gfi'gs’;r.-Tzri ,

ABSTRACT

THE ELECTRICAL CONDUCTIVITY

OF SINGLE CRYSTAL Cr203

by Julian Anthony Crawford

The electrical conductivity of single crystals of Cr203 has been

measured as a function of temperature over a range of 569 to 1416°C
and as a function of oxygen partial pressure over a range of 1 to 10‘-6
atm. .The results show a high-temperature "intrinsic" conductivity that
is independent of oxygen pressure. and a low-temperature defect-controlled
conductivity that varies with oxygen pressure in a manner which cannot
be determined from measurements on single crystals. The very low
mobility deduced from the intrinsic conductivity leads to an interpretation
in terms of the theory of small polarons. The model suggested is that

3+

of charge formation and transport in the localized 3d levels of the Cr

cations .

THE ELECTRICAL CONDUCTIVI TY

OF SINGLE CRYSTAL CI'QO,2

By

Julian Anthony Crawford

A THESIS

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

DOCTOR OF PHILOSOPHY

Department of Physics and Astronomy

1963

ACKNOWLEDGMENTS

I wish to express my appreciation to Dr. Robert W. Vest for his
suggestion of this problem and for his guidance and encouragement
throughout the work. I would also like to thank Dr. Norman M. Tallan
for many stimulating discussions of the problem and its interpretation.

1 wish to thank Professor Donald J. Montgomery for his help and advice
during the preparation of this thesis and Professor Sherwood K. Haynes
for his c00peration and encouragement during my studies. The experi-
mental research reported here was performed in the facilities of the
Metallurgy and Ceramics Laboratory of the Aerospace Research Labora-

tories of the Office of Aerospace Research, United States Air Force.

ii

I.

II.

III.

IV.

VI.

TABLE OF CONTENTS

INTRODU CTION . .

................... 1
EXPERIMENTAL PROCEDURE .......... 6
A. Sample Preparation ................ 6
B . Experimental Apparatus ............... 7
C. Experimental Procedure .............. 21
THEORETICAL CONSIDERATIONS ........... 23
RESULTS AND ANALYSIS ............... . 31

A. Analytical Procedure ................ 31

B. Sample 35C - Unoriented Single Crystal ...... 33

C. Sample ZSC - Unoriented Single Crystal ...... 38

D. Sample 4SC(a) - Oriented Single Crystal ...... 43

E. Sample 4SC(C) - Oriented Single Crystal ...... 48

DISCUSSION ................ , ....... 52

CONCLUSIONS ...................... 59
BIBLIOGRAPHY ..................... 60

iii

Table

IV..

VI.

LIST OF TABLES

Atomic percent of metal impurities in 01.203 single

crystal samples .......... . . . . ......

Lattice parameters for Cr203 at 250C .......
Experimental data for sample 38C .........
Experimental data for sample ZSC . . . . .....
Experimental data for sample 4SC(a) .......

Experimental data for sample 4SC(c) . . . .

Activation energy and pre-exponential for samples

ZSC, 4SC(a), and 4SC(c) in the intrinsic region . . .

iv

23

34

39

44

49

55

LIST OF FIGURES

Figure Page
1. Laue photographs of oriented samples .......... 8
2. Experimental apparatus .......... t ...... 9
3. Sample holder ..... . . . .............. 10
4. High temperature furnace and sample holder ...... 12
5. Vacuum systems ...................... l4
6. ac bridge circuit ..................... l6
7. dc circuit ......................... 18
8. dc lead wire resistance correction ........... . 20
9. The corundum structure ................. Z4

10. Electrical conductivity of sample 35C .......... 37
11. Electrical conductivity of sample ZSC .......... 41
12. Least-squares line for sample ZSC ............ 42
13. Electrical conductivity of sample 4SC(a) ......... 46
14. Least-squares line for sample 4SC(a) .......... 47

15. Electrical conductivity of sample 4SC(c) . . . . . . . . . 50
16. Least—squares line for sample 4SC(c) .......... 51

17. Intrinsic conductivity of samples ZSC. 4SC(a), and
4SC(c) ........................... 57

I. INTRODUCTION

There has been increasing interest recently in the electrical
transport pr0perties of inorganic compounds which are insulators when
very pure and stoichiometric but are quite good conductors when impure
or non-stoichiometric. In particular the transition-metal oxides have
received considerable attention owing to the variable-valence property
of the cations. Verwey1 has suggested for these materials the mechanism
of "controlled valency. " which refers to the change in valence of a small
fraction of the cations brought about by the introduction of a small amount
of aliovalent impurity. Heikes and Johnstonz and van Houten3 have used
this technique to study the properties of lithium-substituted transition-
metal oxides. Their findings indicate that the electrical transport in
these materials can be explained by assuming that the charge carriers
are localized on the cations and that conduction occurs via a thermally-
activated diffusion (or hopping) process. This mechanism is charac-
terized by rather low values of electrical conductivity and by a low
carrier mobility that increases exponentially with increasing temperature.
The activation energy associated with the mobility is a consequence of
the self-trapping of the charge carrier by its own polarization field.

Morin4. in an earlier approach. had analyzed data on

0 and NiO in terms of the usual semiconductor band theory. and

“Fez 3

had found that the mobility of the charge carriers varied between 10"5

and 10"1 cm2/ volt-sec. He interpreted this mobility as being due to

Z
conduction either in a very narrow d-band or in localized d-levels .
He subsequently made a survey5 of the magnetic. electrical. and opti-
cal properties of the 3d metal oxides in an effort to detect evidence of
energy bands in these compounds. His conclusion that the non—bonding
3d electrons in the oxides of scandium. titanium, and vanadium over-
lap sufficiently to form a 3d conduction band was verified experimen-
tally6 for the lower oxides of titanium and vanadium, which show a
metal-to-insulator transition at the Néel temperature. He further
concluded that the 3d electrons do not overlap in the remaining 3d metal
oxides. but that they are in isolated energy states; hence electrical
transport occurs by electron exchange between cation neighbors and
involves an activation energy.

Attempts to derive this type of activated transport mechanism
from first principles have resulted in the theory of polarons. The term
"polaron" is used to describe a quasi-particle consisting of an electron
surrounded by an accompanying cloud of lattice polarization and moving
as an entity through the lattice either by tunneling (polaron band) or by
phonon-activated jumps (hopping process). The lattice polarization
surrounding the electron arises from the strong electron-lattice inter-
action characteristic of ionic materials. Because of this strong coupling
effect. the usual band theory does not apply.

There are two different theoretical approaches to this problem

in the literature. The first7 is an extension of the usual effective-mass

3
approximation starting with Bloch functions and utilizing a variational
technique to find the energies of low-lying polaron states. This method.
however. applies only for weak coupling and low temperatures. The
second method, which yields results applicable to metal oxides at high
temperatures. is frequently referred to as "small polaron theory".
The theory as developed by Yamashita and Kurosawas. Sewe119,
and Holstein10 takes the Heitler-London or tight-binding approximation
as a starting point. with the electron localized on one lattice site and
the centers of the lattice vibration harmonic oscillators suitably dis?
placed by an amount depending on the strength of the electron-lattice
coupling. In this theory the relative roles of the periodic potential and
the electron-phonon interaction are reversed; the electron-phonon
interaction is considered in zero order (to produce self-trapping). and
the periodic lattice potential is treated as a perturbation which gives
rise to a wandering of the polaron through the lattice. The eigenstates
of the system are constructed as linear combinations of localized
states. At low temperatures a polaron band is formed. which narrows
as the temperature increases. At a transition temperature of the order
of half the optical-phonon temperature. the. polaron band width becomes
comparable with the breadth of a localized state and the band picture
breaks down. Above the transition temperature the electron apparently
hops from site to site with an activation energy. Apart from impurity

scattering. as the temperature is increased the mobility decreases at

4
low temperatures when the band model applies. but increases exponen—
tially with an activation energy above the transition temperature.

Most of the theoretical work done on small polarofis applies only
to the pure, perfectly periodic lattice. On the other hand. most of the
experimental evidence for hopping transport in the transition metal
oxides has come from impure, doped samples at temperatures where
the conductivity is defect controlled. Perhaps important also is the
fact that most of the data were taken on polycrystalline samples for
which interpretation may be somewhat doubtful. In the light of these
observations it would be useful to have some more precise data from
single-crystal samples taken under well-defined experimental condi-
tions in the temperature region covered by the theoretical assumptions.

The transition metal oxide chosen for this study was chromium
sesquioxide. Cr203. One of the primary factors in its choice was the
availability of single crystals. Moreover, a search of the literature
revealed only a scanty amount of conflicting data on the electrical conduc-
tivity at high temperatures. Hauffe and Block11 measured the electrical
conductivity of a sintered sample of pure Cr203 from 400 to 800°C by
a two-terminal dc method. They observed an oxygen-pressure depen-
dence too small to be explained by the usual defect theories. This
observation led them to postulate the existence of a conductivity

mechanism in chromia which is independent of oxygen pressure.

2
Fischer and Lorenzl extended these measurements to 1750°C with

5

samples doped with Cu 0 and TiOZ. and concluded that there was no

2
oxygen-pressure dependence from 600 to 17500C. They observed a
break in the curve at lZSOoC and calculated an activation energy of 2. 5
ev for the high-temperature part. More recently Hagel and Seybolt13
used a four -termina1 dc method to measure the conductivity of sintered
chromia samples in air. argon. and nitrogen. They calculated a high-
temperature sIOpe of 1.7 ev and a low -temperature sIOpe of 0. 36 ev.
Their data showed a very slight pressure dependence below lZSOoC.
Measurement of the electrical conductivity of single-crystal
samples of (31-203 in the temperature range from 600 to 14000C should
provide the answers to some of the questions raised by previous work.
Controlled variation of the oxygen pressure around the sample, and the

use of oriented single crystals should lead to better-defined data which

will aid in interpretation of the results in terms of the available theories.

II. EXPERIMENTAL PROCEDURE

A. Sample Preparation

 

Single-crystal samples of pure CrZO3 were obtained from Linde
Company” in the form of cylindrical rods approximately 2 cm long
and 1/ 2 cm in diameter. The crystals were grown by the flame-fusion
method from powder of unspecified purity, but presumably of labora-
tory grade. Spectrochemical analyses on crystalline fragments of the

boules were performed by Battelle Memorial Institute. Columbus.

Ohio. The results are shown in Table I.

Table I - Atomic percent of metal impurities in Cr 0 single crystal

 

 

 

2 3
samples.

Metal Unoriented Oriented
Impurity Samples Samples
Fe <0. 001(N) 0. 003
A1 0. l 0.02

Cu <0. 001(N) < 0. 001(N)
Si 0. 002 0.. 003
Mg <0. 001(T) 0. 003
Ca 0. 002 0. 00?
Na <0. 001(T) < O. 001(N)
V <0.005(N) 0.02

(N) Not detected.

(T) Trace.

Samples for measurement of electrical conductivity were cut
in the form of thin plates with the apprOpriate geometry for parallel-
plate capacitance measurements. Two samples were cut directly from
a boule with no attempt) at orientation. One boule was oriented by x-ray

diffraction methods and two samples were cut from it. one with faces

perpendicular to the c-axis and one with faces parallel to it. Figure 1
shows a back-reflection Laue diffraction pattern for each of the oriented
samples.

The samples were cut from the boules with a diamond saw. and
the cut faces were polished on 240-grit wet-or-dry metallographic paper.
The samples were cleaned thoroughly in acetone. given a final rinse in
ether, and fired in air to approximately 800°C in a muffle furnace.

The samples were then electroded by painting a thin layer of Engelhard
#6082 platinum paste15 on the polished faces. One face was coated with
a single continuous film, and the Opposite face was given a smaller
circular inner electrode surrounded by a guard-ring electrode. The
paste was fired slowly in air to approximately 450°C to remove the
organic vehicle, and then heated to 800°C to fire the platinum binder.
The electrode films prepared in this way had a resistance of less than

one ohm.

B. Experimental Apparatus

An overall view of the experimental apparatus for the measure-
ment of electrical conductivity is shown in FigureZ. Two furnaces were
built to Operate with a single vacuum system so that two experiments
could progress simultaneously.

A drawing of the sample holder is shown in Figure 3. The por-
tion in the furnace is constructed entirely of recrystallized alumina

(McDanel AP35 and Morganite Triangle RR) and platinum-rhodium alloys.

 

"C" - Sample

 

"A" - Sample

Fig. 1 - Laue photographs of oriented samples

 

Fig. 2 - Experimental apparatus

10

IL
I
L;
E

 

Recrystallized

Thermo- Alumina

couples

 

Ill/J17’/’/’[’[’fl
I

 

 

  

Guard [-— Guard Ring

Electrode \

 

 

 

 

 

 

 

Sample

 

 

 

 

Fig. 3 - Sample holder

11
The sample area is enclosed by a platinum-coated crucible. The bottom
of the sample holder is supported from three alumina rods. one of which
carries the lower thermocouple leads. The upper thermocouple is
contained in another alumina rod which is spring loaded at its upper
end and is situated in the center of the assembly. The guard lead was
made from platinum wire and contained in a small alumina rod. also
spring loaded at the tOp. The two spring-loaded electrodes served to
hold the sample in place against the lower electrode.

The sample holder and three radiation shields were suspended
from a large brass plate which constituted the t0p plate of the high-
temperature furnace. Figure 4 is a schematic of the furnace assembly
with the sample holder in place. The furnace tube which separates the
sample chamber from the furnace is McDanel AP35 recrystallized
alumina which is vacuum tight up to 1800°C. The tube is cooled at each
end by large water-jacketed c0pper collets. and is made vacuum tight
by O-ring scale. The furnace element is made of tungsten wire wound
on an alumina core with molybdenum and tantalum radiation shields.
Power is supplied from a 0-140v. 25a variable autotransformer. The
furnace power is regulated by a Minneapolis-Honeywell R 7086A Potenti-
ometer Controller modified to achieve a sensitivity of approximately 5
microvolts. The controller responds to a platinum + 6% rhodium vs.
platinum + 30% rhodium thermocouple located next to the furnace core.
The temperature at the sample is held constant to within 0. 5C0 above

looo°c.

 

Fig. 4 - High temperature furnace and sample holder

13

Figure 5 is a schematic diagram of the three vacuum systems
built into the apparatus. The tungsten furnace element is protected by
a vacuum of better than 10'.5 torr. The sample chamber has two separate
vacuum systems to cover different ranges. The low-vacuum system.
which serves as a roughing line for the high-vacuum system. operates
with a mechanical pump down to less than 10 microns. Pressures from
one atmosphere to 1000 microns are read on a Wallace and Tiernan
Precision Dial Manometer FA 145 having a range of 0 to 30 inches of
mercury and an accuracy of 0. 1% of full scale. From 10 to 1000 microns
a Veeco Thermocouple gauge is used. The high—vacuum system will
produce a vacuum of approximately 5 x 10“5 torr at the sample. as
determined by a Veeco type-RC 75 ion gauge located at the top of the
furnace.

The high-temperature furnace is designed so that the atmosphere
surrounding the sample can be maintained at a total pressure of from
one to 10.6 atmosphere. and at an oxygen partial pressure of from one
to approximately 1002‘0 atmosphere. Various oxygen partial pressures
may be obtained by (1) blending helium-plus-water-vapor with hydrogen.
(2) blending carbon dioxide with carbon monoxide. (3) mixing helium with
oxygen. or (4) reducing the total oxygen pressure over the sample.

In the present investigation only the latter two methods were used to
vary the oxygen pressure around the sample and thereby establish the

region of pressure dependence.

14

Left Right
Furnace Furnace

Ion Ion
=®=Ji . 1% Gage Gage w . IF=®=
Gas Gas
In In

 

 

 

Discharge
Gage _.

 

 

 

 

F J

 

Vent Vent

 

 

 

 

 

 

 

 

 

 

 

 

L: m L_____J)
v 65
(iii Gage Thermocouple
Gage

 

 

FP 3
Vent

Fig. 5 - Vacuum systems

15

Electrical-conductivity measurements were made using both ac
and dc methods in different but overlapping ranges. In the low—conductivity
region. ac measurements were made on a transformer ratio-arm bridge
(Wayne Kerr Universal Bridge B221) with a conductance range of 10.9
to 10"1 mho (stated accuracy 1 0. 2%). The bridge circuit is shown in
Figure 6. An internal oscillator adjusted to 1592 j; 1% cps (w = 104 rad/ sec)
provides the source voltage. A buffer amplifier isolates the oscillator
from the bridge circuits. to which four-terminal connections can be
made. The detector is a tuned two-stage amplifier with a double-shadow
"magic eye" associated with each stage. Balance of the unknown impedance
is made against standards of conductance and capacitance in parallel.
Tappings on the two bridge transformers. connected to decade controls.
permit measurements to be made accurately over a wide range of
impedance. Because of the unique design of the bridge. the impedance
of the sample leads is automatically eliminated. At balance. no poten-
tial exists across the detector primary; hence it is possible to connect
an impedance between the right-hand terminal of the unknown and neutral
without affecting the accuracy of the measurement. Similarly. connecting
an impedance between the left-hand terminal of the unknown and neutral
merely reduces the voltage supplied to both the unknown and the standards
in prOportion to the turns ratio. It is possible therefore. to measure

the sample impedance insitu without having to make corrections for

 

parallel lead impedance. Series resistance of the leads does introduce

£50.30 «woman as .. o .mfm

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

- 231332
H omdflm
. O
0 s
H 3.350 Z
M o o uvndvddum
H oocfluscsoo
o OIII o
F 0|] 0
1 LG) 0
Ollll O
nouooaofl 0|] 0
C Lwl. H
/ x54 Toll!
. 0 condom
deduouxm mpuspsdum a a o
oodefiodasu 2 s o
o2 0
£336 socmpoaga o
swag 550:de H a .H

 

 

 

 

 

 

.53. 82 .

17
an error. but this error is negligible for sample resistances greater
than 100 ohms.

The Wayne Kerr Universal Bridge was also used for measuring
electrical conductivity at frequencies from 50 to 20. 000 cps by connect-
ing an external source and detector. The principle of operation is the
same. except that the internal bridge oscillator and detector are
switched off. The external frequency source was a Hewlett-Packard
Audio Oscillator Model 201B having a frequency range of 20 to 20. 000
CPS and a calibration accuracy of i 2%. A General Radio Tuned Ampli-
fier and Null Detector. type 1232-A. having the same frequency range
as the audio oscillator, was used as‘ a null detector for the signal.

The conductivity of all of the single crystal samples was found to be
independent of frequency in the temperature range of interest.

At high temperatures. where the sample impedance became
less than about 100 ohms. it was necessary to use a guarded dc method
to measure the conductivity. Figure 7 is a diagram of the dc measuring
circuit. The source of constant voltage is a Keithley model 240 regulated
high-voltage supply with a range of _-l_~_ 0 to 1000 volts in one-volt steps
(stated accuracy within 1%. above 10 volts). The series resistor R8 is
a one-megohm precision resistor. Since the sample resistance during
dc measurements is always less than 100 ohms. the current in the circuit
is constant to better than one part in 104. The voltage drop across the
sample is recorded on a Leeds and Northrup Speedomax H Adjustable

Zero-Adjustable Range (AZAR) Recorder having a range of 0.67 to 100

18

 

 

m [Constant
JL flTrigger 1 Voltage
6 _.L

R =10 fl, .—
s ..
H

 

   

 

 

 

 

 

 

 

 

 

R >>R
s u

I=v/Rs

Fig. 7 - dc circuit

19
millivolts full scale and a zero suppression of _+_ 50 millivolts (stated
limit of error 0. 3% of range span). The X1 amplifier is a Kintel model
llZA-B unity-gain plug-in instrument with an input impedance of 101
ohms and a gain stability of O. 001%. Its purpose is to keep the guard
electrode at the same potential as the high side of the sample. thereby
causing RH to appear as an infinite impedance. This procedure insures
that all of the current flowing through Rs flows through the sample Ru'
and that the voltage at the X1 amplifier is just the voltage across the

sample. The resistances RH and R represent leakage resistance from

L
the upper and lower electrodes to the guard electrode.

The lead resistance R . part of which is in the hot zone of the

l
furnace, was measured as a function of temperature by bringing the
upper and lower platinum electrodes of the sample holder into contact

(no sample) and determining the resistance by the method just described.
Figure 8 is a plot of the lead-wire correction over the temperature range
400 to 14oo°c. The data points are for equilibrium temperatures and
are accurate to i 0. 02 ohm.

The sample temperature was measured by means of two platinum
vs platinum + 10% rhodium thermocouples built into the sample holder.
The thermocouples were made from Engelhard standard-grade 0. 020-
inch thermocouple wire. A reference temperature of 0°C was main-
tained by a mixture of crushed ice and water. The thermocouple emf

was measured on a Leeds and Northrup 8686 Millivolt Potentiometer

having a range of -10.1 to +100.1mi11ivolts and a limit of error of

20

003

cofioouuoo 00930.30.” 0.33 603 up .. w .mfim

AOov 0usudu0a508

ooE 00.2 003 com coo

 

 

ON A

on 4

Ca. A

Agnov
00:03? 0m

cm A

cc 4

21

_i: 0. 05% of the reading 1 3 microvolts. The sample temperature was
taken as the average of the readings of the upper and lower thermocouples.

An attempt was made to calibrate the thermocouples relative to
each other at the same time that the dc lead-wire calibration was run.
However. the results of two consecutive runs showed a small drift in
the relative calibration. Walker 511126 have found evidence of thermo-
electric instability of some noble—metal thermocouples operating for
long times at temperatures above 10000C. They attribute this instabili-
ty to contamination of the thermoelements by impurities. predominantly
iron. from the ceramic protection tubes. Since this instability is
cumulative with respect to both temperature and time. a relative cali-
bration of the thermocouples cannot be considered dependable over long
periods of time. Fortunately this uncertainty will not significantly affect
the electrical-conductivity measurements since the analysis is made in
terms of the absolute temperature and the relative error is small.
For the same reasons. the absolute values were determined directly
from a Leeds and Northrup thermocouple conversion table without
calibration. On the basis of Walker' s results and the experimental

procedure followed here. the error is assumed to be negligible.

C. Experimental Procedure

 

Before any electrical measurements were made. the samples
were heated to approximately 1200°C in the experimental apparatus for

a period of two hours at an oxygen pressure of one atmosphere in an

22
effort to reach equilibrium with the surrounding gas. The results of
the equilibration attempts are described in part V.

The temperature dependence of the electrical conductivity at
constant pressure was measured by holding the oxygen partial pressure
constant in the sample chamber and varying the sample temperature
from 600 to 1400°C. The oxygen-pressure dependence of the conducti-
vity was observed by repeating this procedure at several different
partial pressures. The samples were equilibratedat 1200°C each time
the oxygen pressure ("was changed. Measurements were made with the
dc apparatus when the sample resistance was less than about 100 ohms.
The data for the temperature variation of .the conductivity at constant
pressure were quite reproducible. The magnitude of the conductivity

at constant temperature and variable pressures was less reproducible.

III. THE ORE TICAL CONSIDERA TIONS

Chromium sesquioxide. Cr203. has the corundum structure
with the point group symmetry class .3- (Z/m) of the rhombohedral
crystal system. This structure can also be described in the hexagonal

system for ease of calculation. The hexagonal lattice parameters at

room temperature”' 18 are given in Table II.

Table II 4 Lattice parameters for CrZO at 259C.

3
e 2 _[__c a
4. 963 13. 593. 2.74

Geometrically. the structure can be visualized as oxygen ions arranged
in hexagonal.-close-packed layers with two-thirds of the octahedral inter-
stices between each layer containing chromium ions. Located along the
threefold axis are pairs of distorted. cation-occupied octahedra that
share a common face. In the basal plane these octahedra share a common
edge with three similar' octahedra. The corundum structure is shown
schematically in Figure 9.

Chromia is antiferromagnetic below about 35°C with the spin
directions alternating from one cation layer to the next (vis. + - + -
along the c-axislg). However, the Neel temperature is low enough that

even 'with the assumption of short-range order above T the influence

N9
of spin coupling at 1000°C can be neglected.

The model assumed for the electronic structure of a 3d transition-

metal oxide is similar to that of the usual semiconductor in that an empty

23

24

 

 

 

Fig. 9 - The corundum structure

25

conduction band is assumed to arise from the cation 48 levels. and a
full valence band to arise from the anion (in this case oxygen) 2p levels.
In addition. however. the 3d energy levels exist with some of them pre-
sumably located in the band gap. Although the possibility exists that
the 3d wave functions may overlap sufficiently to form a very narrow
band. there is no evidence for this overlap at present. Hence the 3d
states will be discussed in terms of localized levels. In the case of
chromia. the Cr3+ ion has three 3d electrons remaining outside the last
closed-shell configuration. Electronic transport may result from the
motion of charge carriers in the bands or in the localized levels or in
both simultaneously.

The possible "intrinsic" mechanisms for production of electronic

charge carriers in CrZO3 are

02- ... 01- + 4s electron (111-1)
02’ + Cr3+~ 01" + Cr2+ (111-2)
Cr3+ .. Cr4+ + 48 electron (III-3)
ZCr3+ _. Cr2+ + Crla:+ “11"“

These reactions represent (1) the formation of an electron-hole pair by
excitation of an electron from the valence band to the conduction band.
(2) the formation of a hole in the oxygen 2p band and an electron in the
3d levels. (3) the excitation of an electron to the 4s band leaving a hole
in the 3d levels. and (4) the formation of an electron-hole pair in the

3d levels. The term "intrinsic" will be used here to describe those

26
processes which are unaffected by external influence. in particular by
clanges in the defect structure brought about by deviations from
stoichiometry.
In addition to the intrinsic reactions mentioned. there are a
large number of possible defect reactions which could be written to
account for the electrical conductivity. Of the reactions available the

most likely one is

4 3-
(3/4)oz‘3)~ (3/2)olat + vCr + 33) (111—5)
where 02(8) = gaseous oxygen
olat = oxygen ion on a lattice site

3-
vCr = triply-ionized Cr vacancies

3 = positives-charge carrier (hole).

20
From the theory of the thermodynamics of defects the concentrations

of the various quantities can be expressed by the equilibrium equation
3- ’\ 3 3/4
+ =
h... 1le ma K

- 3-
where LVCr J = concentration of Cr vacancies
[C9] = concentration of holes

P0 = partial pressure of oxygen
2

K = equilibrium constant.

Since there are three holes for every vacancy the concentration of holes

27

will be given by

01'

[3:] = K' 13023/16.

The electrical conductivity is related to the concentration of charge

carriers by the relation

0 = le ”[9] (111-6)

where 0‘ electrical conductivity

(D
II

electronic charge

1:
ll

mobility.

If equation (III-5) were the predominant source of charge carriers. then

the relation

0 0C 130 3/16

2

(111-?)

should hold. As mentioned previously. this pressure dependence has

not been observed in conductivity eXperiments. However. recent measure-
ments21 of the self-diffusion of radioactive Cr51 in single crystals of Cr203
which were equilibrated with various damp hydrogen atmospheres at
1300°C have shown an oxygen-pressure dependence very close to that of

equation (III-7). This evidence suggests that the reaction (III-5) is the

correct defect reaction and that the ionic motion is not intrinsic at 13000C.

28
These considerations eliminate the possibility of a predominant intrinsic
mechanism due to charge production by thermally generated and ionized
point defects (Schottky or Frenkel effect).
For an intrinsic mechanism of electron and hole production

at equilibrium

[(9] [3)] CC eXP (-E/kT) ‘

and since [0] = ['9] = n. the concentration of one type of carrier.

\_/

then

n = A exp (~E/ ZkT) (III-8)

€
:2!"
m
H
m
m
u

activation energy

k = Boltzmann's constant

H
u

ab s olute tempe rature .

and the constant A involves the change in entrOpy in the reaction. The

electrical conductivity in this case can be written

a = lelue[3] + leluh[®]
or using equation (III-8)

o = (e I A (file + uh ) exp (~E/ ZkT). (III-9)

29

The temperature dependence of the mobility depends very strongly
on the conduction mechanism. The three intrinsic mechanisms described
by equations (III-1). (III-Z), and (111-3) involve charge transport in either
the 2p or 48 band or both. Morin5 suggests that the 2p band in oxides
is between 10 and 20 ev wide. and that the mobility of holes in a band of
this width should be quite high. Similar information for the 4s band is
not available. but the mobility of electrons here should also be high; and
for either case. of broad-band conductivity it generally decreases as
some negative power of the temperaturezz. For equation (III-4). which
describes charge motion between the localized 3d levels. the mobility
would be quite low and would increase exponentially with temperature.
Experimental evidence for most transition-metal oxides leads to a

mobility relation of the form
p oc exp (~h/kT).

The activation energy. h. is assumed to arise from the self-trapping

of the charge carrier by the polarization it induces in the lattice around
itself. For the case of intrinsic conductivity the charge carrier is
assumed to exist as an excited state of a cation. The existence of an
activated mobility leads to the concept of "hopping" of the charge carrier
between equivalent lattice sites.

If it is assumed that the electron and hole mobilities are equal.

30

equation (III-9) reduces to
o a leIA'exp[-(E/2 + h)/kT]. (III-10)

It has been suggestedz' 3 that "hopping" transport be treated as a diffusion
process; in this case the constant A' would have a l/T or perhaps even
more complicated temperature dependence. For the temperature range
of interest here. the exponential temperature dependence completely
overcomes any simple linear term. Hence the analysis will be carried

out in terms of equation (III-10).

IV. RESULTS AND ANALYSIS

A. Analytical Procedure

 

The data shall be described in the form
a .-. 00 exp (-U/ kT) (IV-l)

. . . . -l
where a = electrical conduct1v1ty in (ohm-cm)

C.
u

activation energy in ev

Boltzmann' a constant (8. 622 x lO-Sev/ OK)

K
II

o
T = absolute temperature in K.

The data are taken in the form of an ac conductance G. or a dc voltage

and current which. through Ohm' 3 law, yield a resistance R. The sample

is most conveniently considered as a resistance in parallel with a capaci-

tance, but the bridge read-out is in terms of reciprocal resistance or
conductance. The relationship between the ac conductance and the dc

re sistance is simply

= 1 . -2
GP /Rp (1v )

The specific conductivity is related to the conductance by the relation

2
o = Gt/A = Gt/1r(d/2) (IV-3)
where t = sample thickness in cm
A = electrode area in cm2

d = electrode diameter in cm.

31

32
Taking the natural 10garithm of both sides of equation (IV-l)

yields
lno =8 lnoo - U/kT
or in terms of common logarithms
log a = log 00 - U/ 2. 3 kT. (IV-4)

Plotting log 0 vs l/ T will give a straight line with.lepe-U/2. 3k and
intercept 10g 00 . In practice log a is plotted against 1000/ T for
convenience.
Z3 , , . .
The method of least squares is applied to obtain the numerical
values of U and Go from the straight-line portions of the data. Equation

(IV-4) has the form
y = a + bx
for which
ba(§§7-§§')/(SP -§7) and azy-b; (IV-5)

where the barred quantities in equations.” (IV-5) are arithmetic means.

The probable errors are calculated from the relations

.. 2
Pb = rej(n/D) and pa .. max/(2xi /D)

33
. - ‘, a -
where re - 0.6745 J{2di/(n 2)]
D = n2x.2 - (2):)“
1 i

d = yi - (a + bxi).

The experimental data for four single-crystal samples are
presented in Tables 111 through VI. The abbreviated column headings

for the tables of data are defined as follows:

UTC = upper thermocouple reading in mv

LTC = lower thermocouple reading in mv

Temp = average temperature in 0C

103/T = reciprocal absolute temperature in 0K-1 x 103
I = dc current in “a

V = dc voltage in pv

G = conductance in millimho (1/ ohm) x 10'3

B. Sample 35C - Unoriented Single Crystal

Table 111 contains the eXperimental data for the electrical conducti-
vity of an unoriented single crystal sample of pure (31-203 over the tempera-
ture range 569 to 1215°C for two oxygen pressures. The sample thickness

was 0.156 cm and the electrode diameter was 0. 37 cm. The log of the

conductivity is then. by equation (IV-3).
log 0 = log G + 0.162.

Figure 10 is a plot of log ovs l/ T for sample 35C.

34

Table 111 — Experimental data for sample 38C.

 

UTC LTC Icnw uP/T G logo
(mv) (mv) 1°C) PK)“ 1mm
P02 8 1 atm
10.124 10 057' 1045 0.759 9.057 -l.881
9.341 9.286 978 0.799 5.000 -2.139
8.654 8.611 918 0.840 3.311 -2.318
8.219 8.181 879 0.868 2.706 -2.406
7.473 7.447 812 0.922 2.036 -2.529
6.655 6.643 737 0.990 1.494 -Z.664
5.874 5.874 663 1.068 0.9786 -Z.847
4.905 4.920 569 L188 0.4255 -3.209
5.300 5.309 608 1-135 0.6157 -3.049
6.199 6.194 694 1.034 1.190 -2.762
6.938 6.922 763 0.965 1.672 -2.615
9.732 9.669 1011 0.779 6.634 -2.016
10.469 10.393 - 1074 0.742 11.93 -1.761
8.302 8.264 887 0.862 2.818 -2.388
10.279 10.176 1057 0.752 9.869 -1.844
11.028 10.906 n19 0.718 17.96 ‘ -1.584
_ -s
1802 .. 10 atnn

12.182 12. 050 1215 0. 672 40. 20 -l. 234

Table III - Continued

35

 

UTC LTC Temp 103/1“ G log 0

(mv) (mv) 1°C L 1°10" (mm

10.214 10.127 1052 0.755 8.880 -1. 890
9.038 8.972 951 0.817 3.609 -2.281
7.720 7.679 834 0.903 1.790 -2.585
6.458 6.431 718 1.009 0.9375 -2.866
5.445 5.428 621 1.119 0.4196 -3.215
4.990 4.974 576 1.178 0.2650 -3.415
5.900 5.879 665 1.066 0.6221 ~3.044
7.010 6.978 769 0.960 1.277 -2.732
8. 030 7. 985 862 0.881 2.064 -2. 523
8.662 8.604 918 0.840 2.856 -2.382
9.599 9. 520 999 0.786 5.394 —2.106
10. 493 10. 396 1075 0. 742 11.18 -1. 790
7.392 7.356 804 0.929 1.606 o2.632

POz = latm

7. 398 7. 361 805 0.928 1.624 -2.627
8. 586 8. 522 911 0.845 2. 835 -2. 385
9. 500 9.416 990 0.792 5.130 -2.128
7.336 7.298 799 0.933 1.603 -Z.633
10.113 10. 015 1043 0.760 8.396 -1. 914

11. 885 11. 745 1190 0. 684 33. 87 -1. 308
8.761 8.688 926 0.834 3.228 -2.329

LTC

36

4 Table III - Continued

 

UTC Temp lOa/T G log 0
(mm) (mv) 1°C) 1°10“ (mM)

12.092 .950 1207 0.676 39.17 -1.245
8.745 .675 925 0.835 3.175 -2.336
7.263 .225 792 0.939 1.530 -2.653
6.n4 .094 685 1.044 0.8225 -2.923

P02 10 atm

12.191 .060 1216 0.672 40.92 -1.226
9.424 .353 984 0.796 4.733 -Z.163
8.363 .310 891 0.859 2.380 -Z.461
7.357 .315 801 0.931 1.446 -Z.678
6.208 .185 694 1.034 0.7342 -2.972
5.029 .015 580 1 172 0.2650 -3.415
5.594 .574 635 1.101 0.4452 -3.189
6.977 .943 766 0.962 L187 -2.764

37

 

 

 

 

 

1000 800 600

-1. 5 r I r ' f U f

__ 0 Temperature (0C)

0
lst 2nd
'- o 0 A -- P = latm
02
.20 _- 0 E1 0 --P =10-aatm
0
Z
>- a
‘0
log 0 .. 0
(mho/era),

1-

-2. 5 -
O
-3. 0 L
a o

L- € 0

L-

b D
- h -1

3° 5 1000/T (OK)
I l 1 l J A
0.70 0.80 0.90 1.00 1.10 1. 20

Fig. 10 - Electrical conductivity of sample 35C

’ 38

C. Sample 25C - Unoriented Single Crystal

 

Table IV contains experimental data for the electrical con-
ductivity of a second single-crystal sample of Cr203 cut from the
same boule and having the same unspecified orientation as sample
3SC. These data cover a temperature range of 686 to 14160C for
two oxygen pressures. The sample thickness was 0. 155 cm and

the electrode diameter was 0.37 cm. The log of the conductivity

is given by
log 0: log G + 0.158.

Figure 11 is a plot of log avs l/T for sample ZSC.

The data in Table IV which are marked with an asterisk were
used in a least-squares analysis to obtain the values of E and Go for
equation (IV-1) in the high-temperature, straight-line region of the con-

ductivity curve. The results for sample ZSC was
a = (1.48 i 0.10)x10‘expr-(1.63 +_ 0.01 ev)/kT:l

in the temperature range 1115 to ”16°C. Figure 12 is a plot of the

experimental points on the least-squares line.

39

Table IV - Experimental data for sample ZSC.

 

UTC LTC Temp 10°/'r 1 v G 16g 0
1m) 1m) 1°C) 1°10" 111a) 111v) 1li
POZ = latm
14. 549 14.500 1416 0. 592 50. 0 432 141. 0 -0. 693*
13.974 13.930 1368 0. 609 40. 0 454 102. 0 -0. 833*
13.473 13.431 1326 0. 625 30. 0 442 75. 8 -0. 962*
13.043 13.016 1291 0. 639 20. 0 380 57. 2 -1. 085*
12. 354 12.323 1234 0. 664 20. 0 581 36. 3 -1. 282*
11.657 11.634 1176 0. 690 20; 0 938 22. 0 -1. 500*
10. 923 10. 909 1115 0. 720 10. 0 816 12.5 -1. 745*
10.141 10.135 1049 0.756 ---- ~-- 6. 735 -2. 014
8.959 8.970 947 0.820 ---- --- 2.892 -2.381
8. 080 8.102 869 0. 876 ---- --- 1. 850 -2. 575
7. 276 7. 312 797 0. 935 ---- --- 1. 266 -2. 740
6. 083 6.141 686 1.043 ---- an 0. 5770 -3. 081
6. 960 7. 011 768 0. 961 ---- --- 1. 065 -2. 815
7. 914 7. 952 855 0.887 ---- --- 1.718 -2. 607
8.795 8.822 933 0.829 ---- --- 2.634 -2.421
9.364 9.379 983 0 796 ---- --- 3.750 -Z.268
9.812 9.815 1021 0. 773 ---- --- 5.200 -Z.126
10.360 10. 350 1067 0. 746 ---- --- 7. 992 -1. 939
11.053 11.036 1126 ,0. 715 ---- --- 13.79 -1. 702*

Table IV ~ Continued

40

 

UTC LTC Tgmp 183/3 1 v G log 0

(mv) (mv) ( C) ( K) (pa) (uv) (li
11.900 11.873 1196 0. 681 20. 0 796 26.1 -1. 425*
12.610 12.572 1255 0. 654 20. 0 497 42. 8 -1. 211*
13.584 13.537 1335 0. 622 30. 0 419 80. 5 -0. 936*
11.492 11.467 1162 0. 697 10. 0 533 19. 3 -1. 557*
8.652 8.670 920 0.838 --—- --- 2.355 -2.470

0 1:: 10"3 atm
2

13.624 13.555 1338 0.621 30. 0 422 79. 8 -0. 940
12.636 12.581 1256 0.654 20. 0 499 42. 7 -1. 212
11.579 11.536 1169 0. 693 15.0 773 20. 0 -1. 541
10.666 10.634 1092 0.733 ---- --- 9.720 -1. 854
9.728 9.706 1013 0.778 ---- --- 4.628 -2.177
8.706 8.696 924 0.835 ---- --- 2.417 -2.459
7.749 7.758 839 0.899 ---- 1.583 -2.643

41

 

 

 

1400 1200 1000 800
G\ Temperature (0C)

-1. 0 -

- G ‘ ' ’ ’ P = 1 atm

0
2 -2
- El -- - - P = 10 atm
0
5X9 2
-1. 5 1 19
El
log a ’
0
(mhb/cm) - \
l3
Q

l- B\

P O
-2. 5 . O

1-

O
-1
' 1000/'1‘ (°1<) \Q
0.60 0.70 0.80 0. 90

Fig. 11 - Electrical conductivity of sample ZSC

42

0mm wagon as so: mougvntuedoau .. NA .wmh

mbd

cod

 

   
  

350m mouddvmtuumfl -113

350d dducogmuomxo 1.. t 0

d 4 -

1326a H \ooS

 

o .7

v.7.

N .7
abo\onev
b mod

o4:

m .o:

43
D. Sample 4SC(a) - Oriented Single Crystal
Table V contains experimental data for the electrical conducti-
vity of a single-crystal sample of Cr203 oriented so that the current
passes through it in a direction perpendicular to the c-axis. These data
cover a temperature range of 887 to 1392°C for two oxygen pressures.

The sample thickness was 0. 269 cm and the electrode diameter was

0. 35 cm. The log of the conductivity is given by
log a = log G + 0.447.

Figure 13 is a plot of log ovs l/ T for sample 4SC(a).
The data in Table V which are marked with an asterisk were
used in a least-squares analysis for the high-temperature. straight-line.

region of the conductivity curve. The result for sample 4SC(a) was
a = (1.96 .t 0.16) x 10‘ exp [-(1. 68 i 0. 01 ev)/ kT]

in the temperature range 1278 to 13920C. Figure 14 is a plot of the

experimental points on the least-squares line.

44

' Table V - Experimental data for sample 4SC(a).

 

UTC LTC Tgmp 133/3 1 v G log 0
(mv) (mv) ( C) 1 K) (11a) 111v) (li
P02 = latm

14.202 14.002 1380 .605 30.0 588 55.4 -0.809
13.987 13.794 1363 .611 30.0 661 48.8 -0.8 64
13.666 13.480 1336 .622 30. 0 790 40. 3 -0. 947
13.381 13.197 1313 .631 25. 0 779 33.8 -1. 025
13.079 12.903 1288 .641 20. 0 742 28.1 -1. 104
12. 720 12.552 . 1258 .653 15.0 690 22.5 -1. 201
12.158 12.004 1212 .673 10. 0 642 15.9 -1. 350
11.764 11.620 1180 .688 ---- --- 12.72 -1. 449 '
11.299 11.163 1141 .707 ---- --- 9.939 -1. 556
10. 880 10. 758 1107 .725 ---- --- 8.164 -1. 641
10.354 10. 242 1062 .749 ---- 6.537 -1. 738
9.172 9.089 962 .810 an --- 4.268 -1.923
8.319 8.252 887 .862 ---- --.. 2.596 -2.139
9.291 9.199 972 .803 ---- --— 3.940 -1.958
12.017 11.865 1201 .678 10. 0 715 14.3 -1. 398
12.545 12.383 1244 .659 15.0 793 19.5 -1. 264
12.957 12.787 1278 .645 20. 0 829 25.0 4.1541:
13.249 13.072 1302 .635 25.0 863 30. 3 -1. 072:)-
13.563 13.. 38?- 1328 .625 30. 0 855 37.1 -0. 984*

Table V - Continued

45

 

UTC LTC Temp 103/7“ 1 v G 10g a

1m) 1m) 1°C) 1°1<)‘1 111a) 111v) 1mM)
13.889 13.700 1355 0.614 30.0 705 45. 5 -0.895*
14. 343 14. 145 1392 O. 601 30. 0 551 59. 5 -O. 779*
13.996 13.803 1364 0.611 30. 0 667 48.4 -0. 869*
13.675 13.488 1337 0.621 30. 0 804 39.6 -0. 955*
13. 392 13.210 1314 0.630 25.0 786 33.4 -1. 029*
13. 078 12. 902 1288 0. 641 20. 0 759 27. 5 -1.114*
12. 628 12. 461 1251 0. 656 20. 0 980 21.1 —1. 230
11. 935 11. 791 1194 0. 682 ---- --- 12. 83 -1. 445

P0 = 10"° atm
2

12.579 12.417 1247 0.658 ---- --— 19.64 -1.260
12.227 12.074 1218 0.671 ---- --- 15.21 -1. 371
11. 693 11. 548 1174 0. 691 ---- --- 10. 69 -l. 524
11. 258 11.130 1138 0. 709 ---- --- 8. 053 -1. 647
10. 809 10. 695 1101 0. 728 ---- --- 6. 076 -l. 769
10.323 10.221 1060 0.750 --—- --- 4. 586 -l.892
9.397 9.316 982 0.797 ---- --- 2.905 -2.090
8.319 8.259 887 0.862 --~- --- 1.798 -2.298

46

 

 

 

1400 1200 1000
U I I I
o
- Temperature ( C)
" o
o -
-1.0 '-
0-..- P = latm
0
1. 2
cf... P = 10" atm
0
log a _ b 2
(mho/cm) \
o
o
-1.5 .-
O
O
D
’ o
O
-2.0 -
1000/T (°K)'1
-2.5 .-
l 1 J l
0.60 0.80

Fig. 13 - Electrical conductivity of sample 4SC(a)

47

“309.0 035.3 you on: moumdvmuunmod t 3 .mfm

cod «no.0 No6 cod

 

a u . d d a d a

13262 a. \ooS

   
   

350m mundavmtumdua 1111 8

350a dducoewuomxn. 1....) O

 

abo\o€bv
b m3

w.o..

48
E. Sample 4SC(c) - Oriented Sigle Crystal
Table VI contains experimental data for the electrical conducti-
vity of a single crystal sample of CrZO3 oriented so that the current
passes through it in a direction parallel to the c-axis. These data cover
a temperature range of 1087 to 1381°C for one oxygen pressure. The

sample thickness was 0.195 cm and the electrode diameter was 0. 28 cm.

The log of the conductivity is given by
log a = log G + 0.501

Figure 15 is a plot of log ovs VT for sample 4SC(c).
The data in Table VI which are marked with an asterisk were
used in a least-squares analysis for the high-temperature, straight-line

region of the conductivity curve. The result for sample 4SC(c) was
a = (1.37 :t 0.15) x 10‘ exp [-11. 59 _+_ 0.02 ev)/kT:l

in the temperature range 1247 to 1334°C. Figure 16 is a plot of the experi-

mental points on the least-squares line.

49
Table VI - Experimental data for sample 4SC(c).

 

UTC LTC Temp 103/ T 1 v G 103 a
1m) 1m) 1°C) 1°K)’1 111a) 111v) 1mM)
PO2 = latm

13.959 13.785 1361 0. 612 30. 0 593 54. 9 -0. 760
13.625 13.456 1334 0.622 30. 0 724 44. 3 -0. 853*
13.313 13.149 1308 0. 633 30. 0 872 36. 3 -0. 939*
13.060 12.903 1287 0.641 25. 0 839 31. 2 -1. 005*
12.672 12.523 1255 0.654 20. 0 844 24. 6 -1.108*
12.197 12.058 1216 0.672 15.0 840 18. 4 -1. 235
11.787 11.659 1182 0. 687 10. 0 712 14.3 -1. 342
11.299 11.179 1142 0. 707 an --- 10. 80 -1. 466
10. 640 10. 534 .1087 0.735 mm --- 7. 300 -1. 636
11.198 11.074 1133 0. 711 -—-- --- 9.792 .1. 508
11.724 11.590 1177 0. 690 ---- --- 13.29 -1. 375
12.042 11.902 1203 0. 678 10. 0 634 16.2 -1. 291
12.578 12.431 1247 0. 658 15.0 683 22.7 -1.143*
12.996 12.841 1282 0. 643 20. 0 708 29. 5 -1. 029*
13.271 13.108 1304 0.634 25.0 754 34.9 -0.956*
13.553 13.387 1328 0.625 30. 0 769 41. 5 -0. 881*
13.929 13.757 1359 0. 613 30. 0 641 50. 5 -0. 796
14.203 14.027 1381 0. 605 30. 0 543 60. 5 -0. 718
13.884 13.714 1355 0. 614 30. 0 652 49. 5 -0. 804
13.570 13.405 1329 0. 624 30. 0 781 40. 8 -0. 888

13.276 13.115 1305 0.634 30.0 914 34.6 -O.961

 

 

‘1!
[I
III-Ill

50

 

 

 

 

1400 1300 1200 1100
T V f If
Temperature (CC)
-0.8 -
-1. 0 -
log 0
(mho/cm) 2
'1. 2 P
-l.4 P
-l.6 .
o -1
1000/T ( K)
J l l _l
0.60 0.65 0.70 0.75

Fig. 15 - Electrical conductivity of sample 4SC(c)

q

51

E0? 035d» you on: mougweuummod .. 3 .mfm

he .o me .o

med 3.0

 

1 q u 1

13C H. \ooS

 

350m moundvntumdofi a..- B

350a Aflaoewuoaxo )1- 0

+ 4 d

N41

0 .H)

A50\o£Ev
b m3

 

m6)

 

 

ill'lri'l lull)

V. DIS CUSSIOhL

The experimental results clearly show that the electrical conducti-
vity of pure Cr203 exhibits intrinsic behavior above a characteristic
temperature which varies from one sample to another, and is dependent
upon oxygen pressure below that temperature. For the unoriented samples
this characteristic temperature is approximately 1100°C, and for the
oriented samples it is approximately 125006. Figures 10 and 13 show
this effect for samples 38C and 4SC(a).

The conductivity in the defect-controlled region (low‘tempera-
tures) is apparently quite complicated. The possible sources of conducti-
vity in this region are (l) the defect structure of the host crystal, (2)
impurities. (3) an intrinsic reaction and (4) ionic conductivity. The first
and fourth mechanisms mentioned will be dependent upon oxygen pressure
whereas the third will not. The contribution from impurities may also
vary with oxygen pressure if the predominant impurities can easily change
their valence state with change in oxygen pressure. Hence the pressure
dependence could be a complicated function of several mechanisms.

The contribution from the ionic conductivity will be several
orders of magnitude below the total conductivity13 at all temperatures
of interest here, and will not be considered further. Since the intrinsic
contribution will be small in the defect-controlled region there remains
only the impurities and the host defects. The effect of impurity concentra-
tion can be seen in the different temperatures at which the oriented and

unoriented samples become intrinsic. Table I shows that the predominant

52

 

 

 

 

I ! Iii! 1.! I! ll III .II' III. I lllf '1

53
impurity in the unoriented samples was aluminum. which has been shown1
to go into CrZO3 substitutionally as Al3+ ions on normal Cr3+ sites and
since it has a closed-shell configuration it should not contribute to the
conductivity. On the other hand the oriented samples contained relatively
large amounts of both aluminum and vanadium. Since vanadium forms
a V203 compound with the corundum structure it may also go into Cr203
substitutionally as a V3+ ion. The V3+ ion differs from the Al3+ ion,
however. in its ability to change valence readily. Thus the vanadium may
be the source of the higher conductivity of the oriented samples in the ’
defect-controlled region.

The oxygen-pressure dependence of the oriented and unoriented
samples would probably be different because of the variable-valence
impurity present in the former. However, it is not practical to determine
the pressure dependence using single-crystal samples. Figure 10 shows
the results of an attempt to determine the pressure dependence of the
conductivity for sample . 38C. The lack of reproducibility is due to the
very long equilibration time necessary for single crystals. A rough
calculation based on Hagel' 313 diffusivity values at 1300°C gives an
equilibrium time of the order of 104 hours. It is obvious that the samples
were not reaching equilibrium with the surrounding gas in the defect—
controlled region. The conductivity of the oriented samples was observed
to change slowly with time at constant temperature and pressure. at

, temperatures just below the intrinsic region. This effect can be seen

in Figures 13 and 15. The conductivity of the unoriented samples under

54
the same circumstances showed no drift since the approach to equilibrium
had become extremely slow at 11000C.

It is obvious from the foregoing discussion that very little signi- '
ficant information can be obtained from the conductivity of single crystals
of CrZO3 in the defect-controlled region. For this reason the measure-
ments were not extended to lower temperatures. On the other hand the
conductivity in the intrinsic region does not depend on equilibration with
the atmosphere or on impurity content. but should be interpretable in
terms of one predominant mechanism.

Figures 10, 11. and 13 show the electrical conductivity of samples
3SC. ZSC. and 4SC(a) respectively over a combined temperature range of
569 to 141600 with oxygen pressure as a parameter. The 1ack of pressure
dependence in the intrinsic region was well enough established with these
samples that sample 4SC(c) was studied at only one pressure, as shown
in Figure 15. Prior to any measurements it was found necessary to
"pre-condition'l the samples for roughly two hours at about lZOOOC. As
previously mentioned. the samples did not come to equilibrium with the
surrounding gas phase. but a certain amount of change was observed for
a short time in the intrinsic region. This change is attributed to a slight
loosening of the painted platinum electrodes at their edges. After this
initial pre-conditioning period the data in the intrinsic region were quite
reproducible if the measurements were restricted to temperatures below

about 1400°c. Above 1400°c the measured conductivity was observed to

 

Illllll'llllallllt I‘llllll'lllllll’l
{.llllllvllvl’ 17.1.1“.

55
drift steadily to lower values at a rate dependent upon the temperature.
this phenomenon was found to be due to volatilization of the sample at
these temperatures with a consequent decrease in the electrode area.
Restricting the experimental measurements to temperatures below
1400°c made this effect negligible in all samples except 4SC(c). which
appeared to lose its electrode surface more readily than the others.
Since all of the samples were prepared and pre-conditioned in the same
way. the most likely explanation for this appears to be preferential
thermal etching of the surface.

Figures 12, 14, and 16 show the electrical conductivity in the
intrinsic region for samples ZSC. 4SC(a). and 4SC(c) respectively along
with a plot of the least-squares line obtained for each sample. Table
VII is a tabulation of the results of the analysis for U and 0o for these
samples.

Table VII - Activation energy and pre-exponential for samples ZSC.
4SC(a). and 4SC(c) in the intrinsic region.

 

Sample U (ev) go (mho/ cm)
4SC(a) 1. 68 i 0. 01 (l. 96 i 0.16) x 10‘
ZSC l. 63 1; 0.201 (1. 48 i 0.10) x 10‘
4SC(c) 1. 59 i 0. 02 (1. 37 _+_- 0.15) x 10‘

The deviations quoted are the probable errors calculated from the scatter
of points about the least-squares lines. The electrode problem previously

mentioned is presumably responsible for the increased scatter of points

l.{‘lslll|lllill|llllllllu l.1l.ll|llllls'll ills It'll-Is! 11‘s...
‘I I 1 Ill-Ills! [[1 in.lll lull- ...Itlltl ll

56
and consequent larger uncertainty in the activation energy for sample
4SC(c).

Figure 17 shows the intrinsic lines for all three samples.
Because the electrode geometry was known to change slightly, the rela-
tive positions of the three lines probably are not very significant. Although
the data used in the analyses were quite reproducible. the apparent
anisotropy in the activation energy is so small that very little signifi-
cance can be attached to it. It is difficult to estimate from the compli-
cated crystal structure just what effect the crystalline anisotropy would
have on a given conduction mechanism.

The interpretation of the experimental conductivity data would
be considerably easier if some information concerning mobility were
available. Unfortunately it is not experimentally possible at present to
determine the mobility for transition metal oxides at high temperatures
with any reliability. However, some limits on the mobility can be deduced
from simple arguments. From the observed variation in magnitude of
the extrinsic conductivity attributed to impurities, a lower limit for the
concentration of charge carriers is estimated to be approximately 5 x
10'.17 cm'3. This corresponds to about ten parts per million which is
roughly equivalent to the smallest impurity concentrations observed.

At the other extreme the carrier concentration is certainly no greater
than 5 x 1022 cm‘3. which is approximately the number of cations in the

lattice. From equation (III-6) and the experimental results for the

57

303 one .303 .03 nonsense do sessososoo 025.35 - S .mE

 

  

mod wed «sod No.0 cod
4 W s q 4 q d u u
o as a \
H. oood
A... O . V41
0
. NA)
.an\o£EV
b m3
0 . 9.7
350a mousswmtumdou 1110

3.09... .. 4
303 1-0
03 -10

 

58
~conductivity at 12000C, the corresponding mobility range would be 1 to
10‘ cm2/ volt-sec at that temperature. This range of low mobilities
is indicative either of conduction in a very narrow band or of charge
transport by hopping of the carriers between localized sites. In either

case, of the four intrinsic equations suggested the relevant mechanism

is probably

2
2 (:r3+ .. Cr + + Cr4+. 1111-4)

The distinction between a very narrow :1 band and hOpping transport cannot
be made without further experimental evidence. if it can be made at all.
The possibility exists of a continuous transition from very narrow bands
to isolated states with a small change of lattice spacing. It is not at all
clear how this situation could be studied experimentally.

The intrinsic mechanisms described in equations (III-2) and
(III-3) cannot be ruled out completely, but they seem less likely since
the conductivity would be dominated by the band component which would
have a high mobility. Morin5 calculated a mobility of the order of
100 cm2/ volt-sec for N10 on the basis of conductivity by 2p holes in.the
valence band. This process would completely dominate any contribution
from electrons in the 3d levels. A similar argument holds for electrons
in the 4s band and holes in the 3d levels. The possibility of band-to-
band transitions as described by equation (III-l) can probably be ruled
out by the large energies observed optically for the absorption edges

of the metal oxides.

VI. CONCLUSIONS

Measurements of the electrical conductivity of single crystal
CrZO3 as a function of temperature and oxygen partial pressure have
clearly shown that there is a dependence of the conductivity on oxygen
pressure below a characteristic temperature which appears to depend
upon the impurity content of the material. The functional dependence
of the conductivity on oxygen pressure probably depends upon the type
and concentration of impurity as well as the properties of the host
crystal, but this cannot be determined from measurements on single
crystals owing to the excessive equilibrium time involved.

The conductivity measurements have also shown that there is
an "intrinsic” type of charge transport at high temperatures which is
independent of oxygen pressure. Because of the low mobility deduced
from the measurements. the usual band theory of semiconductors does
not seem as appropriate to use for interpretation as does the theory of
small polarons. The experimental results are best interpreted in terms
of charge transport on the cation lattice, which is just the model for
which the theory of small polarons was devised. Although the evidence
is not conclusive in itself, it is strongly in favor of the interpretation

suggested. Some means of separating the charge concentration and the

mobility is needed to complete the argument.

59

10.

11.

12.

13.

Bibilography

E.J.W. Verwey, SemiconductiflMaterials, p. 151, Butterworths.

 

" London (1951).

R. R. Heikes and W. D. Johnston. "Mechanism of Conduction in
Li-Substituted Transition Metal Oxides. " J. Chem. Phys. 26. 582
(1957).

S. van Houten. "Semiconduction in LixNi(1_x)0. ” J. Phys. Chem.
Solids EL. 7 (1960).

F. J. Morin. "Electrical Properties of 01 Fe 0 anda Fe 0 Containing
Titanium, " Phys. Rev. _8_3_. 1005 (1951); "Electrical PrOpert es of
aFe 0 . " Phys. Rev. _93, 1195 (1954); "Electrical Properties of N10, "

Physz. ev. 33. 1199 (1954).

F. J. Morin. "Oxides of the 3d Transition Metals, " Bell System
Tech. J. _3_7_, 1047 (1958).

F. J. Morin. "Oxides Which Show a Metal-to-Insulator Transition
at the Néel Temperature. " Phys. Rev. Letters _3. 34 (1959).

T. D. Schultz. "Slow Electrons in Polar Crystals: Self-Energy,
Mass, and Mobility. " Phys. Rev. 116 . 526 (1959).

J. Yamashita and T. Kurosawa, )‘On Electronic Current in N10, "
J. Phys. Chem. Solids _5_, 34 (1958).

G. L. Sewell, "Electrons in Polar Crystals." Phil. Mag. 36. 1361
(1958).

T. Holstein. "Studies of Polaron Motion Part I. The Molecular-
Crystal Model. " Ann. Phys. _8__, 325 (1959); "Studies of Polaron Motion
Part II. The 'Small' Polaron. “ Ann. Phys. _8_, 343 (1959).

K. Hauffe and J. Block. "Fehlordnungsmodell eines Eigensto'rstellen-
Halbleiters am Beispiel des Chromoxyds. " Z. Physik. Chem. 198 .
232 (19510. j. .. "'-

w. A Fischer and G. Lorenz, "Die Sauerstoffdruckabhfingigkeit
der elecktrischen Eigenschaften des Chrom(m)oxydes. " Z. Physik.
Chem. NF. 11. 308 (1958).

W. C. Hagel and A. U. Seybolt. "Cation Diffusion in CrZO . "

J. Electrochem. Soc. 108, 1146 (1961). 3

60

1 .111) i. 1 11
1 II I 1111‘ 11' Ill. 1

14.

15.

16.

17.

18.

19.

20.

21.

22.

23.

61

Linde Co. . Crystal Products Dept. . 4120 Kennedy Ave. . East Chicago.
Indiana.

Engelhard Ind. . Hanovia Liquid Gold Div. . 113 Astor St. . Newark
2. N. J.

B. E. Walker 3511. ”Thermoelectric Instability of Some Noble Metal
Thermocouples at High Temperatures, " Rev. Sci. Inst. 33. 1029
(1962).

J. Graham. "Lattice Spacings and Colour in the System Alumina-
Chromic Oxide. ” J. Phys. Chem. Solids _11. 18 (1960).

R.W.G. Wyckoff. Crystal Structures. Vol. I. Interscience Publishers.
New York and London. (1948).

 

B. N. Brockhouse. "Antiferromagnetic Structure in Cr203. " J. Chem.
Playmfl. 961(1953).

F. A. Krfiger and H. J. Vink. "Relations between the Concentrations
of Imperfections in Crystalline Solids. " Solid State Physics Vol. 3.
307. Academic Press. New York.(1956).

L. C. Walters and R. E. Grace. ”Self-Diffusion of Chromium in
Single Crystals of Cr 03. ” J. Electrochem. Soc. 110. 192C.
Abstract No. 194 (1963).

C. Kittel. Introduction to Solid State Physics. p. 352. 2nd Edition.
Wiley. New York. (1956).

H. Margenau and G. M. Murphy. The Mathematics of Physics and
Chemistry. p. 519. Van Nostrand. New York. (1956).

 

lllll‘.‘