuration thermal contact with the sample was assured,
heat
conduction
thermocouple
through
temperature
the
was
thermocouple
then calibrated
read on the laser power meter (accuracy +/parameters
leads
was
against
the
one with the
the laser on continuously.
laser
This
power
as
0.5 watt ) while other laser
such as current and gas mixture were noted.
sets of data are shown:
minimal.
and
In figure 4.5 two
laser in a pulsed mode and one with
The curve is not linear when plotted power (q)
vs T4 because the radiation is not normal radiation heat loss as mentioned
earlier.
sample
used
A
used
separate
for
to calibrate
calibration
spectra
was
done
acquisition.
the optical
pyrometer
The
serve
-
as
thermocouple
reading
pyrometer (see figure 4.6).
- 88 -
to
a
check on
each
temperature was
of a microfocus
optical
All
factors that could
kept constant.
change
the heat
controlled,
all
variables
and reproducible
this was obtained
900
OC
sample were
The setting of the laser power was
kept
constant,
alignment on the sample checked regularly.
reasonable
from the
The sample size, geometry, sample polish, the sample stand
and heat shielding were kept constant.
carefully
loss
temperature were
other
exactly.
the
laser
beam
With all of these precautions,
from the reflectivities
overlaid each
and
attained.
taken;
The
A cross
check on
reflectivities
taken at
accuracy
of
the
temperature
measurement was +/- 25 OC.
4.4 Conclusions
A
high
designed,
beam
temperature
vacuum ultraviolet
spectrophotometer
built and used for optical measurements on Al2 03
spectrophotometer
capable
of
measuring
allows
A focusing beamsplitter
measurements
in
the
range
from
been
It is a dual
polarization
absolute reflectance and transmittance on high temperature
normal incidence.
has
dependent
samples at near
is used and a windowless source
6.5
to
15
eV.
Nude
electron
multipliers are used as solar blind detectors, and lock-in techniques are
used
to
improve
the
signal
to
noise
ratio.
Spectra
are
acquired
by
computer, 2000 points per 100 nm, and to calculate the reflectivity a four
file method is used that compensates for the variation in the base line of
the
spectrophotmeter
absolute
resolution
resolution
is +/-
due
to
in energy
0.01
eV.
aging
is +/The
of
the
optics
0.1 eV,
system's
- 89 -
while
ability
and
the
to
detectors.
relative
remove
The
energy
the highly
dynamic
lamp
signal
from the calculated
features over 0.3 eV wide were
reflectivity
dictated
that
only
considered to arise from the sample.
The
reflectivity is accurate to within +/- 12 %. Spectroscopy can be performed
on samples up to 1100 OC,
without contamination.
using
a CO2
The accuracy
laser to
cleanly heat
in temperature measurement
OC.
- 90 -
the sample
is +/-
25
3.0
14
-
Energy (eV)
10
12
-
2.8
8
I
I
2.6
2.4
2.2
Volts
2.0
1.8
I.e
1.6
1.4
1.2
1.0
.9
.6
.4
.2
I
0.0
-- 0 -
0
-0
d
d~
g;
nm
FIGURE 4.1 H2 Lamp spectrum.
CO
2
100 pm slits, 0.4 nam resolution.
LASER
H2
for
diff. pump
LAMP
SAMPLE C3-AMR
I det +
1beam
/
r det
MONO
ted
-shuttersample
/
ros
choper
view port-,
heat shield-
'-tct
/
VACUUM
SYSTEM
/
4-
FIGURE 4.2 High Temperature Spectrophotometer.
sample chamber built by Acton Research.
-
91 -
Monochromator
and
14
Energy (eV)
10
12
8
1.6
1.4
1.2 -
Volts
1.6
.4
.2
_j
0.01
nm
FIGURE 4.3a Two file baseline.
14
Energy (eV)
10
12
R = rb
8
1.20
1.15
Volts
1.10
1.05
1. 00
. 95
4
li
.1
%11
.90
.90
.75
.70
.65
.60
.55
.50
.45
.40
.35
.30
.25
.20
.15
.10
.00
0.0
o
0
00
0
0
0
0
nm
FIGURE
4.3b
Baseline
of
a
four
file
background
corrected
reflectivity. Taken in transmission position with no sample, therefore T
= 1.
No offset is evident.
Four file reflectivity calculated by R
(r/rb) (ib/i)
- 92 -
I
I
I
I
I
I
I
I
0.16
500am
0.12
c
0
. 0.08
200p m
-
0
100j m
0.04
*
0
6
7
8
FIGURE 4.4 Absolute
slit widths.
30
9
10 11
E (eV)
energy
12
13
14
resolution versus
energy
co nt inuous ..
-
15
for different
-
.e
0
0
.*
-
420
.*
0
puls0
s'
10
0 C
2
4
6
8
10
12
14
16
T (OCx102)
FIGURE 4.5 Laser power versus thermocouple
curve
- 93 -
temperature
calibration
12001100 .*
U
.1000 -,.
0
0'
900 -
00
90
800
7
'
9
11
13
15
I
17
a
19
-2
Real Temp. (oCxlO
FIGURE 4.6 Optical pyrometer
temperature calibration curve.
- 94 -
'temperature'
versus
thermocouple
Chapter 5
Experimental Results
Temperature dependent reflectivities were
taken on samples of Al 203
using
the
High
Temperature
Vacuum
Ultraviolet
Spectrophotometer.
The
spectra were taken from 83 nm to 210 nm, corresponding to energies from 15
This energy
eV to 6 eV.
130
to
230 nm.
temperature
up
corresponds
to
experimental
absolute
+/-
0.01
The
to
eV,
samples
1100
the
were
eC.
absolute
limits.
accuracy
range is
The
the
reflectance
value
of
is
width
80 to 180 nm and
contamination from room
or
the
+/- 0.1 eV,
minimum
in two scans,
heated without
reflectivity
in energy is
and
The
covered
transmittance
property
accurate
to
measured
measured
+/-
within
%,
12
the
while the relative accuracy is
for
a
spectral
feature
to
be
attributable to the sample is 0.3 eV.
5.1 Room Temperature Results
The samples used in this study, unless otherwise noted, were single
crystal A1 2 03 grown by the Czochralski method by Union Carbide. 9
high
quality,
optical
scattering centers.
grade
crystals
with
no
bubbles,
striations
The dislocation density is typically 10 /cm2,
- 95 -
They are
or
and the
samples
are
loosely
stated
to
be
99.99 %
pure.
The
samples
used
for
reflectivity measurements are polished on one side to a 1/4 micron finish
with the back
back
side.
surface
side of the wafer
The
samples
used
roughened to reduce reflection from this
are
wafers
with
of the wafer and will be referred
orienting
the
sample
in
the
the c
to here
axis
as
spectrophotometer,
'c
lying
in the
in plane'.
reflectivities
By
both
perpendicular and parallel to the c axis can be taken, due to the inherent
polarization
compared
to
of
the
spectrophotometer.
spectra
of high purity
In
(
<
addition
spectra
70 ppm cation
taken were
impurities
)laser
float zone grown and zone refined sapphires produced by Morris.95
5.1.1 Reflectivity of Sapphire
Reflectivities
sapphire
laser
and
float
a
laser
taken
at
float
zone
room
temperature
sapphire
zone crystal had a natural basal
sample surface).
spectrometer
so
and
a 13
face
(c
Union
in figure
that
the predominate
electric
eV peak that
The
vector
features
consists of a peak at
has not been used in the analysis because
'spectroscopy
notch'
(see
this feature is low.
for
results are
the
section
Carbide
5.1.
axis normal
The
to the
oriented in the
E of the incident
observed are
13
eV and
There is also a suggestion of a shoulder at 10.4 eV,
shoulder.
identical
shown
a
The spectra were taken with the sample
light is perpendicular to the c axis.
peak
are
on
a 9 eV
a 12
eV
but this
this is the area of the 10.3 eV
4.3.2.1);
therefore,
confidence
in
One can see from these spectra that the features are
two
intrinsic
samples.
This
demonstrates
that
the
reported
to Al203 and independent of sample origin or trace
- 96 -
impurity content.
natural,
as
dislocation
In addition,
grown,
basal
density
or
comparison of a basal
since the laser float grown crystal has a
face,
the
polishing
face sample
results
damage
are
in
the
not
controlled
crystals
to a c in plane
sample
used.
by
The
also supplies
useful information on the polarization efficiency of the spectrometer.
can be
seen
from
the
Union
Carbide
spectra,
the
polarized to excite only transitions perpendicular
light
is
As
sufficiently
to the c axis,
leading
to results that are comparable to those from a true basal face.
In
room
temperature
Carbide sapphire,
9 eV peak
samples
and
without
reflectivities
on
7
samples
of
Union
the major features were an absorption edge below 9 eV,
a broad 13
eV peak.
These
regard to orientation,
the geometrical center of the peak.
for samples
taken
of one orientation,
a
features were common to all
and were
measured by
determining
The variability of the peak positions
reported
as the
standard deviation u n-l'
was +/- 0.025 eV for the 9 eV peak and +/- 0.1 eV for the 13 eV peak.
5.1.2 Orientation Dependent Features
In
figure 5.2 room temperature reflectivities from UC c in plane
sapphire are shown.
while
curve
show the
a,
Curve a was taken with E perpendiculaT
b corresponds
to
orientation dependent
excitations
features
for E perpendicular to the c axis,
parallel
to the
to the c axis
c axis.
observed for sapphire.
These
In curve
there is superimposed on the broad
peak a small peak at 13.1 eV and a shoulder at 12 eV.
In curve b, for E
parallel to the c axis, the broad peak has two small peaks superimposed on
it at
12.9
eV and 12.3
eV.
Looking
- 97 -
at
curve b
one
can see
that
the
structure
shoulder
peak.
is
more
complicated
suggesting
a third
than
peak
summarized
at
higher
Due to the emphasis on temperature
here;
there
is
a
energy in the
13
eV broad
dependent
features are only noted in passing in this study.
slight
spectroscopy, these
Such fine features will
broaden and disappear with increasing temperature.
5.1.3 Transmissivity of Sapphire
Measuring
the
transmissivity of a sample
determine the energy of the absorption edge.
the
absorption
transmissivity
coefficient
(clear)
rapidly
sapphire is
shown.
One can see T
and then a strong decrease
There
sample
the
is
still
some
is opaque.
region
leads
r
57.52
to
0
to
a
transition
(opaque).
from
In figure 5.3
high
the
to the c axis) Union Carbide
decreasing
slowly from 7.5
to 8.5 eV,
in Tr starts, leading to Tr = 0.01 at 8.8 eV.
transmission present
up
to
9.5
eV, whereupon
the
The absorption edge energy is determined by projecting
of abrupt decrease
absorption process
(1102,
technique
In transmission a change in
to low transmissivity
transmissivity Tr of R plane
is a useful
responsible
in Tr, corresponding
for the edge,
gives an absorption edge energy of 8.8 eV.
pm (0.020 inch) thick.
to the onset of the
to Tr = 0. In this case this
The sample used here was 500
The value of the absorption coefficient a could be
determined by measuring Tr for samples of varying thickness, but this was
not done.
-
98 -
5.1.4 Other Forms of Al
203
It
effects
is
of
structure.
interesting
structural
to
at
other
differences
Reflectivities
polycrystalline
look
alumina.
have been
The
on
the
below 600 OC,
amorphous
to the a phase.
It
therefore
is believed,
(<
and
of
to
see
electronic
or y Al203
Al203
was
5.5
shows
a
and
formed
the phase
transformation
from x-ray diffraction measurements,
to
In the reflectivity shown in
one can see that the absorption edge has shifted to 8 eV,
energy peak which is at 12 eV,
the
200 *C) and then dried at
inhibiting
a peak is not observed above this edge.
Figure
Al203
spectra
sample96
be amorphous with very small y crystallites.
figure 5.4,
of
taken on amorphous
through a sol-gel process at low temperature
temperatures
forms
and
Also the center of the broad high
is shifted 1 eV from the crystalline case.
reflectivity
taken
from
a
sample
of
Lucalox,
a
polycrystalline Al203 that is doped with MgO so as to yield a fully dense
translucent material.
shifted to 8 eV,
there
Here again one can see that the absorption edge has
is a peak at 9 eV,
and the broad high energy peak
is centered at 13 eV.
5.1.5 Other Materials
To
confirm
that
the
Al203 correspond to unique
features
observed
characteristics
in
the
reflectivities
of
of A1 2 03 , reflectivities taken
with the spectrophotometer on other materials are shown.
The reflectivity
of
one
single
crystal
MgO
is
shown
in
figure
5.6.
Here
can
absorption edge starting at 7.4 eV, increasing to a peak at 7.6 eV.
- 99 -
see
an
There
is a series of peaks up to 11.5 eV,
and then a single large peak at 13.5
eV.
of single
Figure
gradually
5.7
is
the reflectivity
increasing
reflectivity up to
short broad peak at 14.2 eV.
on
Al203
are
unique
to
crystal LiF.
a single
This shows a
peak at 12.7
eV, and a
These examples demonstrate that the results
the
material
and
not
a
function
of
the
spectrometer.
5.2 High Temperature Results
Temperature dependent reflectivity measurements were taken on Union
Carbide c in plane sapphire crystals at temperatures up to 1100 *C. These
measurements
were
taken
with
the
polarization of the spectrometer.
c
axis
parallel
to
the
predominant
This orientation was chosen so that the
13 eV peak would have the doublet structure and broaden into a symmetrical
peak,
and the problem of having
to measure
the
shift of an asymmetrical
peak and shoulder as would occur for spectra taken perpendicular to the c
axis,
could be
above
800 OC,
close
to overload,
Spectra
taken
showed no
avoided.
The
and at 1100 OC,
which
samples
were
glowing
red
at
temperatures
the Lock-in amplifiers were operating very
required
at room temperature
substantial
on
change in the reflectivity,
not controlled by changes in polishing
smoothing
samples
before
demonstrating
of
and
that
damage or surface
the
after
spectra.
heating
the results
are
structure due to
annealing.
Spectra
taken at many temperatures are
they have been displaced
in R to
show the
- 100 -
shown in figure 5.8 , where
temperature
dependent
shifts.
All of these curves, with the exception of the 900 OC curve, were taken as
one set
in two major runs over two weeks.
low experimental
uncertainty.
absorption
shifting
being
edge
below
intensity
with
From these
to
8 eV at 1100
lower
OC.
increasing
The
is still
'notch'
that
with
increasing
9 eV peak broadens
temperature.
The
9
eV
8.8 eV
temperature,
and decreases
peak
to
has
in
shifted
10.5 eV shoulder, which
to
lies in the
and has been attributed to a spectroscopy aberration,
observable
reconfirming
spectra one can see the
energy
approximately 8 eV at 1100 OC. The
area of the
Therefore they are subject
at high
it
is
temperature,
and has
a machine dependent
which for this sample orientation
not
shifted
feature.
has a doublet
The
structure,
13
in energy,
eV peak,
broadens with
increasing temperature and shifts to lower energy.
More
detailed
reflectivities.
analysis
In figure 5.9
is
possible
by
looking
a
reflectivity taken at
The doublet structure of the 13 eV peak is still
shifted to 12.7 eV.
figure 5.10)
13
to 12.4eV.
to 12.3
eV,
and decreased
ev peak
is
at
At
900 OC
(see
At 700 OC (see
figure 5.11)
the 9 eV peak to 8.2 eV,
in intensity.
At 1100
12.2 eV, the 9 eV peak
is
temperature the absorption edge is below 7.7 eV.
5.3 Calculated Results:The Temperature Coefficient
- 101 -
unstacked
visible,but the peak has
the 9 eV peak has broadened and shifted to 8.3 eV,
peak has shifted
the
the
300 *C is shown.
The 9 ev peak has shifted to 8.75 eV.
13 eV peak has shifted
has broadened
at
P
while the
the 13 ev
and the 9 eV peak
OC
(see
figure 5.12)
at
7.9 eV.
At
this
The
high
temperature
to the
temperature
reflectivities
shift coefficient
functional
of
Efeature(T),
dependence.
analyzed
for the three major
fact that no theoretical models
dependence
were
the
For this
was
analysis
edge were determined by two techniques.
the explicit
analyzed
the
extract
spectral features.
predicted
data
to
to
energies
Due
functional
determine
of
a
its
the peaks and
For the peaks, see
figure 5.13,
the geometrical center or center of mass of the peak was determined.
This
was done by drawing a line under the peak, determining the half maximum of
the peak,
and measuring
peak at half maximum.
reproducible
manner
the
To determine
of
needed.
Geometrical
Instead
an energy for
energy of the midpoint
the
did
a
not
characteristic
yield
edge was determined
set of runs where the magnitude
of R was
a
The
eV peak,
energy (eV)
versus
temperature
energy.
from a closely controlled
accurate
within +/- 2.5 %.
In
figure 5.14) was used
(K)
for the 3 features,
spectra is given in figures 5.15 through 5.17.
and 9 eV peaks,
shift rate of the peak center,
average
edge
was
This was midway up the absorption edge.
peak position for the 13
is
energy
a
the 13
the 9 eV peak and the absorption edge from all of the temperature
dependent
This
edge
systematic
these the energy at which R on the edge was 0.2 (see
as an edge energy.
width of the
the energy of the absorption edge,
determining
methods
of the
most
appropriate
transition
broadening
and
the
of
shift
shift are
the
minimum
In
value of 0 corresponds
to
and ignores the effect of peak broadening,
for determining
energy.
the
By measuring the
the
included
case
the
of
temperature
the
absorption
in the calculated
transition
energy.
Due
P,
to
reproducibly measuring the shift of the absorption edge,
edge were determined only up to 700 OC.
- 102 -
shift
of
edge,
the
both
corresponding
the
difficulty
to
of
P values for the
But due to the fact that the
P of
the edge
is similar to that of the peaks,
in unison,
the result
for the
edge
is
and the three features
stated
to hold for
shifted
the complete
temperature range studied, up to 1100 OC.
The
complete
data
shows
temperature
a
predominantly
range.
A linear
linear
relationship
regression fit of
over
these
the
data was
performed to yield an equation of the form:
Efeature (T) = Ef(T=OK)
Here E f(T=OK)
the
linear
-
3 T.
is the energy position of the feature at T =0 K,
temperature coefficient of
edge calculation, E (T=0K)
was
done
below,
with
along
correlation
the
coefficient
demonstrating
In the
case of
the
is not a relevant number since the calculation
an arbitrary
with
the feature.
while 0 is
energy on the
correlation
in
all
edge.
The
coefficient
cases
was
results
of
quite
the
are given
data.
The
+/-
1,
that for a linear model the data was highly correlated;
in
close
all cases the probability that the data is uncorrelated
to
is less than 1 %
and in most cases less than 0.1 %.
TABLE 5.1
Linear Regression Fit Of Feature Energy To Temperature
13 eV peak
E(T) = 13.18
-
7.3 x 10-4 T
c.c. = -.93
9 eV peak
E(T) = 9.32
-
9.9 x 10
T
c.c. = -.98
abs. edge
E(T) = 8.56
-
1.1 x 10-3 T
c.c. = -.97
The
standard
deviations
aen-1
determined by calculating the value of
of
f
for
the
3
features
were
P for each high temperature spectra
relative to room temperature and then determining the standard deviation
of these results.
From these values a standard error can be determined,
- 103 -
which considers
points for
points
the number of data points N used to determine
the 13 eV peak,
14 data points
for the
for the absorption edge were used.
as: standard error =
a- /N1 /2
9 eV peak and 4 data
The standard error
is defined
The values are given below.
9
Standard Deviation And Standard Error Of A
TABLE 5.2
an-1 (eV/K)
Standard Error (eV/K)
13 eV peak
1.5 x 10~4
4 x 10-5
9 eV peak
1.2 x 10-4
3 x 10-5
abs. edge
1.9 x 10~4
1 x 10-4
To
P. 13 data
confirm
the
linear
relationship
between
feature
temperature the data was fitted to a second order polynomial
the magnitude of the second order term.
energy
and
to determine
This was done for the 13 and 9 eV
peaks:
13 eV peak
E(T) = 12.93 - 7.4 x 10-4 T + 6 x 10-8 T2
9 eV peak
E(T) = 9 - 8.7 x 10
-4
T - 7 x 10
-8
T2 .
In each case the second order term is very small,
there
believe
Additional
that
the
relationship
given by considering
to earlier.
is
second
order.
is no reason to
support
is
the data from the one set of two major runs referred
These spectra were very closely controlled and in figure 5.18
we can see that the data fall on a straight line with the exception of one
Additionally,
point.
data
is
measurement
attributable
figure
to
techniques used,
5.18
demonstrates
systematic
errors
that
the
arising
'spread'
from
the
in the
peak
not from actual variability of the properties
of Al2 03
- 104 -
From this one can conclude
measured
that
the energy of the three features
is linearly dependent on the temperature from 300 to 1400
the linear temperature coefficient for each feature
0
C. 0,
is given in the table
below.
TABLE 5.3
$ (Temperature Coefficient) Values For Al 203
+- 1 x 10
eV / K
1.1 x 10
Absorption Edge
+/-3 x 10-5
1.0 x 10-3 eV /K
9 eV Peak
7 x 10
13 eV Peak
+/-4 x 105
eV /K
5.4 Summary of Results
Reflectivities were measured of single crystal sapphire, with light
polarized both parallel
and perpendicular
oriented with the c axis parallel
spectrophotometer
observed
to the
c
axis,
and on samples
to the predominant polarization
from room temperature
to
1100
OC.
The
major
at room temperature are an absorption edge at 8.8 eV,
the absorption edge at 9 eV,
and a broad peak at 13 eV.
of the
features
a peak on
The 13 eV peak
excitation parallel
to
the c axis there is a doublet peak, while with excitation perpendicular
to
shows
orientation dependent
the c axis,
there
features,
for optical
is a higher energy peak, and a shoulder evident at lower
increasing
temperature
energy.
With
energy.
At 1100 OC the absorption
all
three
features
shift
edge has shifted to 7.6 eV,
- 105 -
to
lower
the 9 eV
peak decreased in intensity and has shifted to 7.9 eV,
shifted to 12.2 eV.
The relationship between the energy of the
and temperature is found to be linear.
A for
these
and the 13 eV peak
three features
The linear temperature coefficient
are given in table 5.3.
edge is 1.1 x 10-3 eVK.
- 106 -
features
P for the absorption
111111111111111111
illilillillI
jill
jill
liii
1111111111111111111111111
.15
I I I
Ref 1.
I
I I I
I I I I
I I I I I I I
I I
I I I I
I I I I I I I I I I
10
111111
0
0
liii
0
Ln
0
IllilIll
_______________________________________________________
0
Ln
0
0
0
0
cyi
(*Ti
0
0
CJ
0
in
0
0
LA
0
LA
0
0
C;
d
0
0
LA
0
0
0
LA
r-
C6
eV
Energy
FIGURE 5.1a Reflectivity of laser float- zone grown sapphire:
natural basal face, E perpendicular to the c axis, T equals 21 OC.
-
.30
-
.25
Ref 1.
20
15
0
0
Ii
LA
LA
0
0
0
LA)
o
0
o
LA
Ct;
8
S
d
8
0
LA
0
LA
0
0
0
LA
CS
eV
Energy
FIGURE 5.1b Reflectivity of Union Carbide
perpendicular. to the c axis, T equals 21 *C.
- 107 -
sapphire:
c
in plane,
E
1111111111111
i i il
I
I I
1111
1
1
1 111111T
1 1
1
1
1
1
I I I I I
.30
.25
Refi.
-I
_J
JJL
.20
15
0
8
0
0
'u4
'i
(Vi
rri
(Ij
0
6;;
..
8 0
0
C5
0;
0
C5
0
0
0
-V
Energy
FIGURE 5.2a Reflectivity of Al 203
LXiS, T equals 21 OC.
the c
c in plane,
E perpendicular
to
.30
.25
Refi.
.20
.15
SI
0
C3
I II I I I
C)
0)
C)
0
U)
I I I I
I
0
0
1
C)
(Vi
1 1 1 1 1 1 1 1 1 1 1 1 1 1 AI I I A A I I
8
WV
Energy
FIGURE 5.2b Reflectivity of Al 203:
axis, T equals 21
2C.
-
108 -
0
U)
8
63
0
I
86
I
I . .
C)
..
0
0
U)
0
0
rl
c in plane, E parallel to the c
1.00
. 95
. 90
. 85
. 80
. 75
. 70
. 65
. 60
. 55
Ii I
I
I I I I I I I I I I i I I I I I I I I I I I I I I IIIi
i
i i
I I
r
-
50
Trans ..
. 45
. 40
. 35
. 30
.25
. 20
. 15
.10
. 05
0. 0
1 I
-
I I I I
0:
0
0
Un
0
0
-4
1
ui
I I
I I I I
0
In
I I I
0
0
I I I
I
J
0
J
J
A
0
0
UO
I
0
1
0
0
U)
Pi
c4
J
I I
I
i
0
I
0
0
U)
0
0
0
U)
d~5 6
-
-
0
0
0
U)
--
0
In
r-
-
ev
Energy
FIGURE 5.3 Transmissivity of A12 0
T
I
1
II
II I
I I II
I 1 III
II
III
I I
I
I I
I I
I I
I I
R plane, T equals 21 OC.
Iti
i
i
a aa a
i1 1
u
1
1
a I
a
.01[
Ref1.
a a
I
8
a aa aa aA
8
53
8
a a a a -
E
8
A
5?
d
a.
I
8
5?
8
49
8
8
a;
Cd
Energy
FIGURE 5.4 Reflectivity of gel-amorphous Al203: T equals 21 OC.
- 109 -
lllllllllllllllllllllllllllilllIllllIllllllillllliIlllil
111111111111111111
.15
Reft.
.10
till
.05
0
I
LI
0
0
0.
0I
U*O
I
* I I I
0
0f
0
in
(1;
I
I I I
80
I I I I I
8
ev
I I I
0
In)
CS
I I
0
0
I I I
I I I I
0
0
0
0)
0
U.)
0
0
a;
a;
0
U.)
Energy
FIGURE 5.5 Reflectivity of Lucalox polycrystalline
A2 03: T equals
21 OC.
.15
Ref I.
.10
.05
0
LI
U*)
8
U,
C,;
59
8
O
59
8
8
U)
Cd
0
U')
Energy
FIGURE 5.6 Reflectivity of single crystal MgO: T equals 21 OC.
- 110 -
0
0
.30
.25
.20
Ref 1.
. 15
.10
.05
C,
*0
0'
1~-
0
0
0)
ev
Energy
FIGURE 5.7 Reflectivity of single crystal LiF: T equals 21 *C.
-
111
-
p
11000C
900 *C
\
50 O*C
Ref I.
3 0 O*C-
20-r
.05
1
I
8
. . . . . . . . . . . . . . . . . . . . . . , . . .I .IIIIIIIIII
r ai eaaa a a a 11
I I I I I I I I I I II I I A
8
0
Energy
8
8
CS
Ci
8
0
eV
FIGURE 5.8 Temperature dependent reflectivities of Al 0
T (from
bottom to top of figure) equals 21, 300, 500, 700, 900, and A&" OC. Each
division on R axis equals 0.01.
- 112 -
..................
.35
. 30--
.25
Ref 1.
.20
.15
0
iLl
0
0
0
I
0
0
a;
cS
0
I
0
..
.-.
0
0
d
dto0;
ton0
n
0
CS
Bn
CA
d
0
MO
d
0
0
0
WE
r-
0
0
Energy
FIGURE 5.9 Reflectivity of A1 2 0 3 : c
axis, T equals 300 OC.
.30
in plane,
E parallel
to the c
-
.25
Ref 1.
.20
.15
8
o
8
S
8
5
8
5
8
69
8
59
8
5
Iav
Energy
FIGURE 5.10 Reflectivity of Al203:
axis, T equals 700 OC.
113 -
c in plane,
E parallel to the
c
.25
.20
.
i i i i i i i i i i i
i i i I i i i i i i i
l
Ref I.
i i i i i i i i ii0
iiI
iii
iii
iIi
iiii!
iiiiI____
iiii
iiiii
.15
0
in
8
U)
0T
8
"3
U)
8
O)
my
Energy
FIGURE 5.11 Reflectivity of Al 0
axis, T equals 900 Oc.
23
8
d
C9
6
C6
CS
0
0
0;
U)
r-
E parallel to the c
: c in plane,
.30
.25
l
-
~
l
l
1
1
1
1
1
1
I1
1
1
1
1
1
1
1
1
5
1
1
1
1
1
5
1
1
1
1
5III5
1
1
-
.20
Ref 1.
.15
-
.10
-
-
8
0
U)
8
U)
CT
8
0
in
0
0
0'
0
0
CD
U)v
In
d
8
c;
0
0
M)
8
0
U)
CD
0
ai
Energy
FIGURE 5.12 Reflectivity of Al203:
axis, T equals 1100 OC.
- 114 -
c in plane, E parallel to the c
.25
1lt~
9
1aI I
l11 mm191
I
Im Il
I
v
......... ...aIIMIIIIIIII911
,.
1
11111 fill 1
11,111
1
.20
I
--
Ref 1.
.15
. 10
IIIIfIIIIIII I II IIII III I I I~
S
8
53 4
8
8
Cd
8
S
Lon
0
8
d
04
2
8
of
d
0a
di
59
g
8
Od
Energy eV
of the method used to determine
Demonstration
5.13
FIGURE
0
energy of peaks: peak energies are 12.9 and 9 eV, T equals 21 C.
8
1-Z
the
.30
.25
Ref 1.
.20
.15
amaaI.,at
9
b;
0;
Ia
53
amI
am.aIGa a IIIa&IIaI
9 8 5 8
8
C3
,:
It-
am
5E
II..II...II,
S
4d
Energy
II
a I ImmIaIm
8
5
(d
V; V
4V
Iam.
8 5
1 1
m iia m
8
I
a
in
s ,
A 161111kIIII a
8
0
vi
eV
FIGURE 5.14 Demonstration of the method used to determine an
temperature
arbitrary - energy
on the absorption to determine the
coefficient: at R equals 0.20, energy is 8.7 eV, T equals 21 OC.
-
115 -
I
IIII
-
1300
1100
I
a
IIII
0
-
900
T
700
S
C
500
S
300
*0
100.
Mel
I
12 .0
FIGURE
temperature.
5.15
12.4
of
Energy
-
12.8
Energy eV
the
'13
eV'
13.2
peak
as
a
function
of
1300
0
-
1100
-0
0
-
900
-0
T
-0
700
-S
cc
-
-
500
-S
e '0
-
300
-
-
100
7.4
FIGURE
temperature.
5.16
7B
Energy
of
9.0
8.2
Energy eV
the
- 116 -
'9
eV'
peak
as
a
function
of
I
1300
I
I
I
I
I
I
11001-
900
T
700
-0
cc
500
300
-
.
.
100
8.
7.2
7.6
8.0
8.4
Energy eV
FIGURE 5.17 Energy of the R = 0.20 point on the absorption edge as
a function of temperature.
1300
-
-I
1100
900
-
-
T
700
*C
500
300
100
7.4
7.8
FIGURE
5.18 Energy
of
temperature.
This data is from
the linearity of the results.
8.2
8.6
9.0
Energy eV
the
'9 eV' peak
as a function of
a single closely
controlled run; note
- 117 -
Chapter 6
Discussion
6.1 Introduction
In
this
chapter
reflectivity of
Insight
into
structure
the
measurements
single crystal
V~
the electronic
with
of
Al 0 will be
2 3
structure
temperature will be
and
the
vacuum
compared
the
developed
to the
changes
using
of
the
ultraviolet
literature.
the
electronic
results
of
the
temperature dependent reflectivity measurements.
The aim is to understand
the
to
electronic
considering
crystal
structure
the
relationship
lattice,
contribute
temperature.
to
at
to develop
the
Using
observed
this
temperatures
between
a
the
picture
electronic
of
up
electronic
the
OC
and
structure
electronic
conductivity
information, some
1500
of
at
the
that
high
conductivity results will
analyzed to extract information on carrier character and mobility.
- 118 -
and
carriers
Al203
then,
be
6.2 Room Temperature Optical Spectra and Electronic Structure
6.2.1 Comparison of Optical Spectra to the Literature
The
room temperature
orientation
intrinsic
dependent
features
spectra
taken
8.8
leading
eV,
shows
at 12 eV,
at
12.3
features
the
taken
which
optical
from 7 to 15 eV,
dependent
to the
c
on
axis
have
spectra
these
the
of
These
been
demonstrated
single
orientation
(figure 5.2a) there
orientation
study have
crystal
shown
to
a Al 20
absorption
be
In
edge
at
At 13 eV there is a broad peak which
while for E parallel to the c axis
eV.
in this
show a prominent
into a peak at 9 eV.
features
perpendicular
of
reflectivities
dependent
of
the
sample,
is a shoulder
(figure
features
5.2b)
have
for
E
in the peak
there
is a peak
not been reported
previously in the literature.
In
energy
the
range
results.
literature
and taken
most
point by point,
Still,
there
is
results
reported
here;
the
changes
of
the
reflectivity
for E perpendicular
9 eV,
good
is
found
agreement
caused by
to the c axis
in this
have
reducing
are
systematic
(figure
2.18)
been
our
between
only discrepancies
12 eV and 13 eV features,
shoulder
reflectivities
the
over
a
confidence
literature
attributable
errors.
Loh's
limited
in these
and
to
the
shape
results
are in agreement with the
except that he shows a 12 eV peak, where a
work..
In
leading to a 10.6 eV shoulder that
addition he
shows
is not reproduced here,
- 119 -
a prominent
edge
and below the 9
eV peak
edge.
the
reflectivity
only
a gradual
decrease
instead
of
an
a 9.5
eV
This is probably caused by the chromium doping of his sample.
Arakawa
peak,
shows
(figure
2.20)
does
show a prominent
a 11.6 eV shoulder and a 12.8 eV peak.
edge below
Their results don't go to
low enough energy to see the absorption edge below the 9 eV peak, but they
hypothesize
that
don't
the
show
it
is
there
prominent
from
edge
at
transmission
10
shoulder peak structure they observed,
with E perpendicular
(figure
2.21)
shows
perpendicular
to c,
parallel to c,
doublet
to the c axis,
peak
a
9.2
eV
eV that
measurements,
Loh
observed.
observed
this
From
in agreement with this work.
peak,
an
edge
at
10
the
work.
He
also
eV,
Rubloff
and
at 12.3 eV.
a single broad peak at 12.8 eV,
in
they
I believe their sample was oriented
a peak at 13.2 eV and a shoulder
he shows
and
does
for
For E
not showing
not
B
extend
the
his
measurements to low enough energy to determine whether the absorption edge
starts below the 9..2 eV peak.
a small shoulder at 10.4 eV,
Abramov (figure 2.22) shows a peak at 9 eV,
a peak at 11.7
and a broad peak at 12.8 eV
for a sample oriented with E perpendicular to c.
this work shows a shoulder and peak.
He shows a doublet where
He shows a prominent absorption edge
below the 9 eV peak.
All of
these results are
this study,.or
in cases
where
in agreement with the
there
is
disagreement,
spectra
some
taken
in
other spectra
show features similar to those observed here.
6.2.2 Room Temperature Electronic Structure
Using
the
band
structure
and
atomic
- 120 -
cluster
models
of
a Al 203
reviewed
in section 2.3.1,
optical
spectra
is
Al203
an
in terms
ionic
it
of
material
is now possible
the
room
with
to
interpret
temperature electronic
carriers
localized
in
retaining much of their atomic character and origin.
where
the
formation of
The
expected.
excitons,
localized
nature
rise to the polaronic nature
observed
in
yet
of
identified
reflectivity
as forming
to be definitively
localized
the
hydrogenic
structure
been
structure.
space
and
This is a material
electron hole
electronic
has previously
pairs,
also
is
gives
The 9 eV peak
attributed
to
exciton
in this material; both MgO and SiO2 have
excitons.
identified
Still,
as
in Al2 03
arising
require high resolution low temperature
the
real
of the free carriers in Al 20
This seems plausible
formation.
been
the
or
the observed
(10 K)
from
the
9 eV peak has
excitons;
this would
optical studies to document
levels of the excitons.98 A separate
possible
explanation
was put forth by Ciraci that the excitons are due to surface state bands,
which does not seem plausible due to the low density of such states,
does
demonstrate
that
the
spectral
features
are
still
open
but
to
interpretation.
Pending future work to identify or refute the excitonic
nature
9
of
the
eV
peak,
the
9
eV
peak
is
attributed
to
exciton
formation.
The
existence
of the 9 eV exciton peak has made
the absorption edge difficult.
covered
the
showing
an
exciton
reflectivity of
where the
Previously
peak
of
In addition most optical studies have not
the full energy range from 7 to 15 eV,
structure present.
identification
below
the
a
10
spectra
eV
sapphire at room temperature
exciton peak has broadened
therefore omitting some of
has been
edge.
If
interpreted
we
and 900 *C (see
and decreased
compare
as
the
figure 5.8)
in intensity
(due
to
the difficulty of forming the exciton with such a high phonon density), we
- 121 -
can
that
see
there
is
a
large
still
edge
absorption
It
present.
has
shifted to about 7.8 eV but is the same edge that at room temperature lies
at 8.8 eV.
the
Now the question arises whether,
exciton to be
interpreted,
In
formed below
the
for this material,
fundamental
band edge,
as previously
or do the exciton states superimpose themselves
both MgO99
and
SiO2100
the
exciton
superimposed on the absorption edge.
fundamental
absorption
edge below
states
are
to expect
on the edge?
observed
as
a peak
Therefore the observation of a large
the
exciton peak,
which becomes
fully
exposed at high temperature, is reasonable.
Further confirmation of the existence of the fundamental
edge
at 8.8 eV can be gained from looking at the
(see
figure
energy
5.3).
and
then
Here we
decreases
transmissivity of A12 03
see
the transmissivity
to
0.01
between
absorption
8.5
equal
and
to
8.8
0.75 at
eV.
The
low
edge
positiQn determined in this work is given as 8.8 eV +/- 0.25 eV.
The
absorption
corresponds
to
the
edge
at
8.8
fundamental
eV
in
Al203
absorption
edge
at
of
room
the
temperature
material.
It
In
demonstrates the onset of valence band to conduction band transitions.
Al203 this absorption edge corresponds
at
the
top
conduction
than the
of
band.
the
valence band
The
intensity
low density of
corresponds
confirmation
states of
the
to Al
or
higher energy features
to transitions from the 0 2p bands
3s
height of
observed
Al
bands
the
in the
3s bands.
this
is
interpretation
photoconductivity results of Kuznetsov
101
,
temperature photoconductivity to occur at 9 eV.
this
work
corresponds
to
an
optical
- 122 -
absorption
8.8
gap,
is
by
found
the
smaller
due to the
eV absorption
in the material.
found
who
the bottom of
reflectivity
The
to the band gap at room temperature
of
in
edge
Further
considering
the
onset
of
the
room
The band gap measured
since
it
was
measured
in
at
relatively
low temperature
do not
allow
In addition
the
the
it
determine
relaxation processes
phonons
that
to
change
the
indirect or direct;
absorption
shape
edge
to conclude
possible
to
k
involved
optical
of
and by optical
in a thermal
the low temperature
the
measurement
It
electron.
that
The
uncertainty
Calcul.ating
the
this would
which
at room temperature.
Kronig
permit
)
D
means that
gap process.
does
of
is
not supply
the
not
indirect
possible
with these measurements whether the minimum fundamental
Al203 is
due
0
to
is a direct gap because
necessary
transitions
(compared
in
the
accuracy of
this
the
or
constant
scope
of
this
work.
gap of a Al203
direct
separating
in
involve detailed analysis of the
beyond
the optical
dielectric
analysis would more
is
gap
to
number
exciton
index of
accurately determine
is
0.25
from
refraction
is
is at 8.8 eV
low (+/-
peak
It
the
eV),
edge.
using
Kramers
but
was not
the band gap,
performed in this work.
The
third
major feature
13 eV peak which shows
the
most
intense
peak
structure
calculation
states
lie
bands.
to
Al
lying
in the
bands
orientation
found
eV above
the
the
this
peak
reflectivity
energy
in
conduction
is
features.
dependent
in
observed
in the
the
range.
The 13 eV peak
Batra
conduction
band
edge,
the broad
is
in his band
band
density of
arising
from Al
3p
With a band gap of 8.8 eV, the Al 3p bands should lie at 12.8 eV,
corresponding
the
4
observed
3p
states
states
joint
to
sub-band,
to
excitations
in
the
from
the
conduction
top
of
band.
the valence
Other
in the valence band are also occurring
density of states
the Al
3p bands
due
is dominated
to the
- 123 -
transitions
(0 2p)
from
to
lower
at 13 eV, but the peak
by transitions from
high density
giving rise to the 13 eV peak.
band
of
states
of
the 0 2p
the
0
2
p
6.2.2.1 Summary
Through
the
use
of
the
electronic
structure
models
for
a
Al2 03 ,
measured reflectivity spectra at room temperature have been
The
9
eV peak is attributed
evidence
The
is presented at the present
absorption
fundamental
time to substantiate
lies
edge
at
8.8
interpreted.
though no
exciton formation,
to
eV,
and
the
definitive
the assignment.
to
corresponds
transitions from the 0 2p states at the top of the valence band to the Al
3s states at the bottom of the conduction band.
Al203 is
The optical direct gap of
even though no statement
8.8 eV at room temperature,
is made
as
to the direct or indirect nature of the minimum band gap in the material.
The 13 eV peak observed corresponds to transitions to the Al 3p states in
the conduction band.
6.3 Temperature Dependent Optical Spectra and the High
Temperature Electronic Structure
6.3.1 Optical Results
Optical reflectivities taken from room temperature to 1100 OC,
7 to 15
temperature.
shoulders
showed
eV on single crystal a Al 0
2 3
were
The
detailed
lost,
due
shape
to
the
of
peaks
effects
- 124 -
the
following
changed,
of
phonon
for
from
changes with
example
broadening
small
on
the
electronic states.
The peak centers, or centers of mass, shifted to lower
energies with increasing
temperature.
the bands leading to a decrease
peaks also broadened,
This corresponds
to
the
in the band to band energy gap.
corresponding
shift of
The basic
to spreading of the bands.
Measuring
.the temperature coefficient of the peak center, as was done for the 13 and
9 eV peaks,
the
determines
temperature
coefficient
shift and spreading.
interest
and
the
temperature
of
the
coefficient
absorption
for peak shift,
edge
combines
while
both
band
For the band gap the minimum transition energy is of
both band
measured results,
between
the temperature
spreading
and
shift
should be
considered.
The
taken on many samples showed a very linear relationship
temperature
and
coefficients
the
energy
determined
of
a
spectral
in this work
feature.
The
are given in table 5.3.
The absorption edge temperature coefficient is 1.1 x 10-3 eV/K.
To compare to the literature
will
be
used
since
The
experiments.
the
work
edge
of
the absorption edge
shift
Gilles,
as
is
what
was
analyzed
shift coefficient
measured
in previous
by Harrop,
was
largest temperature range, and yielded a value of 1.5 x 10 -3
is the value of
for Al 20
figure
improved
The
value
spectra
show
statistics,
the present
reported
from which
associated
a
nonlinear
energy versus
relationship.
one could determine
seems
the
reasonable,
data
was
This
change
in
the
even
where
absorption
nonlinearity and the magnitude of the result.
- 125 -
temperature
more
samples
is in disagreement
but
though
extracted are
the Cr peak in the area
gross
With
if this
finding of a linear relationship,
existence of
eV/K.
the
P that has been referenced in the conductivity literature
The results, when plotted as
2.24)
over
this
and
with
is not possible.
high,
cause
(see
but
for
the
actual
concern.
The
the data was taken, and
the
edge
the
shape
may
explain
The rest of the data discussed here was analyzed by this author to
extract
values
demonstrates
the
of 0
for comparison.
the variability
importance
of
in A's
error bars
taken over a small
This was
calculated
in these data.
temperature range,
of values from 1.4 to 2 x 10-3 eV/K for P.
1.3
x 10-3
eV/K,
it
from one
set
The data
of Yakovlev was
of data and
and
These lead to a wide range
Laufer's data suggests a value
Hunter and Rubloff each suggest 1.75 x 10-3 eV/K and
and
report
that
temperature.
The latter value
eV/K measured
in this work.
which is very low,
because
as was all the subsequent data,
had noticeable absorption edge shape changes.
of 1.9 x 10-3 eV/K.
instructive,
the value
of
A decreases below room
is quite close to the value of 1.1 x 10
Blum determined
a value of 1.4 x 10-4 eV/K
but he was working over a very limited energy range and
therefore it is hard to determine what feature was shifting.
Most of these values are of the same order as the results reported
The values of A extracted
here.
few
runs,
and
variability
of the
demonstrated.
analysis
do
of
demonstrated
the
advantage
peak
the absorption
which
ignores
in
good
edge
broadening.
the order
of
statistical
the
center
internal
expect
confidence
have
results due to arbitrary
One
both
not
from the literature
of
shift
consistency
shift
to be
The
and
of
the
larger
of
work
Also
decisions
has
absorption
calculated
magnitude
qualities.
analysis
present
are taken from very
has been
been
edge
measurements.
that
the
shift
has
One
would
than a peak center
shift
literature
the effect while
values
the
give
superior
statistics of the present work give confidence in the reported values.
- 126 -
the
6.3.2 Temperature Dependent Electronic Structure Models
It is useful to review the findings of the work on the temperature
dependence
control
the
structure
of
of
the
electronic
electronic
techniques have
interaction of
structure
to
structure
at
addressed
electrons
develop
high
an
idea of
what
temperatures.
the problem,
and phonons.
considering
effects
Only
band
in
terms
it
This work was done mostly
for the covalent semiconductors which show delocalized electron and phonon
states
as
opposed
to the bond
distortions of Al 203
lattice
localized
oriented
Therefore
effects
electrons
Due
to
the
localized
and
lead
The
that,
for
example,
to
polarons.
structure in determining the
electronic properties will be discussed
complexity of
lattice
they are unable to address the gross
interaction of the lattice and the electronic
high temperature
and
in section
electron-phonon calculations,
they have
6.4.
never
been attempted for Al2
03
band structure
The
E (T).
g
of the
decreasing
yield
three basic
is the effect on the electronic
The first
expansion
models
lattice;
this will
tend to
the band width of the valence
two terms produce effects that are
the
exponential
shift.
it.
It
behavior
Debye
far
temperature,
above
it.
The
to E (T)
g
that
as has
The other
larger than that of thermal expansion.
linear
above
It is proportional
this
temperature,
electron self-energy terms
the 3 terms
been
determine
the band gap by
and conduction bands.
They are exponential below the Debye temperature
is plausible
that
structure of the thermal
increase
The Debye-Waller term leads to band broadening.
below
terms
found
can
in
- 127 -
superimpose
this work
to T
and
lead to band
and linear above
to
give a linear
from below the Debye
temperature to above it.
thermodynamic
The
importance
Here
of
the
local
effects
approach
effects
of
bond
that are
the valence
band
(anti-bonding
orbitals),
change
effecting
the
process
will
of
be
corresponding
(bonding
the
electronic
greater
problem
the
demonstrates
the
not well treated in a covalent case.
breakage,
electron from
thereby
to
orbitals)
phonon
energy
to
to the
in
The
of
an
conduction band
properties
levels.
importance
excitation
of
the
effect
materials
with
system
of
this
localized
electronic structures.
The
atomic
appropriate
been
done
molecular-orbital
to localized materials
using
electronic
localized
cluster
this
method
structure.
electronic
periodicity
electronic
and
Because
to
such as Al 203,
study
the
structure
averaged
orbitals
to
the
in
temperature
they
the
of
changing
expansion
the
bond
and
atomic
experimental data.
the time
a
that
in
in
Thermal
band
the
gap
accompanies
allow
states.
With
formation
cluster
can
expansion
be
bond
localized
This
electronic
of polarons,
In addition,
effects
the bond
of
thermal
directly
lengths,
from
but,
in
atomic vibrations can lead to
length,
information
the
a
the
introduced
increases
excitation,
instantaneous
energy.
the
the
vibrations
frame of the electronic
decrease
reduced
lengths
the
translational
relaxation is a noticeable portion of the electronic energy.
by
of
allow the creation of
can
excited
more
dependence
the requirements
structure,
seems
even though no work has
cluster models
without
relax
approach
leading
about
the
the
to
a
atomic
relaxation
thermally
relaxation
energy
gained
due to polaron formation could be calculated directly.
The
interpreting
fact
the
that bond
length
temperature
is
a useful
induced
- 128 -
changes
method
in
for
optical
qualitatively
spectra
and
electronic
structure can be confirmed by considering
the
effects
of bond
length changes on samples at room temperature.
Other forms of A12 03 which
have varying bond
crystal
grained
lengths
polycrystalline
relative
a
Al2 0 3,
to
single
amorphous
derived A1 2 03 , and anodized films of Al 20
these materials
length
is
centers.
lucalox
will
different
lead
to band
from
single
figure
crystal
5.5) and the
y
are fine
crystallite
sol-gel
The strained bonds present in
spreading
This effect has been observed
(see
or
a Al203
a
and,
if
the
shift
A12 03 ,
average
bond
of
band
the
in this work for polycrystalline
(see
sol-gel Al203
figure
5.6),
which
show absorption edges at about 8 eV.
It is useful
to apply the
of Al 2 03 . This
is
fitting
which
model
empirical model of Ravindra to the case
a model with
only
no
depends
essentially linear with temperature.
fundamental basis;
on
the
eV
while
for
0D
equal
to
is
a curve
temperature
and
is
Calculating the decrease of the band
gap with temperature up to 1500 OC, for a 0
1.33
Debye
it
1000
K,
D of 695 K, the gap decreases
the
decrease
Experimentally from this work the gap is predicted
is
1.53
to decrease
eV.
1.65 eV
over 1500 K. So, Ravindra's model underestimates the shift, but then it is
the
only predictive model
available
and
is not
grossly
inadequate.
The
fact that such a naive model can predict the temperature dependence of the
band gap
suggests that
not essential
the complexity of
the electron-phonon modeling
to understand the effect, and that the
simplicity
of
is
the
localized cluster approach which considers bond length changes may be more
appropriate.
6.3.3 High Temperature Electronic
Structure of a Al 203
- 129 -
the
From
the temperature coefficient
and determines
the broadening
and
features.
The
shift of the 0 2p
the minimum transition energy for valence
band to conduction band transitions.
of the band gap.
structure
of three electronic
absorption edge shift demonstrates
and Al 3s bands
optical measurements we determine
temperature dependent
It gives the temperature
dependence
The shift of the 9 eV peak demonstrates the temperature
coefficient of the average energy of the exciton state.
The shift rate of
the 13 eV peak demonstrates the rate of shift, without broadening,
for the
0 2p and Al 3p bands.
It
is
now
of
interest
measured;
temperature.
Al203
around
The
no
change
next
determine
The data was linear
function of temperature.
range
to
of
mode
fundamental
was
change
the
band
*C,
so
there
temperature
coefficient
confidently
use
the
up
is
no
to
these
P value
of
reason
the
lattice
to
x
10-3
as
a
ev/K
the
Debye
properties
of
lattice vibrations
expect
temperatures.
1.1
near
observed
in
energy
over the total temperature
has been observed to be the onset of anharmonic
1500
gap
a
change
Therefore
as
the
in
the
one
can
temperature
coefficient of the band gap energy up to 1500 *C. The room temperature gap
To determine the 0 K value of the gap
determined in this work was 8.8 eV.
we can use A to project back to 0 K.
Hunter found, the 0 value
determine
E (T=OK)
g
may be
is
the
and,
one
more
importantly,
that would be
As
but decreases
Therefore the application of a constant
incorrect.
But with the emphasis
work on the high temperature conductivity of sapphire,
small,
of 9.1 eV.
is linear above room temperature,
to zero below room temperature.
to
This gives a value
the
of
P
this
the error would be
linearly extrapolated T = 0 K band gap
determined
in a
- 130 -
high temperature
conductivity
In this work the
measurements.
linearly
extrapolated
T = 0 K bnad gap
to as the apparent T = 0 K band gap.
will be referred
Combining
this
information, with energy in eV and temperature in K yields:
E [T(K)] = 9.1 eV
g
while
This
and,
It
E [0 K]
=9.1
eV
E [273 K]
=
8.8
eV
E [1500 K]
g
=
7.15 eV.
is the minimum optical energy gap of the electronic
since
thermal
1.1 x 10-3 T
-
it
was
experimentaly
measured,
expansion, Debye-Waller effects
is
an
optical
band
gap
because
automatically
and the
of
the
can be
From these
indirect,
then at high temperatures,
optically measured
gap
is
indirect,
the
minimum band
it
if
direct
with
gap will
the
or
large
is
of
optical
reduce the gap
measurements,
determine
present,
the
seen.
for
self-energy.
frequency
measurements; no effects of lattice relaxation, which may
energy,
accounts
electron's
high
structure
not
possible
but
if
it
to
is
numbers of phonons
be equivalent
to
the
indirect
gap.
6.4 The Role of High Temperature Electronic Structure and
Lattice Properties in Determinini
Electronic Carrier Character
and Creation Energy
Up to this point the temperature dependence of the optical band gap
- 131 -
has been determined,
producing
a value for
the T = 0 K optical band gap
The carriers that
and the linear temperature coefficient of the band gap.
are created optically are free band electrons created in a static lattice,
which
can
then
undergo
lattice
application of these results
relaxation
to understand
forming
polarons.
high temperature
For
the
conductivity,
the thermal band gap and the properties of the carriers with their lattice
relaxation
fields must be
considered.
This is because
the time
scale of
the carrier creation in a thermal process will be that of the lattice, and
the energy required to create the resulting polaronic carrier will be the
thermal
gap energy.
determines
the
the
properties
of the
interaction
In other words
electronic
creation
carrier
of
the
it
is
not
carriers
at
the band
high temperature,
structure with the lattice
energy,
the
carrier
structure
that
and
character
that
but
the
determines
the
carrier
mobility.
6.4.1 Polaronic Nature of The Electronic Carriers
The first
major effect the lattice has on the electronic
structure
is in determining the character and mobility of the carriers: whether they
are
band
electrons
polarons.
sapphire
many
with
The polaron coupling
is
years
Colbourn
or,
constant
a material that forms
has
been
calculated
polarons.
strong
The
suggesting
the
large
that
relative
the
electron-lattice
a equals
or
literature.
- 132 -
of
are
small
in sapphire
that
suggesting
small polarons.
carriers
stability
idea of polaron formation
2.7,
interaction,
Kroger
polarons,
and
large
is accepted
for
while
hole
in the
The
carrier mobility
is
another
good
indicator
of
the
polaronic
nature of the carriers, but we have such information only for electrons in
For electrons the measured mobilities
sapphire.
to 100 cm2/V sec
suggesting that the
lie in the range
electrons are
of 0.1
large polarons.
hole may be a small polaron with a thermally activated
The
hopping mobility.
These results are very similar to those found in amorphous and crystalline
SiO2 where the holes are small polarons with a mobility of 10-5 cm 2/V sec
while electrons
are
large polarons with
a mobility of 20 cm2/V sec.
have no information from the band structure calculations
masses
*
of
m
electrons in
the
carriers,
but
from
naive
band
on the effective
curvature
*
the bottom of the conduction band have m
holes in the top of the valence band m
< m
arguments
,
while for
*
> me due to the very flat bands.
**
The polaron effective mass m
We
*
1.75 m .
=
In the
rest of this work the
band effective mass will be taken to be m
e
6.4.2 Thermal versus Optical Band Gaps
The
other
major
electronic properties
to
lattice
effect
of
the
lattice
on
the
high
is the reduction of the carrier creation energy due
stabilization
of
the
carrier.
This- is
embodied
difference between the thermal and optical band gap energies.
materials
direct
gap
which do
or vertical
corresponds
materials
not
form
transitions
to
temperature
polarons,
in the
phonon-assisted
any assumed equivalence
and indirect gaps is not correct.
the
optical
band structure,
indirect
gap
in
the
In covalent
corresponds
while
transitions.
of optical and direct gaps,
the
In
to
thermal
ionic
and thermal
In materials with localized electronic
- 133 -
structures,
the
optical gap corresponds
electron, while
the
polaron directly.
thermal
gap
is
If we define A
to the energy to create
the
energy
(analogous
required
to
the band
create
to the Stokes Shift)
the
to be
the difference between the optical and thermal gap energies, then A is the
lattice
stabilization energy imparted by the
forming the polaron.
lattice to the band carrier,
The Franck Condon Model then allows us to show the
relationship between optical and thermal gaps,
A as in figure 6.1.
the band electron,
and lattice relaxation and
In the fast optical process, E 9 is required to create
g
which then relaxes to the polaron, while in the slow
thermal case, the lattice relaxes during the time period of the excitation
requiring only E t to create the polaron directly.
g
Kroger estimated A for Al203 to be 0.65 eV +/-
From experimental data
2.1
eV while Colbourn
theoretically calculated the value of 2.8 eV.
In
energy,
an
attempt
which
to
determine
encompasses
both
A
the
true
and
A,
Colbourn's calculation of A for Al 20
the causes
thermal
it
is
carrier
useful
creation
to
consider
In Colbourn's T = 0 K calculation,
of A are electron shell relaxation after excitation for large
polarons,
and
Therefore
one
additionally,
can
see that
temperature
dependent
demonstrate
the
atom
motion
for
the
small
polaron.
this calculation is very different
from the
electronic
differences
core
structure
between
the
calculations.
causes
of
A,
It
serves
which
are
averaged thermal bond length changes leading to increased orbital
to
the
overlap
and energy level shifts and A which is due to localized lattice relaxation
around
the carrier,
after excitation,
the thermal carrier creation energy.
leading
to an overall
reduction
in
Colbourn found that A was 2.8 eV for
band to band transitions, and 1.2 eV for optical versus thermal excitation
of Ti
3+
to Ti
4+
. In the paper they compared their value of a thermal gap
- 134 -
(9.5
2.8 = 6.7 eV) to a value of 6.5 to 7 eV computed by Kroger102 as
-
This was
the gap of Al203 at 1600 K, and stated that there was agreement.
because what they compared was E
in error,
-A;
P term.
they neglected the
0
A to E
(T=OK) -
g
A.
from
Therefore using their value
this
E t(T=1500K)
g
work,
=
OT
the
t(T)=E 9(T=OK) - PT
of A,
and the value of E
4.35
eV.
value
This
(T=OK) -
P and A,
considering both
Instead,
high temperature carrier creation energy is given by E
-
0
g
0
g
(T=OK) and
of
the
P
carrier
creation energy will be evaluated in the next section.
6.5 Intrinsic Electronic Conductivity in Al 0
-3
How can the
Al2 03
be
used
conductivity?
103
temperature dependence
to
understand
First
determined to illustrate the
terms
that
Finally
are
the
or
controlling data
and
alumina's
role
the
of the electronic
of
high
E (T)
g
in
roles of E (T=OK),
g
the
the
mobility (p)
E (T)
values
6.5.1.1 Carrier Concentration
- 135 -
electronic
temperature
controlling
P and A.
terms -
will
be
c
will
be
Then the other
will be discussed.
used
information on the mobility of electronic carriers in Al 203
6.5.1 Factors That Control Conductivity
structure of
to
extract
To
determine
the
conducitivity aelec(T)
role
of
E (T),
from 1400 to 1600 OC,
It is easier to consider the product,
ahe
2
a~n
h e
pe~h
consider
where aelec
the
=
electronic
nqj e + pphe
h
Se Ih q2Nc Nv,-Eg/kT
*
where setting me=11
*
=e
then N N =(2nmk/h2 ) 3T3
c v
and putting in
E (T)=E9(T=0K)-PT-yT2_A
g
g
one arrives at
3
(a ah )/T 3
2
2 3
h q (2nmk/h )
-A)/kT
e0/k eyT/k e-(E
Plotting this function as log (a ah/T ) versus 1/T one can extract
useful information about the carrier creation energy.
note is that
slope
E
0
g
of
the
the activation
curve,
is
E
g
The first thing to
energy for carrier creation,
t(T=OK) the
thermal
band
gap
as given by the
at
zero
K,
or
(T=OK) - A the zero K optical gap minus A.
The effect of the nonlinear term y in the temperature dependence of
the band gap is considered next.
the
conductivity
can
have
The only way that an Arrenhius plot of
noticeable
curvature
is
due
to
a
nonlinear
P term factors out front as a temperature independent
term.
The linear
term.
The y term does give a temperature dependent exponential term, but
in the case of Al2 03 this term is too small to give noticeable curvature.
Using
the results of the nonlinear analysis of the 9 eV peak,
in yT 2 from 1400 to 1600
to the
experimental
0
the
change
C is only 0.04 eV, much too small in comparison
inaccuracies
to be observed
- 136 -
as curvature.
Therefore
due
to
the
T2
insignificant
conductivity
component
of
should plot as a straight
E (T),
g
one can
that
the
line in this type of analysis.
As
analysis of E (T)
g
stated earlier, the nonlinear
is
say
rejected,
and only a
linear dependence is found for E (T).
g
The true effect of E (T)
g
in
the
slope
(or
pre-exponential
sensitive
E crea)
of
fl/k
term.
E
is due to the A term and is not observed
the
conductivity
equals
3.5 x
used
leads
exponential
to
2
orders
making
factor,
5
a large
,
instead
of
in the
factor and very
P. A's role, in the pre-exponential,
to the value of
is similar
A one third change in the value of
to that of the carrier mobility terms.
A
10
plot but
magnitude
change
in
of meaningful
extraction
the
value
of
the
mobility information
very difficult.
for E
Good values
value
of
A
is
estimated to be
that
should
0
g
not
confirmed;
.65
eV.
have
only
and A
(T=OK)
it
is
are presented
calculated
small
effect.
to
be
2.8
eV,
The
and
for the effective masses but
Values are lacking
a
in this work.
The
only modeling
assumptions
employed up to this point are in the use of semiconductor theory, which is
well
accepted,
to determine
the
number
carriers
of
There
created.
has
been no need to make any choices between competing models.
6.5.1.2 Carrier Mobility
There is a lot of controversy about ge and gph. Measured values of g
for
electrons
determine
suggest
whether holes
large
polarons
are large
but
polarons
or
no
small polarons.
lack hard experimental evidence of the mobility of electronic
- 137 -
exists
information
to
Since we
carriers,
it
seems
to use
fruitful
a data
and E (T) data to
g
the
elucidate
carrier
mobilities.
To extract information on mobility we must invoke models,
not necessarily well-founded,
the
holes,
different
the models
functional
for
for the polaron mobilities
small
and
dependencies,
while
not demonstrate
polaron model does
large
polaron
for
the
which are
In the case of
.
mobility
show very
electrons,
the
large
observed temperature dependence.
the
The large polaron mobility is given by a fundamental equation that depends
first
two constants
strongly affect p, while the
third
is unknown and the last
agreed
polaron mobility,
on
D
D,
,
m ,and
upon.
a.
For
exists,
only
depends
on
polaron
model
a
the
The
small
the
phenomenalogical
type
equation
fitted
p sp
and
for
value
of
electrons,
and
are
no
fits
that
can
can derive
equation
to
data,
assume
the
and
is at least
fundamental
Ehop. We
then we
not well-known
a
and
large
parameters
for
hole mobility.
In
controlled
summary,
by
concentration
mobility
the
carrier
terms
high
temperature
concentration
are
controlled
terms are controlled by
0
and
by
E
conductivity
electronic
mobility
,
g
,
D, w , a, p,
terms.
A,
m *
e
h, me
The
and
is
carrier
.
m
and mh.
The
Most
of these parameters are not well-known at the present time but progress is
being made.
6.5.2 Calculation of a mg
e
Using
calculated
the
available
a from 1400
information,
to 1600
an
OC in Al203
- 138 -
attempt
to
will
compare to
made
to
the data
of
be
Kroger
on
nonequilibrium
titrated with hydrogen.
104
105 These measurements
minimum,
or compensation point,
1600
(see
OC
figure
6.2).
measurements
titration
of
Ti
doped
showed the conductivity
from 1400 to
as a function of temperature
The
point
compensation
sapphire
can
then
be
plotted
versus l/T to give the total activation energy for conductivity(see figure
6.3).
Since
data and a model exist
with the Fermi energy
correspond
other
to
words,
for p
the
,
electronic
a mg will be calculated.
at mid-gap
conductivity
This may not
a at the compensation point if L e does not equal ph. In
at
the
compensation
the
point,
Fermi
energy
is
not
necessarily mid-gap.
For the calculation of the carrier concentration, both A = 0 eV and
A = 2.8 eV are used.
These correspond to thermal gaps of 7.15 eV and 4.35
eV at 1500 *C. For the electron mobility values of
K, with the related values of w , are used.
temperature,
OC,
in
D of 695 K and 1000
With this spread in the Debye
the calculated mobilities were 12.1 or 1.6 cm 2/V sec at 1500
the same range as the experimental values.
this range of input parameters are given below.
TABLE 6.1
emg
695
(0cm)
A=2.8 eV
A=0 eV
0 D (K)
e
1000
695
1000
-5
14000C
8x10 -9 lxlO -9
lxlO -4 lxlO
15000C
5xlO8 7x10~9
4x10~4 5x10-5
16000C
3x10 -7 4x10 -8
1x10 -3 2x10
- 139 -
-4
The values of
a mg for
6.5.2.1 Evaluation of A
Comparing the calculated
to
the
measured
2x10-10 (16000C)
of magnitude
values
),
values of a mg computed using A = 2.8 eV
of Kroger
(
-1
a in ((cm) )
=
5x10
-12
(14000C),
one sees that the calculated values are 7 or 8 orders
too high.
If we make an extreme case calculation where ph
=
0, then the creation energy for electrons will be E , and the Fermi energy
g
will lie at the top of the valence band.
In this case, with A = 2.8 eV,
to 10-10 (0cm)~ , which are still
the calculated values decrease to 10
larger than the measured values.
extreme,
but
Therefore
it
even
then
The assumption that p h = 0 is obviously
a values cannot be brought
the
seems that the value of 2.8 eV for A,
the optical and thermal gap energies,
to fit
the available data.
into
agreement.
the difference between
is too large,
and it
cannot be used
From now on in this work A will be assumed to
be zero for lack of any reasonable estimate.
6.5.2.2 Derivation of ph
e
The amin measured by Kroger is determined by the condition ae =
This will correspond to a mg only if Ai
=
h
f
h
a.
then EF <E /2,
ge
i.e., a larger number of low mobility holes must be created to balance the
(higher)
electron conductivity.
By comparing the a mg values to
e
value of the conductivity at the compensation point,
do not agree.
Depending on the choice of 0
of -magnitude of disagreement.
the
hole
mobilities
must
be
The value of
lower
than
- 140 -
the
ammn
.,
the
we can see that they
D, there are 2
to 3 orders
a mg is too high, therefore
electron
mobilities,
as
is
expected
the
from
literature.
The
degree
of
level
Fermi
lowering
directly controlled by the ratio of the hole and electron mobilities,
is
and
is manifested by the difference between o mg and c . . The mobility ratio
e
min
and amount of Fermi level lowering are given by the following:
(E /2)
For
o / mg)2
e=
Ah
e
-
F = -.5kT ln (Ah/
*
D = 695 K, the mobility ratios at the three temperatures are
on the order of
10-5 . Since these calculations are admittedly crude, and the value of 10on the order of 10
seems
more
while for 0
D = 1000 K,
the results
reasonable,
for
they are
0D = 1000 K
(assumed) will
be
shown.
TABLE 6.2
Temp0 C
h
e
E /2 - EF
g (eV)
These
Calculated Mobility Ratios and Fermi Energy Lowering
1400
1500
1600
1.8X10-5
3x10-5
4.7x10-5
.788
.796
.805
mobility
calculations,
ratios,
strongly suggest that
thermally activated mobility.
the literature.
even
considering
the holes
the
are
inaccuracies
in the
small polarons with a
This is in agreement with information from
In addition the degree of the Fermi level shift,
on the
order of one volt, is consistent with one's feeling that in the large band
gap
ionic materials,
it
is very difficult
- 141 -
to shift
the
Fermi
level
far
for
which
hopping
eV T =
9.1
the
in
observed
is
1.4
eV.
eV activation
between the 10.5
and
ph (T)
Upon evaluating
from mid-gap.
is
This
0 K band
gap
of
the
difference
added
the
Therefore
is
the
in his measurements
observed
Al 20 3
conductivity measurements
the
just
obviously
energy Kroger
energy
the activation
one can derive
activation
slope
energy
for
hopping of the holes.
A
product
that
this
E .
result
that
Assuming
g
a
the holes
may
be
intercept
in the
contained
of
values
on
check
the
of
we
deriving
Kroger's data
electrons
polaron model,
small
by
gotten
the
using
no
mobility
assumed
a large polaron model,
fit
the data
can plot
according
and
to
the following:
1/2
a . /T
mmn
o
o
) x
= 1540(p p
sp ip
e-(Eg+Eh) /2kT
The
intercept of the log versus
which can be used,
are
found
these
to be
values
l/T curve gives a mobility product
with a suitable p e values, to derive values of p h.These
.4
and 3
do not
cm 2/V
support
the
sec.
is
There
previously
a
contradiction
derived
mobility
here,
ratios,
as
and
are on the borderline of small to large polaron mobilities.
6.5.2.3 Summary
These
calculations
our understanding
been
expected
have been performed
of the electronic
to provide
to
serve
as
conductivity of Al2 0
exact numbers
an overview
of
They have not
for the mobilities of carriers but
- 142 -
instead
to
fundamental
determine
parameters
difficult.
electronic
More
derive
required
The
making
is
and conductivity,
information
on
the
Knowledge
does
results.
about
serve
the
to
This is:
to definitively understand
many
of
the
mobility.
conductivity,
mobility
show
a
of
the
reasonable
to use the carrier
both reasonably well
carrier
of
further numerical analysis more
needed
calculation
evaluate conductivity
concentration
to
conclusions.
is lacking,
information
carriers.
approach to
plausible
founded
More
but
at present,
information
the knowledge
is
of A
and E (T=OK) allows us to know the carrier concentration and increases our
g
understanding of electronic conductivity in this material.
- 143 -
-------
c.b.
Relationship
6.1
FIGURE
E 0
gap
bard
optical
between
and lattice
thermal band gap E
stabilization A enerlies.
t
Et
EO
9 EE
1
v.b.
unrelaxed
state
relaxed
state
band
polaron
carrier
ICm
t
FIGURE
6.2
Electronic
conductivity
versus
partial
pressure
of
H
showing
the
electronic and ignic compensation
points. From Kroger, reference 104.
z
Txm
--
-0-
*4bk
NO"
-X.-
111iI
1111
I
I
Iii
10
-4-P
(PO)
l
I1
I I
10P
1q0
-w-TO C
ism0 "at)
1450
1400
169
FIGURE
6.3
Compensation
.'.'
point conductivity plotted versus
l/T to give the activation energy
From Kroger,
conductivity.
for
reference 104.
I
-
5.3
- 144 -
5.4
5.5
--.
I
5.6
5.7
10/T.( K)
I
5.8
5.9
6.0
Chapter 7
Conclusions
A High
built
that
samples
Temperature
is capable
Vacuum Ultraviolet
of measuring
the
Spectrophotometer
reflectance
from 6 eV to 15 eV (210 nm to 85 nm)
and
has been
transmittance
of
in the VUV on samples heated
without contamination from room temperature up to 1100 OC. The temperature
dependence
of
structure
the electronic
of
single
crystal
Al203
has been
studied with this facility.
7.1 Reflectivity Measurements at Room Temperature
At room temperature,
9 eV exciton peak,
energy,
of
the
shows three main features:
superimposed on a 9 eV absorption edge,
a 13 eV peak.
orientation
the reflectivity
The 13
sample.
parallel to the c axis,
eV peak shows
A
doublet
peak
and,
a
at higher
features
that depend on the
appears
for
light
incident
while a peak and a shoulder are observed for light
perpendicular to the c axis.
The absorption edge is due to transitions to
the Al 3s sub-band of the conduction band,
while
the main peak at 13 eV
corresponds to transitions to the Al 3p sub-band of the conduction band.
The
valence band has
a
lower
lying
Al-0
- 145 -
sub-band,
with the
top of the
valence band being of 0 2p character.
Transmission measurements show the
This corresponds to
absorption edge to lie at 8.8 eV at room temperature.
This measurement provides
a direct (8.8 eV) band gap at room temperature.
no
information
about the nature
(direct or indirect) of the minimum band
These results are in agreement with band
gap of the material.
structure
and atomic cluster models of the T = 0 K electronic structure of Al 203
7.2 Temperature Dependent Reflectivity Measurements
With increasing
to lower
are
shifts
These
energies.
in the reflectivity shift
the features
temperature
due
This may be explained
energy level bands and shifts of the band centers.
either
the
from
instantaneous
temperature
linear
bond
sub-bands
different
of
effects
superimposed
range
with
thermal
lengths
have
studied,
temperature,
in
expansion
the
different
and
atomic
experimentally determined values of
arising
It
temperature
dependencies.
order
was
found
were
3s abs. edge) = 1.1 x 10-3 eV/K
j (9 eV Exciton Peak) = 1.0 x 10-3 eV
P (13 eV Al3p Peak) =
7 x 10
- 146 -
eV/K
on
the
that
the
Over
the
negligible.
temperature coefficient
expressed in eV/K are:
p (Al
the
the shift was found to be
terms
the linear
from
vibrations
material.
(25 OC to 1100 OC),
second
as
or
interaction,
electron-phonon
of the electronic
to spreading
+
1 x 104
+-
3 x 10-5
+-4 x 10-5
The
J,
7.3 High Temperature Electronic Structure
To understand
the
role of the
lattice
must be
carriers
the high temperature
interaction
The
lower
electronic
carriers
can
mobility.
Mobility
measurements
their
structure
creation
the
of
of Al 203
the electronic
of
the properties
in determining
considered.
electronic
with
lattice
and
energy
affect
the
their
polaron
coupling
constant point to the formation of polarons which are electronic
carriers
surrounded by a stabilizing
and
the
value
of
field due to lattice
the
relaxation.
The degree
of electron-phonon coupling determines whether large polarons with a band
type mobility
or small polarons with
a thermally
activated mobility
are
The lattice stabilization imparted to the carrier also determines
formed.
the difference (A) between the energies of the optical gap and the thermal
gap.
The
that
create
determine
energies
reported
carriers
in this work correspond
without
relaxation.
lattice
to optical
In
an
attempt
the thermal band gap energy a theoretically calculated
value of A was used in an analysis of conductivity measurements.
to conductivities
orders of magnitude
larger
than observed
energies
to
(2.8 eV)
This led
in Al 203
No
reasonable values exist for A.
The room temperature band gap
measured
temperature coefficients,
extrapolated
to T = 0 K,
using
the
gives 9.1 eV for the T = 0 K apparent
optical band gap of A12 0 3 . Ignoring
the contribution
of A,
this
is the
carrier creation energy that will appear in the exponential factor of high
- 147 -
temperature
The carrier creation
conductivity equations.
energy
at 1500
OC is 7.15 eV, as determined by extrapolation from these measurements.
7.4 Implications for Conductivity
Intrinsic
carrier
models
electronic
concentration
with
energy.
the
to
large
is
Al203
controlled
Previously,
have
to
been
determine
deduced
or
the
from
a
by
the
theoretical
which assume the carrier type
scattering mechanism (thermal
Electron mobilities
the
in
carrier mobility.
conductivity measurements
mobility measurements,
Due
and the
for the carrier mobility,
small polaron) and
used
conductivity
(large
impurity),
carrier
or
were
creation
small
number
of
but the mobility of holes has yet to be observed.
number
of
carrier
mobility
models
and
the
lack
of
experimental data, it is misguided to analyze conductivity measurements in
Al203 assuming a specific mobility model and thereby to deduce the carrier
creation energy.
From the present work we know the
carrier
energy
creation
electronic
conductivity,
dependence
A
which
pre-exponential
information
term
(the
band
and
the
due
of
has been used
to
gap)
to
the
activation
energy
value
of
the
band
temperature
its
the
contribution of the
linear
character
conductivity
equation.
with
the
conductivity
appears
is
results
form
of
We
tentatively conclude
the
this
Kroger
to
The mobility of
found to be much higher than hole mobilities,
with mobility measurements.
in
Therefore,
derive insight into the electron and hole mobility terms.
electrons
gap
for
in agreement
that the
electrons
large polarons with a band-type mobility while the holes form small
- 148 -
polarons with a thermally-activated mobility.
high temperature mobility measurements.
- 149 -
This could be confirmed by
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- 156 -
Biographical Note
The
author was born in Rochester
graduating
Before
from
High
Brighton
semester of Mechanical Engineering
School
at LaSalle
in
1975,
Science
and
University in Mexico
graduated
He
Engineering.
with
1957.
one
attended
he
He then entered Cornell University School of Engineering where
in Materials
29,
New York on December
City.
he majored
Bachelor
a
of
Science degree with distinction in 1979.
He then entered Massachusetts Institute of Technology in 1979 where
he pursued graduate work in Materials Science with Prof. R. L. Coble.
Currently a member of the American Ceramics
Physical
Society,
Society,
he
Al 20 3
the Optical Society of America
has presented
three
papers
on
the
Society,
the
American
and the American Vacuum
Electronic
Structure
of
In addition during his undergraduate years he published two papers
on crack propagation in aluminum epoxy bonds.
These were titled 'Use
of
Holographic
Interferometry to Study the Crack Propagation in Metal Plastic
Composites'
for the TMS-AIME
Study Crack Propagation
and
'Use
of the Double Torsion Technique
in an Adhesive Layer'
and Evaluation.
- 157 -
for the Journal
to
of Testing
-
158 -