Oxide Thermoelectrics The Role of Crystal Structur
Oxide Thermoelectrics The Role of Crystal Structur
Oxide Thermoelectrics The Role of Crystal Structur
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A dissertation presented
by
to
Doctor of Philosophy
in the subject of
Applied Physics
Harvard University
Cambridge, Massachusetts
May 2012
© 2012 – Taylor David Sparks
Correlated Spinels
Abstract
and electrical and thermal transport properties of a variety of strongly correlated spinels.
General structure property relationships for electrical and thermal transport are discussed.
However, the relationship between thermopower and features of the crystal structure such
as spin, crystal field, anti-site disorder, and structural distortions are explored in depth.
The experimental findings are reported in the context of improving existing oxide
The need for improved n-type oxide thermoelectric materials has led researchers
report herein that the LiMn2O4 compound reaches the relatively large n-type
thermopower of -73 μV/K which is three times larger than the value observed in other
iii
manganese oxides, -25 μV/K. The cause of this increase in thermopower is shown to be
the absence of a Jahn-Teller distortion on the Mn3+ ions in LiMn2O4. By avoiding this
structural distortion the orbital degeneracy is doubled and the Koshibae et al.‘s modified
Heikes formula predicts a thermopower of -79 μV/K in good agreement with the
experiment. Altering the Mn3+/4+ ratio via aliovalent doping did not affect the
Kobayashi et al. The role of anti-site disorder was further examined in FexMn1-xNiCrO4
x=0, ½, ¾, 1 spinels but the effect on thermopower was inconclusive due to the presence
of impurity phases.
a means whereby the Wu and Mason‘s 30 year old model for using thermopower to
calculate cation distribution in spinels could be revisited. We report evidence that Wu and
Mason‘s original model using the standard Heikes formula and considering octahedral
models are evaluated considering Koshibae et al.‘s modified Heikes formula and
accounting for tetrahedral site contributions. Furthermore, the effect of a possible spin
iv
Table of Contents
2.4 MODIFIED HEIKES FORMULA AS A SCREENING TOOL FOR NEW MATERIALS ................................... 39
v
3.1.4 Current Assisted Pressure Activated Densification ............................................................... 53
SPINELS ...................................................................................................................................................... 81
5.2.1 Cation Distribution from Thermopower Measurements: Inversion Only ............................ 136
vi
5.2.2 Cation Distribution from Thermopower Measurements: Inversion and Spin Unpairing .... 144
5.4.1 Magnetic Structure of Samples Quenched From High Temperatures ................................. 149
vii
List of Figures
Figure 1.1 Diagrams for thermoelectric devices in power generation and refrigeration
mode ............................................................................................................................ 4
Figure 1.2 Band diagrams for a thermoelectric device before and after the metal and
semiconducting regions are joined. ............................................................................. 5
Figure 1.3 Fermi-Dirac distributions corresponding to hot and cold sides of a metal ........ 7
Figure 1.4 Band diagrams for a thermoelectric power generation device in a thermal
gradient before and after steady-state equilibrium is reached..................................... 9
Figure 1.6 Thermoelectric power generation efficiency plotted against , heat source
temperature, for different figure of merit values....................................................... 18
Figure 1.7 Typical trends in thermoelectric properties and figure of merit as a function of
carrier concentration for different classes of materials. ............................................ 20
Figure 2.2 Thermopower calculated from Koshibae et al.‘s modified Heikes formula
plotted against carrier concentration. ........................................................................ 36
Figure 2.4 Electronic degeneracy of Co3+ and Co4+ ions in an octahedral crystal field with
low, intermediate and high spin states. ..................................................................... 38
viii
Figure 2.5 The spin blockade phenomenon. ..................................................................... 40
Figure 2.6 The crystal fields for a variety of different coordinations. .............................. 41
Figure 2.7 Low and high spin states for d3 and d4 ion combinations in a
tetrahedral/cubic/dodecahedral crystal field. ............................................................ 43
Figure 2.8 Crystal field for f shell orbitals in an octahedral crystal field. ........................ 44
Figure 3.3 Illustration of X-ray diffraction from lattice planes in an ordered material. ... 58
Figure 3.4 Dispersion factors, and , for the element iron using Cr (left) and Cu
(right) radiation sources....................................................................................... 62
Figure 3.6 Photograph of a neutron diffractometer with a cryostat (D1B at Institut Laue
Langevin). ................................................................................................................. 65
Figure 3.8 Example of a stick diagram to match measured peaks to candidate structures
on file. ....................................................................................................................... 67
Figure 3.9 Rayleigh and Raman (Stokes and anti-Stokes) scattering diagram. ................ 72
ix
Figure 4.2 Jahn-Teller distortion examples for d4 cations under axial compression and
elongation along the z-axis. ...................................................................................... 85
Figure 4.3 Undistorted and distorted LiMn2O4 crystal structures with A and B cation sites
labeled. ...................................................................................................................... 86
Figure 4.4 Spin and orbital degeneracy values for d3 and d4 cations................................ 88
Figure 4.5 Thermopower for a variety of mixed ion manganates compared to calculated
high temperature thermopower limits corresponding to distorted and undistorted
Mn3+ ions................................................................................................................... 89
Figure 4.6 X-Ray diffraction pattern and Rietveld refinement results for LiMn2O4 at 298
K with an inset showing the (440) reflection at different temperatures.Figure 2.6 .. 91
Figure 4.7 Fraction of the A (tetrahedral) and B (octahedral) sites in the LiMn2O4 spinel
occupied by Li and Mn ions plotted against temperature. ........................................ 94
Figure 4.9 Thermopower versus temperature for LiMn2O4 from this work compared with
other Mn compounds. ............................................................................................... 99
Figure 4.10 Mean oxidation state of manganese ions and Li/Mn ratio for different Li-Mn-
O compounds. ......................................................................................................... 103
Figure 4.11 X-Ray diffraction Mg1-xLixMn2O4 x=0, 1/5, 2/5 samples. .......................... 104
Figure 4.12 Thermopower and electrical conductivity of Mg1-xLixMn2O4 x=0, 1/5, 2/5
compounds. ............................................................................................................. 106
Figure 4.13 In-situ high temperature X-Ray diffraction pattern, measured in argon, of
LiMn2O4 after CADPro. .......................................................................................... 107
x
Figure 4.14 Electrical conductivity and thermopower of LiMn2O4 pressed in a reducing
atmosphere. ............................................................................................................. 109
Figure 5.1 X-Ray diffraction for mixed phase Co3O4 and CoO samples obtained by
intermediate cooling rates, pure Co3O4 obtained by slow cooling rates and pure CoO
obtained by rapid quenching. .................................................................................. 123
Figure 5.3 X-Ray diffraction of Co3O4 from room temperature up to 1273 K. .............. 126
Figure 5.4 Temperature dependence of the lattice parameter of Co3O4 and CoO
determined from in-situ X-ray diffraction. ............................................................. 127
Figure 5.5 Raman shift plotted as a function of temperature for Co3O4 samples.Figure 2.6
................................................................................................................................. 129
xi
Figure 5.6 Arrhenius plot of electrical conductivity for Co3O4 showing three
characteristic regions each with a different activation energy. ............................... 131
Figure 5.8 Fractional occupancy of Co3+ ions on the octahedral sublattice in Co3O4 as a
function of temperature calculated from the thermopower measurements of Co3O4 in
air according to different models described in detail in the text. ............................ 138
Figure 5.9 Thermopower plotted against fractional occupancy of Co3+ ions on the
octahedral sub-lattice according to examine stoichiometric consistency. .............. 140
Figure 5.10 Fractional occupancy of Co3+ ions on the octahedral sublattice in Co3O4 as a
function of temperature accounting for different cobalt ion spin states. ................ 145
Figure 5.11 Fractional occupancy of Co3+ ions on the octahedral sub-lattice in Co3O4 as a
function of temperature calculated from ionic radii and tetrahedral and octahedral
bond lengths. ........................................................................................................... 148
Figure 5.12 Neutron diffraction patterns for Co3O4 powder quenched from 1173 K
measured at 300 K and 1.5 K. ................................................................................. 151
Figure 5.14 Neutron diffraction patterns measured at 1.5 K for Co3O4 samples quenched
from various temperatures. ..................................................................................... 153
Figure 5.15 Ratio of peak intensity for magnetic (200) and nuclear reflections (220) and
(224).
xii
Acknowledgements
I feel so incredibly grateful to the many people and institutions who made this
PhD possible. It‘s cliché, but I truly couldn‘t have done this alone. To start, I owe my
wife, Jodi, an enormous debt of gratitude. When I told her I had a dream to do a PhD and
that meant moving out of state she supported me completely and without complaint. Jodi
earned this every bit as much as I did. She went to bed alone while I worked on
homework, did the lion's share of the work around the house, lived in Shanghai for a
summer (!), cared for our son, Atticus, and even moved clear across the country. In doing
so she sacrificed her own education goals and aspirations to make mine a reality, and I‘ll
I‘m indebted to my parents for raising me with love, discipline, hard work and
humor. It was the example of my dad who, though a high school dropout originally, went
on to change his life, outlive a ―terminal‖ brain tumor for 25 years and graduate college
as a 40 year old in engineering that made me truly believe it when he told me I could do
anything I wanted in life. It was the daily efforts of my mom who raised us in happy,
Though Santa Barbara is as perfect a place as you could imagine to live and to do
materials research, the reason I selected UCSB for graduate school was to work with my
xiii
fantastic advisor, David Clarke. I knew right from the beginning that he was somebody
who I wanted to be like and who would help me achieve great things. I honestly didn‘t
even know what state Harvard University was located in, but when he told me he was
moving there, I knew I‘d come along. I can‘t overstate the significance of his mentorship
confidence in me time and again to come up with my own research ideas and carry them
out. I hope he knows how grateful I am for all this and for his patience in helping me
write (and rewrite) papers and for putting up with the occasional explosion in the
laboratory.
Some of the most gratifying experiences of my PhD were realized while working
with diverse individuals from all over the world. I express thanks to my coworkers
Maged Bekheet and many, many others with whom I carried out experiments. The Clarke
group, too, over the years has been a great source of friendship, learning and I
acknowledge my good friends Bill, Xin, Sam, Andi, Yang, Jiangshui, Mor, Ashwin,
Michael, Mary, Jesse, Roger, John, Jenny, Liguo, Sebastien, and Matt and I will certainly
extended international research stays. Among these are Prof. Pan Wei at Tsinghua
University, China, Prof. Gu Hui at the Shanghai Institute of Ceramics, Chinese Academy
xiv
of Science, China, Dr. Laetitia Laversenne and Dr. Aleksander Gurlo at the Institut Laue-
are many things one simply can‘t learn but by having experienced them.
I‘ve been privileged to receive training from world class experts. Professors like
Ram Seshadri, Frans Spaepan, Shriram Ramanathan, Evelyn Hu, Nicola Spaldin, Omar
Saleh and others at UCSB and Harvard provided exceptional coursework that had an
actual impact on my research. I also recognize the enormous efforts of the International
Centre for Diffraction Data (ICDD) as well as other beamline scientists and neutron and
X-ray diffraction experts who taught me Rietveld refinement. This skill has been
incredibly useful and I owe Joel Reid, Jim Kaduk, Brent Melot, Laetitia Laversenne,
Emma Suarde, John Faber, Cyrus Crowder, Matt Suchomel, Scott Speakman, Eric
Toberer, and Yan Gao my gratitude for all the help, facilities and free advice they offered
me.
Foundation that made my PhD possible. In particular the World Materials Network
xv
Glossary
Boltzmann constant kB
Work function M
Chemical potential
xvi
Fundamental charge q
Distance x
Electric potential V
Electric field
Peltier coefficient
Time t
Area (cross-sectional) A
Lorenz constant L0
xvii
Transport coefficients (l=1,2; m=1,2) Llm , M lm
Number of particles N
Transfer integral t
Superexchange interactions J
Volume of phase α V
Total spin
Activation energy EA
Gas constant R
xviii
Wavelength
Atomic coordinates u, v, w
Density of phase α D ,
Electric potential V
Thermal diffusivity
Porosity P
xix
Ionic radius (subscript indicates site; A, B, O. Superscripts indicate ion) r
Tolerance factor t*
xx
Dedication
I dedicate this dissertation to my father whose hard work and eternal optimism gave me
the self-confidence to reach for lofty goals; to my mother who sacrificed and worked,
often thanklessly, in helping me achieve my goals; and to my family who motivate me,
xxi
Chapter 1. Fundamentals of Thermoelectricity
Fossil fuels such as coal, petroleum and natural gas have been the principal source
of energy fueling mankind‘s development since the Industrial Revolution. Because fossil
fuels are the result of anaerobic decomposition of dead organisms accumulating over
millions of years there is only a finite amount of fossil fuels available. With a finite
amount of fuel the rate of discovery of new deposits decreases with time while the rate of
Because demand outpaces supply, the result is a peak in fossil fuel production, termed the
―Hubbert Peak.‖ 1 The decline in production leads inevitably to increased cost and market
instability. Furthermore, fossil fuels are most commonly combusted and have the
detriment of giving off pollutants such as carbon dioxide, sulfur dioxide and nitrogen
Due to the disadvantages of fossil fuels there is strong appeal for renewable
sources of energy such as wind, solar, hydroelectric, geothermal, tidal, wave and biofuels.
In 1949 roughly 1/3 of all the domestically produced electricity came from renewable
sources.2 Since that time there has been a general decrease in the US reliance on
renewable sources of energy and most recently, in 2010, only 10.4% of electricity in the
United States was from a renewable source.3 The two largest contributors are
hydroelectric power (6%) and wind power (2.3%).4 Because energy demands continue to
1
increase it is critical to invest in developing technology for clean, sustainable and
renewable sources of energy. The purpose of this chapter is to explain the role that
first place it is, perhaps, shocking to realize that ~58% 5 of all initial energy is given off
as waste heat. For example, in automobiles less than 20% of fuel energy is used for
propulsion because 30% is wasted as exhaust heat, 40% in radiator heat and 10% to
etc. is essential, but greater benefit would be attained if even a small fraction of waste
Thermoelectric electrical generators are devices than can harness waste heat, say,
reverse by applying an external voltage to direct a current through the material where the
Peltier effect creates a temperature gradient for solid-state refrigeration with no moving
technologically important and widely used as heat pumps and temperature controllers for
thermoelectric electrical generators are not yet as widely used but radioisotope
2
thermoelectric generators10,11 have found application in deep space probes and cardiac
pacemakers.
The schematic of a simplified device is shown in Figure 1.1 in both power generation
mode (left) and refrigeration mode (right). In both modes there is a hot (red) and cold
(blue) side with a temperature gradient across the two thermoelectric elements. The two
elements differ in carrier type; one relies on holes (purple, p-type) and the other, free
electrons (green, n-type) for electrical conduction. A metal interconnect layer (orange)
connects the two legs and makes a circuit such that the legs are connected in series
electrically but in parallel with respect to the thermal current. The voltage across a single
thermoelectric couple, also called a thermopile, is normally low whereas the required
output voltage is large so actual devices consist of many pairs of thermoelectric couples
connected in series.
thermal gradient that can be produced by waste heat to drive a current through the
thermoelectric couple and produce a potential available to drive an external load. To best
examine the device shown in Figure 1.1 and observe that there are essentially five
regions.
3
Figure 1.1. Diagrams for thermoelectric devices in power generation mode (left) and
refrigeration mode (right). In both cases heat is carried from the top side to the bottom
side. An electrical current is obtained automatically in power generation mode but must
be powered by an external voltage in refrigeration mode.
Starting from the bottom left the electron must travel through a metal to a p-type
across a device, for these five regions are shown in Figure 1.2 before (a) and after (b) the
Let us consider first the case of thermal equilibrium throughout the device. In a
metal the work function, , is the work required to move an electron from the Fermi
level to vacuum. In a semiconductor the Fermi level lies in a region of energy between
the top of the valence band and the bottom of the conduction band where no electronic
states exist called a band gap, . For simplicity in creating figures we select p and n-type
4
semiconductors with equal band gap and electron affinity and metals with the same work
Figure 1.2. Band diagrams for a thermoelectric device before (a) and after (b) the
regions are joined (no temperature gradient). The work functions, vacuum level, electron
affinities, band gaps and built in (barrier) potentials are shown.
5
The Fermi level (shown as a dashed line in Figure 1.2) for an intrinsic or undoped
semiconductor would lie in the middle of the band gap. In doped semiconductors,
however, the Fermi level is closer to the valence band for p-type and closer to the
conduction band for n-type semiconductors. The Fermi levels, , for non-degenerate
ND 1.1
E F Ei k BT ln where N D N A , ni
ni
NA 1.2
Ei E F k BT ln where N A N D , ni
ni
where and are the Fermi level and carrier concentration of the intrinsic
When the five regions in Figure 1.2 are brought into contact and equilibrium is
reached band bending at the junctions occurs such that the Fermi level is constant
throughout. Before we assess the effect of a temperature gradient in this complete device
Figure 1.3.
6
Figure 1.3. “Soft” and “hard” Fermi distributions corresponding to hot and cold sides
of a metal. Figure adapted from Föll.12
1.3
f E , T
1
E / k BT
e 1
where is the energy of the particle and is the chemical potential. At low temperatures
the Fermi level. Therefore, a temperature gradient leads to a concentration gradient and
diffusion of electrons from the hot side to the cold side occurs for two reasons. First,
electrons move to lower energy states on the cold side and in doing so remove the
7
concentration gradient. Second, the electrons on the hot side have a larger momentum
than those on the cold side and therefore diffuse faster towards the cold side than cold
regions also results in a concentration gradient. With increasing temperature the Fermi
level moves closer to the band edges (in agreement with Equations 1.1 and 1.2) because
carriers are thermally activated. This results in further distortions of the bands (see Figure
1.4) and diffusion of electrons and holes across the semiconductor from the hot side to
As higher energy electrons and holes diffuse towards the cold side they leave
behind their charged donors and acceptors respectively. This displacement of charged
carriers leads to the development of an electric field across the semiconductor causing
of current driven by the electric field and diffusion current driven by the concentration
gradient. The governing equations are derived from the Drift-Diffusion model.13 The
dV dn 1.4
j n qnvn qDn
dx dx
dV dp 1.5
j p qpv p qD p
dx dx
8
Figure 1.4. Band diagrams for a thermoelectric power generation device in a thermal
gradient before (a) and after (b) steady-state equilibrium is reached. Holes and electrons
in the valence and conduction bands are shown in white and black respectively. Negative
charges (ionized acceptors, displaced electrons) and positive charges (ionized donors,
displaced holes) are shown in blue and red respectively.
where is the fundamental charge, and are the electron and hole carrier
concentrations, and are mobilities of electrons and holes, and are diffusion
9
coefficients for electrons and holes respectively. The first term of Equations 1.4 and 1.5
represent the drift term, or how a charged particle moves due to an applied electric field,
and the second term represents diffusion or how a particle moves due to a concentration
gradient.
properties do not change with position then we can rewrite Equations 1.4 and 1.5 using
the temperature gradient instead of the concentration gradient and replacing the diffusion
dV dT 1.6
j n qn qS n
dx dx
dV dT 1.7
j p qp qS p
dx dx .
The Seebeck coefficient is the electric field, , or voltage differential across a temperature
gradient
V 1.8
S
T T .
The sign of the Seebeck coefficient is defined as positive if the electric field is in the
same direction as the temperature gradient such that the higher temperature end of a
10
Figure 1.4 clearly shows that as an electron travels from left to right through the
device heat must be absorbed from the middle metal region (hot side) and released at the
outer metal regions (cold side). This is consistent with the energy band diagram. For
example, an electron travels from the hot metal into the conduction band of the n-type
from the hot side to the cold side of the semiconductor they carry heat with them and
when they leave the conduction band to re-enter the metal they go from high energy to
in refrigeration mode. These so called Peltier coolers rely on the Peltier effect defined as
the heat loss per unit time, as current, , travels through a junction between two
materials A and B, or
1.9
A B I
dQ
dt
where and are the Peltier coefficients of the two materials. The Peltier coefficient
ST . 1.10
11
The band diagram for a thermoelectric cooler, shown in Figure 1.5, is instructive
to see how the same device used for power generation can also be used for active cooling
when an external potential is applied across the device. From left to right, an electron
must release heat at the metal/p-type junction, absorb heat at the p-type/metal and
metal/n-type junctions, and finally release heat at the n-type/metal junction. The result is
that the outer metal regions act as heat sinks and active cooling is achieved in the central
metal region. This heat transfer makes sense physically because the p-type material
valance band is at a lower energy level than the metal and the n-type material conduction
Figure 1.5. Band diagram for a thermoelectric cooler. An external potential across the
device directs a current through the material where carriers absorb heat from the central
metal region and release it on the outer regions to achieve active, solid-state cooling.
Figure redrawn from Tamer et al.14
12
1.3 Thermal and Electrical Transport
Figure 1.1 with two thermoelectric elements (n and p-type) of length and cross-
14–17
sectional area . Following the treatment by others, the heat flow rate across the
respective elements is
dT 1.11
Qn S n IT n A
dx
dT 1.12
Q p S p IT p A
dx
where and are the thermal conductivities of the n and p-type materials. The first
term represents the heat transported by the electrical carriers themselves and the second
term represents the heat transported due to the temperature gradient. In the refrigeration
mode, contrary to power generation mode, the heat flow from electrical carriers is
opposite that of thermal conduction, hence the opposite sign. Additionally, due to Joule
heating there is actually heat being generated within the elements themselves at a rate per
d 2T I 2 n 1.13
n A
dx 2 A
13
d 2T I p 1.14
2
p A
dx 2 A
where and are the electrical resistivities of the two elements. Applying the boundary
respectively, solving the differential equations and substituting back into the original heat
p ATH TC I 2 pL 1.16
Q p S p ITC
L 2A .
The total rate of heat transferred across the couple (the cooling power at the heat
sink) is the combination of the heat transferred across n and p-type elements. Summing
p A p I 2 L 1.17
Q SITC TH TC
n n
L 2A
where . Recall that the legs are connected in parallel thermally but in series
electrically. Therefore the thermal resistance, , and the electrical resistance, , can be
written as
14
n A p A 1.18
L L
n L pL 1.19
A A .
I 2 1.20
Q SITC TH TC
2 .
We can derive the power, , consumed in each leg in much the same way
I 2 n L 1.21
Pn S n I TH TC
A
I 2 pL 1.22
Pp S p I TH TC
A
and the total power for the thermoelectric couple is the sum
P SI TH TC I 2 1.23
.
15
between heat and electricity. A high cost, low efficiency thermoelectric device may not
Stirling engines which tend to be more efficient though bulkier and featuring moving
parts.20 The total device efficiency for a cooler, often called the coefficient of
performance, is simply the heat transported divided by the power supplied (Equations
I
2 1.24
SITC TH TC
Q
2
P SI TH TC I 2 .
Heikes16 showed that by taking the derivative of Equation 1.24 with respect to current
determined;
0
I 2 S TC TH 4 I TC TH 2S TC TH
2
1.25
I 2 I 2 I S TC TH
2
16
TH 1.27
1 ZTm
Q TC TC
P TH TC 1 ZTm 1
S2 1.28
ZTm Tm
S 2 1.29
ZTm Tm
e L
where σ is the electrical conductivity and and are the electronic and lattice
power generation by dividing the power extracted by the total heat energy absorbed to
obtain
P TH TC 1 ZTm 1 1.30
Q TH T
1 ZTm C .
TH
17
limit such that as ZT approaches ∞ Equation 1.27 and 1.30 reduce to the Carnot
efficiency for a heat pump and power generator respectively. An illustration of this can
be seen in Figure 1.6 where the efficiency is plotted against , the heat source
temperature, for different ZT values as well as other energy sources such as geothermal,
solar, coal and nuclear using different thermodynamic engine cycles (Rankine, Stirling,
Brayton etc.).21,22 However, despite this potential for high efficiency the best ZT values,
Figure 1.6. Efficiency plotted against , heat source temperature, for different figure of
merit values. For comparison the efficiency of other energy sources and engine cycles
are included. Figure redrawn from Vining.20
18
Achieving increases in ZT are notoriously difficult as evidenced by the
remarkably slow improvement over the last 60 years since Goldsmid et al. initially
demonstrated thermoelectric refrigeration using Bi2Te3;23 an event that began the modern
era of thermoelectric materials research. The majority of thermoelectric devices still use
The reason for such low ZT values is because the three materials properties
involved, electrical conductivity, thermal conductivity and Seebeck coefficient are not
entirely independent from one another. This interdependence can be visualized in Figure
1.7 where representative curves for the properties are depicted against carrier
concentration from insulating to metallic. If, for instance, the carrier concentration is
increased to enhance electrical conductivity this will reduce the Seebeck coefficient and
increase the electrical contribution to thermal conductivity such that there may be no
overall improvement in ZT. Typically, the highest figure of merit values are not obtained
For metals, the electrical and thermal conductivity are related through the
Wiedemann-Franz law
2 kB
2 1.31
L0 e
T 3 q
19
Figure 1.7. Typical trends in thermoelectric properties and figure of merit for different
classes of materials. The abscissa roughly corresponds to a carrier concentration range
from 1018-1021 cm-3. Figure redrawn from Snyder et al.25
where is the Lorenz constant equal to 2.44×10-8 W Ω K-2. Using this relation we can
1.32
S 2
1
ZT
L0 1 L .
e
With the electrical conductivity essentially dictating the electronic contribution to thermal
conductivity this leaves only the possibility of reducing the lattice contribution or
20
In an effort to maintain high electrical conductivity but reduce the lattice thermal
conductivity, or heat transferred via phonons, Slack proposed the ―Phonon Glass and
Electron Crystal‖ (PGEC) concecpt.26 Since the mid 90‘s virtually all the major progress
techniques for reducing thermal conductivity are extensive and the detailed methods are
beyond the scope of this dissertation. A few examples include using ―rattling‖ structures
such as clathrates27,28 or skutterudites, 29–32 structures with point defects, 33–38 anisotropic
applications, efficiency and potential we can now focus on a more detailed look at
21
Chapter 2. Thermopower in Oxides
thermal barrier coatings, optical devices, biomaterials and many more. Nevertheless,
oxides were initially overlooked in the search for high ZT materials because, typically
insulators, they exhibit very low electrical conductivity and most also have a low average
atomic mass relative to many traditional thermoelectric materials and therefore a higher
atomic vibration frequency and thermal conductivity. In this chapter the potential
advantages for oxide thermoelectric materials will be highlighted and a case will be made
for their use in place of traditional semiconductor materials based on the large Seebeck
coefficient values that can be obtained. A survey of the relevant literature will be
presented and the fundamentals governing thermopower in band conducting and small-
production is PbTe at 2-4 tonnes annually. The reliance on these materials is troublesome
partly because of their toxicity (Sb, Se, Pb), which will limit some applications, but more
so because of the scarcity and cost of tellurium which at 1 ppb in the earth‘s crust is even
22
rarer than platinum (37 ppb). The USGS classifies tellurium as one of the nine rarest
metals70 and the DOE lists it as a critically scarce energy material.71 Tellurium is not
scarcity and increasing use in technology, such as CdTe solar panels, the cost of tellurium
has risen dramatically from 3.86$/lb in 2000 to over 150$/lb now.72 Many traditional
benefit from being non-toxic, relying on more abundant transition metals, having very
good high temperature stability and corrosion resistance and facile synthesis.
and clathrates, exhibit thermal conductivity nearing the theoretical minimum thermal
consists of displaced atoms and a wavelength shorter than the interatomic distance is not
are approaching their limit. Furthermore, the nanostructured features necessary for many
phonon scattering approaches are unstable at high temperature and prone to coarsening
23
2.2 Band Conductors
the Seebeck coefficient is characteristically linear with respect to temperature. For oxides
fitting these criteria it is possible to apply the thermopower expression for a single band
metal. To do so we rely on the Mott formula73 relating thermopower to the energy partial
differential of the electrical conductivity evaluated at the Fermi energy. This expression
can be derived from the basic electrical and thermal current transport equations involving
We notice at once that these equations are simply generalized forms of Equations 1.6 and
1.7 that were derived in the last chapter. Thermopower is the ratio of the matrix elements
L21 2.3
S
L11
where
L11 2.4
24
2 k B T 2 E 2.5
L21 T L12
3q E E EF
such that
2 k B T ln E 2.6
2
S
3 q E E EF .
The electrical conductivity of a single band metal with parabolic band approximation is
given by 74
2q 2 df E 2.7
E N E E dE
3m dE
where is the effective mass of the carrier, is the carrier scattering time, is the
Equation 2.7 in terms of the density of states and carrier concentration, , we have 75
2 k B 2T N E 2.8
S
3 q n E EF .
The inverse dependence on carrier concentration alluded to in Figure 1.7 is now clear.
25
nq n , pq p . 2.9
The carrier mobility itself depends on the effective mass of the carrier and the carrier
q n q p 2.10
n
mn , mp .
p
Examining Equations 2.8 and 2.9 and acknowledging the inherent performance
power factor ( ) for band conducting oxides must rely on one of three approaches:
increasing the density of states in the vicinity of the Fermi level, increasing the time
likely is that the nanostructuring (multiple quantum wells devices of Eu doped PbTe for
The strong ionic character of oxides resulting from the large difference in
electronegativity between cations and oxygen has meant low carrier mobility (short
scattering times, narrow bands, high effective mass) relative to traditional semiconductors
26
for degenerately doped semiconducting oxides.81 Taken together with the lower average
atomic mass (higher vibrational frequency and therefore thermal conductivity) of oxides
relative to traditional materials it is not surprising that the ZT values of even the best
materials.
conductivity (0.2 mΩ cm) and, shockingly, had the large value for thermopower of 100
μV/K. Of equal surprise was that the Seebeck coefficient didn‘t have the characteristic
27
Figure 2.1. Thermopower plotted as a function of a temperature for polycrystalline and
single crystal NaCo2O4 samples. Figure redrawn from Fujita et al.110
This large thermopower at high temperatures (~800K), taken with the metallic
oxide thermoelectrics and one that made oxides competitive with traditional
thermoelectric materials. The origin of this thermopower in a material with such a large
carrier density (~1022 cm-3) was at the time completely unknown. For example, the
16
original high temperature formula for thermopower proposed by Heikes in 1961 and
28
kB y 2.11
S ln
q 1 y
expression was unable to adequately explain both the magnitude and temperature
materials, or ones where the charge carrier interactions must be included, the physics of
Heikes formula which accounted for the often overlooked spin and orbital contributions
to entropy and therefore thermopower. The modification also considered the large
exchange of entropy between the two ions involved in conduction on sites (I) and (II) and
k B g ( I ) y 2.12
S ln
q g ( II ) 1 y .
This modified formula not only explained the large magnitude of thermopower in
NaCo2O4, but also accounted for a low-spin (LS) to high-spin (HS) transition in spin state
(mitigated by increasing thermal energy) which provided the alteration in spin and orbital
29
of Koshibae et al.‘s modified Heikes formula will be detailed in Section 2.3.1 and its
NaCo2O4 many other strongly correlated oxide systems were examined. To date,
Koshibae et al.‘s modified Heikes formula has described the thermopower of many
where only the entropy from charge is considered ignoring the fact that the carriers also
have spin and orbital degrees of freedom. Including these components is key to Koshibae
Before we begin discussing the role of spin and orbital degeneracy we must
response expressions shown in Equations 2.1 and 2.2 do not satisfy Onsager‘s reciprocity
30
relation, . Following Koshibae et al. 114 we consider an irreversible system with
a net generation of entropy and in addition to electric potential and temperature gradient
we include the chemical potential as particles move from the hot to the cold side. When
particles and energy is transferred from the hot to cold side of a thermoelectric
system the change in entropy on the cold and hot sides are given by
E T 2.13
scold N
T T
E T T q 2.14
shot N.
T T T T
On the right hand side of Equation 2.15 we see the particle and energy current, dn/dt and
d(ΔE)/dt, multiplied by coefficients allowing us to express the driving forces for particle
q q 2.16
T x
T T T T x x T
T 1 2.17
x .
T 2
x T
31
With these generalized driving forces we are now able express the dynamical
q 1 2.18
j1 M 11 M 12
T x x T x T
q 1 2.19
j2 M 12 M 22
T x x T x T
where and represent particle and energy flux densities. These expressions now
satisfy Onsager‘s reciprocity relation, . At steady state there are electric and
diffusive forces but there is also an interference between the heat and electric currents. 114
The expression for thermopower in steady state ( ) can be determined as the ratio of
1 1 M 12 T T 2.20
S
x q x qT M 11 eT x x
therefore
1 M 12 2.21
S
qT M 11 qT .
The first term on the left hand side of Equation 2.21 has kinetic coefficients which
are experimentally determined. Fortunately, we will now show that these kinetic
coefficients are approximately constant at high temperatures meaning the first term
32
approaches zero. Using the Kubo formalism, an application of an external perturbation on
c 2.22
H t *
I c II c II* cI J s I s I 1
I , II I
where ̅ is the transfer integral of an electron between neighboring sites, cI and c II are
*
d Tre
1 / k BT 2.23
M 1b dt H N / k BT
j1 t i jb
0 0
where N is the number of electrons, Tr is the trace, or summation over a complete set of
reduces to
2.24
S
qT .
33
s 2.25
T N E ,V .
where is entropy and N is the number of particles we can rewrite Equation 2.24 as
1 s 2.26
S
q N E ,V .
k B ln g 2.27
S
q N E ,V .
The degeneracy is calculated as follows. Each site has two orbitals that can either be
occupied or doubly occupied; these conditions will be labeled (I) and (II) respectively.
The total number of ways to arrange these two kinds of sites is given by 112
g N L!L2! L N !
g g I
2 L N
II
N L 2.28
where and are the degeneracies of sites (I) and (II) and is the number of
34
ln g I ln g II k B ln y
kB 2.29
S
q q 1 y
k B g I y 2.30
S ln
q g II 1 y
which is the same as Equation 2.12; Koshibae et al.‘s modified Heikes formula. Equation
2.30 demonstrates fraction of conducting sites, y/(1-y) plays an important role in both the
magnitude and the sign of a material‘s thermopower (see Figure 2.2). Equally important,
35
Figure 2.2. Thermopower calculated from Koshibae et al.’s modified Heikes formula
plotted against carrier concentration.
however, is the ratio of the electronic degeneracies for the ions involved in conduction. A
large ratio in degeneracy yields a large contribution to thermopower from spin and orbital
entropy.
The crystal structure of NaCo2O4 was identified in 1974 by Jansen and Hoppe.141
The compound crystallizes in the P63/mmc space group and consists of a naturally
divided unit cell with layers of edge shared CoO2 octahedra (CdI2 type, triangular lattice)
separated by filled or partially filled layers Na atoms (see Figure 2.3). The sodium atoms
donate their valence electron to the CoO2 layer explaining the large carrier density (~1022
cm-3).
36
Figure 2.3. Crystal structure of NaCo2O4. Yellow atoms are sodium, blue-red octahedral
are CoO6.
For the stoichiometric compound the average oxidation state of the Co ion is +3.5.
Since ions can‘t have half of an electron this really implies that there is equal numbers of
Co3+ and Co4+ or Co ions with 6 and 5 d shell electrons, respectively. Because of electron
shell overlap from the 2p orbitals of surrounding oxygen atoms (octahedral crystal field)
the 5 d shell orbitals for the cobalt ion split into two bands: a lower energy, triply
( and ). The different ways in which Co3+ and Co4+ can arrange their electrons
in these bands are shown in Figure 2.4 along with the electronic degeneracy calculated as
and
g spin 2 1 2.32
where is the total spin and is simply the number of configurations possible for
37
Figure 2.4. Electronic degeneracy of Co3+ and Co4+ ions in an octahedral crystal field
with low, intermediate and high spin states. The total spin, number of configurations and
electronic degeneracy are calculated.
Consider electron hopping between cobalt ions in the low spin (LS) state. The
band of the Co3+ ion is completely filled but the Co4+ ion has an unfilled band and acts as
calculated thermopower is
k B 1 0.5 2.33
S ln 154 V / K
e 6 1 0.5
which agrees fairly well with the low temperature experimental thermopower of 100
μV/K.42 Notably, as an electron hops from a Co3+ to a Co4+ the sites exchange a large
entropy and the electron current flows opposite the current of entropy. At high
temperatures magnetic measurements suggest a spin state transition in Co4+ ions from
low spin to high spin (HS).142,143 If both LS and HS states are available to Co4+ ions then
38
the total electronic degeneracy is and the calculated thermopower
would be
k B 1 0.5 2.34
S ln 214 V / K
e 12 1 0.5
in good agreement with the experimental thermopower at high temperatures (200 μV/K).
in NaCo2O4 and derived the governing equations for thermopower accounting for both
spin and orbital degeneracy in strongly correlated systems. A number of new oxides
related to NaCo2O4 have been identified with large ZT values, particularly single crystals,
but efficiencies are still too low in bulk, polycrystalline ceramics. Additionally, some
authors have suggested that a ‗spin blockade‘ phenomenon (shown in Figure 2.5) can
counterparts.24 Spin blockade occurs when the transport of an electron would lower the
current.144,145 Finally, a relatively small number of ion pairs, such as Co3+ and Co4+, have
been examined and almost all compounds have featured ions in octahedral crystal fields.
Heikes formula to screen for new high efficiency oxide materials looking, in particular,
39
Figure 2.5. The spin blockade phenomenon. Redrawn from Hébert and Maignan.24
Structures exist where these polyhedra are corner shared (perovskites), edge shared
(cordierite) and even face shared (sapphire).146 Nevertheless, many other coordinations
are possible depending on the size and bonding of the cation. Some examples include,
etc. The crystal fields for a few of these alternative coordinations are determined by
In Section 2.3.1 we explained why it isn‘t sufficient for a material to have large
values for and , but rather, the ratio of must also be large in order to
achieve an enhancement in thermopower. However, not all crystal fields are equally
40
Figure 2.6. The crystal fields for a variety of different coordinations. From left to right
(a) distorted cubic, (b) cubic, (c) dodecahedral, (d) tetrahedral, (e) octahedral, (f)
trigonally distorted octahedral, (g) tetragonally distorted octahedral and (h) highly
distorted six-coordinate. Figure adapted from Burns.147
suited to yield large ratios in thermopower. The octahedral crystal field, for example,
benefits largely from the triply degenerate band. For a LS d6 ion the band is filled
and spin is zero yielding the lowest possible degeneracy value, 1, but if an electron is
either added or removed the total spin becomes ½ and the orbital degeneracy increases to
2 or 3 for the and bands respectively. This advantage is also present in tetrahedral,
cubic and dodecahedral crystal fields which can be thought of as the opposite of
octahedral field; the band is now the lower energy band and the band is higher. Note
that tetrahedral coordinations lack a center of symmetry so the subscript in the and
41
Another distinction is that the crystal field stabilization energy (CFSE),
sometimes called 10dq, Δ, and defined as the energy distance between the bands is
different for tetrahedral, dodecahedral, cubic and octahedral crystal fields. In general the
the result of a competition between Hund‘s Rule coupling and CFSE. The implication is
that relative to the octahedral crystal field, less thermal energy is necessary to allow ions
to stabilize in intermediate or high spin states. This, too, could be advantageous for
tetrahedral, cubic or dodecahedrally coordinated ions. If multiple spin states are stabilized
then the degeneracies of each state would sum together and could potentially result in
very large total electronic degeneracies. This was the case for NaCo2O4 at very high
temperatures where both LS, ( ) , and HS, ( ) , Co4+ ions are stabilized
and the total degeneracy was effectively doubled, 12. One prospective combination of d
(LS) with a degeneracy ratio of 16/1 and a calculated thermopower of -240 μV/K (see
Figure 2.7). A complete list of candidate ion combinations and their degeneracy values,
calculated thermopower and possible crystal structures will be presented in the next
section.
42
Figure 2.7. Low and high spin states for d3 and d4 ion combinations in a
tetrahedral/cubic/dodecahedral crystal field. Spin, configurations and degeneracy are
calculated. (for cubic and dodecahedral the nomenclature and are used.)
elements because the bonding electrons in the actinides and lanthanides are f shell rather
than d shell. The presence of two additional orbitals could potentially lead to
enhancement in electronic degeneracy ratios. The crystal field for f2 and f3 electrons in an
octahedral crystal field is shown in Figure 2.8. At first glance there appears to be a good
opportunity for large degeneracy ratio; LS 4f2 & LS, HS, or LS+HS 4f3 is 1/6, 1/12 or
1/18 respectively. However, due to shielding from 5s and 5p shells the 4f shell electrons
do not strongly overlap with the orbitals of surrounding ions and therefore the crystal
43
2
Figure 2.8. Crystal field for f shell orbitals in an octahedral crystal field. The 4f (LS)
3
and 4f (LS and HS) configurations are shown to demonstrate the large ratio in spin and
orbital degeneracy possible. The band is ( ), ( ), and ( ) orbitals,
Identifying the electronic configurations such as 3d3 and 3d4 with potential for
large degeneracy ratios is only the first step in screening for new oxide thermoelectric
materials. Transition metal ions typically have multiple oxidation states and, therefore, a
achieve 3d3 and 3d4 configurations, for example, combinations of either Cr2+/3+ or Mn3+/4+
would be satisfactory.
ionic radii. Many experiments have been performed to accurately determine ionic radii in
different coordinations and oxidation states and extensive tables of values now exist.148–
44
152
If a radii value has not been reported for a given coordination and oxidation state it
may be an indicator that the arrangement is not thermodynamically favorable. Table 2.1
reports the ionic radii values for candidate ion pairs and we observe that ionic radii for
the ions in dodecahedral sites are not available (NA) indicating they are too small for this
site. In fact, only the largest fit in a cubic site. Rather than searching fruitlessly for a
Table 2.1. Ionic radii values (Angstroms) for ion pairs of a few select electronic
configurations. NA indicates that no values are available.
LS d6 & LS+HS d7
Fe2+ / Co2+ 0.63/0.58 0.61/0.65 0.92/0.9 NA
g(I)/g(II) = 16/1
d0 & d 1
Ti3+ / Ti4+ NA/0.42 .67/0.61 NA/0.74 NA
g(I)/g(II) = 6/1
45
Table 2.2. Various ion pairs in different crystal fields with their electron degeneracy
ratio, predicted thermopower and possible crystal structures for candidate materials.
(Si1-xAlx)[Co]2O4
Co2+ / Co3+
LS d 6 & LS+HS d 7 16/1 -240 (Si)[CoFe]O4
Fe2+ / Co2+
(Co)[Co]2O4
Octahedral
(A)[Ti]2O4
LS d 0 & LS d 1 Ti3+ / Ti4+ 6/1 -154
(A=Li,Na)
Cr2+ / Cr3+
d3 & d4 16/1 -240 (Mn)[Ni1-xCrx]2O4
Mn3+ / Mn4+
Tetrahedral
Cubic or
Fe3+ / Fe4+
Dodecahedral 4 5
d & d Mn2+ / Mn3+ 1/12 214 (Mn1-xFex)[NiCr]O4
Mn3+ / Fe3+
Once the appropriate ions have been selected a crystal structure hosting these ions
in the desired site can be identified. Table 2.2 lists candidate crystal structures likely to
benefit from enhanced thermopower using Koshibae et al‘s modified Heikes formula.
describe our experiments to further explore thermopower with respect to its structure,
46
Chapter 3. Synthesis and Characterization: Tools
and Techniques
efficiency and materials guidelines and then went on to explain in detail the origin and
correlated systems. Finally, the effect of spin and orbital entropy on thermopower was
illustrated and a screening methodology was developed to identify oxide compounds that
could exhibit large thermopower. This chapter describes the synthesis routes and
materials. The methods used for materials synthesis, densification and characterization
are quite standard and many detailed texts describing them are available. Nevertheless, an
overview of the basic principles involved as well as the details regarding application of a
Let us begin with the synthesis of new materials. The materials science
with improved or unique properties, specific end-shapes, or complex crystal structure and
ceramics category with diverse synthesis techniques ranging from wet chemistry
47
approaches to polymer derived ceramics to chemical or physical vapor deposition and
many more. We will begin with one of the simplest and most common routes to prepare
Once a composition of interest has been identified one of the first methods used to
make a sample is the solid-state method because of its relative simplicity. If the incorrect
phase or impurities are obtained then more difficult approaches can be considered. The
basic principle is to mix stoichiometric amounts of the constituent oxides in powder form
for a given composition and then to heat the mixture and allow diffusion to achieve
intimate mixing. This so-called ―shake and bake‖ process relies on diffusion of elements
LD 4 Dt 3.1
EA 3.2
RT
D D0 e
determined value that can range from 10-8 m2/s for small molecules moving interstitially
48
through a metal to 10-20 m2/s for large molecules in organic materials. Oxides have a
rather low melting point coefficient of diffusivity between 10-12-10-14 m2/s and a
Substituting these values into Equation 3.1 and 3.2 and assuming the mixture is held near
the melting point for a week we observe that typical mass diffusion lengths could be as
The short diffusion length underscores the importance of aggressively mixing the
precursor powders to minimize the diffusion length and reduce the particle size. Powders
can be ground by hand in a mortar and pestle or they can be dispersed in ethanol and
milled in a ball, attrition or planetary mill with milling media such as zirconia or alumina.
If the powders are calcined and still aren‘t the correct phase or pure the powder can be
milled and the process repeated. For our experiments we used the solid-state method to
However, samples of sufficient purity were only obtained for y=1 composition.
coprecipitation. In this approach the ions are dissolved entirely into a solution and
techniques is that the ions are dissolved at a specific pH value, normally acidic. The
solution is then mixed dropwise to a buffer solution of the opposite pH such that the
49
dissolved ions are no longer thermodynamically stable and precipitate immediately into
the solid form leaving behind an aqueous solution of the acid or base. In order to assess
whether the desired ions should be dissolved in a basic or acidic solution a Pourbaix
diagram can be referenced such as the one shown in Figure 3.1 for iron. A Pourbaix
washed several times in alcohol and deionized water before it is dried and calcined to
increased thermodynamic driving force for crystallization upon heat treating the sample.
Figure 3.1. Pourbaix diagram for iron. Figure used with permission from Früh.154
50
The majority of compounds are made by dissolving salts such as nitrate
precursors in water or water buffered with HNO3. The dissociation reaction is of the form
species. Oxide precursors with the nominal formula can also be used if nitric
acid is used to help dissolve the ions via the elimination reaction (dehydration)
Commonly used bases include NaOH or NH3OH and the final precipitation reaction
coprecitation or reverse coprecipitation refers to whether the ions in solution are added to
the buffer solution dropwise or whether the buffer solution is added to the ion solution
dropwise. One downside to the technique is dealing with insoluble ions, using immiscible
liquid precursors or using multiple ions that precipitate at incompatible pH ranges as was
the case for our experiments. We attempted to synthesize the FexMn1-xNi2-yCryO4 x=0, ¼,
½, ¾, 1 and y=0,1,2 series via coprecipitation but we were unsuccessful because the
elements Fe, Mn, Ni and Cr only had a very narrow window of pH values (7.5 < δ < 8.8)
51
Table 3.3. Insolubility ranges at 0 volts in pH for Fe, Mn, Ni and Cr elements and
coprecipitation.155,156
Fe Mn Ni Cr Coprecipitation range
<6 > 7.5 < 8.8 > 5 and < 11.5 7.5 < δ < 8.8
where solids of each element would precipitate (see Table 3.3). Despite repeated
attempts, after each coprecipitation was complete there still remained ions in solution
The final synthesis route that I‘ll describe in this work is combustion synthesis.
Combustion synthesis combines the relative simplicity of the solid-state method with the
intimate mixing achieved through coprecipitation routes. The idea is to start with
precursors that must decompose during a heat treatment and the remaining elements can
form the desired composition. Consider the combustion of lanthanum carbonate and
or even gas-phase synthesis. Typical starting materials for SHS are organometallic
temperatures range. For solution combustion synthesis, however, metal salt (nitrate)
52
precursors mixed with urea or glycerine as a fuel source are more commonly used and
crucible and the entire crucible is then put in an oven and the temperature is slowly raised
first to evaporate any liquid and then to combust the materials. A regrettable consequence
of decomposition is the inherent gas byproducts which, for the case of nitrates in
particular, can be dangerous if not performed inside a fume hood. Whether NO2 or N2 is
formed depends on the ratio of NO3- to fuel; if sufficient fuel is present then N2 will form.
¾, 1 and y=0,1,2 series the solid-state and coprecipitation methods were unable to
produce dense, crystalline pure phase materials but solution combustion synthesis was
successful. Iron, nickel chromium and manganese nitrate solutions were mixed in a
fume hood. We also employed combustion synthesis using solid precursors (lithium,
materials.
However, it is often necessary to sinter, or densify the compound into a solid body in
53
pellet shaped material (10mm diameter, 1-2mm thickness) and rectangular bars are
2mm width and thickness). There are a variety of methods whereby powders can be
consolidated and sintered into a dense body. The most simple and widely used approach
is to use a die to compact the powders uniaxially into a green, or unfired pellet and then
and solid-gas interfaces acts as the driving force for sintering. The driving force can be
increased, however, when a pressure is applied to the compact during heating. One
specific variant of this process is the so-called spark plasma sintering (SPS) technique. In
fact, SPS is a misnomer as there is no spark and no plasma involved causing some to
refer to the technique as current assisted densification (CADPro). The ceramic powders
are compacted under a load into a die normally made of carbon though some higher
strength materials such as SiC are used occasionally. The powder is heated under vacuum
by applying a DC current across the plungers of the die. As the current passes through the
die, or, in the case of electrically conductivity powders, through the powder compact
itself there is Joule heating. Due to the large currents involved very fast heating rates up
to 1000 K/min can be achieved and the applied pressure can significantly lower the
temperatures required for densification. Faster heating rates paired with lower
growth is suppressed. Originally it was expected that a pulsed DC current would create a
54
Figure 3.2. Custom built SPS or CADPro sintering chamber. The chamber is located
within an Instron frame allowing a mechanical load and displacement to be applied and
monitored.
strong electrical field diffusion effect but the actual role of such an effect is still
pressureless sintering was unable to produce sufficiently dense samples so CADPro was
material. There are many structural characterization tools ranging from macrostructural to
55
meso and microstructural and even probes to examine materials on the atomic scale!
Brandon and Kaplan 160 point out that no single characterization technique is suitable for
examining a structure at all length scales, but rather a compilation of tools applied in
parallel is necessary. They argue that the techniques with the most widespread use are
optical microscopy, X-ray diffraction and electron microscopy which rely on visible
light, X-ray radiation and high energy electron beams respectively to probe a material. In
this work we concern ourselves with the details of the crystal structure. Accordingly, we
rely most heavily on diffraction using X-ray radiation and neutrons, a probe of long range
order as it relates to the crystal structure. However, in a secondary role we examine short
range order using Raman microscopy. Lastly, we go beyond looking at structural features
to examine the response of a material to a thermal gradient and an electrical field; namely
electrical and thermal conductivity and thermopower. A discussion of the structure and
properties determined from these techniques will be described in the following sections.
Since the initial discovery of X-rays in 1895 and its first application to
entire discipline with many techniques and capabilities. A more complete treatment of the
161–163
subject is found elsewhere Nevertheless, it is instructive to at least describe the
incident X-ray strikes a material, scatters and the diffracted beam intensity is measured as
a function of incident and diffraction angle. Modern X-ray powder diffractometers use X-
56
rays produced by colliding a beam of electrons with a metal such as copper, molybdenum
or cobalt and a portion of the energy released is given off as X-rays. The wavelength of
the X-ray depends on electron transitions of inner electron shells and for this reason X-
ray wavelength depends strongly on the anode metal used. The X-ray wavelength can be
When an incident X-ray beam strikes a material with long range order, such as a
crystal, there is scattering from the lattice planes (see Figure 3.3). If the incident and
diffracted beams are in phase then destructive interference is avoided and a strong signal
interplanar spacing, , can lead to a path difference between the incident beam and the
beams reflected from two parallel planes. In order for these beams to be in phase the path
difference must be equal to an integer multiple of the X-ray wavelength; this is the basis
n 2d sin . 3.7
where is the X-ray wavelength, is the incident angle and n is an integer (1,2,3…).
57
Figure 3.3. Illustration of X-ray diffraction from lattice planes in an ordered material.
The diagram illustrates the relation between incident angle and lattice plane
separation known as Bragg’s law. Figure used with permission.164
function of angle. In a crystal there are multiple planes of atoms that can cause
diffraction- these are known as Miller indices. The interplanar spacings are a function of
these Miller indices, or what planes are diffracting, and the lattice parameter of the
1
2
1
2 C11h 2 C 22 k 2 C33l 2 2C12 hk 2C 23 kl C31lh .
3.9
d V
58
The volume, V, of the unit cell is defined as
with lattice parameters a, b and c and unit cell angles α, β and γ. The constants are
given by
Substituting Equations 3.10 and 3.11 into Equation 3.9 gives the planar spacings. For
to
1
h2 k 2 l 2 3.12
d2 a2 .
crystal and depends on a number of factors. Intensity has the general form
59
1 cos2 2 cos2 2
where k is a scaling factor, | | is the structure factor,
sin 2 cos 1 cos2 2
is the
2 B sin 2
reflecting plane, is the absorption correction and exp is a temperature
factor. The complete details of these expressions are beyond the scope of this work, but it
is worth noting that the structure factor, | |, takes into account both the spatial
dictates the individual scattering amplitudes for an atom. For a given reflection the
N
3.14
Fhkl f n exp2ihun kv n lwn .
n 1
B sin 2
2
3.15
f 0 exp f ' f ' '2
2
f
2
where and are anomalous dispersion factors. The atomic scattering factor is
proportional to the atomic number of an element because X-rays interact primarily with
the electron cloud which increases in size with atomic number. If multiple atoms with
sufficiently different atomic scattering factors at a given angle occupy the same
60
X-ray diffractometers feature a number of optics to improve the resolution, purity
and intensity of the signal. For example, Soller slits and curved (Göbles) mirror ensure
the beam is parallel and collimated, divergence slits determine the area of sample to
illuminated by the X-ray beam and filters and monochromators remove undesired
There are many different geometries and conditions for measuring XRD. For the
most part our samples were loose powders placed on a stationary holder made of a
polymer or a zero background holder (ZBH, silicon cut along a non-diffracting plane).
scan where both the incident and diffracted beam angles were equal; i.e. the source and
¼, ½, ¾, 1 and y=0,1,2. and Co3O4 required the use of a cobalt anode at 40 kV and 45
background (see Figure 3.4). High quality scans were obtained by scanning from 20o to
100o with a step size of 0.017o and a rate of 1.4 s/step. Fixed divergence and scattering
slits of 0.5o, 0.02 radian Soller slits and filters and a germanium detector were used.
The height of the sample in the furnace was zeroed by splitting the beam and the tilt was
61
Figure 3.4. Dispersion factors, and , for the element iron using Cr (left) and Cu
(right) radiation sources. When copper radiation is used it is strongly absorbed and
fluoresced by iron. Images made using FPRIME software.165
corrected by rocking the sample. For high temperature data collection an Anton-Paar
HTK1200N furnace was employed to heat the samples. The temperature fluctuation
within the furnace was less than 1 K and the sample was heated slowly at 5 K/min and
allowed 30 minutes to equilibrate before each scan was performed. Measurements were
carried out in an air atmosphere with the exception of the reduced LiMn2O4 samples
The X-ray diffraction experiments reported in this work relied on both basic
The technique is the same except that for the intensity and wavelength of the incident
between different shells a synchrotron is a particle accelerator that uses bending magnets
62
Figure 3.5. Schematic diagram of a synchrotron. The outer ring is a particle accelerator
that has bending magnets (shown in red) used to change the direction of the electron
(radial acceleration) and emit electromagnetic radiation directed into tangentially
situated diffractometers. Image used with permission from EPSIM 3D/JF Santarelli.166
diffractometers meaning excellent signal to noise ratio for even small samples.
Laboratory X-ray diffractometers were used to study all of the samples in this work, but
for the samples in the FexMn1-xNi2-yCryO4 x=0, ¼, ½, ¾, 1 and y=0,1,2 series synchrotron
diffraction was also used in an attempt to determine cation distribution. Powders were
prepared in our lab and then sent to Argonne National Laboratory 11-BM beamline and
the diffraction data was then returned for us to perform Rietveld refinement analysis.
63
3.2.2 Neutron Diffraction
Because of this difference neutron diffraction, first performed in 1945,167 can extract
different information from a material while using essentially the same technique as X-ray
diffraction. In contrast to X-rays, which scatter primarily off the electron cloud, neutrons
interact directly with the nucleus of a material. This has several implications. First, the
penetration depth of neutrons in a material is much greater than X-rays and larger sample
sizes are typically required. Second, since nuclei don‘t have nearly the size range as
electron clouds the atomic number dependence of neutron diffraction is removed. The
atomic scattering factor for neutrons is more random and even isotopes of the same
element are typically different. Third, neutron diffraction intensity does not decrease with
scattering angle as it does in XRD allowing the observation of high angle peaks and very
Perhaps the most celebrated and useful aspect of neutron diffraction is that
neutrons, having spin but no charge, are sensitive to magnetic moments and are therefore
measured for a magnetic material in a cryostat above and below its magnetic ordering
temperature additional peaks corresponding to diffraction from the magnetic structure are
observed. The magnetic moment in an atom derives from the electron cloud however, so
magnetic peaks decrease in intensity at high angle just as XRD peaks do. In our work the
64
magnetic moment for Co3O4 samples air quenched from different temperatures between
K and 1.5 K.
interact less strongly with materials the diffracted signal is much weaker than XRD. This
weak signal means longer scans are necessary. One way to reduce the scan time is to use
an area detector, such as the one shown in Figure 3.6, to measure over a large diffraction
range at the same time. Area detectors can also often measure outside the plane of
diffraction increasing the signal intensity but with a slight reduction in resolution. An
example of this is shown in Figure 3.7 where the Laue cone is visible at low and high
Figure 3.6. Photograph of a neutron diffractometer with a cryostat (D1B at Institut Laue
Langevin).
65
Figure 3.7. Neutron diffractometer area detector results, (left) the entire pattern, high,
low and intermediate angles are shown for a measurement, (right) whole detector (blue)
and center region only (red) demonstrating tradeoff between resolution and intensity.
The majority of XRD users rely on the technique for phase identification of a
material only. A sample is measured and the 2 position and intensity of the observed
peaks are identified and then a search and match software such as Phillips X‘Pert
Highscore, MDI ―Jade‖, or DIFFRACplus is used to match the phase or phases present to
sample is unknown and no chemical restrictions can be applied to the database then it can
be extremely difficult to correctly identify the phase using only a stick diagram (position
present it is much more difficult to accurately determine the phases present and
impossible to ascertain the phase fractions. An example of this can be seen in Figure 3.8
66
Figure 3.8. Example of a stick diagram to match measured peaks (top pattern) to
candidate structures on file (bottom three patterns).
formulated by Hugo Rietveld 168 in 1969 can not only verify that phase identification was
done correctly but also uncovers an enormous amount of additional information that X-
ray diffraction can tell a user about a material. The Rietveld method can be applied to
accurately determine lattice parameters, atomic positions, occupancy of atoms per site,
content, preferred orientation, grain size and strain. Additionally the method can account
for sample and instrument errors such as displacement or shift, transparency, zero of the
goniometer and impurity wavelengths. The reason that Rietveld refinement can extract so
much additional information is because it relies on whole profile fitting, rather than
67
simply search and match based on a stick diagram. Just as there are many search and
match software packages available, there are many Rietveld refinement software
packages available but the most widely used are GSAS (General Structural Analysis
data, calculates all the reflected planes, assigns each peak an intensity based on Equation
3.13 and a shape and then refines attributes of the crystal structure or sample/instrument
corrections using a least squares algorithm until the observed diffractogram best matches
a calculated diffractogram. The least squares algorithm minimizes the following function
1
2 3.16
M Wi I iobs I icalc
i c
where is a weighting factor (higher intensity values, such as peaks are weighted more
heavily than background values), is the intensity value at each point . The variable is
a scaling factor such that . The quality of the fit is given by the weighted
residual
M 3.17
Rwp .
W I
i
i i
obs 2
For a given Bragg lattice we can calculate which planes will have
reflections. To do so we solve for all , and values such that Equation 3.14 has a non-
68
zero result for atoms located at . Peak shapes can be Gaussian in nature as
neutron diffraction peaks are, or they can have a significant Lorentzian component. The
correct profile function, or peak shape, is the one that best fits the data. If the instrument
beam has a large amount of vertical divergence, then asymmetric peaks will be observed,
challenging. The phase abundance can be determined by indirect methods such as the
172
Bogue method where the total chemical composition is determined and assuming a
chemical composition for each phase an estimation can be made. Phase fraction can also
be measured by direct methods relying on some property specific to each phase such as
Quantitative phase analysis using powder diffraction is another such direct method. Phase
fraction is determined by ―single peak‖ methods such as Reference Intensity Ratio (RIR)
173
or by whole pattern methods such as the SMZ method.174,175 More accurate and
precise results are obtained by the SMZ method because overlapping peaks are accounted
for. In Rietveld refinement the intensity of a reflection (or group of reflections), , can be
reduced to
W 3.18
I i Ci
D, *
m
69
where is a constant for reflection(s), , in phase , is the weight fraction, is
the density and is the mass absorption coefficient. The intensity is proportional to the
I i S 3.19
and the constant is inversely proportional to the square of the unit cell volume
1 3.20
Ci .
V 2
ZM 3.21
D ,
V
where is the number of formula units in the unit cell and is the molecular mass of the
formula unit. Substituting Equations 3.19-3.21 back into Equation 3.18 and solving for
we obtain
S ZMV m* 3.22
W
K
where K is a scaling factor necessary to put on an absolute basis. For multiple phases,
70
S ZMV 3.23
W n
.
S ZMV
k 1
k k
phases.
Microscopy. Unlike diffraction, Raman scattering of a light source with a material relies
on an inelastic process. Another contrast is the wavelength of light used. XRD of single
crystals can use white light, but as described in the previous section, powder XRD relies
on monochromated X-rays whereas Raman spectroscopy uses laser light in the visible,
near ultraviolet or near infrared range as an excitation source. As with diffraction, a more
section with a brief overview of the different ways light can interact with matter via
scattering.
Light interacts with the electron cloud and bonds in a material. The majority of
light that is absorbed will cause a molecule to be elevated to a virtual energy level and
when the molecule returns to this state light is reemitted with the same energy. This type
Rayleigh scattering. In some cases, however, there is an exchange of energy between the
71
light and the material (see Figure 3.9). If energy is absorbed by the material, as in Stokes
scattering, then the emitted light must have less energy than the absorbed light (red
shifted). Alternately, if energy is lost by the material, as in anti-Stokes scattering, then the
emitted light is more energetic than the absorbed light (blue shifted). Typically, the
Raman shift in energy, , is on the order of 200-4000 cm-1, the conventional unit of
1 1 3.24
E .
absorbed emitted
Figure 3.9. Rayleigh and Raman (Stokes and anti-Stokes) scattering. Rayleigh is eleastic
light scattering while Raman scattering is inelastic with a shift in energy between
absorbed and emitted light. Image used with permission.179
72
As can be seen in Figure 3.9, some molecules in a material will be in the ground
state to start with (Rayleigh and Stokes scattering) and some molecules will be in an
excited vibrational state (anti-Stokes scattering). For a given energy difference between
the excited vibrational state and the ground state, , we can rely on a Boltzmann
distribution to determine the number of molecules in the excited and ground states
and :
ground state bands. The implication of Equation 3.25 is that there is always a greater
number of molecules in the ground state than in the excited vibrational state. Therefore,
since scattering is related to the number of initial molecules in a state the intensity of
Materials have unique bonding and symmetry so the Raman scattering from the
vibrational modes are material specific and can be used to characterize materials. As
diffraction techniques is very well understood insomuch that calculated patterns can even
be fitted to observed patterns with very good agreement. Vast databases of indexed
patterns exist for a hundreds of thousands of inorganic and organic compounds including
73
Raman spectroscopic characterization of materials is not yet as developed. Despite
efforts, such as the RRUFF project,180 to develop Raman spectra databases similar to
those available for diffraction data the majority of Raman spectroscopy users rely on the
one another.
especially when paired with optical microscopy. Raman microspectroscopy features good
resolution (<1 μm), very fast data collection and little or no sample preparation. In
length scale than diffraction. Vibrational modes observed by Raman spectroscopy are
laser beam (Kimmon Electric US, Ltd. IK Series He-Cd LASER Englewood, USA) to
temperatures in air were obtained using a Linkam thermal stage (Scientific Instruments
Ltd., Waterford Surrey, England). Spectra were taken from room temperature (298 K) up
to a temperature of 1223 K and vice versa during the heating and cooling ramp (10
74
3.3 Electrical and Thermal Transport
electrical and thermal conductivity. Although the emphasis of this work is to highlight the
electrical and thermal conductivity add a degree of context and insight. We discuss the
have coplanar faces and placed between the top and bottom electrodes of the
measurement system. Thermocouple probes are carefully set into contact with the face of
the sample and the distance is measured via camera calibrated to the sample width.
The entire fixture is enclosed in a furnace and for samples vulnerable to oxidation the
chamber can be evacuated and backfilled with an inert gas, such as helium. The furnace
75
Figure 3.10. Thermopower and electrical conductivity measurement schematic.
Operation described in text.
1.8). The same fixture can be used to measure electrical conductivity. If a constant
current source, , is directed through the sample and the voltage drop is measured across
the thermocouples then we can measure the resistance of the sample between the
V V2 V1 IR 3.26
76
L d1 3.27
R
A A
recalling that ρ is the resistivity, is the distance between thermocouple probes and is
the cross-sectional area. Electrical conductivity is, of course, the inverse of electrical
resistivity.
Steady-state heat flux methods such as comparative method,181 guarded heat flow
Q T 3.28
.
t A x
Devices using these methods are the thermal analog to the electrical conductivity
infamously large uncertainty (>20%) in values though because accurately measuring the
temperature gradient and heat flow rate is difficult. Especially challenging is how to
account for radiative losses at elevated temperatures. An approach used much more
D Cp 3.29
77
where is the heat capacity at constant pressure and is the sample density. Specific
heat can be easily measured but for our experiments we used the Kopp-Neumann law
given by
N 3.30
C CiWi
i 1
where is the total specific heat, and is the specific heat and weight fraction of the
-th component of an alloy containing components. Calculating the specific heat from
sample is heated by a laser pulse on one side and the temperature is measured by an
infrared sensor on the back side of the pellet. The infrared sensor voltage is proportional
to the temperature and is plotted as a function of time. Because the sample is machined
coplanar with diameter much larger than thickness (typically 10mm diameter, 1 mm
thickness) approximately adiabatic boundary conditions exist and the heat flow is roughly
diffusion equation is
T x, t 2 T x, t 3.31
T
t t 2 .
78
c 3.32
T x, t exp ct A sin
c
x B cos x
a a
and assuming adiabatic boundary conditions for a coplanar sample with no heat flux at
nx n 2 2 at 3.33
T x, t Bn cos exp
n 0 L L2 .
For a finite pulse of very short duration the change in heat measured at the back face is
n 2 2 at 3.34
T x L, t T 1 2 1 exp
n
n 1 L2 .
where is the change in temperature at the back face after infinite time. Thus
expression can be rearranged to solve for diffusivity with the following approximation
ln 1 / 4 L2 T 3.35
.
2 t1/ 2 T
where is an important parameter defined as the time required for the temperature to
applied to calculate from the temperature time profile such as Clark and Taylor 188 or
79
To minimize radiative transport through the sample it is common practice to coat
both faces with a thin metallic layer (~1μm of gold) using an Effacoater sputtering
system (Ernst F. Fullman Inc, Clifton Park, NY). An additional coating of colloidal
graphite is applied to ensure good absorption and emissivity. The density of the samples
medium. A final correction to thermal conductivity was applied for compounds retaining
1 P 3.36
0 .
1 P
where is the thermal conductivity of the sample porosity, is the thermal conductivity
of the nonporous material, is the porosity and is a geometrical factor equal to ½ for
80
81
The previous chapters laid out much of the necessary groundwork for
measurement a recurring theme was that the crystal structure, materials processing,
concentration has a dramatic effect on electrical conductivity. The intensity, shape and
composition, crystal structure, texturing, thermal displacement, grain size and more. The
study of these relationships and many others is the fundamental basis of materials science
as a discipline.
structure and disorder will alter the electronic degeneracy ratio and carrier concentration
81
parameters in the Heikes formula for strongly correlated systems. The effect of Jahn-
LiMn2O4 and the effect of anti-site disorder will be examined in LixMg1-xMn2O4 spinels
Structural distortions are quite common in oxides. Take perovskites, one of the
most common oxides, for example. Perovskites have the nominal structure ABO3 and the
ideal structure is cubic (Pm ̅ m) with A site cations surrounded by 12 anions occupying
the cube corners, B site cations surrounded by 6 anions located at the cube center and
oxygen atoms centered on the cube faces. Modeling the ions as space filling hard spheres
with radii , and the ideal structure requires geometrically that the A-O distance
(center of cube face to center of cube, ). The tolerance factor describes the
deviation from the ideal structure by comparing these distances and is given by the
following expression
rA rO 4.1
t* .
2 rB rO
For the ideal perovskite structure and this tolerance factor can be used to predict
crystal structure of ABO3 compounds.191 For tolerance factor values near 1, i.e.
82
, a cubic structure is formed. For larger values a hexagonal
structures will form. Finally, values will result in hexagonal ilmenite type
structures.
distortions in the crystal structure. This can be seen in compounds with smaller and
larger values leading to tilting of the BO6 octahedra in order to fill space. For
orthorhombic structures the tilting is about the b and c axes and for rhombohedral
structures the tilting is about each axis. This tilting leads to a reduction in coordination
number for A, B or both ions (see Figure 4.1). In addition to tilting, cations can also be
Figure 4.1. Examples of ideal and distorted perovskite structures. (left) cubic SrTiO3,
(center) rhombohedral LaAlO3 and (right) orthorhombic CaTiO3. Structures all shown
projected along the b-axis to demonstrate octahedral tilting. Unit cell outline in black.
83
Another common structural distortion in oxides is a Jahn-Teller distortion. For a
octahedral coordination, the overall energy of a system can be lowered by removing the
orbital degeneracy of the bands.192 Figure 4.2 shows this energy lowering for Jahn-Teller
between cation and bonded oxygen anion and therefore higher energy of the orbital. By
convention the axis of elongation is labeled z-axis. The spatial displacement of ions
caused by the Jahn-Teller distortion reduces the symmetry of an ideal cubic perovskites
coordinations are less common because the CFSE for these coordinations is smaller and
The spinel structure differs from the perovskite structure but it, too, can exhibit
symmetry lowering structural distortions such as tilting, cation displacement and Jahn-
Teller distortions. Spinels have the nominal formula unit AB2O4 where the A site, 8(a), -
84
Figure 4.2. Jahn-Teller distortion examples for d4 cations under axial compression and
elongation along the z-axis.
The anion sublattice is a close-packed cubic structure and cations partially fill the
AO4 tetrahedra. The undistorted cubic (Fd ̅ m) and Jahn-Teller distorted tetragonal
85
Figure 4.3. Undistorted (left) and distorted (right) LiMn2O4 crystal structures with A and
B cation sites labeled. Green, purple and red atoms correspond to lithium, manganese
and oxygen respectively.
Before going on, it is necessary to define inversion, a term used frequently with
spinels. A large number of compounds crystallize in the spinel structure. There are 2,3
spinels where the cations have +2 or +3 oxidation states; examples include MgAl2O4,
CuFe2O4, Fe3O4, Mn3O4, Co3O4 and many others. There are also 4,2 spinels where
cations have +4 and +2 oxidation states; examples include TiMg2O4 and SiMg2O4. The
spinel has the A site completely occupied by cations of lower oxidation state as in Co3O4.
Alternately, an inverse, or inverted spinel has no cations of lower oxidation state on the A
site as in Fe3O4. Gorshkov et al. observe that synthetical spinels synthesized at high
temperatures are frequently inverted but naturally occurring spinels are normal.193 The
86
site disorder where the lower oxidation state cations partially occupy both A and B sites
terms ―A site‖ and ―tetrahedral site‖ or the subscript T are all used interchangeably to
denote the 8(a) cation site in Fd ̅ m space group. Similarly, the terms ―B site‖ and
Too small ions can rattle on a site to scatter phonons. Cations shifted off site will also
mean a charge is displaced and a dipole will result. We now discuss how a Jahn-Teller
In Section 2.4 we described how the modified Heikes formula can be used as a
screening tool for identifying new oxide thermoelectric materials. The search for n-type
Mn3+/Mn4+ ion combinations.136 Assuming the ions take a HS undistorted state the
electronic degeneracy ratio in Equation 2.30 is 10/4 and with equal concentrations of
k B 10 0.5 4.2
S ln 79 V / K .
q 4 1 0.5
87
Figure 4.4. Spin and orbital degeneracy values for d3 and d4 cations. Jahn-Teller
distortions in Mn3+ ions reduce the orbital degeneracy by a factor of 2.
If, however, the Mn3+ ions are stabilized by a Jahn-Teller distortion then the electronic
degeneracy is reduced to 5/4 (see Figure 4.4) and the thermopower is reduced to
k B 5 0.5 4.3
S ln 19 V / K .
q 4 1 0.5
113,114
Koshibae et al. predicted that due to the strong likelihood of Mn3+ forming
Massarotti et al. seem to confirm this.136,194 The thermopower for the mixed ion
LiMn2O4 were measured up to 1000 K (see Figure 4.5). Despite the different composition
respectively) each tended to approach the high temperature thermopower limit of -25
88
μV/K in close agreement with the Jahn-Teller distorted limit for thermopower. Notably,
though none of the compounds reached the undistorted limit (-79 μV/K), at very high
temperatures (> 950 K) the thermopower of the compounds increased to -32 μV/K and
seemed to approach the undistorted limit perhaps indicating that enough thermal energy
distortion.
Figure 4.5. Thermopower for a variety of mixed ion manganates. The high temperature
theoretical thermopower limit is shown for Jahn-Teller distorted compounds, -ln(5/4),
and undistorted compounds, -ln(10/4). The compounds agree with the distorted limit until
very high temperatures when the orbital degeneracy recovers. Redrawn from Kobayashi
et al.136
89
Further examination of the thermopower results reported by Massarotti et al. led
195
us to detect an inconsistency; LiMn2O4 is a spinel and previous studies demonstrate
clearly that a Jahn-Teller distorted tetragonal phase is only observed below 280 K, yet the
high temperature thermopower in Figure 4.5 corresponds to the distorted limit. This
apparent discrepancy motivated us to re-examine this compound and clarify the role of
Jahn-Teller distortions, using high temperature X-ray diffraction, between the crystal
The results of the in-situ heating X-ray diffraction confirmed that in the air-
sintered samples the major phase is an undistorted, cubic spinel-type LiMn2O4 (Fd ̅ m,
a=8.2514 Å, Z=8). At room temperature, there was only 1.5 wt% of the tetragonal phase
(Table 4.4, Figure 4.6). The finding of a cubic spinel complements the work of Yamada
et al. who showed that full transition to the tetragonal phase (I41/amd) only occurs below
280 K.195
90
Figure 4.6. X-Ray diffraction pattern and Rietveld refinement results for LiMn2O4 at 298
K. The weighted residual, Rwp, for the refinement was 3.46% and the fraction of the
undistorted spinel LiMn2O4 phase (Fd ̅ m) was 98.5 wt% with only 1.5 wt% tetragonal
(I41/amd) LiMn2O4 phase. The inset shows the XRD pattern of the (440) reflection for the
Fd ̅ m phase for different temperatures. The lattice parameter increases with temperature
evidenced by the peak shifting to smaller 2θ values. Also, the 298 K peak is considerably
wider than the others because the lower symmetry tetragonal phase includes
convolutions of two closely-spaced additional reflections; (224) at 63.5o and (440) at
63.9o.
The Rietveld refinement of room temperature XRD data is presented in Figure 4.6
with an inset showing elevated temperature data. (Table 4.4 has detailed results for all
temperatures) With increasing temperature the XRD patterns visibly changed (see inset
inFigure 4.6) in two ways: (1) the peaks shifted to smaller 2θ values as thermal expansion
91
Table 4.4. Structure refinement details for LiMn2O4 measured in air as a function of
temperature (standard deviations in shift shown beneath in parenthesis)
298 (first) 8.25138(7) 0.26195 (20) 94.7 (6) 97.3 (3) 3.46 1.5
298 (last) 8.25209(7) 0.26177 (20) 95.2 (7) 97.6 (3) 3.22 -
caused the unit cell volume to increase and (2) the peak widths decreased and the
distinction between Cu and increased. The peak widths are greatest at room
temperature because of the presence of the tetragonal phase. The convolutions for
additional peaks from the lower symmetry I41/amd phase produce wider peaks. Al-
though the weight fraction of the tetragonal phase refined to zero for all non-ambient
temperatures, the width of the peaks at high temperatures do seem to very slightly, yet
systematically decrease indicating that perhaps there is a very small amount of tetragonal
phase remaining that is not being accounted for in the Rietveld refinement.
92
The coefficient of thermal expansion (CTE) is calculated to be 93-120 ppm/K
over the range 298-1123 K based on the increase in the lattice parameter for the LiMn2O4
Fd ̅ m phase. The compound also demonstrated slight anti-site disorder upon heating, that
is to say, a fraction of Li+ ions occupied the octahedral B 16(d) site instead of the
tetrahedral A 8(a) site and Mn ions did the opposite (see Figure 4.7). At room
temperature only 2.5% of the B site ions were Li+ but by 1123 K the percentage had
increased to 8.1%. The anti-site disorder can likely be ascribed to strain minimization as
the larger Li+ ions (ionic radius 73/90 pm for tetrahedral and octahedral coordination)
respectively increase their coordination from 4 to 6. The smaller manganese ions (ionic
radius 64/78.5 and 53/67 pm for high spin Mn3+ and Mn4+ in tetrahedral and octahedral
tetrahedral coordination is not available in the literature but was calculated from the
as 124 pm
The anti-site disorder of LiMn2O4 structure with increasing temperature has not
been reported previously. However, the structure of LiMn2O4 is very sensitive to cooling/
heating conditions and can undergo complex phase transformation and cation
rearrangement since the lithium, manganese and oxygen stoichiometry are freely
confirming the occupancies of tetrahedral and octahedral sites by Mn3+ and Mn4+ ions.
93
Figure 4.7. Fraction of the A (tetrahedral) and B (octahedral) sites in the LiMn2O4 spinel
occupied by Li and Mn ions plotted against temperature. The lines represent linear best
fits (R2 = 0.971). Anti-site disorder on the B site is observed to increase linearly over the
temperature range.
0.26168 at 298 K and 1073 K, respectively) and calculated oxygen fractional coordinate
(u) for LiMn2O4 employing different distortion models, i.e. occupancies of A-sites by (i)
94
+
i Mni4+]A[Lii+Mn13+Mn1-i4+]BO4 (u=0.26207 / 0.26084) and (iv) that in undistorted
r B
0.07054
4.4
u 0.25 0.375 0.3876 ,
r A
r A i Li A rtetr Li 1 i Li A rtetr Mn 3 / 4 , 4.6
r B 0.5i Li B roct Li 1 i Li B roct Mn 3 / 4 . 4.7
experimental and calculated structural parameters given above does not provide
unambiguous evidence for preferential site occupancies. Our assessment is that the
fraction of octahedral Mn4+ lies in the range between 0.46 and 0.54 at 1073 K. However,
as we have found that the lattice parameter of the cubic LiMn2O4 increases with
at T > 1053 K LiMn2O4 loses oxygen becoming nonstoichiometric. This is not the case
for our study. Since we also did not observe the decomposition of LiMn2O4, which
95
typically happens at T > 1113 K (XRD) upon heating in air,199 we can exclude the
demonstrated that: (i) single-phase undistorted normal LiMn2O4 spinel is stable over a
broad temperature range (298-1123 K) and, (ii) the inversion of spinel structure is less
than 8% even at the highest temperature studied (1123 K). Therefore, as LiMn2O4 is free
Electrical conductivity
previous measurements on LiMn2O4. For example, Figure 4.8 shows the electrical
3 2 4.8
q 2 aˆk F2
where is the reduced Planck‘s constant, ̂ is the Mn-O-Mn distance (estimated to be 0.4
nm) and is the three dimensional Fermi wave vector.136 The activation energy, , for
96
1/T and setting the slope equal to – (see inset to Figure 4.8). This value agrees
136,194
well with the 0.4 eV reported experimentally for small-polaron conduction in
LiMn2O4 but contrasts with a recent computational study using density functional theory
Figure 4.8. Arrhenius plot of the electrical resistivity. The high temperature value of the
LiMn2O4 series approaches (dashed line) 9 mΩ cm, close to the Ioffe-Regel limit (solid
line) of 2 mΩ cm for a mean free electron path of 2.9 Å. The inset is an Arrhenius plot
confirming small-polaron conduction with an activation energy of 0.36±0.01 eV.
97
when a Jahn-Teller distortion was associated with the high-spin Mn3+ ions. Interestingly,
in the case where there was no Jahn-Teller distortion, the activation energy was
Seebeck Coefficient
194
In contradiction to an earlier thermopower report for LiMn2O4 the
thermopower measured in this work reaches a high temperature value of -73 μV/K.
Interestingly, this value is more than 3 times larger than previously reported in small-
polaron conducting manganates (see Figure 4.9). It is, however, fully consistent with the
high temperature limit of the modified Heikes formula (see Equation 2.30 and Figure 4.4)
Mn4+ ions.
differ either in oxidation state or coordination environment, for example, Mn3+ /Mn4+ on
tetrahedral and octahedral sites) the total thermopower should simply be the weighted
kB
g i ,c y c c g j ,c y d d
, 4.9
S ln
cd
cd ln
g 1 y
q g i ,c 1 y c
j ,c d
98
Figure 4.9. Thermopower versus temperature for LiMn2O4 from this work compared with
other Mn compounds as reported by Kobayashi et al. 136 The dashed lines at -19 and -79
μV/K represent the predictions based on the high temperature limit of the Heikes formula
for Mn3+ /Mn4+ ions in equal concentration with and without a Jahn-Teller distortion
( = 5/4 or 10/4). The measured values reach -73 μV/K in good agreement with
the Heikes formula for an undistorted structure.
Here and are fractions of each ion pair, is the electron degeneracy of the ion
pairs (i,j) on sites (c,d) and is the fraction of Mn4+ on each site. Considering
99
contributions only from Mn3+ /Mn4+ pairs and assuming their equal distribution between
kB 15 10 4.10
S 0.06 ln 0.94 ln ,
q 12 4
Evaluation of this expansion indicates only a slight reduction in thermopower from -79 to
Summarizing, LiMn2O4 samples with the cubic (Fd ̅ m) spinel structure exhibit a
large n-type high temperature thermopower of -73 μV/K. This large thermopower is
larger relative to other mixed valence Mn3+ /Mn4+ compounds studied previously. The
distortion in cubic LiMn2O4 as verified by high temperature in-situ XRD. Relative to the
structure with a Jahn-Teller distortion, the orbital degeneracy is doubled. As a result, the
conduction, Mn3+ & Mn4+, is increased from 5/4 (Jahn-Teller distorted) to 10/4 (Jahn-
73 μV/K is consistent with the prediction of the thermopower from the modified Heikes
formula (-79 μV/K) using this latter value of the electron degeneracy.
100
4.3 Anti-site Disorder in LixMg1-xMn2O4 and MnxFe1-xNiyCr2-yO4
Spinels
In the remainder of this chapter we examine the effect of anti-site disorder on
compounds provide an excellent segue and to compare Jahn-Teller distortions and anti-
and y=0,1,2 spinels addressing specifically the contribution from tetrahedral sites to
thermopower in spinels. We will return to this subject in greater detail in the following
In Section 2.3.1 we derived the modified Heikes formula and observed that the
thermopower depends strongly on the carrier concentration. This can be seen clearly in
Figure 2.2. When the majority of oxide thermoelectric materials are doped with aliovalent
ions such that the oxidation state of the conducting ions is altered the carrier
concentration and thermopower are changed as well.24,136,139,140 For example, the oxygen
neutrality is accomplished by altering the mean oxidation state of the Co ions. When the
observed.140
101
An unusual deviation in the modified Heikes formula is observed for mixed
valence oxide manganates.136 In what the authors termed universal charge transport, the
ions shifting from +3.25 to +3.5 the thermopower of all the compounds approached the
same high temperature limit, -25 μV/K, corresponding to an average oxidation state of
selected candidate oxides with different Mn3+ /Mn4+ ratios. To do so we turned to the
closely related field of mixed valence oxides as cathode materials for lithium ion
batteries. In that body of work, a number of lithium manganese spinels have been
identified with variations of the mean oxidation state of manganese. As shown in Figure
4.10, a number of lithium manganese spinels with different mean oxidation states of
manganese and /or site occupancies are possible.197,202 In this work, we also show that it
altering the mean oxidation state of manganese, and therefore the fraction of Mn4+ ions.
In order to alter the Mn3+ /Mn4+ ratio in lithium manganese spinels several
strategies can be employed including: (i) doping with aliovalent ions to substitute for Li+
102
Figure 4.10. Mean oxidation state of manganese ions and Li/Mn ratio for different Li-
Mn-O compounds. Figure modified from Julien et al.202
strategies, (i) will be considered now, and the final strategy (iii) will be described in the
Single phase, dense samples of Mg1-xLixMn2O4 x=0, 1/5, 2/5 were obtained by
combustion of oxalates and CADPro sintering. The X-ray diffraction patterns are
presented in Figure 4.11. The scans were not optimized for high resolution (fast
collection times, large slits) but the peaks seem to correspond well with the cubic
structure and only the x=2/5 sample has significant peak broadening. Curiously, the peaks
contradiction to expectations based on the change ionic radii for Li+ and Mg2+ ( =0.59
103
Å, =0.57 Å, =0.76 Å, =0.66 Å). However, this observation could be
explained by a small shift displacement in the sample height during diffraction (higher
quality patterns are needed before Rietveld refinement can determine the shift
displacement).
Figure 4.11. X-Ray diffraction Mg1-xLixMn2O4 x=0, 1/5, 2/5 samples. The x=2/5 sample
has significantly broader peaks which could be due to a secondary phase.
104
An Arrhenius plot (see Figure 4.12) of the conductivity shows little change in the
slope for different Mg1-xLixMn2O4 samples so the activation energy is not changed
significantly. The x=1/5 and x=2/5 samples have comparable electrical conductivity
compared to undoped LiMn2O4, but the the x=0 sample, MgMn2O4 is lower.
In all three Mg doped samples, there is a reduced oxidation state for the
manganese cation relative to LiMn2O4. Reducing the oxidation state means the fraction of
Mn4+ is reduced which, in turn, should lead to positive thermopower values (>300 μV/K
for x=0, 110 μV/K for x=1/5 and 40 μV/K for x=2/5). However, as seen in Figure 4.12,
the thermopower of the x=0 and x=2/5 compounds reach -20 to -25 μV/K at high
temperatures in very good agreement with the distorted limit observed for other
manganese oxides whereas the x=1/5 compound reached -70 μV/K in good agreement
with the undistorted limit. It isn‘t clear why the x=1/5 compound reaches the undistorted
limit while the other two correspond to a distorted limit. High resolution optimized X-ray
diffraction is still needed, particularly at high angles, to probe for evidence of tetragonal
Jahn-Teller distortion in the x=0 and x=2/5 compounds. Nevertheless, these intriguing
results are a second evidence of the universal charge transport phenomenon first observed
concentration.
105
Figure 4.12. Thermopower (left) and electrical conductivity (right) of Mg1-xLixMn2O4
x=0, 1/5, 2/5 compounds. The thermopower of the compounds approached either the
Jahn-Teller distorted or undistorted limits despite the variation in Mn3+/Mn4+ ratio due
to doping. The activation energy is similar for all compounds, including undoped,
stoichiometric LiMn2O4.
Rietveld refinement of in-situ high temperature XRD performed in argon indicates a three
c=9.462Å, Z=4). From room temperature up to 950 K the majority phase (60-70 wt%) is
the tetragonally distorted LiMn2O4 but this phase decomposes to a mixture of Mn3O4 (60-
40 wt%) and LiMnO2 at temperatures above 950 K (see Figure 4.13). Upon cooling back
106
Figure 4.13. In-situ high temperature X-Ray diffraction pattern, measured in argon, of
LiMn2O4 after CADPro. The sample is a mixture of spinel, Mn3O4 and LiMnO2 below 973
K, at high temperatures the spinel phase decomposes entirely leaving only Mn3O4 and
LiMnO2.
to room temperature the only phase present is, again, the undistorted cubic spinel
dramatic n-type to p-type shift in thermopower of approximately 400 μV/K (see Figure
107
4.14). Above 950 K the thermopower of this specimen (measured in He) approaches
large positive values of about 400-500 μV/K agreeing well with thermopower measured
for pure Mn3O4 in air. The origin of large positive thermopower in tetragonal Mn3O4 (i.e.
assuming that the tetrahedral concentrations of Mn3+ /Mn4+ are negligible, we obtain from
Equation 2.30 the Mn4+ fraction as being <0.01. Upon cooling, the sample transformed
completely to cubic LiMn2O4. The shift in thermopower between points 1 and 4 in Figure
4.14 is attributed to the removal of the Jahn-Teller distortion after heating the reduced
material above 950 K. When the chamber is vented with air and the same sample is
remeasured the thermopower reaches the high temperature value of -79 μV/K.
provides strong evidence for the influence of mean oxidation state of manganese on
108
Figure 4.14. Electrical conductivity (left) and thermopower (right) of LiMn2O4 pressed in
a reducing atmosphere. The sample was measured in He first (circles) and upon cooling
the chamber was filled with air and the thermopower was re-measured (squares). The
numbers 1-4 are points at which quantitative X-ray phase analysis was performed to
determine weight fraction of phases. At points 1 and 2 the phase fraction is a majority
(60-70 wt%) LiMn2O4 (I41/amd) with Mn3O4 and LiMnO2 impurities. At point 3 the
LiMn2O4 has decomposed entirely to 60 wt% Mn3O4 and 40 wt% LiMnO2. The
thermopower of Mn3O4 (diamonds) is shown for comparison. Upon cooling to point 4 the
sample converts entirely to cubic spinel LiMn2O4.
the octahedral sites are considered, ignoring possible contributions from tetrahedral sites.
Whether or not this is a valid assumption that is appropriate for all spinels is a central
question that will be discussed in the next chapter. In this final section of Chapter 4,
however, we describe our initial attempt to observe a change in thermopower due only to
109
anti-site disorder on the tetrahedral site in FexMn1-xNi2-yCryO4 x=0, ¼, ½, ¾, 1 and
y=0,1,2 series.
The B site cations, Ni2+ and Cr3+, were selected as ions to occupy the octahedral
site since having 3 and 8 d-shell electrons, respectively, they would either half fill the
band or fill the band and half fill the band. These electronic configurations are
expected to be very stable and should, therefore, minimize inversion of the spinel
Therefore, if the Ni and Cr ions remain on the B site but the A site is altered from pure
contribution.
The compounds designed to have only Ni or only Cr on the B site (y=0 and y=2)
resulted in mixed phase materials with (Cr,Fe)2O3 or NiO impurities (see Figure 4.15 and
Figure 4.16).
110
Figure 4.15. X-Ray diffraction of FexMn1-xNi2O4 x=0, ¼, ½, ¾, 1 compounds. NiO
impurities are observed in all compounds.
phase spinel (see Figure 4.17). Samples featuring A site disorder (x=½, ¾) had reduced
electrical conductivity and thermal diffusivity measurements relative to the pure end
111
Figure 4.16. X-Ray diffraction of FexMn1-xCr2O4 x=0, ¼, ½, ¾, 1 compounds. (Cr,Fe)2O3
impurities are observed in all compounds but x=1.
112
Figure 4.18. Electrical conductivity (left) and thermopower (right) of FexMn1-xNiCrO4
x=0, ½, ¾, 1 compounds.
113
The reflections from FexMn1-xNiCrO4 x=0, ½, ¾, 1 spinels and NiO impurities
overlap strongly so to ensure that pure phase spinels were obtained synchrotron
diffraction was measured for the series. The results of the synchrotron diffraction, shown
in Figure 4.20, demonstrate that the samples, while being a majority spinel phase, contain
significant amounts of NiO and unreacted Ni impurities. The fraction of NiO impurities
decreases linearly from ~22% to ~10% across the series with increasing iron content (see
Figure 4.21). An unfortunate consequence of the impurities is that the composition of the
114
Figure 4.21. Weight fraction of NiO impurity in FexMn1-xNiCrO4 x=0, ½, ¾, 1
compounds determined by quantitative phase analysis using synchrotron diffraction. With
increasing iron content the NiO fraction decreases. The line is a linear best fit.
spinel phase is no longer known. For example, if the original cation ratio for MnNiCrO4is
[ ] [ ] [ ] but some of the Ni is used up to form NiO and Ni then the ratio for
the spinel has been altered [ ] [ ] [ ] and a nickel deficient spinel is obtained.
As a result of the impurities, deviation from spinel composition and unknown cation
distribution we can draw no firm conclusions in this section regarding the role of
crystal field and anti-site disorder were assessed. An argument was presented that larger
115
Another evidence of universal charge transport was observed in Mg-doped LiMn2O4
samples. Regardless of the mean oxidation state of manganese ions the thermopower of
the samples corresponded to an equal fraction of Mn3+ and Mn4+ with a Jahn-Teller
xNiCrO4 spinels were synthesized to probe whether or not anti-site disorder on the
tetrahedral site would alter thermopower. However, due to the formation of significant
NiO and Ni impurities, the chemical composition and cation distribution of the remaining
spinel phase was not known. As a result, we cannot come to any firm conclusions yet
116
117
cation distribution. This quantity, along with the spin state of the ions, is critical in
determining many optical, electrical and magnetic properties in a material and results
from energetic considerations of ionic size, charge of the cations and crystal field
determine site occupancy for atoms, but atomic scattering factors for elements with a
small difference in atomic number are very similar and greater contrast is needed.
Neutron diffraction doesn‘t involve the electron cloud so scattering lengths are more
random, but certain pairs of atom will still have too little contrast for differentiation.
Still more complicated are compounds featuring only different oxidation states of
the same element. For instance, the iron (II, III) and cobalt (II, III) oxides, Fe3O4 and
Co3O4 are amongst the most studied group of the 2,3 spinel-type oxides. Despite similar
117
compositions and elements, they form completely different structures in terms of cation
(Co2+)A[Co3+]B2O4 with Co2+ (S=3/2) and low-spin (LS) Co3+ (S=0) cations on the
tetrahedral A and octahedral B sites, respectively. On the other hand, Fe3O4, is an inverse
spinel (Fe3+)A[Fe2+Fe3+]BO4 with Fe3+ and Fe2+/Fe3+ cations on the tetrahedral A and
octahedral B sites, respectively. In practice, however, the cation distribution over the
tetrahedral and octahedral sites strongly depends on the specimen history, including
addition, the degree of cation inter-mixing between the tetrahedral and octahedral sites
increases with increasing temperature, making the correlation between the oxide structure
conditions.
The difficulty in directly measuring the cation distribution in spinels has led some
enthalpies of formations etc. are collected for pure elements and then Gibbs energy
expressions based on entropic, magnetic or other interactions are best fit to the
thermochemical data. Using these expressions the phase diagram, structure, heat capacity
and composition of binary, tertiary and even more complex compounds can be calculated.
Chen et al.,205 for example, use such a thermodynamic assessment to estimate the cation
distribution for the difficult to measure compound Co3O4 described in the previous
118
paragraph. Chen et al. did not consider the possibility that the octahedral Co3+ ions
transitioned from a LS to a HS state. This assumption may not be valid and will be
Wu and Mason206 will be analyzed and alternate models will be proposed and evaluated
using Co3O4 as a case study. Thermopower analysis will be complimented by X-ray and
neutron diffraction, Raman scattering and TGA/DTA techniques. At the end of this
chapter I will briefly describe future work and a few other possible characterization
techniques.
the cation distribution in spinels. The basic principle is that, for small-polaron
kB y 5.1
S ln .
q 1 y
When this approach was first introduced in 1981 the real innovation was recognizing that
one oxidation state in particular acts as a dopant, donating or accepting an electron, fixing
119
it, therefore, to the fraction of conducting sites in the Heikes formula. In their seminal
Fe3O4, to argue that the structure goes from inverse at low temperatures to a disordered
The Wu and Mason model makes two important assumptions. The first was that
the thermopower and electrical conductivity of the spinel oxides originates solely from
the ions occupying the octahedral B site. The second assumption was that the orbital
calculating the degeneracy term, β, outright, it has commonly been used as a variable,
fitting parameter, ranging between 1 and 2. Non-integer values have been justified by
ignoring the vibrational entropy term associated with ions surrounding the polaron site.207
One report 209 even showed that the value would have to change from β=1 at low carrier
arising from spin and orbital degeneracy as well as the degeneracy ratio between
conducting sites.111 Despite the success of Koshibae et al‘s modified Heikes formula
120
distribution in spinels: (i) incorporating the ratio of spin and orbital degeneracy terms for
each ion, and (ii) accounting for the thermopower contributions from ions on both
tetrahedral A and octahedral B sites in the spinel structure, ie in the mixed spinels.
applying four different variations of the model to examine the role of tetrahedral vs
octahedral site contributions, electron degeneracy terms and change in spin states. The
results of these calculations will be compared with cation distributions calculated from
205
equilibrium free energy thermodynamic arguments by Chen et al as well as from a
veracity of the different models because the cation distribution in Co3O4 is strongly
heat capacity and lattice parameter measurements suggest there is a high temperature
structural anomaly. A consensus has not yet been reached as to whether this high
and structure evolution from room temperature up to almost 1200 K and these results will
121
be presented in the next section prior to analyzing thermopower models for cation
distribution.
decomposes at elevated temperatures to first CoO and then to Co. The region of Co3O4
stability depends strongly on oxygen partial pressure as well as the size of Co3O4
decreasing crystallite size and increasing oxygen partial pressure. Bulk Co3O4
decomposes to CoO at 1242 and 909 K at p(O2)=1 and 10-6 bar, respectively. 213,214 There
are two studies that report significantly lower temperatures. Results of an in-situ heating
215
Infrared Emission Spectroscopy have indicated the disorder of the spinel structure
begins at 873K and its conversion to the rock-salt CoO occurs at 923 K. There is also a
high temperature X-ray diffraction study that reports that the transformation of Co3O4 to
mixture of spinel Co3O4 and rock-salt CoO. To find the optimum conditions to produce
phase-pure Co3O4, we varied the sintering conditions and different heating and cooling
rates. We found that dense samples of phase-pure spinel Co3O4 could be reproducibly
prepared by sintering at 1148 K and then slowly cooling at 1K/min. CoO was produced
if, instead of cooling slowly, the samples were quenched from 1148 K. These results are
122
Figure 5.1. X-Ray diffraction for mixed phase Co3O4 and CoO samples obtained by
intermediate cooling rates (a), pure Co3O4 obtained by slow cooling rates (b) and pure
CoO obtained by rapid quenching (c). The poor signal-to-noise ratio for CoO is due to
fluorescence from using Cu Kα radiation.
123
summarized in Figure 5.1 (a)- mixture Co3O4 and CoO after 1373 K sintering, (b)-
obtained by thermal analysis, high temperature XRD and Raman microscopy. The
thermal analysis data indicates that Co3O4 decomposition in air starts at about 1165 K and
is completed at about 1270 K (see Figure 5.2). The observed weight loss of 6.42%
corresponds well to the theoretical weight loss of 6.64% for the decomposition:
about 1084 K, and is completed by about 1200 K with the same weight loss. Upon slow
cooling, the increase in the specimen weight is observed at T < 990 K, which could be
explained through the incomplete oxidation of CoO to Co3O4 by some oxygen residuals.
The results of the XRD characterization agree very well with the results of the
thermal analysis. According to analysis of the X-ray diffraction patterns, the spinel Co3O4
remains phase-pure from room temperature up to 1163 K upon heating in air (Figure 5.3).
A non-linear increase of the lattice parameter (Table 5, Figure 5.4) above 900 K was
216
observed in agreement with reports by others who attributed this phenomenon to the
124
0 heating
A) cooling 0.20
-1
-2
Weight loss % 0.15
DTA [%/°C]
-3
0.10
-4
-5
0.05
-6
in air
0.00
-7
B) 0
heating
0.20
-1
-2
0.15
Weight loss %
DTA [%/°C]
cooling
-3
0.10
-4
-5
0.05
-6
in Ar
0.00
-7
125
Figure 5.3. X-Ray diffraction of Co3O4 from room temperature up to 1273 K. The sample
converts entirely to CoO between the scan at 1163 K and 1273 K.
126
Figure 5.4. Temperature dependence of the lattice parameter of Co3O4 (left-hand side y
axis) and CoO (right-hand side y axis) determined from in-situ XRD. The scale of both y-
axes are the same, but whereas the CoO lattice parameter increases linearly through all
temperatures the Co3O4 lattice parameter rapidly increased above 900 K.
(see Figure 5.3). This decomposition was reversible: upon cooling in air the CoO
oxidized back to phase-pure spinel Co3O4. The results of the quantitative Rietveld
analysis results for phase fraction, lattice parameter and oxygen fractional parameter, u,
are recorded in Table 5. Specimens quenched to retain the rock-salt CoO phase remain
127
Table 5. Structure refinement details for Co3O4 as a function of temperature (standard
deviations in shift shown beneath in parenthesis)
stable in air on heating from room temperature until 950-1188 K, when they are oxidized
to Co3O4, and then above 1188 K Co3O4 again decomposes reversibly to 100 wt% CoO.
The results of the Raman spectroscopy (Figure 5.5) are also consistent with the
microstructure of the specimens. The Co3O4 exhibit four Raman active modes: A1g (689
cm-1), F2g (619 and 521 cm-1) and Eg (481 cm-1).217 The high frequency A1g mode is due
to vibrational mode of octahedral cations, whereas Eg and F2g modes are determined by
both tetrahedral and octahedral cations. With increasing temperature on heating from
128
1.7E4
A) 2000
400
1500
1300
1200
Co3O4 1000
600
1800
800.0
525.0
800
1600 250.0
120.0
dis-Co3O90.00
[K]
4
Tempearture [ C]
1200
1400 80.00
0
30.00
Temperature
1000 dis-Co3O4
800
600 Co3O4
400
973 K
873 K
773 K
673 K
573 K
473 K
373 K
298 K
Figure 5.5. Raman shift plotted as a function of temperature for Co3O4 samples. The five
peaks resulting from the spinel modes disappear above 800 K before transformation to
CoO rock-salt which has no Raman active modes.
129
room temperature, the Co3O4 bands shifts towards lower energies and broaden, consistent
with thermal expansion. Above about 850 K the peaks undergo significant broadening
possess a different Raman response due to a difference in length and covalency of Co-O
CoO, a NaCl type crystal having no first-order Raman active modes. Despite this, several
reports 218–220 observe peaks at 475, 515, 555, 600 and 680 cm-1 yet, in our study, even at
the highest temperatures these peaks are not observed. These processes are reversible (in
air) and the temperature range of hysteresis is the same as observed by X-ray diffraction.
(418-540 K), 0.15±.05 eV (540-890 K) and 2.28±.06 eV (890-1188 K). These are
reasonable agreement with the reported hopping energy value 0.17 eV and optical
130
Figure 5.6. Arrhenius plot of electrical conductivity for Co3O4 showing three
characteristic regions each with a different activation energy. Above 1188 K Co3O4
decomposes to CoO, a band conductor.
and helium) is shown in Figure 5.7 together with that of CoO (measured in helium only).
A key feature of the graph is that the spinel Co3O4 samples measured in air and helium
both have a maximum in thermopower of ~600 μV/K around 850 K. Above this
temperature, the thermopower decreases almost linearly with increasing temperature until
it abruptly increases with a slope of ~250 μV/K in air and ~400 μV/K in helium. The
131
region of approximately linear decrease correlates to inversion of the spinel structure as
will be discussed in the next section. The discontinuity in thermopower (1188 K in air
132
and 1150 K in He) is attributed to the decomposition of the spinel Co3O4 to rock-salt CoO
decreasing thermopower indicates that electron hopping (with much higher mobility)
At low temperatures the octahedral sites in Co3O4 are occupied almost entirely by
Co3+ ions, consistent with the normal spinel structure. Neutron diffraction 223
and
224
magnetic susceptibility measurements suggest that the d 6 electrons from Co3+ pair up
7
to completely fill the octahedral t2g triplet whereas the d electrons from Co2+ on the
tetrahedral site leave 3 unpaired electrons on t2g orbitals that account for the observed
These structural findings have been attributed to two main effects. The first is a
Co3+ and octahedral sites by Co2+. The consequence of this being that electron exchange
(hopping) can occur between octahedral Co2+ and Co3+ ions, resulting in increased
transfer from Co2+ to Co3+ (hopping) which is also involved in the explanation of
133
CoA2+ (eg4t2g3) + CoB3+ (LS, t2g6eg0) ↔ CoA3+ (LS, eg4t2g2) + CoB2+ (LS, t2g6eg1) 5.3
The effective change of the lattice parameter due to the reaction in Equation 5.3 is not
obvious and required detailed calculation which will be performed below. The question
arises also about the spin state of Co2+/Co3+ on tetrahedral and octahedral sites.
The second effect that could, in part, explain our structural findings is an unpairing of the
with possible formation of an intermediate state, IS- CoB3+, with t2g5eg1, S=1.221
The HS and IS CoB3+ ions have a larger ionic radius compared to that of LS
CoB3+. The result is an increase in the lattice parameter as well as in the thermal
thermopower and electrical conductivity require detailed analysis which will be described
below. Due to the partially filled eg level, the IS state is Jahn-Teller active, creating
distortions of (CoO6) octahedral (see Figure 4.2), associated with lattice distortion and
that the tetrahedral sites will be occupied by both Co2+ (eg4t2g3) and Co3+ (eg4t2g2) cations,
whereas the octahedral sites will be occupied by LS Co2+ (LS, t2g6eg1), and Co3+ (d 6) in
134
different spin states. Furthermore, as disordering begins at the lower temperatures, it is
most probable that the spin state changes from LS (t2g6eg0), to HS (t2g4eg2) at the highest
135
5.2 Alternate Thermopower Cation Distribution Models
elevated temperatures through the distribution of Co2+/Co3+ cations in different spin states
thermopower measurements made on Co3O4 sample measured in air (circles, Figure 5.7)
that Co3O4 changes from a normal spinel below 850 K to a disordered spinel, CoO3
=0.64, (complete disorder would be 2/3 ≈ 0.67) at the highest temperature before the
onset of the spinel to rock-salt transformation. The gradual inversion of the structure is
accompanied by transferring an electron from the B to the A site. The result is that on the
6 7
octahedral site there is a mixture of low spin d and d electrons with the Co2+ ion
behaving as an electron donor because of its unpaired electron occupying the eg orbital
(see Table 6). The results from these calculations will be compared to the predicted
later.
136
(i) The original model proposed by Wu and Mason,206 in which only hopping
both orbital and spin degeneracy ratio for Co2+ and Co3+,
Case (i). According to Wu and Mason‘s model, the thermopower for small-
concentration CoO3 . Combining the spin degeneracy, 2S+1, and orbital degeneracy, 2,
the degeneracy term of this electron donor should have the value of β = 2*(2*1/2+1) = 4.
inconsistent results for the cation distribution. If we treat β as a fitting parameter, as some
previous authors have done, and select its value arbitrarily to be 2, then there is closer
would be reasonable if there were a Jahn-Teller distortion of the B site since the orbital
degeneracy of the eg levels would be lifted and the resulting degeneracy would, in fact, be
2. However, there is no evidence of such a Jahn-Teller distortion either in our own XRD
results or in the literature so the β=2 assignment appears to have no physical basis. It is
also appropriate to mention that the distribution determined by Chen et al. using free
energy arguments assumes equilibrium conditions but our samples might not have been
137
Therefore, the low temperature deviation could be the result of kinetically limited anti-
site disorder. At higher temperatures the thermopower determined distribution and the
inflection.
Figure 5.8. Fractional occupancy of Co3+ ions on the octahedral sublattice in Co3O4 as a
function of temperature calculated from the thermopower measurements of Co3O4 in air
according to case (i) (squares), case (ii) (circles) and case (iii) (diamonds) described in
detail in the text. The solid line is the distribution calculated thermodynamically from
free energies according to the procedure given by Chen et al.205 Good agreement with
Chen et al.’s prediction was found for case (iii), proposed in this work with no variable
parameters. Case (i) also had good agreement, but only when an unphysical value of 2
was selected for β.
138
Case (ii). Following Koshibae et al.‘s treatment, the spin and orbital degeneracy
of both octahedral Co2+ and Co3+ ions are needed to calculate the thermopower. The Co2+
ion degeneracy is the same as in case (i), 4. The Co3+ d 6 electrons are low spin and fully
occupy the t2g orbital so the degeneracy is 1*(2*0+1)=1 and the degeneracy term β is 4/1.
The thermopower expression is the same as Equation 5.1. This distribution result is not
stoichiometrically consistent though because it requires more Co2+ ions on the octahedral
site than are initially present in the formula unit; the CoO2 fraction cannot be larger than
minimal oxygen vacancies is a good approximation for Co3O4 because the Seebeck
coefficient and electrical conductivity in Co3O4 do not depend on oxygen partial pressure
222
( PO ).
2
Furthermore, thermal analysis data do not show any weight loss below ~ 1100
K in Ar and the oxygen loss at the CoO transformation agrees very well with the
occurs, the fraction of CoO2 calculated from thermopower according to case (ii) was
already slightly greater than ½ (see Figure 5.9) and it‘s possible, even likely, that the
Interestingly, Mocala et al., using heat capacity measurements at different oxygen partial
temperature.211 They found that in air the transformation took place at 1180 K whereas at
1 bar of oxygen it had increased to 1240 K. Furthermore, Mocala et al. calculates that in
139
Figure 5.9. Thermopower plotted against fractional occupancy of Co3+ ions on the
octahedral sub-lattice according to cases (i), (ii) and (iii) described in detail in the text.
The vertical dotted line represents the minimum possible value for CoO3 in Co3O4. As
thermopower decreases due to inversion both cases (i) and (ii) approach the limit of
Co =0, in violation of stoichiometry in the spinel structure, but case (iii) remains
3
O
consistent.
20 bars of oxygen, the transformation would increase up to 1350 K. Because our values
of the thermopower decreased linearly with increasing temperature up until the abrupt
would continue to decrease further if larger oxygen pressures can suppress the
140
transformation. Then, both case (i) and case (ii), based on the thermopower contribution
from the B site only using Wu et al.‘s and Koshibae et al.‘s models, respectively, would
be stoichiometrically inconsistent. This finding, shown in Figure 5.9, is perhaps the most
significant in the entire chapter because it provides evidence that Wu et al.‘s original
Case (iii). In this third case, we consider the thermopower contributions from both
A and B sites in the spinels using Koshibae et al.‘s thermopower expression. An earlier
226
analysis by Dieckmann et al. also considered octahedral, tetrahedral and octahedral-
As described in case (i), inversion creates electron donors on the octahedral site.
Interestingly, as suggested by Koumoto and Yanagida, 222 it is also possible that inversion
simultaneously hole dopes the tetrahedral site. Tetrahedral Co2+ ions are high spin with 3
unpaired electrons in the t2g triplet, but tetrahedral Co3+ ions have only 2 unpaired
electrons in the t2g triplet. The contribution to thermopower from multiple, parallel
O S O T ST 5.5
S total
total
141
where O , T , S O and S T , are the contributions to electrical conductivity and
us take Fe3O4 (magnetite) case. The results of Mössbauer spectroscopy of Fe3O4 at low
temperatures have been attributed to electron hopping along the octahedral sub-lattice.227
A later detailed analysis by Dieckmann et al. 226 separated octahedral sub-lattice hopping
contributions to electrical conductivity from other mechanisms and found that octahedral
conduction exists at high temperatures. We attempted a similar analysis for Co 3O4 , but is
not straight forward because Co3O4, unlike Fe3O4, which is a half-metal, is a large band-
gap semiconductor and the onset of thermally activated carriers masks the contribution of
proportional to the site fraction such that 1/3 of the conduction occurs via tetrahedral
conduction and 2/3 via octahedral conduction, then we can use Equation 5.5 to express
142
S total 2 S O 1 S T 5.6
3 3
The thermopower due to conduction on the octahedral sites would be the same as in case
(ii) and the thermopower due to the tetrahedral sites would depend on the degeneracies of
high spin Co2+ and Co3+ in tetrahedral coordination, 4 and 10 respectively. Because the
carriers are now holes, instead of electrons, the sign is also changed to give
kB 4 y 5.7
ST ln
q 10 1 y
with x now being CoT3 .
When we calculate the cation distribution for case (iii) we can make two
observations. The first is that it agrees with Chen et al.‘s at least as well as case (i) which
relied on an arbitrary β parameter. The second is that, even if the thermopower were to
continue to decrease with temperature, as we predict it will for samples measured in high
Co would approach the high temperature limit of 0.5, in agreement with Chen et al.‘s
2
O
prediction of the site occupancies. To illustrate this important point, consider Figure 5.9
where we have plotted thermopower against CoO3 based on each of the three cases
considered here. The lowest value for thermopower measured in air on our samples was
109 μV/K and the lowest permissible thermopower values according to Koshibae et al‘s
and Wu and Mason‘s models are 119 and 60 μV/K, respectively. In contrast to both of
143
these we see that in our model the CoO2 fraction approaches the physically sensible
Finally, we discuss what the change in spin state could play in determining the
accounts for the change in spin state from low-spin (LS) to high spin (HS) of Co4+ ions in
111
NaCo2O4 with increasing temperature There has been extensive debate regarding the
spin-state of Co3+ in Co3O4 at high temperatures. Chen et al. argue that measurements as
diverse as lattice parameter,212,216,228 EMF of oxygen potential,212,229 and heat capacity 211
all indicate a high temperature anomaly at 1120 K, just before Co3O4 transforms to CoO.
The general consensus is that a second-order transition from LS-HS in Co3+ ions takes
place at this temperature. However, the effect this would have on cation distribution is
still debated. Kale, for example, suggests an inverse spinel,212 Liu and Prewitt favor a
disordered spinel and Mocala et al. suggests a normal spinel with only 5-10% disorder.216
A change in spin-state alters the spin and orbital degeneracy and would, therefore,
affect the thermopower analysis of case (ii) and (iii) in equation 2. For example, the
octahedral Co3+ degeneracy term increases from 1 to 15 with a LS→HS transition. The
cation distribution calculated according to case (ii) and (iii) assuming different spin-states
for octahedral cobalt cations is shown in Figure 5.10. We note that when a Co3+ LS→HS
144
Figure 5.10. Fractional occupancy of Co3+ ions on the octahedral sublattice in Co3O4 as
a function of temperature accounting for different cobalt ion spin states in case (ii) (a)
and case (iii) (b). In case (iii) a fixed high spin state for Co2+ and Co3+ ions on the
tetrahedral site was assumed.
case (ii). For case (iii) a stoichiometrically consistent cation distribution is found
regardless of the spin state of Co3+, but the scenario closest to Chen et al‘s prediction is if
Figure 5.10 also shows that neither case (ii) nor case (iii) can account for the
thermopower produced by a spin state transition alone; both cases show that a minimum
of ~20% anti-site disorder on the B site must be present. Mocala et al. showed the
anomaly in heat capacity, reportedly due to Co3+ LS→HS, begins at 1000 K and
continues until decomposition. Figure 5.7 shows that thermopower decreases at an almost
continuous rate from 900 K until transformation to CoO with no abrupt change that could
be attributed to a high temperature change in spin state. For these reasons, we cannot
145
conclude definitively that a change in spin state occurs; it remains possible that the
magnetic susceptibility and determination of the magnetic structure over the temperatures
850-1200 K. Unfortunately, such measurements have not yet been performed for Co3O4
are underway. In their absence, other than Chen et al.‘s prediction of the ion distributions,
which is limited to the assumption that no change in spin state occurs, there are very few
remaining methods whereby we can compare and assess the three cases described here. In
this section we describe one method to evaluate the thermopower cation distribution
determine what combination of Co2+ and Co3+ ionic radii could account for the observed
bond lengths.
The fraction of Co3+ on the octahedral site, CoO3 , was calculated using the bond
lengths of tetrahedral and octahedral ions 216 ( and ) as well as the coefficient of
230
thermal expansion reported previously for Co3O4 up to high temperatures. The ionic
149
radii for Co2+, Co3+ and O2- were used at room temperature and the coefficient of
146
thermal expansion for Co3O4 was applied for non-ambient calculations. The distribution
and
rOCo 2 (T ) rB O (T ) r O 2 (T ) 5.9
Co 3
O
(T ) B O bond
r Co 2
(T ) r Co 3
(T )
O O
where rTCo,O2 and rTCo,O3 are the ionic radii for Co2+ or Co3+ in tetrahedral or octahedral
coordination respectively (HS rTCo 2 , LS rOCo 2 , HS rTCo 3 , LS rOCo 3 ), rAO and rBO are
for O2-. Both sets of data show the same trend in approximate agreement with the cation
distribution calculated in cases (i), (ii) and (iii); i.e. little or no inversion at lower
temperatures and the onset of inversion above ~1000 K. At 1200 K the octahedral bond
length data suggests complete inversion while the tetrahedral bond length data indicates a
random distribution of cations ( CoO3 =62%). However, due to the large variation in
bond lengths, we are unable to use the bond length data and the analysis of the ionic radii
147
Figure 5.11. Fractional occupancy of Co3+ ions on the octahedral sub-lattice in Co3O4
as a function of temperature calculated from ionic radii and tetrahedral (brown squares
with circles) and octahedral (squares with triangles) bond lengths. The cation
distribution calculated from thermopower measurements according to cases (i), (ii) and
(iii) are shown for comparison. The low precision in the bond length data is due to
variation in the bond lengths determined by the Rietveld refinement of high temperature
X-ray diffraction data.
In addition the inconclusive bond lengths analysis of cation distribution there are
a few other techniques and measurements that could offer insight to cation distribution
and the high temperature anomaly in Co3O4. Measuring the magnetic moment via
148
vibrating sample magnetometry (VSM) could be performed in-situ at high temperatures
or on rapidly quenched samples from a range of high temperatures in hopes that the high
temperature electronic and crystal structure would be preserved. The electron binding
energies of Co2+ and Co3+ ions in octahedral and tetrahedral sites are not the same so X-
ray photoelectron spectroscopy (XPS) could be used to measure the kinetic energy and
number of electrons from a material being irradiated by X-rays. The spectra could ideally
be deconvoluted and fractional occupancy of Co2+ and Co3+ ions on octahedral and
absorption fine structure (EXAFS) or X-ray magnetic circular dichroism (XMCD) could
rapidly quench samples and then to perform neutron diffraction and examine the
magnetic structure. The preliminary results from this last method will be described in turn
Refining the magnetic structure from neutron diffraction data provides a very
clear measure of the magnetic moment on each site in Co3O4. The magnetic moment, in
turn, is tied to the occupancy of Co2+ and Co3+ ions on a site. For example, Co2+ on a
tetrahedral site has 3 unpaired electrons but introducing a high spin Co3+ ion on the
tetrahedral site increases the magnetic moment because it has 4 unpaired electrons. As a
result, determining the magnetic structure of Co3O4 as the structure undergoes inversion
could be used to quantify the cation distribution. However, the temperatures where
149
inversion of the structure occurs are very high (800-1200 K) and Co3O4 transitions to an
Above the Néel temperature, cobalt oxide is paramagnetic and no magnetic structure
reflections are observed; thus, the magnetic structure can only be determined below 40 K.
Our approach to this problem was to heat a series of Co3O4 powders to various
temperatures (298 K, 773 K, 823 K, 873 K, 923 K, 973 K, 1023 K, 1073 K, 1123 K and
1173 K). The samples were then rapidly quenched in hopes that the high temperature
electronic and crystal structure would be preserved and the magnetic structure of the
sample above (300 K) and below (1.5 K) its magnetic ordering temperature are shown in
Figure 5.12 to exhibit the additional scattering from the magnetic structure.
The increase in peak intensity below the Néel temperature is not uniform for all
reflections because the magnetic structure and the crystal structure belong to different
space groups (F-43m and Fd3m symmetry respectively, see Figure 5.13). For example,
the (200) and (420) peaks are principally magnetic reflections, (220) and (224) are
principally nuclear reflections and the remainder have contributions from both. Normal
spinel Co3O4 has no magnetic moment on the octahedral B site because Co3+ is low spin
and has no unpaired electrons. The magnetic structure arises only from the unpaired
150
Figure 5.12. Neutron diffraction patterns for Co3O4 powder quenched from 1173 K
measured at 300 K and 1.5 K. Nuclear, magnetic and combined reflections are indicated.
151
Figure 5.13. (left) Paramagnetic structure of Co3O4. Though both A and B sites have
cobalt ions, for clarity, the B site octahedra are shown in blue and A site tetrahedra are
shown in teal. (right) Antiferromagnetic structure of Co3O4. B site octahedra are omitted
because they contain no magnetic moment. Red and blue arrows indicate up and down
spins.
Refinement of the magnetic structure for these neutron diffraction patterns is not
yet complete. Nevertheless, when the patterns of samples quenched from all temperatures
are compared, as in Figure 5.14, there are two important observations. First, the patterns
from samples quenched at high temperatures have sharper peaks and, second, the ratio of
former can be understood by the powders coarsening at elevated temperatures. The latter
observation could be evidence of inversion and is quantified in Figure 5.15 where the
152
Figure 5.14. Neutron diffraction patterns measured at 1.5 K for Co3O4 samples quenched
from various temperatures.
ratio of the magnetic (200) reflection to the nuclear (220) and (224) reflections is plotted
As was portrayed in Table 6, inversion of Co3O4 introduces high spin Co3+ ions
featuring four unpaired electrons on the tetrahedral site. This extra unpaired electron
relative to high spin Co2+ ions normally on the site should increase the magnetic moment
and, therefore, the magnetic scattering. While anti-site disorder qualitatively describes the
153
Figure 5.15. Ratio of peak intensity for magnetic (200) and nuclear reflections (220) and
(224). The ratios increase as a function of temperature possibly indicating that the
magnetic moment on the A site is increasing with temperature.
trend in Figure 5.15, inversion would also require that Co2+ ions with unpaired electrons
occupy the octahedral site which should result in a magnetic moment on the B site. The
effect of a small B site magnetic moment on the low temperature neutron diffraction
patterns will be considered in detail when the Rietveld refinement of the magnetic
structure is complete.
154
Clearly there remain ample opportunities for further research in both the high
5.4.2 Conclusions
diffraction and Raman scattering of Co3O4 spinel are reported from room temperature up
to 1273 K. The electrical conductivity was consistent with other large band gap
mechanism. The thermopower reaches a maximum value (~600μV/K) around 850 K and
above this temperature it decreased almost linearly with temperature to 109 μV/K at
which the structure transformed to CoO rock-salt. The temperature of this transformation
The thermopower correlates to partial inversion of the Co3O4 structure, but the
change in spin state is considered using either Wu and Mason‘s original model case (i),
or one based on Koshibae et al.‘s modified thermopower expression, case (ii). When
doped tetrahedral sites are included, taking into account both spin and orbital degeneracy
as well as the ratio of these degeneracies, case (iii), a stoichiometrically consistent cation
distribution is observed with or without a change in spin state. This case agrees well with
Chen et al.‘s cation distribution calculated from free energies as well as the distribution
155
calculated from bond lengths and ionic radii. Neutron diffraction of Co3O4 powders
quenched from elevated temperatures indicate that the magnetic moment on the A site is
temperature. In agreement with previous studies, the spinel lattice parameter is unusually
large at values above 900 K. While this has been attributed to a change in spin-state of
the Co3+ ions by other authors, the thermopower results in the current work show no
156
157
The Seebeck coefficient is a direct probe of the carrier type and entropy per
correlated systems in particular, the strong coupling between charge, orbital, spin and
lattice degrees of freedom means that thermopower is intimately related to the crystal
structure in a way that neither electrical or thermal conductivity alone would be. The
relationships, but also to apply them towards developing better oxide thermoelectric
accomplishment resulting from our experiments was identifying that Mn oxides need not
oxide, LiMn2O4, was identified that had thermopower values over 3 times greater than
is still prohibitively large for practical thermoelectric application this study represents an
157
performing p-type counterparts. There is outstanding potential to use thermopower as a
probe of crystal structure in materials and to apply the modified Heikes formula as a
For over 30 years thermopower has been used as a valuable tool to characterize
high temperature structural anomaly of Co3O4. The experiments offer evidence that Wu
et al.‘s model could be altered to include Koshibae et al.‘s modified Heikes formula and
have demonstrated in this thesis, the complex structural and magnetic transitions (anti-
site disorder, spin unpairing etc) possible in spinels make use of thermopower to
158
References
1. Hubbert, M. K. Nuclear Energy and the Fossil Fuels. Spring Meeting of the Southern
District, American Petroleum Institute, Plaza Hotel, San Antonio, Texas, March 7–8-
9 (1956).
5. Lawrence Livermore National Laboratory Estimated U.S. Energy Use in 2010: 98.0
Quads. (2011).at
<https://flowcharts.llnl.gov/content/energy/energy_archive/energy_flow_2010/LLNL
USEnergy2010.pdf>
10. Robert D. Abelson et al. Enabling Exploration with Small Radioisotope Power
Systems. 04-10, (2004).
159
14. Tamer F., R., William S., L. & Russell J. De, Y. Temperature Control of Avalanche
Photodiode Using Thermoelectric Cooler. (NASA Langley Technical Report Server:
1999).
17. Snyder, G. J. & Ursell, T. S. Thermoelectric Efficiency and Compatibility. Phys. Rev.
Lett. 91, 148301 (2003).
21. Desai, A. V., Jhirad, D. & Munasinghe, M. Nonconventional energy. (New Age
International: 1990).
22. Ginley, D. S. & Cahen, D. Fundamentals of Materials for Energy and Environmental
Sustainability. (Cambridge University Press: 2011).
25. Snyder, G. J. & Toberer, E. S. Complex thermoelectric materials. Nat Mater 7, 105–
114 (2008).
27. Toberer, E. S., May, A. F. & Snyder, G. J. Zintl Chemistry for Designing High
Efficiency Thermoelectric Materials. Chemistry of Materials 22, 624–634 (2009).
28. Nolas, G. S., Poon, J. & Kanatzidis, M. Recent Developments in Bulk Thermoelectric
Materials. MRS Bulletin 31, 199–205 (2006).
160
29. Sales, B. C., Mandrus, D. & Williams, R. K. Filled Skutterudite Antimonides: A New
Class of Thermoelectric Materials. Science 272, 1325 –1328 (1996).
30. Feldman, J. L., Singh, D. J., Mazin, I. I., Mandrus, D. & Sales, B. C. Lattice
dynamics and reduced thermal conductivity of filled skutterudites. Phys. Rev. B 61,
R9209–R9212 (2000).
31. Caillat, T., Fleurial, J.-P. & Borshchevsky, A. Preparation and thermoelectric
properties of semiconducting Zn4Sb3. Journal of Physics and Chemistry of Solids 58,
1119–1125 (1997).
32. Nolas, G. S., Slack, G. A., Morelli, D. T., Tritt, T. M. & Ehrlich, A. C. The effect of
rare‐earth filling on the lattice thermal conductivity of skutterudites. Journal of
Applied Physics 79, 4002–4008 (1996).
33. Callaway, J. & von Baeyer, H. Effect of Point Imperfections on Lattice Thermal
Conductivity. Physical Review 120, 1149–1154 (1960).
36. Qu, Z., Sparks, T. D., Pan, W. & Clarke, D. R. Thermal conductivity of the
gadolinium calcium silicate apatites: Effect of different point defect types. Acta
Materialia 59, 3841–3850 (2011).
39. Shen, Y., Clarke, D. R. & Fuierer, P. A. Anisotropic thermal conductivity of the
Aurivillus phase, bismuth titanate (Bi4Ti3O12): A natural nanostructured
superlattice. Applied Physics Letters 93, 102907 (2008).
40. Sparks, T. D., Fuierer, P. A. & Clarke, D. R. Anisotropic Thermal Diffusivity and
Conductivity of La‐Doped Strontium Niobate Sr2Nb2O7. Journal of the American
Ceramic Society 93, 1136–1141 (2010).
41. Koumoto, K., Terasaki, I. & Funahashi, R. Complex Oxide Materials for Potential
Thermoelectric Applications. MRS Bulletin 31, 206–210 (2006).
161
42. Terasaki, I., Sasago, Y. & Uchinokura, K. Large thermoelectric power in NaCo2O4
single crystals. Phys. Rev. B 56, R12685–R12687 (1997).
43. Wan, C., Sparks, T. D., Wei, P. & Clarke, D. R. Thermal Conductivity of the
Rare‐Earth Strontium Aluminates. Journal of the American Ceramic Society 93,
1457–1460 (2010).
45. Gascoin, F., Ottensmann, S., Stark, D., Haïle, S. M. & Snyder, G. J. Zintl Phases as
Thermoelectric Materials: Tuned Transport Properties of the Compounds CaxYb1–
xZn2Sb2. Advanced Functional Materials 15, 1860–1864 (2005).
46. Zhu, P. et al. Enhanced thermoelectric properties of PbTe alloyed with Sb2Te3.
Journal of Physics: Condensed Matter 17, 7319–7326 (2005).
53. Hsu, K. F. et al. Cubic AgPbmSbTe2+m: Bulk Thermoelectric Materials with High
Figure of Merit. Science 303, 818 –821 (2004).
54. Caylor, J. C., Coonley, K., Stuart, J., Colpitts, T. & Venkatasubramanian, R.
Enhanced thermoelectric performance in PbTe-based superlattice structures from
reduction of lattice thermal conductivity. Applied Physics Letters 87, 023105–
023105–3 (2005).
162
55. Hochbaum, A. I. et al. Enhanced thermoelectric performance of rough silicon
nanowires. Nature 451, 163–167 (2008).
56. Beyer, H. et al. High thermoelectric figure of merit ZT in PbTe and Bi2Te3-based
superlattices by a reduction of the thermal conductivity. Physica E: Low-dimensional
Systems and Nanostructures 13, 965–968 (2002).
58. Cahill, D. G., Watson, S. K. & Pohl, R. O. Lower limit to the thermal conductivity of
disordered crystals. Phys. Rev. B 46, 6131–6140 (1992).
60. Cook, B. A., Kramer, M. J., Wei, X., Harringa, J. L. & Levin, E. M. Nature of the
cubic to rhombohedral structural transformation in (AgSbTe2)15(GeTe)85
thermoelectric material. Journal of Applied Physics 101, 053715–053715–6 (2007).
63. Kim, W. et al. Thermal Conductivity Reduction and Thermoelectric Figure of Merit
Increase by Embedding Nanoparticles in Crystalline Semiconductors. Phys. Rev. Lett.
96, 045901 (2006).
64. Yao, T. Thermal properties of AlAs/GaAs superlattices. Applied Physics Letters 51,
1798–1800 (1987).
66. Kim, H. J., Božin, E. S., Haile, S. M., Snyder, G. J. & Billinge, S. J. L. Nanoscale α-
structural domains in the phonon-glass thermoelectric material β-Zn4Sb3. Phys. Rev.
B 75, 134103 (2007).
67. Winter, M. R. & Clarke, D. R. Oxide Materials with Low Thermal Conductivity.
Journal of the American Ceramic Society 90, 533–540 (2007).
163
68. Kanatzidis, M. G. Structural Evolution and Phase Homologies for ‗Design‘ and
Prediction of Solid-State Compounds. Acc. Chem. Res. 38, 359–368 (2004).
69. Kauzlarich, S. M., Brown, S. R. & Snyder, G. J. Zintl phases for thermoelectric
devices. Dalton Trans. 2099–2107 (2007).doi:10.1039/B702266B
70. Gordon B. Haxel, James B. Hedrick & Greta J. Orris Rare Earth Elements—Critical
Resources for High Technology | USGS Fact Sheet 087-02. U.S. Geological Survey
Fact Sheet 087-02 at <http://pubs.usgs.gov/fs/2002/fs087-02/>
71. U.S. Department of Energy U.S. Department of Energy Critical Materials Strategy.
(2010).
72. Michael George USGS Minerals Information: Selenium and Tellurium. USGS 2010
Minerals Yearbook (2010).at
<http://minerals.usgs.gov/minerals/pubs/commodity/selenium/>
74. Mott, N. F. & Jones, H. The Theory of the Properties of Metals and Alloys. (Dover
Publications: 1958).
75. Nolas, G. S., Sharp, J. & Goldsmid, J. Thermoelectrics: Basic Principles and New
Materials Developments. (Springer: 2001).
77. Hicks, L. D., Harman, T. C., Sun, X. & Dresselhaus, M. S. Experimental study of the
effect of quantum-well structures on the thermoelectric figure of merit. Phys. Rev. B
53, R10493–R10496 (1996).
164
81. DiSalvo, F. J. Thermoelectric Cooling and Power Generation. Science 285, 703 –706
(1999).
83. Tsubota, T., Ohtaki, M., Eguchi, K. & Arai, H. Thermoelectric properties of Al-
doped ZnO as a promising oxide material for high-temperature thermoelectric
conversion. Journal of Materials Chemistry 7, 85–90 (1997).
84. Cai, K. F., Müller, E., Drašar, C. & Mrotzek, A. Preparation and thermoelectric
properties of Al-doped ZnO ceramics. Materials Science and Engineering: B 104,
45–48 (2003).
85. Bhosle, V., Tiwari, A. & Narayan, J. Metallic conductivity and metal-semiconductor
transition in Ga-doped ZnO. Applied Physics Letters 88, 032106 (2006).
86. Singh, S. et al. Structure, microstructure and physical properties of ZnO based
materials in various forms: bulk, thin film and nano. Journal of Physics D: Applied
Physics 40, 6312–6327 (2007).
87. Guilmeau, E., Maignan, A. & Martin, C. Thermoelectric Oxides: Effect of Doping in
Delafossites and Zinc Oxide. Journal of Electronic Materials 38, 1104–1108 (2009).
88. Jood, P. et al. Al-Doped Zinc Oxide Nanocomposites with Enhanced Thermoelectric
Properties. Nano Letters 11, 4337–4342 (2011).
89. Maignan, A., Martin, C., Damay, F., Raveau, B. & Hejtmanek, J. Transition from a
paramagnetic metallic to a cluster glass metallic state in electron-doped perovskite
manganites. Phys. Rev. B 58, 2758–2763 (1998).
90. Hejtmánek, J. et al. Interplay between transport, magnetic, and ordering phenomena
in Sm1-xCaxMnO3. Phys. Rev. B 60, 14057–14065 (1999).
91. Raveau, B., Zhao, Y. M., Martin, C., Hervieu, M. & Maignan, A. Mn-Site Doped
CaMnO3: Creation of the CMR Effect. Journal of Solid State Chemistry 149, 203–
207 (2000).
165
94. Flahaut, D., Funahashi, R., Lee, K., Ohta, H. & Koumoto, K. Effect of the Yb
substitutions on the thermoelectric properties of CaMnO3. 25th International
Conference on Thermoelectrics, 2006. ICT ’06 103–106
(2006).doi:10.1109/ICT.2006.331291
97. Park, J. W., Kwak, D. H., Yoon, S. H. & Choi, S. C. Thermoelectric properties of Bi,
Nb co-substituted CaMnO3 at high temperature. Journal of Alloys and Compounds
487, 550–555 (2009).
98. Populoh, S., Trottmann, M., Aguire, M. H. & Weidenkaff, A. Nanostructured Nb-
Substituted CaMnO3 N-Type Thermoelectric Material Prepared in a Continuous
Process by Ultrasonic Spray Combustion. Journal of Materials Research 26, 1947–
1952 (2011).
99. Okuda, T., Nakanishi, K., Miyasaka, S. & Tokura, Y. Large thermoelectric response
of metallic perovskites: Sr1-xLaxTiO3 (0<~x<~0.1). Phys. Rev. B 63, 113104
(2001).
100. Muta, H., Kurosaki, K. & Yamanaka, S. Thermoelectric properties of rare earth
doped SrTiO3. Journal of Alloys and Compounds 350, 292–295 (2003).
102. Ohta, S., Nomura, T., Ohta, H. & Koumoto, K. High-temperature carrier transport
and thermoelectric properties of heavily La- or Nb-doped SrTiO3 single crystals.
Journal of Applied Physics 97, 034106–034106–4 (2005).
104. Edwards, P. P., Porch, A., Jones, M. O., Morgan, D. V. & Perks, R. M. Basic
materials physics of transparent conducting oxides. Dalton Transactions 2995
(2004).doi:10.1039/b408864f
105. Bizo, L., Choisnet, J., Retoux, R. & Raveau, B. The great potential of coupled
substitutions in In2O3 for the generation of bixbyite-type transparent conducting
oxides, In2−2xMxSnxO3. Solid State Communications 136, 163–168 (2005).
166
106. Bérardan, D., Guilmeau, E., Maignan, A. & Raveau, B. In2O3:Ge, a promising n-
type thermoelectric oxide composite. Solid State Communications 146, 97–101
(2008).
107. Berardan, D., Guilmeau, E., Maignan, A. & Raveau, B. Enhancement of the
thermoelectric performances of In2O3 by the coupled substitution of M2+/Sn4+ for
In3+. Journal of Applied Physics 104, 064918–064918–5 (2008).
108. Guilmeau, E. et al. Tuning the transport and thermoelectric properties of In2O3
bulk ceramics through doping at In-site. Journal of Applied Physics 106, 053715–
053715–7 (2009).
109. Sakai, A., Kanno, T., Takahashi, K., Yamada, Y. & Adachi, H. Large anisotropic
thermoelectricity in perovskite related layered structure: SrnNbnO3n+2 (n=4,5).
Journal of Applied Physics 108, 103706–103706–5 (2010).
111. Koshibae, W., Tsutsui, K. & Maekawa, S. Thermopower in cobalt oxides. Phys.
Rev. B 62, 6869–6872 (2000).
112. Koshibae, W. & Maekawa, S. Effects of Spin and Orbital Degeneracy on the
Thermopower of Strongly Correlated Systems. Phys. Rev. Lett. 87, 236603 (2001).
113. Maekawa, S. et al. Physics of Transition Metal Oxides (Springer Series in Solid-
State Sciences). (Springer: 2004).
115. Tajima, S., Tani, T., Isobe, S. & Koumoto, K. Thermoelectric properties of highly
textured NaCo2O4 ceramics processed by the reactive templated grain growth
(RTGG) method. Materials Science and Engineering: B 86, 20–25 (2001).
117. I., T. Transport properties and electronic states of the thermoelectric oxide
NaCo2O4. Physica B: Condensed Matter 328, 63–67 (2003).
167
119. Park, K., Jang, K. U., Kwon, H.-C., Kim, J.-G. & Cho, W.-S. Influence of partial
substitution of Cu for Co on the thermoelectric properties of NaCo2O4. Journal of
Alloys and Compounds 419, 213–219 (2006).
127. Wang, D., Chen, L., Yao, Q. & Li, J. High-temperature thermoelectric properties
of Ca3Co4O9+δ with Eu substitution. Solid State Communications 129, 615–618
(2004).
130. Sugiura, K. et al. High electrical conductivity of layered cobalt oxide Ca3Co4O9
epitaxial films grown by topotactic ion-exchange method. Applied Physics Letters 89,
032111–032111–3 (2006).
131. Maignan, A., Flahaut, D. & Hébert, S. Sign change of the thermoelectric power in
LaCoO3. The European Physical Journal B 39, 145–148 (2004).
168
132. Pelloquin, D., Hébert, S., Maignan, A. & Raveau, B. Partial substitution of
rhodium for cobalt in the misfit [Pb0.7Co0.4Sr1.9O3]RS[CoO2]1.8 oxide. Journal of
Solid State Chemistry 178, 769–775 (2005).
133. Kobayashi, W., Hébert, S., Pelloquin, D., Pérez, O. & Maignan, A. Enhanced
thermoelectric properties in a layered rhodium oxide with a trigonal symmetry. Phys.
Rev. B 76, 245102 (2007).
134. Marsh, D. B. & Parris, P. E. Theory of the Seebeck coefficient in LaCrO3 and
related perovskite systems. Phys. Rev. B 54, 7720–7728 (1996).
135. Pal, S., Hébert, S., Yaicle, C., Martin, C. & Maignan, A. Transport and magnetic
properties of Pr1-xCaxCrO3 (x = 0.0–0.5): effect of t2g orbital degeneracy on the
thermoelectric power. The European Physical Journal B 53, 5–9 (2006).
136. Kobayashi, W. et al. Universal charge transport of the Mn oxides in the high
temperature limit. Journal of Applied Physics 95, 6825–6827 (2004).
140. Taskin, A. A., Lavrov, A. N. & Ando, Y. Origin of the large thermoelectric power
in oxygen-variable RBaCo2O5+x (R=Gd,Nd). Phys. Rev. B 73, 121101 (2006).
141. Jansen, M. & Hoppe, R. Notiz zur Kenntnis der Oxocobaltate des Natriums.
Zeitschrift für anorganische und allgemeine Chemie 408, 104–106 (1974).
143. Ray, R., Ghoshray, A., Ghoshray, K. & Nakamura, S. 59Co NMR studies of
metallic NaCo2O4. Phys. Rev. B 59, 9454–9461 (1999).
144. Maignan, A., Caignaert, V., Raveau, B., Khomskii, D. & Sawatzky, G.
Thermoelectric Power of HoBaCo2O5.5: Possible Evidence of the Spin Blockade in
Cobaltites. Phys. Rev. Lett. 93, 026401 (2004).
169
145. Hébert, S. et al. Thermoelectric properties of perovskites: Sign change of the
Seebeck coefficient and high temperature properties. Progress in Solid State
Chemistry 35, 457–467 (2007).
148. Shannon, R. D. Revised effective ionic radii and systematic studies of interatomic
distances in halides and chalcogenides. Acta Crystallographica Section A 32, 751–
767 (1976).
149. Shannon, R. D. & Prewitt, C. T. Effective ionic radii in oxides and fluorides. Acta
Crystallographica Section B Structural Crystallography and Crystal Chemistry 25,
925–946 (1969).
150. Shannon, R. D. & Prewitt, C. T. Revised values of effective ionic radii. Acta
Crystallographica Section B Structural Crystallography and Crystal Chemistry 26,
1046–1048 (1970).
151. Shannon, R. D. & Prewitt, C. T. Effective ionic radii and crystal chemistry.
Journal of Inorganic and Nuclear Chemistry 32, 1427–1441 (1970).
152. Whittaker, E. J. W. & Muntus, R. Ionic radii for use in geochemistry. Geochimica
et Cosmochimica Acta 34, 945–956 (1970).
157. Munir, Z. A., Anselmi-Tamburini, U. & Ohyanagi, M. The effect of electric field
and pressure on the synthesis and consolidation of materials: A review of the spark
plasma sintering method. Journal of Materials Science 41, 763–777 (2006).
170
158. Hungría, T., Galy, J. & Castro, A. Spark Plasma Sintering as a Useful Technique
to the Nanostructuration of Piezo‐Ferroelectric Materials. Advanced Engineering
Materials 11, 615–631 (2009).
159. Tiwari, Basu, B. & Biswas, K. Simulation of thermal and electric field evolution
during spark plasma sintering. Ceramics International 35, 699–708 (2009).
167. Shull, C. G. Early development of neutron scattering. Rev. Mod. Phys. 67, 753–
757 (1995).
168. Rietveld, H. M. A profile refinement method for nuclear and magnetic structures.
Journal of Applied Crystallography 2, 65–71 (1969).
170. Toby, B. H. EXPGUI, a graphical user interface for GSAS. Journal of Applied
Crystallography 34, 210–213 (2001).
172. Bogue, R. H. Calculation of the Compounds in Portland Cement. Ind. Eng. Chem.
Anal. Ed. 1, 192–197 (1929).
171
173. Hubbard, C. R., Evans, E. H. & Smith, D. K. The reference intensity ratio,I/Ic, for
computer simulated powder patterns. Journal of Applied Crystallography 9, 169–174
(1976).
174. Hill, R. J. Expanded Use of the Rietveld Method In Studies of Phase Abundance
in Multiphase Mixtures. Powder Diffraction Journal 6, 74–77 (1991).
181. ASTM Standard E1225 - 09, ‗Standard Test Method for Thermal Conductivity of
Solids by Means of the Guarded Comparative Longitudinal Heat Flow Technique‘
ASTM International, West Conshohocken, PA, 2009. at <DOI: 10.1520/E1225-09>
182. ASTM Standard C177 - 10, ‗Standard Test Method for Steady State Heat Flux
Measurements and Thermal Transmission Properties by Means of the Guarded Hot
Plate Apparatus‘ ASTM International, West Conshohocken, PA, 2010. at <DOI:
10.1520/C0177-10>
183. ASTM Standard C1113 / C1113M - 09, ‗Standard Test Method for Thermal
Conductivity of Refractories by Hot Wire (Platinum Resistance Thermometer
Technique)‘ ASTM International, West Conshohocken, PA, 2009. at <DOI:
10.1520/C1113_C1113M-09>
184. ASTM Standard D5930 - 09, ‗Standard Test Method for Thermal Conductivity of
Plastics by Means of a Transient Line Source Technique‘ ASTM International, West
Conshohocken, PA, 2009. at <DOI: 10.1520/D5930-09>
172
185. Barin, I. Thermochemical data of pure substances. (VCH: Weinheim, Federal
Republic of Germany; New York, 1993).
188. Clark III, L. M. & Taylor, R. E. Radiation loss in the flash method for thermal
diffusivity. Journal of Applied Physics 46, 714–719 (1975).
189. Cape, J. A. & Lehman, G. W. Temperature and Finite Pulse‐Time Effects in the
Flash Method for Measuring Thermal Diffusivity. Journal of Applied Physics 34,
1909–1913 (1963).
195. Yamada, A. & Tanaka, M. Jahn-Teller structural phase transition around 280K in
LiMn2O4. Materials Research Bulletin 30, 715–721 (1995).
196. Bosi, F., Hålenius, U., Andreozzi, G. B., Skogby, H. & Lucchesi, S. Structural
refinement and crystal chemistry of Mn-doped spinel: A case for tetrahedrally
coordinated Mn3+ in an oxygen-based structure. American Mineralogist 92, 27 –33
(2007).
197. Paulsen, J. M. & Dahn, J. R. Phase Diagram of Li−Mn−O Spinel in Air. Chem.
Mater. 11, 3065–3079 (1999).
198. Sickafus, K. E., Wills, J. M. & Grimes, N. W. Structure of Spinel. Journal of the
American Ceramic Society 82, 3279–3292 (1999).
173
199. Thackeray, M. M., Mansuetto, M. F., Dees, D. W. & Vissers, D. R. The thermal
stability of lithium-manganese-oxide spinel phases. Materials Research Bulletin 31,
133–140 (1996).
201. Atanasov, M., Barras, J.-L., Benco, L. & Daul, C. Electronic Structure, Chemical
Bonding, and Vibronic Coupling in MnIV/MnIII Mixed Valent LixMn2O4 Spinels
and Their Effect on the Dynamics of Intercalated Li: A Cluster Study Using DFT. J.
Am. Chem. Soc. 122, 4718–4728 (2000).
204. Metselaar, R., Van Tol, R. E. J. & Piercy, P. The electrical conductivity and
thermoelectric power of Mn3O4 at high temperatures. Journal of Solid State
Chemistry 38, 335–341 (1981).
205. Chen, M., Hallstedt, B. & Gauckler, L. J. Thermodynamic assessment of the Co-
O system. Journal of Phase Equilibria 24, 212–227 (2003).
207. Gleitzer, C., Nowotny, J. & Rekas, M. Surface and bulk electrical properties of
the hematite phase Fe2O3. Applied Physics A Solids and Surfaces 53, 310–316
(1991).
209. Tuller, H. L. & Nowick, A. S. Small polaron electron transport in reduced CeO2
single crystals. Journal of Physics and Chemistry of Solids 38, 859–867 (1977).
174
212. Kale, G. M., Pandit, S. S. & Jacob, K. T. Thermodynamics of Cobalt (II, III)
Oxide (Co3O4): Evidence of Phase Transition. Transactions of the Japan Institute of
Metals 29, 125–132 (1988).
213. Jung, I.-H., Decterov, S. A., Pelton, A. D., Kim, H.-M. & Kang, Y.-B.
Thermodynamic evaluation and modeling of the Fe–Co–O system. Acta Materialia
52, 507–519 (2004).
214. Navrotsky, A., Ma, C., Lilova, K. & Birkner, N. Nanophase Transition Metal
Oxides Show Large Thermodynamically Driven Shifts in Oxidation-Reduction
Equilibria. Science 330, 199 –201 (2010).
215. Bahlawane, N., Tchoua Ngamou, P. H., Vannier, V. & Kottke, T. Tailoring the
properties and the reactivity of the spinel cobalt oxide. Physical Chemistry of
Chemical Physics 11, 9224–9232 (2009).
217. Hadjiev, V. G., Iliev, M. N. & Vergilov, I. V. The Raman spectra of Co3O4.
Journal of Physics C: Solid State Physics 21, L199–L201 (1988).
219. Gallant, D., Pézolet, M. & Simard, S. Optical and Physical Properties of Cobalt
Oxide Films Electrogenerated in Bicarbonate Aqueous Media. J. Phys. Chem. B 110,
6871–6880 (2006).
220. Melendres, C. A. & Xu, S. In Situ Laser Raman Spectroscopic Study of Anodic
Corrosion Films on Nickel and Cobalt. J. Electrochem. Soc. 131, 2239–2243 (1984).
223. Roth, W. L. The magnetic structure of Co3O4. Journal of Physics and Chemistry
of Solids 25, 1–10 (1964).
175
224. Cossee, P. Magnetic properties of cobalt in oxide lattices. Journal of Inorganic
and Nuclear Chemistry 8, 483 (1958).
226. Dieckmann, R., Witt, C. A. & Mason, T. O. Defects and Cation Diffusion in
Magnetite (V): Electrical Conduction, Cation Distribution and Point Defects in
Fe3‐δO4. Berichte der Bunsengesellschaft für physikalische Chemie 87, 495–503
(1983).
227. Kündig, W. & Steven Hargrove, R. Electron hopping in magnetite. Solid State
Communications 7, 223–227 (1969).
176