Polaron Hopping in Olivine Phosphates Studied by Nuclear Resonant Scattering - CaltechTHESIS
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Polaron Hopping in Olivine Phosphates Studied by Nuclear Resonant Scattering
Citation
Tracy, Sally June
(2016)
Polaron Hopping in Olivine Phosphates Studied by Nuclear Resonant Scattering.
Dissertation (Ph.D.), California Institute of Technology.
doi:10.7907/Z95H7D67.
Abstract
Valence fluctuations of Fe
2+
and Fe
3+
were studied in a solid solution of Li
FePO
by nuclear resonant forward scattering of synchrotron x rays while the sample was heated in a diamond-anvil pressure cell. The spectra acquired at different temperatures and pressures were analyzed for the frequencies of valence changes using the Blume-Tjon model of a system with a fluctuating Hamil- tonian. These frequencies were analyzed to obtain activation energies and an activation volume for polaron hopping. There was a large suppression of hopping frequency with pressure, giving an anomalously large activation volume. This large, positive value is typical of ion diffusion, which indicates correlated motions of polarons, and Li
ions that alter the dynamics of both.
In a parallel study of Na
FePO
, the interplay between sodium ordering and electron mobility was investigated using a combination of synchrotron x-ray diffraction and nuclear resonant scattering. Conventional Mossbauer spectra were collected while the sample was heated in a resistive furnace. An analysis of the temperature evolution of the spectral shapes was used to identify the onset of fast electron hopping and determine the polaron hopping rate. Synchrotron x-ray diffraction measurements were carried out in the same temperature range. Reitveld analysis of the diffraction patterns was used to determine the temperature of sodium redistribution on the lattice. The diffraction analysis also provides new information about the phase stability of the system. The temperature evolution of the iron site occupancies from the Mossbauer measurements, combined with the synchrotron diffraction results give strong evidence for a relationship between the onset of fast electron dynamics and the redistribution of sodium in the lattice.
Measurements of activation barriers for polaron hopping gave fundamental insights about the correlation between electronic carriers and mobile ions. This work established that polaron-ion interactions can alter the local dynamics of electron and ion transport. These types of coupled processes may be common in many materials used for battery electrodes, and new details concerning the influence of polaron-ion interactions on the charge dynamics are relevant to optimizing their electrochemical performance.
Item Type:
Thesis (Dissertation (Ph.D.))
Subject Keywords:
LiFePO4; NaFePO4; Cathode; Polaron; Mossbauer
Degree Grantor:
California Institute of Technology
Division:
Engineering and Applied Science
Major Option:
Materials Science
Thesis Availability:
Public (worldwide access)
Research Advisor(s):
Fultz, Brent T.
Thesis Committee:
Fultz, Brent T. (chair)
Rossman, George Robert
Johnson, William Lewis
Faber, Katherine T.
Defense Date:
3 September 2015
Funders:
Funding Agency
Grant Number
EFree, an Energy Frontier Research Center
DE-SC0001057
Record Number:
CaltechTHESIS:10062015-165650934
Persistent URL:
DOI:
10.7907/Z95H7D67
ORCID:
Author
ORCID
Tracy, Sally June
0000-0002-6428-284X
Default Usage Policy:
No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:
9203
Collection:
CaltechTHESIS
Deposited By:
Sally Tracy
Deposited On:
08 Oct 2015 16:52
Last Modified:
04 Oct 2019 00:10
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Polaron hopping in olivine phosphates studied
by nuclear resonant scattering
Thesis by
Sally June Tracy
In Partial Fulfillment of the Requirements
for the Degree of
Doctor of Philosophy
California Institute of Technology
Pasadena, California, USA
2016
(Defended September 3, 2015)
ii
2015
Sally June Tracy
iii
For Matt
iv
There’s no such thing as a perfect piece of writing.
Just as there’s no such thing as perfect despair.
-Haruki Murakami, Hear the Wind Sing
Acknowledgments
I would like to acknowledge my advisor, Dr. Brent Fultz, for his guidance during the years I
spent at Caltech. I hold Professor Fultz’s intellectual taste in high esteem and have been impressed
by his thoughtful approach to science and dedication to a thorough understanding of fundamental
questions. Furthermore, Professor Fultz is skilled in the art of always providing an optimum level
of guidance, giving his students the space to grow into independent scientists but not letting them
get too far off track along the way. I would also like to thank my undergraduate advisor, Professor
George Schmiedeshoff, who first exposed me to scientific research. Without his push, I most likely
would not have had the confidence in my own abilities to pursue research. I would be remiss not
to mention my post-baccalaureate mentor Dr. Jason Cooley, who has always been someone I felt I
could come to with questions. It has always amazed me that whatever the inquiry, even if he doesn’t
have an answer off hand, he will always know where to point you to find what you are looking for.
I am always surprised, but incredibly grateful, that he seems to still take interest in what I am doing
and continues to support me in whatever ways he can. I can only hope that his students since me
have not caused him as much trouble.
I would like to express my gratitude to the faculty on campus who have generously allowed
us access to their lab space, especially Professors Jennifer Jackson, William Johnson, and George
Rossman. The access to equipment to synthesize samples as well as the ability to prepare diamond
anvil cells on campus and collect Raman spectra was invaluable in making the experiments in this
thesis possible. In addition to these faculty, I would like to acknowledge the other member of my
committee, Professor Katherine Faber. I would like to thank all the people involved with EFree and
HPCAT as well as the other scientists at Geophysical Lab and the Advanced Photon Source with
whom I have had the chance to interact. In particular, I would like to express my appreciation of the
technical staff at HPCAT and GSECARS who design and maintain an extensive suite of diamond-
vi
anvil cell preparation tools and sample environments for high-pressure measurements. Access to
this equipment, along with their dedicated user support, was invaluable in collecting the data in this
thesis.
During my time at Caltech, I have had the pleasure of working closely with Dr. Lisa Mauger.
Without Lisa’s help there is no way most of this work would have been completed. During this time,
Lisa has served as mentor, a collaborator and friend. I am continually amazed with her experimental
prowess and her pragmatic outlook of life. I can only hope that the countless sleepless nights spent at
the APS have forged a lifetime bond that will yield fruitful collaborations in the future and a lifelong
friendship. I would also like to mention the other students I had the chance to collaborate with
while at Caltech, particularly my former officemate Dr. Jorge Muñoz, who shared my inclination
for working late at night, and is someone whom I hold in high regard. I would also like to thank Dr.
Hongjin Tan, who is always a jovial labmate and was kind enough to teach me a bit of chemistry.
More recently I also have had the pleasure of working with both Jane Herriman and Dr. Hillary
Smith. I must also mention my other Los Angeles friends, particularly Lizzie Adelman and Amanda
Armstrong. During my years in graduate school, spending time with these two was always a much
needed breath of fresh air. They are really the best friends anyone could ask for.
Finally, I would like recognize my family for their ongoing love and support. I credit my parents
with instilling me with a strong work ethic and creative outlook on the world. Whether helping her
nearly thirty year old daughter edit her resumé, shoe shopping, or grading middle school social
studies reports, my mother tackles the task at hand with what can only be described as an intense
and unrelenting fervor. Regardless of the circumstance, this ardent pursuit of perfection will only
come to a close when she has gone above and beyond the call of duty. I have always admired my
father’s unwillingness to subscribe to other people’s notions of fulfillment and success. I would
like to think I have inherited some of this unique outlook, his intellectual curiosity, and perhaps a
dash of his anti-authoritarian convictions. I would also like to mention my sisters. While at times
I found it difficult to be in the middle of two accomplished and well liked sisters, who were spared
(to varying degrees) my intensely socially awkward tendencies, I feel incredibly fortunate to have
two incredible and generous sisters. I can only hope that the official completion of my many, many
years of schooling will allow me to make my final assent to crown status as the undisputed golden
child. Last but most important, I would like to thank my partner Matt Graham, who has continually
vii
made sacrifices to allow me to pursue my education. His patience with me and tolerance of my
often neurotic behavior continues to amaze me. I am incredibly lucky to have a companion who
challenges me and makes me laugh. Without him I would be lost.
viii
Abstract
Valence fluctuations of Fe2+ and Fe3+ were studied in a solid solution of Lix FePO4 by nuclear
resonant forward scattering of synchrotron x rays while the sample was heated in a diamond-anvil
pressure cell. The spectra acquired at different temperatures and pressures were analyzed for the
frequencies of valence changes using the Blume-Tjon model of a system with a fluctuating Hamiltonian. These frequencies were analyzed to obtain activation energies and an activation volume
for polaron hopping. There was a large suppression of hopping frequency with pressure, giving an
anomalously large activation volume. This large, positive value is typical of ion diffusion, which
indicates correlated motions of polarons, and Li+ ions that alter the dynamics of both.
In a parallel study of Nax FePO4 , the interplay between sodium ordering and electron mobility
was investigated using a combination of synchrotron x-ray diffraction and nuclear resonant scattering. Conventional Mössbauer spectra were collected while the sample was heated in a resistive
furnace. An analysis of the temperature evolution of the spectral shapes was used to identify the onset of fast electron hopping and determine the polaron hopping rate. Synchrotron x-ray diffraction
measurements were carried out in the same temperature range. Reitveld analysis of the diffraction
patterns was used to determine the temperature of sodium redistribution on the lattice. The diffraction analysis also provides new information about the phase stability of the system. The temperature
evolution of the iron site occupancies from the Mössbauer measurements, combined with the synchrotron diffraction results give strong evidence for a relationship between the onset of fast electron
dynamics and the redistribution of sodium in the lattice.
Measurements of activation barriers for polaron hopping gave fundamental insights about the
correlation between electronic carriers and mobile ions. This work established that polaron-ion
interactions can alter the local dynamics of electron and ion transport. These types of coupled
processes may be common in many materials used for battery electrodes, and new details concerning
ix
the influence of polaron-ion interactions on the charge dynamics are relevant to optimizing their
electrochemical performance.
Contents
Acknowledgments
Abstract
viii
Introduction
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
Battery background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
Lithium-ion cathode materials . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.5
Sodium-ion cathode materials . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6
Crystallography of framework oxides . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1
Overview of cathode crystal structures . . . . . . . . . . . . . . . . . . . .
1.6.2
Triphylite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.3
Maricite . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Phase stability of olivine phosphates . . . . . . . . . . . . . . . . . . . . . . . . .
10
1.7
Polaron models
13
2.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
13
2.2
Holstein’s molecular crystal model . . . . . . . . . . . . . . . . . . . . . . . . . .
14
2.3
Semiclassical treatment of the molecular crystal model . . . . . . . . . . . . . . .
15
2.4
The Holstein Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
17
2.5
Adiabatic electron transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
19
2.6
Non-adiabatic electron transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . .
20
xi
The activation volume
24
3.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.2
Previous polaron activation volume measurements . . . . . . . . . . . . . . . . . .
26
3.3
Corrections to the apparent activation volume . . . . . . . . . . . . . . . . . . . .
27
3.4
Pressure dependence of the Holstein activation barrier . . . . . . . . . . . . . . . .
31
3.5
Activation volume for ion diffusion . . . . . . . . . . . . . . . . . . . . . . . . . .
32
Methods
33
4.1
Synchrotron x-ray diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
33
4.2
X-ray diffraction data analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . .
34
4.3
Mössbauer spectrometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
4.3.1
The Mössbauer effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
35
4.3.2
Hyperfine interactions . . . . . . . . . . . . . . . . . . . . . . . . . . . .
36
4.3.3
Mössbauer measurements . . . . . . . . . . . . . . . . . . . . . . . . . .
39
4.3.4
Resistive furnace for high temperature Mössbauer measurements . . . . . .
40
4.4
Nuclear forward scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
40
4.5
High pressure measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
42
4.6
Experimental setup for high temperature, high pressure measurements . . . . . . .
45
4.7
Relaxation effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.7.1
Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
4.7.2
Blume-Tjon model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
48
Nuclear resonant scattering data analysis . . . . . . . . . . . . . . . . . . . . . . .
52
4.8
Polaron-ion correlations in Li0.6 FePO4
54
5.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
54
5.2
Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
57
5.3
Simulational . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
5.4
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
5.4.1
Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
63
5.4.2
Simulational results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
68
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
69
5.5
xii
5.6
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
73
Polaron mobility and disordering of the Na sublattice in Nax FePO4 with the triphylite
structure
74
6.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
74
6.2
Experimental . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
78
6.3
Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
6.3.1
X-ray diffractometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
79
6.3.2
Mössbauer spectrometry . . . . . . . . . . . . . . . . . . . . . . . . . . .
82
6.4
Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
89
6.5
Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
96
Conclusions and future work
98
7.1
Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
99
7.1.1
Pair distribution study of Lix FePO4 . . . . . . . . . . . . . . . . . . . . .
99
7.1.2
Mössbauer study of electron dynamics in maricite-NaFePO4 . . . . . . . . 100
7.1.3
Nuclear resonant scattering study of activation barriers in Li2 FeSiO4 . . . . 100
7.1.4
Activation volume studies . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Bibliography
102
xiii
List of Figures
1.1
Replica of Alessandro Volta’s original cell, composed of a stack of zinc and copper
disks. Figure: GuidoB- CC BY-SA 3.0 . . . . . . . . . . . . . . . . . . . . . . . .
1.2
Essential components of electrochemical cell. . . . . . . . . . . . . . . . . . . . .
1.3
Yearly breakdown of number of Na-ion battery manuscripts in the last fifteen years.
The total number of manuscripts in 2015 has been projected based on the number
published up through July. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
Atom fraction of the chemical elements in Earth’s upper continental crust as a function of atomic number. Major rock-forming elements are shown in green field and
minor rock forming elements are in light green field. The major industrial metals
are shown in bold. Figure courtesy of the United States Geological Survey. . . . .
1.5
Crystal structures of prospective cathode materials, illustrating different dimensionalities of alkali ion diffusion pathways. a) Layered LiCoO2 where colbalt octahedra
are shown in royal blue, Li ions are in light blue and oxygen are in red. b) Spinel,
LiMn2 O4 with Mn ions in magenta. c) Olivine, LiFePO4 , with iron in light brown
and phosphate tetrahedra in grey. d) Na3 V2 PO4 NASICON structure with Na ions
in yellow and vanadium octahedra in red.) Li2 FeSiO4 with silicate tetrahedra in blue.
1.6
Orthorhombic olivine-phosphate structure. (a) Lithiated triphylite structure. (b)
Delithiated heterosite structure. Iron octahedra are shown in brown, oxygen ions
are in red, phosphate tetrahedra are in grey and lithium ions are in blue. . . . . . .
1.7
10
A comparison of the triphylite and maricite crystal structures.The first two rows
show different views of the structures and the last row depicts the differences in the
alkali ion octahedra connectivity. Iron octahedra are shown in brown, oxygen ions
are in red, phosphate tetrahedra are in grey and sodium ions/octahedra are in yellow.
11
xiv
2.1
An electron polaron localized in an iron oxide lattice. Shifts of surrounding ion
cores are shown with arrows. Image Courtesy of Pacific Northwest National Laboratory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
14
Holstein’s Molecular Crystal Model. Here an excess electron is localized on the
central diatomic molecule, depicted in red.
. . . . . . . . . . . . . . . . . . . . .
15
2.3
Energy of molecule as a function of distortion coordinate, xn . Here B ∝ M ω02 . . .
16
2.4
Depiction of polaron hop between adjacent sites on 1D chain illustrating the development of a coincidence event allowing for electron transfer. . . . . . . . . . . . .
2.5
17
Depiction of crossover between low temperature, band conduction and elevated
temperature Arrhenius-type behavior. Here the dotted line illustrates an Arrhenius
fit to the high temperature result. Here θ is the characteristic temperature given by
θ0 = ~ω0 /kB . Figure from Holstein (1959) [1]. . . . . . . . . . . . . . . . . . . .
3.1
Illustration of the local lattice expansion or contraction that occurs during charge
transfer process. Image Courtesy of Pacific Northwest National Laboratory. . . . .
3.2
22
25
Depiction of polaron hop between adjacent sites on 1D chain, illustrating the development of a transient distortion pattern that allows for electron transfer and shows
the local volume change in the transition state. a) Polaron localized on left site. b)
Activated state during electron transfer, illustrating a local expansion of the lattice.
c) Carrier localized on right site. . . . . . . . . . . . . . . . . . . . . . . . . . . .
25
4.1
Typical 2D image file collected with CCD detector. . . . . . . . . . . . . . . . . .
33
4.2
Energy diagram of isomer shift and quadrupole splitting, ∆EQ , for the 57 Fe
3/2→1/2 transition in an asymmetric EFG. Figure: UC Davis ChemWiki- CC BYSA 3.0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
38
4.3
Schematic of furnace used for elevated temperature Mössbauer measurements. . . .
40
4.4
A coherent superposition of wavlets from slightly offset energy levels gives rise to
quantum beats in the temporal evolution of the nuclear decay. Figure from Röhlsberger (2004) [2]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.5
41
Schematic of beamline set up for nuclear resonant scattering. Picture adapted from
Zhou, et al. (2004) [3]. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
41
xv
4.6
Schematic of diamond-anvil cell. Figure: Tobias 1984- CC BY-SA 3.0. . . . . . . .
43
4.7
Photo of Tel Aviv-DACs, shown with stack of quarters for size comparison. . . . .
44
4.8
Photo of Mao-type symmetric cell. . . . . . . . . . . . . . . . . . . . . . . . . . .
44
4.9
Pressure membranes for remote pressure control. . . . . . . . . . . . . . . . . . .
44
4.10 Schematic of furnace used for high temperature, high pressure synchrotron experiments with Tel Aviv-DACs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.11 Photo of furnace used for high temperature, high pressure synchrotron experiments
with symmetric DACs. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
46
4.12 Photo of setup at beamline 16-IDD, including vacuum furnace, online ruby system,
CCD for XRD, and APD for NFS. . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
47
Olivine-type structure of LiFePO4 with chains of Li+ ions (blue), planes of FeO6
octahedra (brown), and phosphate tetrahedra (grey).
. . . . . . . . . . . . . . . .
55
5.2
Schematic of randomly populated 1D coupled Li+ ion and electron chains. . . . .
60
5.3
Six subprocesses describing ion and electron jumps on coupled 1D chains, where
the energy barrier for each subprocess is listed below the schematic. Ep and Ei
are the free polaron and ion activation barriers respectively, Epi is the polaron-ion
interaction energy, and Vi is the activation volume for ion hopping. For Li+ ion
jumps, depicted in the lower frames, the energy barrier depends only on the initial
1NN electron site; the final 1NN site on the electron chain is not depicted. . . . . .
5.4
60
Temperature series of NFS spectra taken at 0, 3.5, 7.1 and 17 GPa. The fits (black
curves) overlay experimental data (red points). Temperatures are listed to the left
of spectra in Kelvin. The x-axis is the delay in nanoseconds after the arrival of the
synchrotron pulse. The spectra have been scaled by their maximum value and offset
for comparison. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5
64
(a) Polaron hopping frequencies, Γ(T, P ), at 0, 3.5 and 7.1 GPa, as determined from
the solid curves of Fig. 5.4. Solid curves are Arrhenius-type fits using a pressureindependent prefactor. (b) Activation enthalpies, ∆H = Ea +P Va , versus pressure,
where Ea = 470 meV. Black triangles are results for fixed prefactor and red circles
are for a pressure dependent prefactor. . . . . . . . . . . . . . . . . . . . . . . . .
66
xvi
5.6
Electron MSD versus time for six series of MC simulations for a pair of coupled
1D ion and electron chains. Units for MSD are site index squared. Time is dimensionless. Each subplot shows the results for a different Epi . Subplots are labeled
with -Epi (0, 50, 150, 250, 350 and 450 meV). In each series, the MSD is shown for
three different pressures: 0 (black), 3 (red) and 7 GPa (green). . . . . . . . . . . .
6.1
70
(a) Triphylite-type structure of MFePO4 (M= Li, Na) with chains of M+ ions (yellow), planes of FeO6 octahedra (brown) and phosphate tetrahedra (grey). (b) Ordered superstructure for x=2/3. Three structurally distinct iron sites are shown in
blue, green and red. The axes on left are for the orthorhombic Pmna cell. Oblique
axes of P21 /n cell are shown in black. . . . . . . . . . . . . . . . . . . . . . . . .
6.2
76
(a) Temperature series of XRD spectra taken between 295 K and 550 K. The Rietveld fits (black curves) overlay experimental data (points). Black tick marks at
bottom of figure indicate locations of superstructure phase in ordered structure. (b)
Enlargement of (200) peak on linear scale.
6.3
. . . . . . . . . . . . . . . . . . . . .
Molar fraction of x=2/3 ordered phase, determined from Rietveld analysis of x-ray
patterns shown in Fig. 6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.4
79
80
Structural parameters determined from Rietveld fits in Fig. 6.2. (a) Volume of unit
cell for ordered and solid solution phases. Volume of ordered (P21 /n) cell has been
normalized by a factor of three for comparison with orthorhombic Pmna cell. (b)
Lattice parameters for orthorhombic solid solution phase. (c) Lattice parameters for
monoclinic ordered phase. The P21 /n a-axis coincides with the Pmna b-axis and the
P21 /n b-axis coincides with the Pmna c-axis. . . . . . . . . . . . . . . . . . . . . .
6.5
81
Temperature series of Mössbauer spectra taken between 295 K and 550 K. The fits
(black curves) overlay experimental data (points). Temperatures are listed to the left
of the spectra in Kelvin.
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
83
6.6
298 K Mössbauer spectra Nax FePO4 (x=0.54, 0.67, 0.97).
. . . . . . . . . . . . .
84
6.7
Down-temperature spectrum, at 298 K, acquired after cooling from 550 K. . . . . .
86
6.8
(a) Polaron hopping frequencies, Γ(T ), as determined from the solid curves of Fig.
6.5. Solid curve is an Arrhenius-type fit. . . . . . . . . . . . . . . . . . . . . . . .
88
xvii
6.9
Relative weight of Fe2+ iron sites as a function of temperature, as determined from
fits to Mössbauer spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
88
6.10 Revised phase Diagram, including two phase region above x=2/3. . . . . . . . . .
89
6.11 Sodium-ion sublattice of the x = 2/3 ordered phase. Sodium ions are shown in
yellow and three structurally distinct iron sites are shown in blue, green and red,
corresponding to iron sites with three, two and one vacancies in the six-fold sodium
coordination shell. The pyramidal outline of this coordination shell in shown in
black for a red-type iron site. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
92
6.12 Temperature evolution of three iron sites in Na2/3 FePO4 structure. Fe3+ ions are
shown in yellow and Fe2+ in purple, where dark purple and lavender depict Aand B-type ferrous sites. Sodium ions are shown in black and sodium vacancies
are shown in white. The pyramidal outline reflects the coordination environment
of the central iron site, corresponding to the colors of the iron ions shown in Fig.
6.11. Red represents a crystallographic iron site with five sodium neighbors, green
represents an iron site with four sodium neighbors, blue represents an iron site with
three sodium neighbors and brown represents the average coordination environment
for an iron site in the disordered solid solution phase. Temperature is listed at the
top of each column along with B/A site ratio. . . . . . . . . . . . . . . . . . . . .
93
6.13 Mössbauer spectrometry of Fig. 6.5, inverted, stacked and normalized for comparison. Dotted lines mark the A- and B-type ferrous absorptions at ∼0 mm/s. . . . . .
95
Chapter 1
Introduction
1.1
Motivation
A global transition to renewable energy sources is of vital importance for the future of our
planet. The challenge lies in finding creative ways to sustainably support a world of nine billion
people while allowing for the continued globalization of modern technology. While I have hope
that attitudes of indifference and the lack of education concerning human environmental impacts
represent trends that are shifting, it is important to recognize that the only surefire way to institute change is through the development new technologies that make the transition to sustainable
alternatives both convenient and affordable.
Energy storage remains a critical challenge in the transition to green energy. Rechargeable
batteries have emerged as a particularly promising storage choice. Batteries can be designed in
numerous shapes and sizes and are adaptable to a wide range of uses. The increasing reliance on
batteries for portable electronics, electric vehicles and even large scale grid storage necessitates
further research aimed at discovering new high-performance battery materials that are affordable,
environmentally friendly, and safe. This goal requires a fundamental understanding of the material
properties that effect electrode performance.
Transport of electrons and ions is a central issue in many energy storage materials. In particular,
battery-cathode materials require relatively facile mobility of intercalation ions as well as reasonable mobility of electronic carriers. The electronic conduction mechanism in many transition-metal
oxide cathodes is small polaron hopping. This mode of electronic transport is characterized by
phonon-assisted hopping of localized carriers between adjacent transition-metal ion sites. The aim
of the work presented in this thesis was to develop a more detailed picture of charge transport in
these framework oxides and better understand how the phase stability and transport properties of
these materials are affected by electron-ion interactions.
1.2
Overview
Candidate cathode materials, specifically LiFePO4 and NaFePO4 , were studied using a combination of synchrotron x-ray diffraction and nuclear resonant scattering for a range of temperatures and pressures. Both ion diffusion and small polaron hopping can be understood as activated
processes having an Arrhenius-type temperature dependence. Consequently, measurements of the
electronic charge hopping frequency as a function of temperature allow for the determination of an
activation energy. Nuclear resonant scattering presents an ideal technique to examine the valence
fluctuations that accompany polaron hopping in iron-bearing cathode materials. These measurements provide a local probe at the iron ion that is sensitive to valence switching within a frequency
range that is well suited for studies of polaron dynamics.
Extending these measurements to elevated pressure using synchrotron techniques allows for
the additional determination of an activation volume. This quantity can provide a window into the
atomic rearrangements that occur during the transient state of a hopping event. The activation energy
is the energetic barrier for the moving species to hop between adjacent sites. The activation volume
quantifies the effect of pressure on this activation barrier, as it gives the local change in volume as
the particle moves through its transition state. The important role of activation energy in setting the
temperature dependence of the transport process is well known, but the activation volume is less
well understood. The experiments in this thesis revealed new information concerning the nature of
the activation barriers for polaron hopping in mixed electron-ion conductors.
Synchrotron x-ray diffraction carried out in the same temperature range gives complementary
information concerning the structural evolution and phase stability of these materials with temperature and pressure. Combined with the nuclear resonant scattering results, diffraction data provides
insight into the relationship between the development of disorder on the alkali-ion sublattice and the
enhancement of electronic mobility. Additionally, the structural information obtained from these
Figure 1.1. Replica of Alessandro Volta’s original cell, composed of a stack of zinc and copper disks. Figure:
GuidoB- CC BY-SA 3.0
measurements gives insights into the relationship between ion ordering and electronic mobility.
1.3
Battery background
With the advent of consumer electronics, global societies are becoming increasingly reliant on
battery technology, driven by an ever-increasing demand for portable power sources. The use of
batteries as devices for electrochemical energy storage and conversion is by no means new technology. The first modern battery dates back to the Enlightenment, when Alessandro Volta, an Italian
professor of natural philosophy, described his “voltaic pile” in an 1800 report to the London Royal
Society. Volta’s device was composed of a stack of zinc and copper disks immersed in a saltwater
brine that served as an electrolyte [4]. Fig. 1.1 shows a replica of the original cell on display at a
museum near Volta’s home in Como, Italy. While the fundamental concepts of the electrochemical
cell remain unchanged, contemporary developments in materials technology continue to make batteries a viable power source for modern electronics, and battery research now represents a diverse
and vibrant field.
Figure 1.2. Essential components of electrochemical cell.
1.4
Lithium-ion cathode materials
During the operation of a conventional rechargeable lithium battery, lithium diffuses out of the
anode material (usually graphite) and is transported through the electrolyte and through an electrically insulating separator layer. The Li-ions are then intercalated into the crystal structure of
the cathode material. During this discharge process, electrons are transferred through an external
circuit, providing usable electronic energy. To recharge the cell, a voltage is applied across the
electrodes forcibly extracting lithium from the cathode material. The ions diffuse back through the
electrolyte and are reinserted into the anode.
The material properties of the anode and cathode materials establish the overall performance of
the electrochemical cell. The amount of lithium that can be reversibly extracted and reinserted into
the cathode’s crystal structure determines the cell’s overall capacity and the relative Fermi energies
of the anode and cathode set the cell voltage. An ideal material has a high energy density and
exhibits good stability on cycling. It is also essential that the cathode is a good ionic conductor and
exhibits reasonable electronic conductivity. Preferably the material is also non-toxic and affordable.
The majority of commercial batteries today contain LiCoO2 cathodes. Cobalt is not only expensive
but is also quite toxic. Consequently, there is a great deal of interest finding viable cathode materials
that contain alternate multivalent elements.
500
450
400
350
300
250
200
150
100
50
2015
2014
2013
2012
2011
2010
2009
2008
2007
2006
2005
2004
2003
2002
2001
2000
Figure 1.3. Yearly breakdown of number of Na-ion battery manuscripts in the last fifteen years. The total
number of manuscripts in 2015 has been projected based on the number published up through July.
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fraction of the chemical elements in Earth’s upper
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as a functionnificant
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major industrial metals are shown in bold. Figure courtesy of the United States Geological
occur in other, low-Th minerals,
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larly bastnäsite.
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the commercialization of color television.
mestic source of REE, the mine a
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Mountain Pass, with an average grade of
tain Pass, California, has operated
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9.3% and reserves of 20 million metric tons capacity and only intermittently. F
Pr, and Nd—constitute 80 to 99% of the to(Mt) REO (at 5% cutoff), remains the only
environmental and regulatory pro
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with the main wastewater pipelin
high grades of the scarcer and more valucontent. Mountain Pass ore is very strongly REE separation (solvent extractio
able heavy REE (HREE: Gd to Lu, Y) and
dominated by the light REE (figs. 3, 6).
was shut down. Mountain Pass cu
Eu are particularly desirable.
Nonetheless, the large quantities of ore pro- produces only bastnäsite concentr
1.5
Sodium-ion cathode materials
Rechargeable sodium-ion batteries were first proposed in the same time period as early work
on lithium batteries [5–10]. Due to the overwhelming successes of Li-ion technology, research
tended to move in this direction and interest in these sodium systems dwindled. In recent years
there has been a resurgence of interest in Na-ion batteries. Figure 1.3 shows the yearly breakdown
of number of manuscripts on sodium-ion batteries in the last fifteen years. While this trend would
be dwarfed by a similar plot of lithium-ion battery studies, the figure makes it apparent there has
been an explosion of interest in the topic within the last few years. This newfound interest can be
attributed mainly to the fact that in the last twenty years our society has rapidly become dependent
on Li-ion technology. Given this rising demand, it has become apparent that lithium itself is a
limited resource. In contrast, sodium is relatively plentiful. The abundance of various elements in
the earth’s crust is illustrated in Figure 1.4. Sodium is situated in the dark green field, indicating it
is one of the major rock forming elements, as opposed to lithium that sits below the light green field
of the minor rock forming elements. Consequently, the prospect of creating sodium analogues to
lithium-ion batteries has become immensely attractive. Furthermore, in Fig. 1.4 cobalt can be found
well below even lithium, motivating the interest in cathodes containing alternative earth-abundant
transition metal redox ions, for example iron, manganese or magnesium.
Sodium-ion batteries have the potential to be significantly cheaper than their lithium-ion analogues. The prospect of designing a more affordable class of batteries enhances the feasibility of
employing batteries for large scale grid applications. Much of the knowledge that has been acquired
through the investigation of lithium systems is relevant to sodium cathodes, but Na-ion chemistry
allows for additional intercalation structures, some of which may not form in their lithium counterparts. Although known sodium-ion systems tend to have lower energy densities and voltages,
these details are surprisingly sensitive to the cathode’s crystallography. An in-depth understanding
of how crystal structure affects the transport properties and performance of these materials is critical
to designing competitive sodium batteries.
1.6
Crystallography of framework oxides
1.6.1
Overview of cathode crystal structures
The performance a cathode is strongly influenced by the details of the material’s crystal structure. An important feature is the availability of diffusion channels for alkali ions. Furthermore,
the dimensionality of these diffusion pathways plays a key role in setting the overall rate capacity.
Figure 1.5 shows the crystal structures for a range of different types prospective cathode materials.
The most well established of these materials are the LiMO2 layered structures, where the transitionmetal ions (M) are generally cobalt or nickel and lithium ions are inserted between sheets of corner
sharing MO6 octahedra. Different stacking sequences of the oxygen planes give rise to different
variants of these layered-type structures. The Natrium Super Ionic CONductor (NASICON) family
(Ax MM’(XO4 )3 , A=Na, Li) is characterized by a three-dimensional scaffolding of metal-oxygen
octahedra (MO6 and M’O6 ) corner-linked with phosphate or silicate tetrahedral units, XO4 . This
open framework forms large interconnected channels for alkali ion mobility. The spinel family is
another popular cathode candidate, most commonly LiMn2 O4 . This structure is composed of a cubic close pack lattice with Mn and Li cations occupying one-half and one-eighth of the octahedral
and tetrahedral sites, respectively. Spinels have the benefit of three-dimensional ion conduction
pathways, wherein alkali ions can move between tetrahedral sites via the unoccupied octahedral
sites in the structure. More recently the orthosilicates have emerged as a promising possibility. The
Li2 MSiO4 family (M= Mn, Fe, Co) presents the potential for a M4+ /M2+ two electron redox reaction. The orthosilicates exhibit complex polymorphism and there is controversy over structural
changes that occur during cycling. Notwithstanding, there is consensus that the general framework
is composed of a distorted hexagonal close packing of oxygen with half of the tetrahedral sites
filled by cations such that two dimensional layers of LiO4 tetrahedra are linked by silicate tetrahedra. While the diffusion pathways in these structures are less well characterized, it is likely that
these LiO4 layers enable two-dimensional ion diffusion. The possibility of a Li2 FeSiO4 cathode is
especially appealing as all the constituent elements are members of the major rock-forming group
at the top of Fig. 1.4.
a)
b)
c)
d)
e)
Figure 1.5. Crystal structures of prospective cathode materials, illustrating different dimensionalities of al-
kali ion diffusion pathways. a) Layered LiCoO2 where colbalt octahedra are shown in royal blue, Li ions
are in light blue and oxygen are in red. b) Spinel, LiMn2 O4 with Mn ions in magenta. c) Olivine, LiFePO4 ,
with iron in light brown and phosphate tetrahedra in grey. d) Na3 V2 PO4 NASICON structure with Na ions
in yellow and vanadium octahedra in red.) Li2 FeSiO4 with silicate tetrahedra in blue.
1.6.2
Triphylite
The focus of this thesis is the olivine-phosphate family, MFe2+ PO4 (M=Li, Na). The olivine
structure can be understood as a hexagonal close-packed array of oxygen atoms, with alkali and iron
ions occupying half of the octahedral sites and phosphate atoms occupying one-eighth of the tetrahedral sites. The orthorhombic Pmna structure is shown in Fig. 1.6 (a). Layers of corner-sharing
networks of canted FeO6 octahedra in the b-c plane are spaced by phosphate tetrahedra, and Li+
cations form one-dimensional chains that run between the FeO6 planes. Within typical olivine notation, iron cations sit in the M1 octahedral sites and the alkali ions occupy the M2 sites. Upon lithium
extraction, Fe3+ PO4 has the same underlying structure with unfilled M1 octahedral voids, shown in
Fig. 1.6 (b). The structures are often referred to by their mineral names triphylite and heterosite, for
the lithiated and delithiated structures, respectively. In contrast to the structures discussed above,
the diffusion path in triphylite is restricted to one dimension, along the b-axis channels. Due to this
reduced dimensionality, ion mobility suffers as a result of channel blockage by defects, specifically
cation Fe-Li antisite defects [11, 12]. It has been established that the rate capacity of the material
can be improved by using nanosized cathode particles [13, 14]. The reduction in particle size will
reduce the length of any given channel, reducing the probability that it is blocked.
1.6.3
Maricite
The ground state structure of the sodium analogue of LiFePO4 is the maricite structure, shown
in Fig.1.7(b). While density functional theory calculations suggest the ground state energies of the
two structures are essentially equivalent [15], it is apparent that the maricite structure is favored
at higher temperatures where the material is synthesized. The only known way to form triphyliteNaFePO4 is through a chemical ion-exchange process using LiFePO4 as the starting material. Once
formed, the triphylite structure is stable up to temperatures of ∼800 K, at which point it will revert
back to the maricite structure [16]. Calculations suggest a lithium analogue of the maricite structure
is unstable and it has not been observed [15].
Maricite draws little to no interest from the electrochemical community as there are no apparent conduction pathways for sodium ions. Compared to the triphylite-type structure favored by
LiFePO4 , the site occupancies of the alkali ion are swapped with the iron cations. The sodium
10
a)
b)
Figure 1.6. Orthorhombic olivine-phosphate structure. (a) Lithiated triphylite structure. (b) Delithiated het-
erosite structure. Iron octahedra are shown in brown, oxygen ions are in red, phosphate tetrahedra are in grey
and lithium ions are in blue.
cations are isolated by the phosphate groups, and the structure is electrochemically inactive. Figure 1.7 shows a comparison of the two structures. In the triphylite structure the iron ions occupy
the larger M2 octahedral sites, having corner sharing connectivity, and the alkali ions occupy the
smaller edge sharing M1 octahedra. This edge type connectivity of the M1 octahedra creates a facile
channel for ion mobility. In maricite the occupancies are switched, putting the alkali ions into the
corner sharing M1 sites where there is no good pathway for ion diffusion. Despite the larger size of
the sodium ions, the sodiated triphylite structure has reasonably good ionic conductivity and shows
noteworthy electrochemical performance compared to other sodium-ion cathode materials [17, 18].
As both polymorphs are variants of the olivine structure, they will be distinguished by using their
mineral names, triphylite and maricte.
1.7
Phase stability of olivine phosphates
Lix FePO4 is known to exhibit two-phase behavior at room temperature, with minimal
Li/vacancy solubility in the end members. Despite the tendency for total phase separation, the
observed viability of the intercalation process implies there is necessarily some amount of solubil-
11
Triphylite*
Maricite*
Figure 1.7. A comparison of the triphylite and maricite crystal structures.The first two rows show different views of the structures and the last row depicts the differences in the alkali ion octahedra connectivity.
Iron octahedra are shown in brown, oxygen ions are in red, phosphate tetrahedra are in grey and sodium
ions/octahedra are in yellow.
12
ity. The idea that the solubility limits are enhanced with reduced particle size has been discussed,
as well as the suggestion of some sort of kinetically stabilized intercalation route [19–21]. There
has been a great deal of interest in better understanding the details of how the lithiation process
proceeds, specifically wether it occurs particle by particle though some sort of “domino cascade”
process, wherein after a nucleation event the phase boundary propagates rapidly perpendicular to
the b-axis diffusion channels, such that at any given time all particles are either fully lithiated or
delithiated [22].
For temperatures above 500 K, the Lix FePO4 phase diagram exhibits a broad solid solution
with a eutectoid around x=0.6 [23, 24]. It has been suggested that the details of the two-body
coulombic interactions between the Li-ions, vacancies, electrons and holes have an influence on the
phase stability, particularly for intermediate compositions where lithium removal gives rise to mixed
valent iron ions. The attractive Li+ /Fe2+ and vacancy/Fe3+ interactions contribute to a tendency
for phase separation while the repulsive Li+ /Li+ and vacancy/vacancy interactions tend to stabilize
a solid solution. The electronic configurational entropy gained from disordering Fe2+ and Fe3+ on
the lattice is thought to account for the stabilization of the solid solution at high temperatures [25].
Although the phase diagram for Nax FePO4 has not been as well characterized as the lithium
system, it is clear that there are some interesting distinctions when Na+ replaces Li+ as the intercalation ion. While these discrepancies in phase behavior are potentially an effect of differences
between Na+ and Li+ interactions in the material or a result of differences in the ion’s electronegativity, they likely arise as a result the size discrepancy in ionic radii. The ionic radius is more than
30% larger for Na+ than Li+ , and full sodiation of the FePO4 lattice results in a nearly 17% volume expansion, compared to the 7% expansion seen in LiFePO4 [16]. For Na concentrations above
x=2/3, there is a stable solid solution phase even at low temperatures. Below x=2/3 there exists an
intermediate ordered phase that only disorders above ∼500 K. The ordered phase is described by a
structure having a vacancy at every third sodium site along the Pmna b-axis [26]. Below x=2/3 the
phase diagram exhibits a two phase region between the heterosite end member and this intermediate
ordered phase.
13
Chapter 2
Polaron models
2.1
Overview
The concept of polaron formation was first proposed in a 1933 manuscript by Lev Landau [27].
Landau put forward the idea that an extra electron will inevitably self-trap in an ionic lattice. In
an inversion of the Born-Oppenheimer approximation, the surrounding ions adjust their positions
in such a way as to lower the potential energy of the excess charge. If these displacements produce
a sufficiently deep potential well, the carrier becomes bound. Polaron formation is energetically
stable when the localized carrier’s binding energy exceeds the strain energy expended in displacing
the surrounding nuclei. Once trapped, the carrier can only move if the local distortion travels with it,
resulting in a slow moving particle with a large effective mass. The resulting “polaron quasiparticle”
is composed of the carrier plus this locally-induced distortion. The quasiparticle is referred to as a
“small polaron” when the spatial extent of the carrier’s wavefunction is on the order of the separation
between ions or molecules in the structure. In contrast, a “large polaron” extends over several unit
cells. The work in this thesis is confined to the study of small-polaron motion. Large polarons
exhibit many different properties than small polarons, and while not the focus of this thesis, are
themselves an active topic of research.
Small polaron theory is a large subfield of condensed matter physics. A combined effect of
narrow bands and strong electron-phonon coupling, small polarons are observed in a wide range
of materials including transition-metal oxides, molecular crystals, mantle minerals, rare gas solids,
fullerenes, high-Tc superconductors, and various glasses. At elevated temperature, small polarons
14
Figure 2.1. An electron polaron localized in an iron oxide lattice. Shifts of surrounding ion cores are shown
with arrows. Image Courtesy of Pacific Northwest National Laboratory.
move by thermally-activated hopping, having a low mobility that rises with increasing temperature.
This contrasts with the behavior of conventional free carriers that have a mobility that tends to fall
off with temperature.
2.2
Holstein’s molecular crystal model
In 1959 Theodore Holstein published two seminal papers about polaron localization and mobility [1, 28]. The first introduces the general theory of his molecular crystal model, and the second
addresses the small polaron more specifically. This work provides the framework upon which much
of later polaron theory is based. Holstein’s model is essentially a tight-binding treatment of an
excess electron in a one-dimensional chain of diatomic molecules. While this molecular crystal
model pertains to an idealistically simple system, it captures the essential physics of small polaron
formation and dynamics and illustrates the presence of two distinct temperature regimes. At low
temperatures, the small polaron carriers are in band states. Because an excess carrier can self trap
at any crystallographically equivalent ion site, the polaron can occupy Bloch states. This is not
unlike semifree carriers, the difference being that the energy of the polaron band is many orders of
magnitude smaller than the band for a free electron, typically far less than 1 meV. Above a threshold temperature, the energy uncertainty of these states exceeds their bandwidth, and the Bloch-type
model breaks down. Above this temperature the polaron states are best treated as localized and the
15
Figure 2.2. Holstein’s Molecular Crystal Model. Here an excess electron is localized on the central diatomic
molecule, depicted in red.
carrier mobility becomes diffusive, similar to the motion of light ions in a crystal. The following
discussion will introduce the underlaying assumptions of the molecular crystal model and present
the relevant results. For a full treatment of the Holstein model, readers are referred to the original
1959 manuscripts.
In its simplest form, the molecular crystal model describes the motion of a single electron (or
hole) in a one-dimensional chain of deformable diatomic molecules, shown in Fig. 2.2. Each
molecule has a fixed orientation and center of gravity and is specified with a position Rn = na, an
internuclear distortion variable, xn , and a reduced mass M, where M −1 = N −1 ions m−1 . Here
the distortion variable represents the deviation of the internuclear separation from its equilibrium
position and a is the one-dimensional unit lattice vector. Within this model, a “lattice vibration”
consists of a breathing mode in the individual internuclear separation (xn ) of the nth molecule,
which affects the electron through a molecular potential U (r − Rn , xn ). Here the electron-phonon
interaction is established through the dependence of this potential of the distortion variable, xn .
2.3
Semiclassical treatment of the molecular crystal model
Allowing for one vibrational degree of freedom per molecule, the positive strain energy is
quadratic in the xn (e.g., the interatomic separation of two ions in a diatomic molecule) with
harmonic oscillator frequency, ω0 , associated with the configurational coordinate of an isolated
molecule. The energy is reduced linearly with xn in proportion to the strength of an electronphonon interaction parameter, A, that characterizes the electron-lattice coupling strength in units of
force.
E = M ω02 x2n − A(xn − x0 ).
(2.1)
16
a)
Bx2n
xn
b)
-‐Axn
xn
c)
Bx2n-‐Axn
xn
Figure 2.3. Energy of molecule as a function of distortion coordinate, xn . Here B ∝ M ω02 .
Minimizing this energy gives,
E0 = Axn − Eb ,
Eb =
A2
2M ω02
(2.2)
Fig. 2.3 illustrates how a combination of the electronic and strain terms can give rise to a bound
state. For Eb > Axn , a stable, self-trapped polaron will form. Once trapped, the carrier can move
by hopping to a neighboring site. To hop, the energy of the carrier must be equivalent on adjacent
sites. This condition is fulfilled when the distortions xn and xn+1 of the initial and final sites are
identical. If this distortion pattern is produced through thermal vibrations of the molecules, the
carrier has the potential to transfer. Fig. 2.4 illustrates how electron transfer might occur through
the development of a coincidence condition on a 1D chain. The transition is called adiabatic if given
the development of this coincidence condition, the carrier will always transfer. Alternatively, if the
carrier has a small chance of transfer despite the coincidence condition, the transition is termed
non-adiabatic. At elevated temperature polarons travel through the crystal via thermally activated
17
Figure 2.4. Depiction of polaron hop between adjacent sites on 1D chain illustrating the development of a
coincidence event allowing for electron transfer.
hopping transitions, giving a mobility,
Γ ∝ P exp(−Ea /kB T ),
(2.3)
where P = 1 for adiabatic hopping and P < 1 for non-adiabatic hopping. The activation energy,
Ea , is on the order of half the polaron binding energy, Eb . This type of hopping conduction is
limited to elevated temperatures. Below a threshold temperature, the zero point energy allows for
band-like tunneling between pairs of molecules .
2.4
The Holstein Hamiltonian
The total Hamiltonian of the molecular crystal system is composed of an electronic component,
an electron-lattice interaction component, and a lattice component. The electron-lattice interaction,
Hint , is a function the electron coordinate as well as the lattice displacements.
H =Hel + Hint + Hlatt ,
~2 2 X
Hel + Hint = −
∇ +
U (r − Rn , xn ),
2M
n=1
N
~2 ∂ 2
2 2
+ M ω0 xn + M ω1 xn xn+1 .
Hlatt =
2M ∂x2n 2
(2.4)
n=1
The final term (xn xn+1 ) couples nearest neighbors, giving rise to dispersion of the vibrational frequencies. Proceeding with a typical tight-binding approach, the total wavefunction is built from a
18
linear superposition of molecular wavefunctions,
Ψ=
an (x1 , x2 ...xn )φn (r, xn ),
(2.5)
where the one-electron wavefunctions, φn (r, xn ), constitute an orthonormal set and individually
satisfy Schrödinger’s equation for an isolated molecule,
~2 2
∇ + U (r − Rn , xn ) φn (r, xn ) = E(xn )φn (r, xn ).
2M
(2.6)
The constituent wavefunctions, φn , are each localized about a particular molecular site and depend
on the internuclear distortion variable of that molecule. Holstein assumed the eigenvalues, describing the energy of an electron on an isolated molecule, depend linearly on xn ,
E(xn ) = −Axn .
(2.7)
Equations for an (x1 , x2 ...xn ) are derived using a time-dependent Schrödinger equation, making use
of customary tight binding approximations,
i~
an (x1 , x2 ...xn ) = [Hlatt − Axn ]an (x1 , x2 ...xn ) +
J(xn , xn±1 )an±1 (x1 , x2 ...xn ), (2.8)
∂t
where J is the two-center overlap integral,
J(xm , xn ) =
φ∗ (r − Rm , xm )U (r − Rm , xm )φ(r − Rn , xn )dr.
(2.9)
If in Eq. 2.8 the internuclear distortion variables are all fixed at single value, x, the molecular
crystal model becomes no different from the standard tight-binding solution with a degenerate set
of localized wavefunctions giving Bloch wave expansion coefficients,
an = eikn ,
(2.10)
Ek = E(x) + (N/2)M ω02 x2 − 2J cos(k),
(2.11)
and an energy band,
19
where J has been set equal to a negative constant.
2.5
Adiabatic electron transfer
For adiabatic electron transfer, the Hamiltonian describes a carrier traveling through nuclei
whose positions are effectively fixed. In the adiabatic case, the electronic overlap energy J exceeds
the phonon energies and electron transfer will invariably occur when the phonon displacements
bring the nuclei into a coincidence position. As a result, the terms in Eq. 2.4 involving derivatives
of the electron wavefunctions with respect to nuclear distortion coordinates, xn , are dropped and
the Hamiltonian reduces to,
n=1
n=1
~2 2 X
H=−
∇ +
U (r − Rn , xn ) +
2M
2 2
M ω0 x n .
(2.12)
Neglecting the lattice coupling terms, Eq. 2.8 can be written,
1X
M ω02 x2n − Axn
n=1
an − J(an+1 + an−1 ) = E(x1 , x2 ...xn )an .
(2.13)
There are two different types of solutions to Eq. 2.13. The first is the band-type solution,
an = eikn /N 1/2 ,
(0)
(0)
(2.14)
E(x1 , x2 ...x(0)
n ) = −2J + Jk ,
(0)
(0)
(0)
where (x1 , x2 ...xn ) are the distortion coordinates that minimize the energy. A second localized
solution to Eq. 2.13 is solved using a perturbation expansion in powers of J. To first order, it can
be shown that the polaron binding energy becomes
EP =
A2
− 2J.
2M ω02
(2.15)
It is apparent the quantity A2 /2M ω02 represents the maximum polaron binding energy, in the limit
of infinitely narrow polaron bandwidth (J = 0). The first term in Eq. 2.15 is half of the polaron
binding energy derived in Eq. 2.2. Furthermore, the adiabatic case results in a transition probability
20
between localized states per unit time,
W =
ω0
exp−(Eb /2−J)/kB T ,
2π
(2.16)
where the activation barrier is half the polaron binding energy and the pre-exponential factor is a
characteristic vibrational frequency.
2.6
Non-adiabatic electron transfer
This adiabatic treatment is no longer valid for systems in which the electronic overlap J is
sufficiently small. When the electronic bandwidth (2J) is small compared to a characteristic energy,
2J <
A2
2M ω02
(2.17)
it becomes appropriate to treat J as a perturbation. For the zeroth-order solution (J=0), the carrier
is localized to a site and the energies become,
E (Nq ) = n (0) + Eb +
Eb ∼ −
1 X
A2
2M ωq2
~ωq Nq +
(2.18)
In the derivation of Eq. 2.18 the x-dependence of the electronic overlap integrals is neglected and all
Js are reduced to a single constant. The quantity n (0) pertains to the energy of the carrier at site n
in a rigid lattice and Nq is the phonon occupation number of the qth normal mode with frequencies
given by the dispersion relation,
ωq2 = ω02 +
ω12 X
cos(q).
2 n
(2.19)
Here q = 2πj/N , the integer j taking values between ±N/2. The second term in Eq. 2.18 is
analogous to the polaron binding energy in Eq. 2.2.
Including a non-zero overlap integral gives rise to fundamental differences in the high and low
temperature behaviors. Applying first-order perturbation theory, it can be shown that as the temper-
21
ature goes to zero the energies become,
E(k, Nq ) = E 0 (Nq ) − 2J cos(k) exp [−S(Nq )] ,
1 X
A2
S(Nq ) =
(1 + 2Nq )
(1 − cos(q)).
N q
2M ωq2 ~ωq
(2.20)
Here S is related to the overlap between the harmonic oscillator wavefunctions on adjacent sites.
Within this limit, transitions involving changes in the phonon occupancy of the system (Nq 6= Nq0 )
are negligible. This low temperature solution describes a polaron band with a bandwidth that depends on the vibrational quantum numbers through S,
∆E(Nq ) = 2J exp[−S(Nq )].
(2.21)
Assuming a thermal average, hNq i = 1/(exp(ω/kB T ) − 1), the bandwidth is maximum at absolute
zero and shrinks exponentially with rising temperature.
Above a certain threshold, the polaron bandwidth becomes so small that this band picture
breaks down, and a localized description becomes more appropriate. At these temperatures, transitions where the initial and final vibrational quantum numbers are changed become dominant. The
crossover between low temperature band conduction and high temperature hopping behavior occurs
when the inverse lifetime of the polaron band states becomes small compared to the bandwidth.
This generally occurs at a temperature, Tt ∼ ~ω0 /2kB . The crossover between these two temperature regimes is shown in Fig. 2.5. Above this threshold, the so called ‘diagonal’ transitions,
where Nq = Nq0 , are negligible compared to transitions where Nq 6= Nq0 . A full treatment of
these ‘non-diagonal’ transitions using standard perturbation theory gives the transition probability
between localized states per unit time [29]. In the high temperature limit, ~ω0 /kB T << 1, this
probability reduces to,
J2
W (p, p ± 1, T ) =
4kB T Ea
1/2
exp−Ea /kB T ,
(2.22)
where Ea is the activation energy for carrier hopping,
Ea =
1 X A2
1 A2
(1
cos(q))
= Eb /2.
2N q 2M ωq2
2N 2M ω02
(2.23)
22
Figure 2.5. Depiction of crossover between low temperature, band conduction and elevated temperature
Arrhenius-type behavior. Here the dotted line illustrates an Arrhenius fit to the high temperature result. Here
θ is the characteristic temperature given by θ0 = ~ω0 /kB . Figure from Holstein (1959) [1].
23
The temperature dependence of 2.22 is characteristic of an activated process with an activation
barrier for polaron hopping between adjacent sites of Ea .
24
Chapter 3
The activation volume
3.1
Overview
Electron transfer between adjacent sites in a crystal lattice occurs in an environment that is
macroscopically at constant pressure. Accordingly, the activation barrier is best described in terms
of an activation enthalpy that can be broken down into an energy component plus a volume dependent term,
Ha = Ea + P Va .
(3.1)
The activation energy, Ea , describes the energy barrier for polaron transfer between adjacent sites
and the second term quantifies the pressure effect on the activation barrier, giving the extra energy
cost due to the finite volume change in the activated state. The activation volume, Va , can be interpreted physically as the local change in volume as the particle moves through its transition state.
Va can be either positive or negative, indicating a local expansion or contraction of the lattice, respectively. Fig. 3.2 illustrates a possible sequence of events for a polaron hop on a one-dimensional
chain. The second frame displays the development of a transient distortion pattern that facilitates
electron transfer and illustrates the volume change in the activated state. In this case, the chain
undergoes a local dilation, indicating a positive activation volume.
Accounting for the theoretical details of the last chapter, a general expression for the polaron
25
V>0 or V<0
Figure 3.1. Illustration of the local lattice expansion or contraction that occurs during charge transfer process.
Image Courtesy of Pacific Northwest National Laboratory.
Figure 3.2. Depiction of polaron hop between adjacent sites on 1D chain, illustrating the development of a
transient distortion pattern that allows for electron transfer and shows the local volume change in the transition
state. a) Polaron localized on left site. b) Activated state during electron transfer, illustrating a local expansion
of the lattice. c) Carrier localized on right site.
26
hopping rate can be expressed as,
Γ0
Ea + P Va
Γ(P, T ) ∼ n exp −
kB T
(3.2)
where n=1/2 in the nonadiabatic case and n=0 for adiabatic case of concern here. Accordingly,
measurements of the temperature dependance of the polaron hopping rate allow for the determination of the overall activation barrier. Measurements of the activation volume require the ability to
alter the pressure of the system and gauge any effect on the activation barrier. For a positive activation volume, there is local expansion in the activated state and pressure will tend to suppress the
polaron hopping rate. Conversely, a negative activation volume describes a local lattice contraction
during electron transfer, in which case applying pressure will tend to enhance the polaron hopping
rate. Activation volumes are usually reported using either cm3 /mol or in Å3 . As the later is more
intuitive, this convention will be used from here on out.
3.2
Previous polaron activation volume measurements
The importance of the activation energy for understanding the kinetics of hopping type mobility
is well established and there have been numerous studies looking at activation energies in polaronic
systems. Meanwhile, measuring activation volumes remains largely unexplored. Despite the wealth
of potential information these type of studies could provide, the experiments tend to be challenging.
Historically there have been a handful of high pressure conductivity studies, mostly focusing on
electrical transport measurements of minerals. These high pressure conductivity measurements are
tricky and sorting out issues that arise due to pressure effects on the electrical contacts is not trivial.
Recent improvements in tools for high pressure measurements are opening up high pressure work to
different scattering techniques and expanding the availability of sample environments. With these
advances, this topic is becoming increasingly accessible. In particular, nuclear resonant scattering
is a technique that allows for a direct measure of the polaron hopping rate and can be extended to
high pressures using synchrotron nuclear forward scattering in an externally heated diamond anvil
cell.
The majority of experimental reports of activation volumes are from the geophysical literature.
These studies generally address the high-pressure conductivty of iron-bearing mantle minerals in-
27
cluding perovskite-(Mg,Fe)SiO3 , olivine-(Mg,Fe)2 SiO4 , and magnesiowüstite-(Mg,Fe)O [30–33].
In these systems, the concentration of polarons is related to the concentration of vacancies and
to oxygen partial pressure. Geophysical models as well as magnetotelluric and geomagnetic deep
sounding methods give credence to a theory of a lower mantle layer with enhanced conductivity [34]. Consequently, understanding the electronic properties of minerals at elevated pressures
and temperatures is relevant to constraining the chemistry profile of the mantle. These measurements of electrical transport properties of oxides under applied pressure gave small, negative values
for Va of a few tenths of a cubic angstrom. This is often interpreted in terms of better overlaps of
electron wavefunctions when the ions are pushed closer together. Authors have gone so far as to
suggest that a small, negative activation volume is inherent to polaron conduction [35]. This metric
has been used to distinguish between ionic conduction and polaron conduction, where it is accepted
that values of Va for ionic mobility are larger and positive by comparison [30].
3.3
Corrections to the apparent activation volume
The activation barrier sets the temperature dependence of the polaron hopping rate. The activation volume is defined in terms of the pressure dependance of this activation barrier,
Va =
∂Ha
∂P
(3.3)
Activation volumes are usually determined by performing a temperature series of a measurement
that relates to the polaron hopping rate for a set number of fixed pressures in order to identify any
shift in activation barrier with pressure. Consequently, the activation barrier is commonly defined
as,
Va ≈ −kB T
∂ ln Γ(P, T )
∂P
(3.4)
As the majority of activation volume data are from conductivity measurements, Va is often expressed
in terms of the electrical conductivity, in which case the polaron hopping rate, Γ(T, P ), in Eq. 3.4
is replaced with the conductivity, σ(P, T ).
Equation 3.4 assumes the prefactor in Eq. 3.2 does not depend on volume. While most studies
use Eq. 3.4, it is possible that both the exponential term and the prefactor could exhibit a pressure
28
dependence. If this is the case, correction terms must be added to “apparent activation volume,”
defined in Eq. 3.4. It is also possible that there is an entropic contribution to the activation barrier.
As a result of the inverse temperature dependance of the exponential in Eq. 3.2, any entropic
contribution results in a constant term that is absorbed into the prefactor. The activation volume
defined in Eq. 3.4 is only rigorously correct if the pressure dependance of this entropic term is
negligible. Furthermore, depending on the experimental technique used to determine the polaron
hopping rate, other pressure dependent terms can enter the prefactor as well.
For adiabatic electron transfer, the prefactor in Eq. 3.2 reduces to a characteristic phonon frequency. Recasting the expression for the adiabatic polaron hopping rate defined in Eq. 2.16 gives
an activation volume,
Va = −kB T
∂ ln Γ
∂ ln ω0
+ kB T
∂P
∂P
(3.5)
It is straightforward to show that the final term can be rewritten in terms of a Grüneisen parameter
and a compressibility,
Va = −kB T
∂ ln Γ
+ kB T γκT .
∂P
(3.6)
Assuming a typical Grüneisen parameter γ ∼ 2, a bulk modulus of 150 GPa, and a temperature
of 300 K, the final term in Eq. 3.6 is on the order of 0.06 Å3 . This term can be understood as an
enhancement of the polaron hopping rate due to pressure induced stiffening of the lattice, resulting
in an overall reduction of the the apparent activation volume. This correction must be added to a
volume determined using Eq. 3.4 to ascertain the true value of Va .
For nonadiabatic hopping, the prefactor depends on the square of the overlap integral, J. The
pressure dependence of the overlap integral can be estimated using an inverse localization length,
α, describing the spatial extent of the wavefunction,
J = J0 exp−αR ,
(3.7)
where R is the distance between two sites. Substituting Eq. 3.7 into the adiabatic polaron hopping
rate defined in Eq. 2.16 gives an activation volume,
Va = −kB T
∂ ln Γ
+ kB T (2αR0 κT /3),
∂P
(3.8)
29
where the pressure dependence of J arises from the reduction of the inter-site distance with pressure,
R = R0 (1 − P κT /3),
(3.9)
and the inverse localization length is assumed to be pressure independent. Taking αR0 = 2, this
correction term becomes ∼ 0.04 Å3 . Similar to the nonadiabatic correction, this factor must be
added to the experimentally determined activation volume, derived from the pressure derivative of
the polaron hopping rate.
Generally, for localized electrons, α > 1/R. However, placing an upward bound on the inverse
localization length is not straightforward without resorting to electronic structure calculations. Even
in cases where calculated electronic density information is available, it is well known that density
functional theory fails to accurately model strongly localized states, where self-interaction errors
become important. Before the widespread availability of first principles calculations, predictive
estimates of wavefunction overlap were often used to explain observations related to electronic
structure.
In his “Solid State Table of the Elements,” Walter Harrison developed a suite of approximation
tools, providing parameters that allow for simple calculations of a range of material properties
[36]. Using Harrison’s method for approximating wavefunction tails, the overlap integral for d-type
wavefunctions goes as the inverse 5th power of the site separation,
J ∝ J0 /R5 .
(3.10)
Substituting this overlap approximation into Eq. 2.16 results in an activation volume,
Va = −kB T
∂ ln Γ
10kT
+ kB T
∂P
3 − κT P
(3.11)
For pressures of of 0−10 GPa, the final term raises the activation volume by an ∼0.09 Å3 , giving a
slighter larger result than the previous overlap approximation, with Eq. 3.8. In both cases, this factor
represents an enhancement of the polaron hopping rate, due to the enlargement of the prefactor from
30
increased wavefunction overlap. The result is an overall shift to a lower apparent activation volume,
− kB T
∂ ln Γ
∂HA
∂HA
10kT
− kB T (2αR0 κT /3) ≈
− kB T
∂P
∂P
∂P
3 − κT P
(3.12)
For cases where the pressure shift of the enthalpy is small, this factor can be responsible for the
observation of negative activation volumes.
Another commonly used expression for the polaron hopping rate is the Mott conductivity equation, originally developed to treat hopping conduction in transition-metal-containing glasses [37].
Given certain approximations of prefactor terms, this treatment is essentially equivalent to nonadiabatic Holstein model. That said, as this formalism is frequently adopted for experimental data
analysis, it is treated separately here. Within this model, the expressions for the polaron hopping
rate and the activation volume become,
Γ(T, P ) = ω0 exp
Va = −kB T
−2αR
exp
Ea + P Va
kB T
∂ ln Γ
+ kB T (2αR0 κT /3) + kB T γκT .
∂P
(3.13)
(3.14)
Applying the same approximations as above, it can be shown at 300 K that the last two terms of the
right hand side of Eq. 3.14 sum to ∼ 0.1 Å3 .
Despite the aforementioned challenges associated with high pressure conductivity measurements, bulk transport measurements remain the most common method to study activation volumes.
While conductivity data are clearly pertinent to the performance in an electrode material, these measurements are not a direct measure of the local polaron hopping rate, Γ(P, T ). The Nernst-Einstein
equation relates the conductivity to the mobility of the moving species,
σ=
cq 2
D,
kB T
(3.15)
where q is the polaron charge, c is the concentration and D is the diffusivity. Furthermore, the
conductivity is related to the polaron hopping rate,
σ=
e2
c(1 − c)R2 Γ(T, P ),
kB T
(3.16)
31
accounting for dependence of the polaron diffusivity on the average number of surrounding open
sites as well as the jump distance between sites. The additional factor R2 in the prefactor of Eq.
3.16 gives rise to an additional correction factor,
∂ ln σ
∂ ln Γ
− kB T
= −kB T
+ kB T
∂P
∂P
2κT
3 − P κT
(3.17)
Again, assuming pressures of of 0−10 GPa, a temperature of 300 K, and a bulk modus of 150 GPa,
this correction factor is ∼ 0.02 Å3 .
3.4
Pressure dependence of the Holstein activation barrier
While the molecular crystal model does not explicitly account for any volume dependence,
an examination of the pressure shift of the Holstein activation barrier can lead to pertinent physical
insights. As detailed in the preceding chapter, for adiabatic polaron hopping at elevated temperature,
the activation barrier in the Holstein model becomes,
Ea = Eb /2 − J =
A2
− J,
4M ω02
(3.18)
where A is an electron-phonon coupling term, M is the reduced mass of the molecule, ω0 is characteristic phonon frequency, and J is the overlap integral. For nonadiabatic hopping, Eq. 3.18 reduces
to half the polaron binding energy, Eb . A pressure derivative of Eq. 3.18 gives,
∂Ea
= Eb
∂P
∂ ln A ∂ ln ω0
∂P
∂P
∂J
∂P
(3.19)
Using Harrison’s approximation for the overlap integral Eq. 3.19 can be expressed,
∂Ea
= Eb
∂P
1 ∂A
− γκT
A ∂P
− κT J.
(3.20)
Unlike the correction terms discussed in previous section that arose from pressure effects on the
prefactor, the terms in this expression are a result of the pressure dependence of the activation
barrier itself. From the first term, it can be seen that a fractional enhancement of the electronphonon coupling with pressure will give rise to a positive term in the activation volume. The final
32
term, due to the enhancement in wavefunction overlap, is often cited as the source of negative
polaron activation volumes.
3.5
Activation volume for ion diffusion
The topic of activation volumes for ion diffusion has received more attention than the polaron
counterpart. Similar to polaron mobility, ion diffusion in solids is treated as a thermally activated
process with an Arrhenius-type mobility. The ionic activation volume is associated with a change
in material volume associated with the migration of an ion. The pressure dependence of ionic
conductivity gives information about the volume relaxation associated with the formation and motion of defects. Although there are reports of some superionic conductors having negligible or even
negative activation volumes, the vast majority of conventional ionic conductors exhibit a strong suppression of ionic mobility with pressure, giving activation volumes ranging from +1to+10 Å3 [38].
Applying a simple hard-sphere model, the migration volume is roughly equivalent to the volume of
the diffusing species. However, experimentally determined activation volumes are usually significantly smaller than this model would imply. Several more sophisticated models assume the free
energy for ion migration can be attributed to the strain energy of the lattice. Treating the surrounding
crystal as a continuous medium, the activation volume scales with the lattice compressibility.
33
Chapter 4
Methods
4.1
Synchrotron x-ray diffraction
X-ray diffraction (XRD) is a powerful technique for determining crystal structures. The spectral
brilliance of synchrotron x-rays allows for the study of small samples confined in a diamond-anvil
cell. In addition, the high intensity of a synchrotron beam provides sufficient resolution to resolve
weak superstructure peaks that would be difficult to observe with a typical lab-diffractometer. Using
a two-dimensional CCD detector, a full pattern can be collected in matter of seconds. Angular
dispersive XRD patterns are collected in transmission mode using a monochromatic synchrotron
beam and a MAR CCD detector. The raw data appear as a set of rings centered around the beam
Figure 4.1. Typical 2D image file collected with CCD detector.
34
stop, with radii related to the crystallographic d-spacings through Bragg’s law,
λ = 2d sin(θ),
(4.1)
where θ is a quarter of the diffraction-cone angle. A typical CCD image is shown in Fig. 4.1. A
diffraction pattern from a CeO2 sample calibration is collected to establish the distance between
the detector and the sample. To transform the image file into typical one-dimensional diffraction,
data in the images are integrated azimuthally using the Fit2D package [39]. The data can then be
analyzed to study the crystal structure of the material.
4.2
X-ray diffraction data analysis
Rietveld analysis of the synchrotron x-ray data was performed to obtain information on phase
fractions and thermal trends of lattice parameters. Given starting structural information, Rietveld
programs calculate a model diffraction pattern, then proceed to iteratively optimize this model by
minimizing the difference between the fit and the experimental data. The iterations refine experimental parameters such as background, peak broadening and lattice constants. This type of refinement requires a reasonably accurate starting structure, and while the method is powerful for honing
in on precise structural details, if the crystal structure is not known, it will not be possible to solve
for it. Given a starting structure, Rietveld refinement allows for accurate determination of unit cell
parameters, strain broadening effects, and qualitative phase analysis. The method is particularly
useful for the evaluation of powder patterns with overlapping peaks.
Rietveld analysis was performed using the General Structure Analysis System (GSAS) [40, 41].
A shifted Chebyschev polynomial was used to fit the background, including between four and six
terms as needed. To capture accurate peak shapes, the fit model employed psedo-Voigt profile
functions. For all samples, both Gaussian and Lorentzian crystallite size broadening were assumed
to be negligible. As low angle data was not considered, profile terms related to axial-divergence
were not included. The angular dependence of the Gaussian variance is set by the Cagliotti Function,
σ 2 = U tan2 θ + V tan θ + W.
(4.2)
35
It was assumed that the instrumental broadening was primarily Gaussian. As such, the Cagliotti
terms (U , V ,W ) were determined from a refinement of a CeO2 standard and fixed for all additional
refinements.
Strain broadening effects were treated as entirely Lorentzian. To reproduce accurate profiles,
anisotropic strain broadening had to be accounted for, where peak broadening varied by reflection
class. To reproduce this anisotropy, profile terms based on Stephen’s anisotropic strain model were
included [42]. Within this semi-empirical model, the strain components are restricted in terms of
the first and second order terms allowed by lattice symmetry. For an orthorhombic cell this gives
rise to six independent strain terms,
Γ2S = S400 h4 + S040 k 4 + 3(S200 h2 k 2 + S202 h2 l2 + S022 k 2 l2 ).
(4.3)
When possible, these anisotropic strain broadening terms were fixed for each phase and not allowed
to vary for patterns collected at different temperatures. In cases where this did not produce good
fits, the terms were refined with strong“damping”, such that only a small fraction of the shift is
applied. In this case, the temperature evolution of the strain terms were carefully monitored to
ensure the trends were physically reasonable. The refinement of strain terms was only necessary
when handling the precipitation of a secondary phase. At low temperature, the phase fraction of the
second phase is low and the effects of strain broadening are readily apparent. As the temperature
is raised and the fraction of the new phase increases, strains are relaxed and the magnitude of the
strain terms falls off. Having established physically reasonable and reasonably reproducible profile
terms, the phase fractions as well as the lattice parameters can be refined. This allows for a study of
the phase stability as well as the thermal expansion.
4.3
Mössbauer spectrometry
4.3.1
The Mössbauer effect
The Mössbauer effect describes the recoil-free absorption and emission of a γ-ray by atoms in a
solid. The γ-ray emission that accompanies the decay of a radioactive nuclei in an excited state has
the potential to excite other similar nuclei. For free nuclei, the efficiency of this process is severely
36
limited by the nuclear recoil during both the emission and absorption processes. As a result, the
energy of the emitted γ-ray is reduced by the kinetic energy of recoil of a free nucleus ER ,
ER =
Eγ2
∼ 2 meV,
2M c2
(4.4)
where Eγ is the photon energy, M is the nuclear mass, and c is the speed of light. The linewidths
of the nuclear excited states are remarkably precise, on the order of 10−9 eV. The recoil of a free
nucleus will prevent an emitted γ-ray from having an energy that falls in the tight window necessary
to excite an additional nuclei. During his doctoral work, Rudolf Mössbauer recognized that this
complication could be effectively circumvented if the recoil was absorbed by an entire crystal rather
than a single nuclei. In this case the mass in Eq. 4.4 is enhanced by a factor equal to the number
of atoms in the crystal and the recoil energy is inconsequential. This will be realized if the recoil
energy is less than the energy of the lowest quantized mode of the crystal. Within the framework
of the Debye model, the probability of a recoil free event is described by the Lamb-Mössbauer
factor [43],
−6ER
f=
kB ΘD
ΘD
2 Z ΘD
dx ,
exp(x) − 1
(4.5)
where kB is Boltzmann’s constant and ΘD is the Debye temperature. To observe the Mössbauer
effect, the recoilless fraction, f , must be large. In general this occurs when ER << ~ωD . Since the
Lamb-Mössbauer factor depends on the γ-ray energy through, ER , the probability of a recoil-free
event is only appreciable for certain isotopes with low lying excited states. Fortunately this includes
57 Fe, making the Mössbauer effect useful for a wide range of studies of iron-bearing materials. In
addition to iron, the Mössbauer effect has been observed in a wide range of isotopes, including but
not limited to 151 Eu, 191 Ir and 119 Sn.
4.3.2
Hyperfine interactions
Recoilless γ-ray emission in 57 Fe occur between a nuclear ground state having spin I=1/2 and a
14.41 keV excited state, I=3/2. Electrons in the vicinity of the resonant nuclei can break rotational
symmetry and perturb the energies of the nuclear states. Measurement of the nuclear transitions then
serves as a probe to study the electronic environment of the resonant atom and its nearest neighbors.
The isomer shift (IS) arises from a Coulomb interaction between the nuclear and electronic charge
37
distributions. The s-electron wave functions have a finite overlap with the nucleus, lowering the
nuclear energy levels. The Coulomb interaction between the s-electron density and the nuclear
charge causes an energy level shift,
δE = πZe2 |Ψ(0)S |2 hRi2 ,
(4.6)
where Z is the atomic number, |Ψ(0)S | is the s-electron density, and hRi is the mean-square radius
of the nuclear charge distribution. Any difference in the s-electron density between the absorber and
the emitter will give an overall shift in the observed resonance. The measured splitting becomes,
δE = πZe2 (|Ψ(0)A |2 − |Ψ(0)S |2 )(hRi2E − hRi2G ),
(4.7)
where the subscripts S, A, E and G refer to the source, absorber, excited and ground states respectively. Screening effects of the 3d electrons can reduce the electron density at the nucleus, thereby
increasing the IS. Consequently, a ferrous ion will show an appreciably larger IS compared to a ferric ion in a similar environment. IS is a relative quantity and can only be determined in comparison
to other materials or as a difference between two crystallographically distinct Fe sites within the
same material.
The electric quadrupole splitting (QS) results from the interaction of the nuclear quadruple
moment with an inhomogeneous electric field. The quadrupole moment, Q, describes the departure
from spherical symmetry in the rest frame of the nucleus. For nuclei with spin quantum numbers
I=0 or 1/2, the nuclei will have spherical symmetry and Q=0. A nucleus with a spin number I>1/2
will have a non-spherical charge distribution where Q>0 describes an oblate nucleus with respect
to the spin axis, while Q<0 indicates a prolate nucleus. When a nonzero nuclear quadruple moment
is exposed to an asymmetric electric field resulting from an asymmetric electronic environment, an
electric quadrupole interaction results in the loss of degeneracy of the nuclear energy levels . The
asymmetry of the electronic environment of the nuclei is characterized by the electric field gradient
(EFG). The result of this interaction is a splitting of the nuclear energy levels corresponding to the
different alignments of the quadruple moment with respect to the principal axes of the EFG tensor,
38
Figure 4.2. Energy diagram of isomer shift and quadrupole splitting, ∆EQ , for the 57 Fe 3/2→1/2 transition
in an asymmetric EFG. Figure: UC Davis ChemWiki- CC BY-SA 3.0.
Vzz . The eigenvalues of the Hamiltonian for the quadrupole interaction are,
1/2
eQVzz
η2
EQ =
[3mI − I(I + 1)] 1 +
4I(2I − 1)
(4.8)
mI = I, I − 1, ... − |I|,
where I is the nuclear angular momentum and η is the asymmetry parameter,
η=
Vxx − Vyy
Vzz
(4.9)
The result for 57 Fe is an excited state with two doubly degenerate sublevels corresponding to the
m =I ±3/2 and mI = ±1/2 nuclear spin states. The energy splitting is given by,
1/2
η2
δE = ± eQVzz 1 +
(4.10)
As the nuclear quadruple moment is fixed, Q ∼ 0.16 for the I=3/2 excited state of 57 Fe [44],
the magnitude of the splitting gives information about the local electric field in the vicinity of the
Mössbauer nuclei. Variation in QS between different Fe-bearing materials is caused by distinct
EFG tensors arising from valance differences and changes in the local symmetry surrounding the
39
resonant ion.
4.3.3
Mössbauer measurements
The experimental set-up for a typical Mössbauer measurement involves a radioactive source,
containing the resonant isotope in an excited state, and a sample, containing the resonant nuclei in
the ground state. For measurements involving 57 Fe, the usual source is 57 Co embedded in a Rh
matrix. 57 Co decays to a metastable 57 Fe state via electron capture which successively decays to
the ground state by emitting a 14.4 keV photon. To study the hyperfine structure, the energy of the
incident γ-ray must scan through a range that covers the spectral splittings. To modulate the energy,
the source is moved relative to the absorber using Doppler drive. The γ-ray source is mounted on a
mechanical transducer which oscillates back and forth giving a shifted energy,
E=
Eγ ,
(4.11)
where v is the source velocity, Eγ is the 14.4 keV photon energy and c is the speed of light. The
Mössbauer spectrum is collected by recording the transmitted photons as a function of source velocity. The resonant absorptions appear as dips in the spectrum with Lorentzian linewidths. The
cross section for resonant absorption is described by the Breit-Wigner formula,
σa (E) =
σ0 Γ2a /4
(E − E0 )2 + 14 Γ2a
(4.12)
where σ0 is the nuclear resonant cross-section and Γa is the line width of the excited state of the absorber. Similarly the emitted γ-rays from the source have a Lorentzian distribution around 14.4 keV
with a linewidth, Γs . Given a sufficiently thin source and absorber, the observed resonance curve is
simply a convolution of the two distributions, resulting in a Lorentzian curve with linewidth Γs +Γa .
For the limiting case where both the source and observer have a natural linewidth, Γn ∼ 0.097 mm/s,
the observed resonance will have a linewidth of 2Γn , where Γn = ~/141 ns, and 141 ns is the lifetime of the first nuclear excited state for 57 Fe.
40
Figure 4.3. Schematic of furnace used for elevated temperature Mössbauer measurements.
4.3.4
Resistive furnace for high temperature Mössbauer measurements
To collect Mössbauer data at elevated temperature, the sample is mounted in a resistive furnace,
depicted in Fig. 4.3. The sample is contained in high temperature kapton and sandwiched between
two aluminum plates, with a window for γ-ray transmission. The temperature is controlled with two
resistors mounted on the aluminum sample holder. Three thermocouples to are affixed to different
positions around the sample to monitor the temperature. The entire assembly is aligned between the
source and the detector to optimize γ-ray transmission.
4.4
Nuclear forward scattering
The high intensity and pulsed structure of a synchrotron radiation source allows for the collection of nuclear resonant time spectra. This time domain analogue to traditional Mössbauer Spectrometry is known as nuclear forward scattering (NFS) or Synchrotron Mössbauer Spectrometry
(SMS). The small line width of the nuclear resonance of 57 Fe, 4.66 neV, necessitates an x-ray
source with high spectral intensity to excite the nuclear transition. The low angular divergence and
41
2.1 Classification of Scattering Processes
E1
E2
E3
k0
E1 + E2 + E3
sample
detector
E1 + E2 + E3
Fig. 2.2. Coherent elastic NRS in forward direction. The superposition of waves
various
levels
leads
to quantum
in theemitted
temporalfrom
evolution
of thehyperfine-split
nuclear decay. Figure
from
Röhlsberger
(2004) beats
[2]. in the temporal
evolution of the decay. This is illustrated by overlaying three wavetrains of slightly
different frequencies, leading to a Moiré pattern that represents the quantum beats
Figure 4.4. A coherent superposition of wavlets from slightly offset energy levels gives rise to quantum beats
condensed matter physics. The basic principles and applications particularly
in the field of thin-film magnetism are treated in Sect. 4.4.
2.1.2 Coherent Inelastic Nuclear Resonant Scattering
Figure 4.5. Schematic of beamline set up for nuclear resonant scattering. Picture adapted from Zhou, et al.
(2004)
[3]. type of scattering is illustrated in Fig. 2.3. The excited nuclear state
This
interacts with lattice vibrations in the sample that transfer energy to the
highreemitted
brilliance ofphoton.
the synchrotron
beam allows
for high-pressure
experiments
to be runas
in aadiamond
The energetic
analysis
of the scattered
radiation
func-
tion of momentum transfer allows the determination of phonon dispersion
relations and the study of vibrational excitations in condensed matter. This
and tunes theelectronic
incident beam
to the
resonant which
energy, has
and been
a highis pre-monochromater
typically done viafilters
(nonresonant)
x-ray
scattering
developed
into a powerful
method
at modern
synchrotron
sources
resolution
monochrometer
further reduces
the energy
bandwidth
to ∼2meV.radiation
The ensemble
of 57 Fe
[1, 2, 3, 4]. Unfortunately, this scattering process is much less favorable in case
nuclei
simultaneously
by a synchrotron
radiation
pulse.
standard
time structure
of are
nuclear
resonantexcited
scattering.
A detailed
analysis
wasThe
given
by Sturhahn
Kohn
[5].
One
reason
is
that
the
lifetimes
of
thermal
phonons
are
very
short
of the synchrotron radiation at the advanced photon source (APS) provides 24 pulses separated by
compared to the nuclear lifetime. Therefore, the coherence of the waves scat153.3
ns, each
having
a duration
∼70ps. The
electronic scattering
occurs
withinshort
femtoseconds
tered
by the
nuclei
in theofsample
is preserved
only during
a very
time.
Then,
in analogy
to lifetime
nuclearofresonant
in isthe
of diffusion
of the
pulse arrival,
while the
the nuclearscattering
resonant state
141presence
ns. This allows
for a clear
−12
s) of coherent
(see Sect. 4.6), one expects an extremely fast decay (∼10
separation
of the
prompt
electronic
scattering
from
the delayed, extremely
resonant scattering
of interest.
The
inelastic
NRS,
which
would
make its
observation
difficult.
A closer
inspection
thisnuclear
type excitation
of scattering
can be
appreciable
when
synchrotron
pulsereveals
creates a that
collective
with coherent
interference
between
emitted
phonon is created upon absorption while during reemission the lattice state
photons
the forward
direction.
If thesince
degeneracy
of the nuclear
levels has
has been
by hyperfine
doesin not
change.
However,
the reemitted
photon
the lifted
nuclear
transition energy,
it de-excitation
suffers strong
resonant
that beat
reason
coherent
interactions,
the phased
of slightly
offsetabsorption.For
energy levels produces
patterns
in transanvil-cell.
42
mitted intensity. The delayed emisson is expressed as a sum over oscillatory terms whose arguments
are the differences in the energies of the nuclear levels superimposed on the exponential decay [45].
T (t) ∼
η2
exp(−t/τ
exp(−iωj,l t) a∗0 Wj Wl a0 .
16Ωτ 2
j,l
Here ~Ω is the energy bandwidth of the synchrotron pulse, W is the normalized weight of the
nuclear transition, ωj,l = ωj − ωl , a0 is the polarization unit vector of the synchrotron radiation,
and η = f σρD is the effective thickness. The effective thickness quantifies the scattering power
and the influence of sample thickness on spectra. In this expression f is the recoil free fraction, σ
is the nuclear resonant cross section, ρ is the density of 57 Fe nuclei, and D is the sample thickness.
A sample with two Fe sites, each with distinct quadrupole splittings and isomer shifts, will have six
component beat frequencies in the transmitted intensity, each with a period that is inversely related
to the difference in nuclear energy levels, ~/∆EHF .
To collect an NFS spectra, timing electronics are used to block the signal for the first several
ns after the pulse arrival. After this deadtime, an avalanche photodiode detector (APD) positioned
in the forward-scattered x-ray beam measures the delayed counts as a function of time after pulse
arrival.
4.5
High pressure measurements
Elevated pressure measurements where carried out using diamond-anvil cells (DACs) to generate quasi-hydrostatic pressure. Pressure is simply the applied force divided by the area over which
this force is distributed: P=F/A. To generate a high pressure with a moderate force, the area must
be small. The flattened faces of gem quality diamonds are ideal surfaces to generate high pressures because diamonds have unparalleled hardness and are optically transparent at typical x-ray
frequencies. In a conventional diamond-anvil cell, depicted in Fig. 4.6, the sample is compressed
between the culets of two gem quality diamonds. Typical culet diameters range from 100-500µm.
For the experiments in this thesis all culets were ∼ 300µm. The sample is contained in a chamber
made from a metal gasket sandwiched between the opposing diamonds. The diamonds are mounted
on tungsten carbide seats with epoxy resin and these seats are screwed into the cell, taking care to
ensure the diamonds are both centered and the culet faces are parallel. To prepare the sample cham-
43
Figure 4.6. Schematic of diamond-anvil cell. Figure: Tobias 1984- CC BY-SA 3.0.
ber, a rhenium gasket is pre-compressed between the diamonds to generate a culet-size indentation
in the center of the gasket. A sample chamber is drilled in the center of the indentation using an
electric discharge milling machine. For 300µm culets the appropriate drill bit is chosen to make a
hole with an ∼ 100µm diameter. After drilling, the gasket is cleaned and placed back into the DAC.
Using the tip of a needle, the sample is loaded into the chamber, along with a few small pieces of
ruby that will be used for pressure measurement. Finally the cell is closed by tightening the screws,
pushing the two sides of the cell together confining the sample between the opposing culets and the
gasket. The cell has several screws around its circumference that push the opposing sides of the
DAC together. Care is taken to maintain parallel alignment, by tightening the cell in small steps
and turning pairs of screws on either side of the cell simultaneously. This is done using a DAC
designed with a combination of right and left-handed screws. Commonly the screws are fitted with
spring-loaded washers to allow for carefully-controlled tightening.
For the measurements in this thesis, two different types of DAC designs were used, a Mao-type
symmetric DAC and a smaller Tel-Aviv DAC, shown in Figs. 4.8 and 4.7 respectively. The smaller
of the two designs is a Merill-Bassett, Tel Aviv-type cell [46]. This design has two opposing plates
that are held together with a series of pins. The symmetric cell has a piston-cylinder design, that has
44
Figure 4.7. Photo of Tel Aviv-DACs, shown with stack of quarters for size comparison.
Figure 4.8. Photo of Mao-type symmetric cell.
the benefit of making it easier to ensure the diamonds remain parallel during operation. To generate
hydrostatic pressure, this sample chamber is filled with a pressure medium if possible. The pressure
medium transmits the uniaxial pressure generated by the opposing diamond culets to the sample by
creating an isotropic fluid-filled pressure chamber that is confined laterally by the gasket material.
Pressurized helium or neon gas are optimal pressure media. To fill the sample chamber with gas, the
cell is placed in a specialized chamber that is first evacuated then filled with pressurized gas. The
cell is then closed using gears within the chamber that are controlled remotely. A system designed
to gas-load symmetric DACs is maintained by the staff at sector 13 at the Advanced Photon Source.
An alternative to using the screws to increase pressure is to use a gas-membrane system. This
Figure 4.9. Pressure membranes for remote pressure control.
45
configuration allows for the pressure to be adjusted remotely, which is particularly useful when conducting experiments in a sample environment (i.e. a cyostat or furnace). Shown in Fig. 4.9, a doughnut shaped gasket that can be inflated with gas is pushed against the back of the cell and clamped in
place flush with the cell. The force that this bladder exerts on the back of the cell substitutes for the
force that would otherwise be applied by tightening the screws. The expansion/contraction of the
bladder is controlled remotely by controlling the gas pressure.
To determine the pressure in the sample chamber, a laser is used to monitor the frequency shift
of the R ruby-fluorescence lines. The behavior of this band at elevated pressure and temperature
is well known. The R1 /R2 doublet shows an approximately linear shift upward with increasing
pressure, providing a good reference for pressure calibration. Using a laser/spectrometer system
that can be moved into the beam path when the x-ray shutter is closed, the pressure of the sample
can be monitored in situ. When using the gas-membrane system to control the sample pressure, this
information can be used as feedback to maintain the desired pressure. This feedback is particularly
important when heating or cooling the DAC assembly, as changing the temperature will often result
in changes in pressure in the cell.
4.6
Experimental setup for high temperature, high pressure measurements
To collect synchrotron data at elevated pressure and temperature, the diamond-anvil cell is
mounted in a resistive furnace. The first iteration of experiments in this thesis where carried out
using inconel Tel-Aviv DACs and a resistive furnace, shown in Fig. 4.10. A continuous flow of
Ar/1%H2 gas was maintained with the aim of protecting the diamonds from damage at elevated
temperature. In later experiments, a symmetric cell was used. In these experiments, a copper-block
furnace was used to heat the cell, shown in Fig. 4.11 This furnace was used in conjunction with a
gas membrane pressure system, allowing for remote monitoring and adjustment of pressure in the
cell. The entire furnace assembly was contained in a vacuum chamber. To monitor the temperature,
three thermocouples were used, one was glued to the side of the diamond, a second was affixed to
the outside of the cell, and a third was attached to the Cu block. For NFS experiments at beamline 16ID-D at the Advanced Photon Source, the vacuum furnace was used in conjunction with a
46
Figure 4.10. Schematic of furnace used for high temperature, high pressure synchrotron experiments with
Tel Aviv-DACs.
Figure 4.11. Photo of furnace used for high temperature, high pressure synchrotron experiments with sym-
metric DACs.
47
Figure 4.12. Photo of setup at beamline 16-IDD, including vacuum furnace, online ruby system, CCD for
XRD, and APD for NFS.
pressure-membrane/online-ruby system. In addition, a CCD detector was used to collect in-situ
XRD using the 14.4 keV wavelength. The full experimental set-up is shown in Fig. 4.12.
4.7
Relaxation effects
4.7.1
Overview
Nuclear resonant scattering provides a particularly effective method to study the dynamics of
charge hopping processes. It is possible to observe the dynamical effects when the fluctuations in
the hyperfine field occur on the same time scale as the nuclear decay process. The quadruple interaction provides a natural time window for sampling the electric field surrounding the Mössbauer
nuclei: τEF G = ~/∆EHF . If the time scale of the charge hopping processes is much slower than
this window, the observed spectrum is a coherent superposition of two static components associated
with distinct 2+ and 3+ Fe valences. If the polaron hopping frequency exceeds this sampling frequency, observable distortions appear in the spectra. With further increase in hopping frequency, the
spectrum begins to resemble that of a single, time-averaged doublet. Altering the valence fluctuation frequency can give rise to rich variations of the shape and symmetry of the quadrupole doublets.
The interesting behavior occurs when the valence of a 57 Fe ion fluctuates between Fe2+ and Fe3+
48
at a frequency between 1 and 100 MHz. In conventional Mössbauer energy spectra the quadrupole
doublets from Fe2+ and Fe3+ merge together, with asymmetric, non-Lorentzian lineshapes for these
intermediate frequencies. For nuclear forward scattering in the time domain, these effects are seen
as a distortion and washing-out of the quantum beat pattern from interference of the nuclear hyperfine levels. When the relaxation is most pronounced, there is a large suppression in the integrated
count rate. The fast fluctuations tend to dephase the photons emitted from separate nuclei, resulting
in a loss of coherence that is seen as an overall reduction in the probability for the excitation of
a nuclear exciton. Although the dynamical information that can be extracted from an NFS spectra is essentially the same as from a conventional Mössbauer spectrum, this strong suppression of
intensity is unique to NFS.
4.7.2
Blume-Tjon model
Drawing on ideas about motional narrowing developed by Kubo and Anderson [47], Blume
and Tjon developed a stochastic theory to describe the effect of randomly fluctuating hyperfine
fields on a Mössbauer spectrum [48, 49]. Within this model, a fluctuating nuclear environment is
treated by providing multiple sets of hyperfine parameters, together with a matrix of transition rates
describing the random jumps between the different sets of hyperfine parameters. This allows for
the refinement of the hyperfine parameters specific to each site, as well as a transition frequency.
For polycrystalline samples, the problem reduces to the simplest case treated by Blume and Tjon in
which the stochastic and quantum mechanical parts of the problem are separable as there is no issue
of non-commutativity of the Hamiltonian at different times.
The nuclear Hamiltonian is written in terms of a stochastic function f (t) = ±1, describing
either the 3+ or the 2+ site at any instant,
Ĥ = Ĥo +
1
(1 + f (t))∆QS 2+ + (1 − f (t))∆QS 3+ (3m2I − I(I + 1)),
12
(4.13)
49
where Ĥ0 is the nuclear Hamiltonian in the absence of hyperfine splittings,
Ĥ0 |I0 m0 i = E0 |I0 m0 i,
Ĥ0 |I1 m1 i = E1 |I1 m1 i,
(4.14)
E1 − E0 = 14.41 keV.
Isolating the time dependent terms, Eq. 4.13 can be rewritten as,
Ĥ = Ĥo + [q1 + q2 f (t)] (3m2I − I(I + 1)),
(∆QS 2+ + ∆QS 3+ ),
12
q2 = (∆QS 2+ − ∆QS 3+ ).
12
q1 =
(4.15)
The probability for a transition between the excited and ground state with the emission of a
photon is Lorentzian in form,
P1,0 (ω) =
| hI0 m0 |Ĥ (+) |I1 m1 i |2
(ω + E0 − E1 )2 + 14 Γ2
(4.16)
where H(+) is interaction Hamiltonian of solid and photon and Γ is the linewidth of the excited
state. Expressing the denominator in integral form, the relationship
−1
Z ∞
= Re
exp [i(ω + E0 − E1 )t − Γt/2] dt,
(ω + E0 − E1 )2 + Γ2
(4.17)
allows for a recasting of Eq. 4.16 to incorporate an explicit time dependence,
Z ∞
Re
exp(iωt − Γt)hI0 m0 |Ĥ (+) |I1 m1 i∗ hI0 m0 | expi−E0 t Ĥ (+) expi−E1 t |I1 m1 idt
Z0 ∞
= Re
exp(iωt − Γt)hI1 m1 |Ĥ (−) |I0 m0 ihI0 m0 |Û † (t)Ĥ (+) Û (t)|I1 m1 idt.
P1,0 (ω) =
(4.18)
Here Ĥ (−) = Ĥ (+)† and Û (t) is the time evolution operator,
Û (t) = exp(−iĤt),
Ĥ (+) (t) = Û † (t)Ĥ (+) Û (t).
(4.19)
50
The observed probability results from averaging Eq. 4.18 over all possible excited states |I1 m1 i
and summing over the ground states |I0 m0 i,
P (ω) =
p1 W0,1 =
0,1
Re
Z ∞
exp(iωt − Γt))[hĤ (−) Ĥ (+) (t)i]av dt.
(4.20)
Here []av indicates an average over the Hamiltonian’s stochastic degrees of freedom. The correlation
function in Eq. 4.20 is defined as,
[hĤ
(−)
Ĥ
(+)
(t)i]av =
[hI1 m1 |Ĥ (−) |I0 m0 ihI0 m0 | exp(i
2I
m1 m0 m1 m0
× hI0 m00 |Ĥ (+) |I1 m01 ihI1 m01 | exp(−i
Z t
Z t
Ĥ(t0 )dt0 )|I0 m00 i
Ĥ(t0 )dt0 )|I1 m1 i]av .
(4.21)
Here it is assumed that the excited m1 sublevels are equally probable, i.e. p1 = 1/(2I1 + 1), and
the operator in Eq. 4.19 has been replaced with a time-ordered operator,
Z t
Û (t) = exp −i
Ĥ(t )dt .
(4.22)
The sum in Eq. 4.21 involves two time-ordered series. For the Hamiltonian defined in Eq. 4.13,
the first of these series is straightforward to evaluate, as the ground state in the 57 Fe I= 3/2 → 1/2
transition is unspilt,
hI0 m0 | exp(i
Z t
Ĥ(t0 )dt0 )|I0 m00 i = exp(iE0 t)δm0 m00 .
(4.23)
The second sum in Eq. 4.21 becomes,
[hI1 m01 | exp(−i
Z t
Z t
Ĥ(t0 )dt0 )|I1 m1 i]av = exp(−iE1 t) exp(−iβt) exp −iα
f (t0 )dt0 δm1 m10
av
α = q2 (3m2I − 15/4),
β = q1 (3m2I − 15/4).
(4.24)
The stochastic average in Eq. 4.24 is evaluated with the use of a transition probability matrix,
51
W, that describes the transition rates between the different sets of hyperfine parameters,
Z t
f (t )dt δm1 m10
exp −iα
av
pρ (σ| exp(−iαF + W)t|ρ).
(4.25)
ρσ
Here F is a matrix with the possible values of the stochastic function f (t) on the diagonal,
1 0
.
F=
0 −1
(4.26)
The elements transition probability matrix, Wρσ , represent the probability per unit time that the
system experiences a transition from state ρ to state σ,
−w3+→2+ w3+→2+
.
W=
w2+→3+ −w2+→3+
(4.27)
The diagonal elements of the transition probability matrix are determined by the off diagonal elements such that,
Wρρ = −
(4.28)
Wρσ .
For charge hopping between Fe2+ and Fe3+ sites, in order to maintain charge balance, W is
expressed using a single frequency weighted by population of the two states,
−ρ2+ w ρ2+ w
.
W=
ρ3+ w −ρ3+ w
(4.29)
Substituting Eqs. 4.23 and 4.24 into Eq. 4.20 and integrating gives the observed profile as an
analytic expression in terms of the transition probability matrix,
P (ω) =
1 X
|hI0 m0 |Ĥ (+) |I1 m1 i|2 Re
pρ (σ|[pI + iαF − W]−1 |ρ),
2Γ m m
ρσ
p = − i (ω − ω0 − β) + Γ/2,
ω0 =E1 − E0 = 14.4 keV.
(4.30)
52
The matrix elements |hI0 m0 |Ĥ (+) |I1 m1 i| determine the polarization and intensity of the individual lines and for polycrystalline samples the first part of Eq. 4.30 can be treated as a constant.
For charge hopping in the phospho-olivines, ferric iron is introduced by removing alkali ions,
3+
Mx Fe2+
x Fe1−x PO4 , such that ρ2+ = x and ρ2+ + ρ3+ = 1.0. The calculation of the second
part of Eq. 4.30 is now straightforward,
P (ω) ∝ Re
4.8
p + 2w + iα(2x − 1)
p + α2 + pw + iαw(2x − 1)
(4.31)
Nuclear resonant scattering data analysis
Analysis of nuclear resonant scattering data was carried out using the CONUSS package (COherent NUclear resonant Scattering by Single crystals) [50]. CONUSS allows for the calculation
and refinement of nuclear resonant scattering spectra both in the time and energy domains. The program enables the user to define a fit model, specifying the hyperfine parameters associated with a
number of iron sites and the phase fractions of these sites. CONUSS requires a starting model that is
reasonably accurate and is best suited for optimizing parameters within a known model. While the
software was originally developed for analysis of time spectra from synchrotron nuclear resonant
scattering, the program can handle conventional Mössbauer data as well. The program incorporates
a full treatment of thickness-dependent dynamical effects that can give rise to “speed-up,” as well
as complicated spectral distortions.
CONUSS offers a choice between static hyperfine interactions and randomly fluctuating hyperfine fields, as described by a Blume-Tjon model [48, 49]. It is also possible to set up a fit model
where certain sites experience fluctuating hyperfine fields while others are static. A static site is
characterized by its weight fraction, an isomer shift, a quadrupole splitting and a linewidth for a
lorentzian quadrupole splitting distribution. When appropriate, it is also possible to include an
asymmetry parameter as well as parameters associated with magnetism and textural effects. To
calculate thickness effects, the program input also calls for the definition of the sample thickness
and a Debye temperature to calculate the recoil free fraction. Given these parameters, the hyperfine
splittings and corresponding eigenvectors are determined by numerically diagonalizing the nuclear
53
Hamiltonian,
Φj0 m0 Hm0 m1 Φm1 j1 = Ej δj0 j1 .
(4.32)
m0 m1
Fluctuating hyperfine fields are treated by providing the same set of hyperfine parameters together with the matrix of transition rates, W , describing the jumps between different sets of hyperfine parameters. In this case, the eigenvalue problem becomes,
ρσ
σν
Lµρ
j0 j1 m0 m1 Am0 m1 m0 m0 Rm0 m0 l0 l1 = Ωjj1 δµν δj0 l0 δj1 l1 .
m0 m1 m00 m01 ρσ
(4.33)
Here L and R are the left and right eigenvectors and the eigenvalues (Ω) are now complex. The
matrix A incorporates the transition probability matrix, W , along with the matrix elements of the
Hamiltonian for the excited and ground states,
Aρσ
= iW ρσ δm0 m00 δm1 m01 + hI1 m1 |Ĥ ρ |I1 m01 iδm0 m00 δρσ − hI0 m0 |Ĥ ρ |I0 m00 iδm1 m01 δρσ ,
m0 m1 m0 m0
(4.34)
where ρ indicates a particular set of hyperfine parameters.
CONUSS uses the solutions to the eigenvalue problems of Eqs. 4.32 or 4.33 to calculate the
coherent nuclear scattering amplitudes. For nuclear forward scattering, the Fourier transformation
of the energy-dependent transmission function allows for the calculation of the time-dependent
intensity. This theoretical model is then compared to the data, and parameters are adjusted iteratively
to minimize the mean square deviation between the fit model and the experimental spectra. This
allows for the refinement of the hyperfine parameters specific to each site as well as the elements
of the transition probability matrix, W , which defines the jump rates between the different sets of
hyperfine parameters. For a typical two-site dynamical model, the ratio of these matrix elements
is proportional to the weight ratio of the two sites. Consequently, to maintain charge balance a
constant ratio must be maintained. CONUSS allows for the grouping of fit parameters such that the
matrix elements can be refined while keeping their ratio fixed.
54
Chapter 5
Polaron-ion correlations in Li0.6FePO4
5.1
Introduction
Lithium-iron phosphate, LiFePO4 , is a new material for cathode electrodes of rechargeable
Li-ion batteries [51]. An important issue, however, is its low electrical conductivity; at low temperatures LiFePO4 is an insulator with a band gap of approximately 3.7 eV [52–54]. LiFePO4 has
the orthorhombic olivine-type structure shown in Fig. 5.1. Layers of corner-sharing networks of
canted FeO6 octahedra in the b-c plane are spaced by phosphate tetrahedra. Li+ cations form onedimensional chains that run between the FeO6 planes. Previous work showed that the Li+ diffusion
pathway is along these b-axis channels [12, 55]. The electronic carrier mobility is expected to be
two-dimensional, occurring within the layers of FeO6 octahedra that are separated by insulating
phosphate groups.
Experimental values of both electrical conductivity and Li+ -ion diffusivity in LiFePO4 span
several orders of magnitude [56–62]. These large discrepancies have been attributed to differences
in samples and experimental technique [63]. It is generally accepted that the Li+ ion diffusivity is
highly sensitive to defects in the one-dimensional channels along the b axis. Less understood is the
scatter in reported values of electrical conductivity, however, which contributed to an early controversy about whether the electronic conductivity can be improved by doping [64]. Measurements of
bulk properties on polycrystalline samples also present challenges in decoupling the intrinsic conductivity from the interparticle conductivity. Nevertheless, a keen interest remains in improving the
intrinsic electrical conductivity of LiFePO4 , and better understanding the transport of Li+ ions and
55
Figure 5.1. Olivine-type structure of LiFePO4 with chains of Li+ ions (blue), planes of FeO6 octahedra
(brown), and phosphate tetrahedra (grey).
electrons.
As with many other transition metal oxides, the mechanism of electrical conductivity in mixed
3+
valent Lix Fe2+
x Fe1−x PO4 is small polaron hopping [65–67]. A small polaron quasiparticle com-
prises an electron or hole localized by atomic displacements of neighboring anions. When an electron transfers between Fe2+ and Fe3+ sites, the local configurations of the FeO6 octahedra must also
transfer. The difference between these atomic configurations in LiFePO4 is large. By removing Li+
ions from the lattice, lithiated Li1 Fe2+ PO4 is transformed into delithiated Fe3+ PO4 with the same
olivine-type structure. As the Fe ions change from Fe2+ to Fe3+ during delithiation, the average
Fe-O bond lengths are reduced by 6% [68].
At moderate temperatures, the motion of a polaron quasiparticle is diffusive and can be understood as an activated process with a jump rate [37, 69],
Ea + P Va
Γ(T, P ) ≃ ν exp(−2αR) exp −
kB T
(5.1)
where T is temperature, P is pressure, kB is the Boltzmann constant, ν is a characteristic phonon
56
frequency, R is the Fe-Fe distance, and α is the inverse localization length of the Fe wave functions.
The activation energy, Ea , describes the energetic barrier for the polaron quasiparticle to transfer
between adjacent iron sites. Previous measurements of bulk electronic conductivity as a function
of temperature gave a wide range of activation energies between 155 and 630 meV [64, 70–73].
Mössbauer spectrometry provides a more direct measurement of the rate of polaron hops between
iron sites, and gives an activation energy around 500 meV [65, 66].
The effect of pressure on the activation barrier is quantified with an activation volume, Va . P Va
is the extra enthalpy required from thermal fluctuations to induce a polaron hop when the material is
under the pressure P . For Va > 0, the activation barrier is effectively raised with pressure, and the
polaron hopping frequency is suppressed. Va is the difference in volume between the material with
the configuration favorable for electron transfer, and the volume in the equilibrium configuration.
It is expected to be local in origin, and is expected to reflect the local expansion or contraction
in the vicinity of the hopping polaron. In accordance with the Frank-Condon principle, these local
atomic distortions bring the electron levels of the initial and final states into coincidence, facilitating
electron transfer. An understanding of the activation volume therefore gives insight into the atom
configurations at the transient state of the polaron hop.
There have been few studies of the activation volume for polaron hopping. Previous measurements of electrical conductivity in geophysically relevant oxides under applied pressure gave small,
negative values for Va of a few tenths of a cubic angstrom [30–33]. It has been suggested that the
dominant effect in these systems was the decrease of R under pressure, allowing the electron to
better sample the final state, therefore enhancing the polaron conductivity [33]. To our knowledge
there has been no measurement of the electronic conductivity of LiFePO4 under pressure.
Unlike the motion of polarons, the diffusion of Li+ ions can be understood classically. Ion jumps
into vacant neighboring sites occur by an activated process that does not sense the ion destination
until after the jump is complete. First principles simulations suggest that Li+ ions diffuse rapidly
along the [010] channels, but there is a high energy barrier to cross between channels [12]. These
calculations do not include defects or electron-ion interactions, however, and other reports suggest
the material is a slow ion conductor [56]. The one-dimensional character of the Li+ mobility results
in an ion conductivity that is highly sensitive to defects that block conduction channels, such as
Li-Fe antisite point defects [11, 12].
57
Here we report new results on the charge dynamics at elevated pressure, obtained by performing measurements on Lix FePO4 heated in a diamond-anvil cell. The 57 Fe valence fluctuations in
Lix FePO4 are strongly sensitive to pressure, giving a large and positive activation volume for polaron hopping that is more characteristic of ion diffusion. We show how this large effect could
result from a correlated dynamics of polarons and mobile Li+ ions. Previous density functional
theory calculations for LiFePO4 gave low activation barriers for polaron hopping compared to experimental results. This discrepancy was attributed to polaron-ion interactions [74]. The concept
of a bound polaron has also been discussed in calculations of polaron migration barriers for lithium
peroxide [75]. These studies assume a rigid lattice during electron transport, however. The authors
state, “...the electron density alone is relaxed self-consistently and atom positions remain fixed for
calculations along the migration path.” [74] In other words, this method employs a linear combination of the initial and final states without allowing for ion rearrangements in the transition state,
so Va = 0. Ion-electron correlations have also been mentioned in reports of NMR and molecular
dynamics studies on LiMn2 O4 and Lix NiO2 [76, 77], for example, but there has been scant experimental evidence to support this concept. With a polaron-ion interaction, the activation enthalpy
for moving a polaron depends in part on the ion motion by a vacancy mechanism. Vacancy diffusion is suppressed by pressure, and activation volumes for ion transport in oxides range from +1 to
+10 Å3 [38]. Because ion transport is suppressed by pressure, polaronic conductivity should also
be suppressed if the polaron-ion interaction energy, Epi , is large. In what follows, we estimate Epi
to be approximately –300 meV, which should have important consequences for the dynamics and
positions of both polarons and ions.
5.2
Experimental
A solid solution of Li+ ions in Lix FePO4 is stable at temperatures above 473 K, and is easily
preserved at room temperature by quenching [23, 24]. Previous x-ray diffractometry measurements
showed that the olivine structures of FePO4 , Li0.6 FePO4 , and Li1 FePO4 are stable to pressures of
at least 30 GPa at 300 K [78]. Solid solutions of Li0.6 FePO4 were prepared by a solid-state reaction
and delithiated as described previously [23, 79]. Powders were loaded into a Merrill-Bassett, Tel
Aviv-type, diamond-anvil cell [46] along with ruby chips for pressure measurement by the ruby
58
florescence method [80]. The cells were prepared using rhenium gaskets and diamonds with 350 µm
culets. The cell was heated in a resistive furnace with an Ar/1% H2 atmosphere and kapton windows
for x-ray transmission.
Nuclear forward scattering (NFS) measurements were performed at beamline 16ID-D at the
Advanced Photon Source at Argonne National Laboratory. An avalanche photodiode detector was
placed in the forward-scattered x-ray beam to measure transmitted intensity as a function of time.
Four sets of measurements were taken at pressures of 0, 3.5, 7.1, and 17 GPa, with temperatures between 298 and 598 K. A high-resolution monochrometer tunes the incident beam to the 14.414-keV
resonant energy and reduces the bandwidth to ∼2 meV. The synchrotron flashes had durations of
70 ps, and were separated by 153 ns. Electronic scattering occurs within femtoseconds of the pulse
arrival at the sample. The relatively long lifetime of the nuclear resonant state (τ = 141 ns) allows
for a clear separation of the prompt electronic scattering from the delayed, resonant scattering.
The 57 Fe nuclei in the sample are simultaneously excited by the synchrotron x-ray pulse, giving
rise to coherent interference between emitted photons in the forward direction. When the degeneracy of the nuclear levels is lifted by hyperfine interactions, the phased de-excitation of slightly offset
energy levels generates beat patterns in the transmitted intensity. Within the kinematical limit, the
delayed emission in the forward direction is expressed as a sum over oscillatory terms whose arguments are the differences in the energies of the nuclear levels, superimposed on the exponential
decay [45],
A(t) ∼ exp(−t/τ )
exp(−iωj,l t) a∗0 Wj Wl a0 .
(5.2)
j,l
Here W is the normalized weight of the nuclear transition, ωj,l = ωj − ωl , and a0 is the polarization
unit vector of the synchrotron radiation. A sample with two iron sites, each with a distinct value
for quadrupole splitting (QS) and isomer shift (IS), will have six component beat frequencies in the
transmitted intensity, each with a period that is inversely related to the difference in nuclear energy
levels.
Nuclear resonant scattering allows for the study of local electron dynamics at iron ions. The
measured spectra are altered when the hyperfine fields fluctuate on the same time scale as the characteristic frequency of the hyperfine interaction energies, ~ω. In Lix FePO4 the frequency of valence
fluctuations, and how this frequency changes with temperature, leads to rich variations of the shape
59
and symmetry of the quadrupole doublets from Fe2+ and Fe3+ . At low frequencies and low temperatures, the spectral components from Fe2+ and Fe3+ remain distinct, and at very high frequencies
the spectrum is a single doublet. The rich behavior occurs when the valence of a 57 Fe ion fluctuates between Fe2+ and Fe3+ at a frequency between 1 and 100 MHz. In conventional Mössbauer
energy spectra the quadrupole doublets from Fe2+ and Fe3+ merge together, with asymmetric, nonLorentzian lineshapes for these intermediate frequencies. For nuclear forward scattering (NFS) in
the time domain, these effects are seen as a distortion and washing-out of the quantum beat pattern
from interference of the nuclear hyperfine levels. Previous conventional Mössbauer energy spectrometry studies on Lix FePO4 reported dramatic spectral distortions at temperatures between 373
and 513 K [65, 66].
5.3
Simulational
The hops of electron polarons are likely confined to the b-c plane, but they would tend to follow
the one-dimensional paths of ions if the interactions between polarons and ions are strong. We
performed a series of Monte Carlo (MC) simulations on a coupled pair of one-dimensional chains.
As shown in Fig. 5.2, one chain contained Li+ ions, and the other contained electron polarons. The
goal of these simulations was to estimate the strength of the polaron-ion interaction by comparing
the simulated electron dynamics under pressure to the valence fluctuations measured by nuclear
resonant scattering.
The hop of a Li+ ion requires an empty site at an adjacent position on the ion chain, so ion
diffusion was assumed to occur by a vacancy mechanism. Likewise, an electron-polaron at an Fe2+
site requires a neighboring Fe3+ on the same chain for the electron to hop, so a vacancy mechanism
was used for the electron dynamics as well. Activated state rate theory was used to calculate jump
probabilities of the ions and electrons. The activation barrier for the ion depended only on the
initial configuration, but in the adiabatic approximation the electron samples the initial and final
state energies before making a transition.
Each chain depicted in Fig. 5.2 had 3000 sites and periodic boundaries. Half of the sites on each
chain were initially populated at random, one with Li+ ions, and the other chain with electrons.
Both species moved along their respective chains by a vacancy-type mechanism. For each step in
60
Figure 5.2. Schematic of randomly populated 1D coupled Li+ ion and electron chains.
Figure 5.3. Six subprocesses describing ion and electron jumps on coupled 1D chains, where the energy
barrier for each subprocess is listed below the schematic. Ep and Ei are the free polaron and ion activation
barriers respectively, Epi is the polaron-ion interaction energy, and Vi is the activation volume for ion hopping.
For Li+ ion jumps, depicted in the lower frames, the energy barrier depends only on the initial 1NN electron
site; the final 1NN site on the electron chain is not depicted.
61
the simulation, every site on both chains was selected in a random sequence. If the site contained
an electron or ion, the probability that a jump will occur was calculated using Boltzmann factors,
described below, for T = 573 K. The time was obtained as a running sum of the inverse of the
Boltzmann factors of the jumps that occurred.
The energies used in the Boltzmann factors are
{Ei , Ep , Epi } ,
(5.3)
where the first two are activation energies for the jump of a bare (noninteracting) ion and a polaron,
respectively, and the third is the polaron-ion interaction energy. For a given event, the relevent site
occupancies of the electron or ion on a site were either 0 or 1, as set by four Kroneker δ-functions.
For a site directly opposite on the other chain the index is 0, to its left –1, or right +1,
{δ0p , δ0i , δ−1i , δ+1i } .
(5.4)
The two Kroneker δ-functions for the vacancy pertain to vacancies on the same chain as the moving
species, which allow the jump to occur to the left or right (±1),
{δ−1v , δ+1v } .
(5.5)
The Boltzmann factors for the four jumps to the left or right by the ion or electron-polaron are
B−1i = δ−1v exp −β(Ei + P Vi + δ0p Epi ) ,
B+1i = δ+1v exp −β(Ei + P Vi + δ0p Epi ) ,
B−1p = δ−1v exp −β[Ep + (δ0i − δ−1i )Epi ] ,
B+1p = δ+1v exp −β[Ep + (δ0i − δ+1i )Epi ] ,
(5.6)
(5.7)
(5.8)
(5.9)
where the ion jump is influenced by pressure, and depends on the presence of an electron-polaron
directly opposite (subscript 0), whereas the electron jump depends on the presence of an ion directly
opposite, but also opposite to its final position after the jump.
The jump probabilities were normalized by the two possibilities that could occur and the possi-
62
bility of no event
Γ−1i =
Γ+1i =
Γ−1p =
Γ+1p =
B−1i
1 + B−1i + B+1i
B+1i
1 + B−1i + B+1i
B−1p
1 + B−1p + B+1p
B+1p
1 + B−1p + B+1p
(5.10)
(5.11)
(5.12)
(5.13)
At each step of the simulation, the state of the chains was used to obtain the Kroneker δ-functions
needed for Eqs. 5.6 through 5.9. The electron or ion under consideration moved left, right or
remained stationary based on a randomly generated number, Q, between 0 and 1. For a given
electron, a left jump occurred when Q < Γ−1p , a right jump when Γ−1p < Q < Γ+1p + Γ−1p , and
no jump when Q > Γ+1p + Γ−1p . Ion jumps were determined similarly. The local change after a
successful jump was used to update the state of the chains, and the inverse of the Boltzmann factor
was added to the time.
For the results shown below, activation barriers were set using previous computational results for
the “free-polaron” activation energy, Ep = 215 meV, and the activation energy for Li+ ion diffusion,
Ei = 270 meV [12, 74] (although many other values were tried). These activation barriers were
altered by a polaron-ion interaction energy, Epi , the strength of the coupling between the Li+ ion,
and the electron polaron when the two are first nearest neighbors (1NN), being at the same sites
on their respective chains. First principles calculations place Epi in the range of –370 meV to –
500 meV, depending on the degree of lithiation, and the authors suggested that the the polaron-ion
interaction could affect polaron dynamics [74]. When a Li+ ion jumps away from a 1NN electron,
the activation barrier for the jump is raised by an amount |Epi |. The quantum character of the
electrons gives an activation barrier that depends on the 1NN on the Li+ chain in both the initial
and final positions. Accordingly, the electronic activation barrier is raised by an amount |Epi | when
the electron jumps away from a Li+ 1NN, and is lowered by an amount |Epi | for a jump into a site
with a Li+ 1NN. The possible jumps are broken down into the six subprocesses shown in Fig. 5.3.
The activation barrier for ion hopping was altered by an amount P Vi , where Vi is the activation
volume for ion diffusion. An activation volume of +5 Å3 was used, typical of activation volumes
63
measured for ion diffusion in similar systems [38]. Assuming LiFePO4 behaves similarly to other
transition metal oxides, we expect the activation enthalpy for the hop of a bare electron polaron to
decrease with pressure [30–33]. Because this effect is expected to be an order of magnitude smaller
than the effect on ionic diffusion, for the purpose of these simulations we treated the activation
barrier for electron hopping as pressure independent.
5.4
Results
5.4.1
Experimental results
The NFS spectra are presented in Fig. 5.4. In the 0 GPa series, with increasing temperature,
especially above 400 K, the quantum beats are broadened and flattened, and the integrated count
rate decreases. This washing out of the spectral features and suppression of count rate comes from
a dephasing of the scattered intensity, consistent with the development of broad, asymmetric energy
spectra. The temperature range of the onset of these effects is consistent with the polaron dynamics
reported by conventional Mössbauer spectrometry [66]. At elevated pressures these large spectral
distortions do not occur until higher temperatures, approximately 100 K higher for 3.5 GPa. Smaller
changes can be seen at lower temperatures, however.
The spectra were evaluated using the software package CONUSS [50, 81]. CONUSS allows for
the calculation and refinement of spectra using the theory of Blume and Tjon for random temporal
fluctuations of the hyperfine field [48, 49]. Drawing on the Kubo-Anderson model of motional narrowing [47], Blume and Tjon used a correlation function, time averaged over the stochastic degrees
of freedom, to evaluate the lineshapes of emitted radiation from a system with a fluctuating nuclear
Hamiltonian. Depending on the relaxation time relative to the lifetime of the excited state, the effective widths of the resonance lines can either sharpen or broaden inhomogeneously and amalgamate.
While the probability for a transition between the excited state and the ground state with the emission of photon is Lorentzian in form, the observed probability results from a sum over the possible
ground states and a stochastic average over the sampled excited states. For polycrystalline samples,
the problem reduces to the simplest case treated by Blume and Tjon in which the stochastic and
quantum mechanical parts of the problem are separable as there is no issue of non-commutativity
of the Hamiltonian at different times.
64
Figure 5.4. Temperature series of NFS spectra taken at 0, 3.5, 7.1 and 17 GPa. The fits (black curves) overlay
experimental data (red points). Temperatures are listed to the left of spectra in Kelvin. The x-axis is the delay
in nanoseconds after the arrival of the synchrotron pulse. The spectra have been scaled by their maximum
value and offset for comparison.
65
The fluctuations from polaron hopping require two sets of hyperfine parameters for the Fe2+
and Fe3+ sites, together with a relaxation matrix of transition rates, describing the random jumps
between the two sets of hyperfine parameters:
−Γρ2+ Γρ2+
.
W=
Γρ3+ −Γρ3+
(5.14)
The elements of the relaxation matrix are weighted by the populations, ρ, of the two sites, maintaining charge balance. This allows for the refinement of a QS specific to each site, a relative IS, and a
polaron hopping frequency, Γ(T, P ).
The Blume-Tjon model was not used for the spectra at 298 K. These spectra were fit with a static
model, allowing for the refinement of the sample thickness as well as the distribution of QS that may
result from disorder in the sample and pressure gradients in the cell. The sample thickness and the
distribution of QS were then fixed for the fits at elevated temperatures, minimizing problems from
correlations between the hopping frequency and the distribution of quadrupole splittings (which
produce similar effects at low hopping frequencies). For fitting a data set at a fixed pressure, after
fixing the thickness and the distribution of the QS at their values for 298 K, four parameters were
varied to fit the spectra at elevated temperatures. The refined fits overlay experimental spectra in
Fig. 5.4. Most of the hyperfine parameters showed gradual changes with temperature and pressure
that we summarize here with linear relationships:
QS of Fe2+ : [2.9 − 2 × 10−3 (T − 298)] mm(s K)−1 ,
QS of Fe3+ : [1.1 − 2 × 10−3 (T − 298)] mm(s K)−1 ,
Relative IS: [0.8 − 10−3 (T − 298)] mm(s K)−1 ,
QS of Fe2+ : [2.9 + 0.04P ] mm(s GPa)−1 ,
QS of Fe3+ : [1.1 + 0.04P ] mm(s GPa)−1 .
The relative IS (IS Fe2+ – IS Fe3+ ) did not show a discernible trend with pressure. These parameters
are consistent with those determined using conventional Mössbauer spectrometry for the same material [65, 66]. The data and fits in Fig. 5.4 are plotted on a logarithmic scale. The fitting algorithm
uses a least squares criterion, so the fit discrepancies in the regions of the lowest count rate (most
notably the third minima) are smaller than they appear and do not significantly affect the quality of
66
Figure 5.5. (a) Polaron hopping frequencies, Γ(T, P ), at 0, 3.5 and 7.1 GPa, as determined from the solid
curves of Fig. 5.4. Solid curves are Arrhenius-type fits using a pressure-independent prefactor. (b) Activation
enthalpies, ∆H = Ea + P Va , versus pressure, where Ea = 470 meV. Black triangles are results for fixed
prefactor and red circles are for a pressure dependent prefactor.
the fits.
Figure 5.5(a) shows the polaron hopping frequencies, Γ(T, P ), determined from the fits to the
spectra of Fig. 5.4. For frequencies below approximately 1 MHz, the spectra are fit equally well with
a static model. In this low-frequency limit, a static spectrum and a dynamic spectrum are identical,
all else held constant. For the 17 GPa series, the hopping frequencies for the entire temperature
series were below this threshold. The suppression of hopping frequencies at moderate pressures
indicates that Va is positive and large. The three data sets in Fig. 5.5(a) were fit simultaneously
with Eq. 5.1 to determine the activation enthalpies and the prefactor. From the ambient pressure
series, the activation energy was found to be 470±50 meV, where the uncertainty arises from the
weighting of the different data points in linear or logarithmic fits, and the choice a prefactor for Eq.
5.1. This is comparable to the values of 512, 550, and 570 meV for activation energies of hyperfine
parameters from the same material measured by conventional Mössbauer spectrometry [65].
The prefactor was first assumed independent of pressure. The result, ∼ 1013 Hz, is typical of optical phonon frequencies measured by inelastic neutron scattering and by Raman spectrometry [82, 83]. For a second set of fits, we calculated the pressure dependence of the prefactor, ν exp(−2αR). We extrapolated the attempt frequency to elevated pressure using a typical
67
Grüneisen parameter, γ = 2, and the compressibility, κT ,
ν(P ) ≃ ν0 (1 + γP κT ) .
(5.15)
The wave function overlap was approximated assuming a pressure-independent localization length
exp[−2αR] ≃ exp[−2αR0 (1 − P κT /3)] ,
(5.16)
where R0 is the ambient pressure inter-cation distance. X-ray diffractometry measurements at
300 K on an olivine Li0.6 FePO4 solid solution at pressures up to 32 GPa gave a bulk modulus of
120±4 GPa [78]. This is somewhat larger than for Li1 FePO4 , with bulk modulus measured as
106±8 GPa, and calculated as 96 GPa [84]. These additional considerations did not significantly
affect the results below for Va .
From Eqs. 5.1, 5.15 and 5.16, Va can be determined from the pressure dependence of the
activation enthalpy. For a given pressure, we determine the activation enthalpy by looking at the
linear part of ln(Γ) as a function of β, where β = 1/(kB T ) [31].
ln(Γ) = −β∆H − 2αR + ln(ν) ,
∆H ≃ −
∂ ln(Γ)
∂β
(5.17)
(5.18)
To account for any pressure dependence of the prefactor of Eq. 5.1, we consider the pressure
dependence of the last two terms in Eq. 5.17. Assuming these terms are independent of temperature,
Eqs. 5.15 and 5.16 can be used to correct the Va obtained from the jump rates, Γ(T, P ),
Va =
Va ≈ −
∂∆H
∂P
ln Γ
∂ ∂ ∂β
∂P
(5.19)
2 αR0 κT
γκT
3β
(5.20)
The dominant source of error in the determination of the enthalpies lies in the choice of a prefactor
for Eq. 5.1. Constraining the prefactor to a reasonable range based on past measurements of optical
phonons [82, 83, 85] gives an error in the magnitude of the activation enthalpies of approximately
68
±10%. The slope of the curve in Fig. 5.5(b) gives an activation volume of +5.8±0.7 Å3 . The
second and third terms of Eq. 5.20 are an order of magnitude smaller than the first term from the
slope of Fig. 5.5(b), but will increase Va above the value of +5.8 Å3 . Our Va is between one and
two orders of magnitude larger than previously reported polaron activation volumes from resistivity
measurements on oxides [30–33].
5.4.2
Simulational results
Polaron jump frequencies were calculated as a function of pressure, assuming these frequencies
were proportional to Boltzmann factors with thermal activations. The activation energies were taken
as the appropriate combination of Ep , Ei , and Epi , depending on their local configurations. The
activation energy for an ion jump was increased with pressure by P Vi , where Vi was +5 Å3 and
P was 0, 3 or 7 GPa. We used values of Ep = 215 meV and Ei = 270 meV as reported in the
literature [12, 74], but we also calculated frequencies using several other activation barriers ranging
from 50% to 200% of these values.
For simulations with |Epi | greater than 100 meV, after a quick initial relaxation, more than 90%
of the electrons were paired to a Li+ across the coupled chains. By inspecting the jump probabilities
of Eqs. 5.10 – 5.13, we found that Epi = –300 meV could account for the experimental trend in the
pressure-induced suppression of the polaron jump frequency at T = 573 K. Nevertheless, values of
Epi from –200 to over –400 meV gave similar results.
In the MC simulations, we monitored the mean-squared displacement (MSD) of both species as
a function of pressure and Epi . Our interest was how the electron MSD was altered under pressure
as a result of suppressed ionic mobility. The simulations varied the ionic mobility while monitoring the effect on the electronic mobility. The activation barrier for electron hopping was pressure
independent, so raising the activation barrier for ion hopping (through pressure) has no effect on
the electron MSD when the ion and electron chains are decoupled (Epi =0). When a coupling is
introduced, an indirect effect on the electron mobility is observed with increasing |Epi |. Figure 5.6
presents typical results of such a series of simulations. The electron MSD increases approximately
as t0.5 . This exponent is well-known when particles cannot pass on a 1D chain and require concentration fluctuations to move forward [86]. A suppression of the MSD with pressure clearly emerges
for values of Epi less than –200 meV and becomes increasingly pronounced as the magnitude of
69
Epi is increased. For a polaron-ion interaction energy of –250 meV at 3 GPa the MSD is suppressed
by 45% and at 7 GPa the MSD is suppressed an additional 40%. For agreement with experiment,
it appears that Epi for LiFePO4 is between –200 and –300 meV. Larger magnitude values are not
ruled out, however. The effects of pressure on the polaron jump frequency saturated when |Epi |
was somewhat larger than Ep . It was also noted that the effects of pressure on the polaron jump
rate became larger as Ei decreased relative to Ep , consistent with a larger role of ion motion in the
overall dynamics.
5.5
Discussion
Holstein’s molecular crystal model captures the essential physics of small polaron formation
and dynamics [1, 28, 87]. A tight-binding model is used to describe an extra electron in an array
of N molecules, each with an internuclear distortion variable, xn , and a reduced mass M, where
M −1 = N −1 ions m−1 . The positive strain energy is quadratic in the xn (e.g., the interatomic
separation of two ions in a diatomic molecule) with harmonic oscillator frequency, ω0 , associated
with the configurational coordinate of an isolated molecule. The electronic energy is reduced linearly with xn in proportion to the strength of an electron-phonon interaction parameter, A, that
characterizes the electron-lattice coupling strength in units of force.
A finite local distortion, xn , results in a reduced potential that effectively pins the electron, so
the localized polaron is favored by the binding energy, Eb , relative to an electron in an undeformed
lattice,
Eb ≈
A2
2M ω02
(5.21)
In the adiabatic limit, the prefactor in Eq. 5.1 reduces to the mean optical phonon frequency and the
activation energy is lowered by an amount J, associated with the d-bandwidth [30],
Ea =
Eb
−J .
(5.22)
The activation energy depends on pressure through the exchange integral, J, as well as any pressure
dependence of the binding energy. Taking the activation volume as the pressure derivative of the
activation energy,
70
Electron MSD versus time for six series of MC simulations for a pair of coupled 1D ion and
electron chains. Units for MSD are site index squared. Time is dimensionless. Each subplot shows the results
for a different Epi . Subplots are labeled with -Epi (0, 50, 150, 250, 350 and 450 meV). In each series, the
MSD is shown for three different pressures: 0 (black), 3 (red) and 7 GPa (green).
Figure 5.6.
71
∂Ea
Va =
≈ Eb
∂P
1 ∂A
1 ∂ω0
A ∂P
ω0 ∂P
∂J
∂P
(5.23)
and using the definition of the compressibility and the Grüneisen parameter, γ, the activation volume
becomes,
Va ≈ Eb
1 ∂A
− γκT
A ∂P
∂J
∂P
(5.24)
The last term, from the increased wave-function overlap, is positive, and tends to destabilize the
localized polaron. This term is believed to be responsible for the negative activation volumes in other
polaronic conductors [30]. Our large, positive Va would be consistent with an effect of pressure on
the electron-phonon interaction parameter, A, if ∂A/∂P > 0, giving ∂Eb /∂P > 0 by Eq. 5.21. In
general, however, we expect destabilization of an electron polaron centered at a Fe2+ ion because
the compressibility of ferrous-oxygen bonds is greater than for ferric-oxygen bonds. First principles
calculations suggest the activation barrier is raised by ∼50 meV under 4% biaxial compression
(along the b and c axes) [88]. The authors attribute this effect to an enhancement of the electronphonon coupling. Frozen phonon calculations for the strained system show the electron phonon
coupling constant increases by more than 20% [88]. These effects on the activation barrier from
standard polaron models are too small, or of the wrong sign, to account for our experimental results.
The electron-phonon interaction could be affected by the electrostatic interaction between the
polaron and a nearby Li+ ion if the ion has a pressure-dependent mobility. We suggest the origin of
our large difference between the activation volume measured for LiFePO4 and previous activation
volumes determined using conductivity measurements on oxides without mobile ions is the strong
coupling between the polarons and the mobile Li+ ions, Epi .
Previous first principles calculations for polaron hopping in LiFePO4 gave activation energies
of 175 and 215 meV for electron and hole polarons, respectively [74]. These results are for freepolaron transport. Measured activation energies, from either Mössbauer spectrometry or conductivity measurements, are two to three times higher than these calculated values. This is consistent with
a tendency for the electrons on Fe2+ sites to remain near Li+ ions.
The effect of pressure on valence fluctuations at Fe sites is indirect, but potentially large. It is
well known that pressure suppresses ionic diffusion by a vacancy mechanism [as in Eqs. 5.6 and
5.7]. The MC simulations show how polaron dynamics are suppressed if the polaron-ion interac-
72
tion energy tends to attract the polaron to immobile ions. The required interaction energy, Epi , is
approximately –300 meV for the pressures and temperatures of interest.
Some discrepancies deserve further investigation. Electron jumps between two sites where both
have ion neighbors, or both have vacancies (middle processes at top of Fig. 5.3), are unaffected
by pressure, and predict a background dynamics that is not found experimentally. In the olivine
structure, electron mobility is likely confined to the b-c plane. The Fe and Li sites are staggered in a
way that each Fe site has two symmetrically positioned Li sites, but within a given FeO6 plane each
Li-site has one 1NN Fe site and one second-nearest-neighbor (2NN) Fe site. A polaron following
the path of closest approach to a given ion chain will necessarily alternate between these 1NN and
2NN-type sites where the Li-Fe bond length is 6% longer in the 2NN site [68]. When pressure immobilizes the Li+ ions, there may be a tendency for electron-polarons to localize in these 1NN-type
sites in such a way that local dynamics are suppressed. Alternatively, the experimental technique
may not be sensitive to certain dynamics, for example, minority processes or dynamics that fall outside the window of sensitivity of frequencies sampled by Mössbauer spectrometry measurements. It
is also possible that pressure suppresses other aspects of polaron dynamics, or the ions and polarons
may form an ordered structure with reduced dynamics.
The generally good agreement between the experiment and simulated dynamics with a reasonable value of Epi , together with a measured activation volume of +5.8 Å3 , consistent with ion
diffusion, indicate a strong coupling between the ions and polarons in Lix FePO4 . A transport of
net charge requires decoupling of the ion and polaron motions, however, so the coupling is not
immutable. Nevertheless, the correlated motions of electrons and ions should suppress electrical
conductivity in LiFePO4 . Furthermore, a large correlation in the motions of polarons and ions can
explain why the electrical conductivity of LiFePO4 is so sensitive to materials preparation. Because
Li+ diffusion in LiFePO4 is essentially one dimensional, Li+ ion mobility suffers as a result of
channel blockage by defects [11, 12]. Blocked channels for Li+ ions then suppress electronic conductivity if polaron-ion interactions are strong. This effect may be common in materials when both
ions and electrons are mobile.
A small polaron quasiparticle comprises an electron localized by atomic contractions from
neighboring anions. Both the charge and distortion of the polaron are large enough to interact
with the charge and distortion around a Li+ ion, altering the formation energy and dynamics of the
73
polaron. The quantum dynamics of small polaron hopping is likely modified by the classical dynamics of ion motion; likewise, the configurations of polarons and ions on the crystal lattice should
also be affected by these interactions.
5.6
Conclusions
Nuclear resonant scattering spectra of Lix FePO4 were measured at elevated pressure and temperature. An analysis of the spectra using the Blume–Tjon model for a system with a fluctuating
electric field gradient gave frequencies of Fe valence fluctuations that correspond to frequencies of
polaron hopping. From measurements over a range of temperatures and pressures, both the activation energy and activation volume were determined for polaron hopping. To our knowledge this
is the first measurement of an activation volume for polarons in a material with mixed ion-polaron
conductivity.
Pressure caused a large suppression of valence fluctuations in Lix FePO4 , giving an activation
volume for polaron hopping of +5.8 Å3 . This unusually large and positive activation volume is not
typical of bare polaron hopping. It indicates a correlated motion of polarons and Li+ ions. From
model calculations and Monte Carlo simulations, the binding energy between the polaron and the
Li+ ion was found to be approximately –300 meV. This strong binding and polaron-ion correlation
should suppress the intrinsic electronic conductivity of Lix FePO4 . It may also affect the diffusion of
Li+ ions. Such coupled processes may be common to other materials where both ions and polarons
are mobile.
74
Chapter 6
Polaron mobility and disordering of
the Na sublattice in NaxFePO4 with
the triphylite structure
6.1
Introduction
Li-ion batteries have been an active subject of research within the last two decades and are
now widely commercialized in consumer electronics and electric vehicles. The effort to design
improved batteries has motivated the investigation of a range of polyanionic framework materials.
Within this context, the phospho-olivines have emerged as particularly promising with their low
toxicity, thermal stability and high energy density. The orthorhombic olivine-type structure (Pmna)
is shown in Fig. 6.1(a). Layers of corner-sharing networks of canted FeO6 octahedra in the b-c
plane are spaced by phosphate tetrahedra, and alkali cations form one-dimensional chains that run
between these FeO6 planes. Previous work has shown that the predominant ion diffusion pathway
is along these b-axis channels [15, 55]. In comparison, the electronic carrier mobility is expected to
be two-dimensional, occurring within the layers of FeO6 octahedra that are separated by insulating
phosphate groups.
With the increasing reliance on Li-ion technology, it has become apparent that lithium itself is
a limited resource. By contrast, sodium is one of the major rock-forming elements in the Earth’s
crust, and consequently is both environmentally abundant and relatively affordable. As a result, the
75
idea of designing sodium analogues to lithium cathode materials is immensely appealing, especially
for large energy-storage systems. The sodium counterpart to LiFePO4 has attracted attention as a
particularly promising sodium cathode [16–18, 26, 89–94]. The ground-state of NaFePO4 has the
maricite structure. Compared to the triphylite structure favored by Lix FePO4 , the site occupancies
of the alkali ion are swapped with the iron cations. The sodium cations are then isolated by the
phosphate groups, and the maricite structure is electrochemically inactive. While density functional
theory calculations suggest the energy of the two polymorphs is similar [15], the maricite structure
is apparently favored at higher temperatures where the material is formed. However, recent results
have shown that it is possible to synthesize a sodiated triphylite structure using an ion-exchange
route [89] starting with LiFePO4 . The triphylite-NaFePO4 exhibits excellent electrochemical performance compared to other candidate sodium-ion cathode materials [17, 18].
The phase diagram for the triphylite-Mx FePO4 (M=Na, Li) framework is altered when Na+
replaces Li+ as the intercalation ion. While the Lix FePO4 phase diagram shows a broad solid
solution above 473 K [23, 24], a solid solution of Na+ ions forms only above x=2/3, but is stable
at low temperatures. Below x=2/3 there exists an intermediate ordered phase, which persists to
high temperatures [16]. The proposed structure of the intermediate phase is shown in Fig. 6.1(b).
This superstructure results from a vacancy at every third sodium site along the Pmna b-axis, giving
rise to three crystallographically distinct iron sites, shown in red, green and blue in Fig. 6.1(b).
These distinctions in phase behavior are likely a result of the larger size of the sodium cation,
having a nearly 30% larger ionic radius than lithium. Full sodiation of the FePO4 lattice results
in a 17% volume expansion, compared to the 7% expansion seen in LiFePO4 [16]. This large
lattice mismatch between the end members in Nax FePO4 may play a role in the stabilization of the
intermediate ordered phase.
As with many other transition metal oxides, the mechanism of electrical conductivity in mixed
3+
valent Mx Fe2+
x Fe1−x PO4 is small polaron hopping [65–67]. A small polaron quasiparticle com-
prises an electron or hole localized by atomic displacements of neighboring anions. When the
carrier transfers between adjacent iron sites, this local distortion must also transfer, resulting in a
slow moving particle with a large effective mass. In contrast to semi-free carriers, polarons tend to
have a low mobility that rises with increasing temperature. At moderate temperatures, the motion
76
a.#
b#(pmna)="a"(P21/n)"#
b.#
c#(P21/n)#
!#
c#(pmna)="b"(P21/n)"#
(a) Triphylite-type structure of MFePO4 (M= Li, Na) with chains of M+ ions (yellow), planes
of FeO6 octahedra (brown) and phosphate tetrahedra (grey). (b) Ordered superstructure for x=2/3. Three
structurally distinct iron sites are shown in blue, green and red. The axes on left are for the orthorhombic
Pmna cell. Oblique axes of P21 /n cell are shown in black.
Figure 6.1.
77
of the polaron quasiparticle can be understood as an activated process with the jump rate [37, 69],
Ea
Γ(T ) ∼ ν exp −
kB T
(6.1)
where T is temperature, kB is the Boltzmann constant and ν is a characteristic phonon frequency.
The activation energy, Ea , describes the energetic barrier for the polaron quasiparticle to transfer
between adjacent iron sites.
Mössbauer spectrometry allows for the study of local electron dynamics at iron ions. Polaron
hopping between Fe2+ and Fe3+ results in 57 Fe ions that experience a fluctuating local environment.
The measured spectra are altered when the hyperfine fields fluctuate on the same time scale as
the 57 Fe nuclear decay, or near a characteristic frequency set by the spectral hyperfine energies,
νHF = ∆EHF /~. Fortuitously, typical quadrupole splittings for 57 Fe fall within a range that is well
suited to the study of polaron dynamics, and the temperature evolution of the valence fluctuation
frequencies gives rise to variations of the shape and symmetry of the quadrupole doublets from
Fe2+ and Fe3+ . At low frequencies and low temperatures, the spectral components from Fe2+
and Fe3+ remain distinct, and at very high frequencies the spectrum approaches a single timeaveraged doublet. More complex behavior occurs between these limits, when the valence of a 57 Fe
ion fluctuates between Fe2+ and Fe3+ at a frequency between 1 and 100 MHz. Within this range,
the quadrupole doublets from Fe2+ and Fe3+ merge together, with asymmetric, non-Lorentzian
lineshapes. Previous Mössbauer spectrometry studies of the lithium analogue, Lix FePO4 , reported
dramatic spectral distortions for temperatures between 373 and 513 K [65, 66, 95].
The structure and stability of the intermediate phase at x=2/3 has been addressed in a handful
of recent papers, but a comprehensive understanding of the crystallographic ordering of Na-ions
and electrons is lacking. The origin of a secondary ferrous doublet in the Mössbauer spectra and
how this relates to the crystallographic ordering of Na-ions and electrons or the presence of fast
electron hopping is unsettled [16, 26]. Here we report new results on the phase stability and charge
dynamics in Na0.73 FePO4 at elevated temperatures, obtained by performing measurements in a
resistive furnace. The evolution of the iron site occupancies with temperature, as determined from
Mössbauer spectrometry, gives new information that helps resolve the nature of the ordering in
the intermediate phase. Mössbauer spectrometry coupled with synchrotron x-ray diffraction shows
78
that the disordering of sodium ions at above 450 K and the onset of rapid electron dynamics occur
simultaneously. We suggest that there is a polaron-ion interaction that affects the dynamics of both,
much as is the case for Lix FePO4 [95].
6.2
Experimental
LiFePO4 was prepared by a solid-state reaction and chemically delithiated using K2 S2 O8 in an
aqueous solution [23, 79]. To chemically insert sodium into the lattice, the sample was refluxed
for 48 hours in an acetonitrile solution of excess NaI [92]. The resulting sample retained a minor
amount of ferric iron, ranging from 3-10%. The sodiated sample was subsequently oxidized with
K2 S2 O8 to give a final composition of Na0.73 FePO4 .
LiFePO4 + K2 S2 O8 → FePO4 + (Li2 SO4 + K2 SO4 ),
3z
FePO4 + NaI → Naz FePO4 + NaI3 ,
(x − z)
(x − z)
K2 S2 O8 → Naz FePO4 +
(Li2 SO4 + K2 SO4 ).
Nax FePO4 +
(6.2)
X-ray diffraction of the final sample shows no evidence of residual FePO4 . The concentration of the
final sample was determined using the spectral area ratios determined from Mössbauer spectrometry.
Synchrotron X-ray diffraction (XRD) measurements were performed at beamline 16ID-D at the
Advanced Photon Source at Argonne National Laboratory using a monochromatic beam with λ =
0.86 Å. Diffraction was measured in transmission geometry using a Mar CCD detector plate while
the sample was held in a resistively heated vacuum furnace.
Mössbauer spectra were collected in transmission geometry using a constant acceleration system
with a 57 Co in Rh γ-ray source. Velocity and isomer shift calibrations were performed in reference
to room temperature α-iron. Elevated temperature Mössbauer spectrometry was performed with the
sample mounted in a resistive furnace for a series of temperatures between 298 K and 550 K. At each
temperature, the furnace was given four hours to equilibrate, after which the spectrum was collected
for 20 hours. After collecting the 550 K spectra, an additional 298 K spectrum was collected and
x-ray diffraction was performed on the retrieved sample.
79
(a)
(b)
553
523
503
463
443
stnuoC
]stnuoC[goL
483
423
403
388
373
343
298
10
12
14
16
18
2θ
20
15.5
16.0
16.5
2θ
Figure 6.2. (a) Temperature series of XRD spectra taken between 295 K and 550 K. The Rietveld fits (black
curves) overlay experimental data (points). Black tick marks at bottom of figure indicate locations of superstructure phase in ordered structure. (b) Enlargement of (200) peak on linear scale.
6.3
Results
6.3.1
X-ray diffractometry
Figure 6.2(a) shows the synchrotron x-ray diffraction (XRD) spectra collected at temperatures
between 298 K and 553 K. The locations of the superstructure peaks from Na+ ordering are indicated with tick marks below the 298 K spectra. Based on the phase diagram by Lu, et al. [16], the
present sample (x= 0.73) should be within the solid solution phase at room temperature, but the
presence of the superstructure peaks suggests that long range order persists well above x=2/3. Upon
heating, the superstructure peaks become increasingly diffuse and are completely gone by 483 K.
This is especially clear for the superstructure peak at ∼ 15 degrees.
As the sodium disorders on the lattice, structural changes can also be seen through a closer
inspection of the fundamental peaks, particularly those peaks that are sensitive to the changes in
the Pmna b-axis. Inserting sodium into the Na2/3 FePO4 superstructure results in a large expansion
of the b- and a-axes. A closeup of the Pmna (020) peak is shown in of the Figure 6.2(b). At
298 K this peak has a clear low-angle shoulder. The higher-angle component corresponds to the
expected d-spacing for the ordered structure. As the temperature increases, the relative intensity of
this shoulder increases, and above 500 K the entire weight of the (200) peak has shifted to the lower-
80
25
esahP deredrO noitcarF raloM
20
15
10
300
350
400
450
500
550
Temperature (K)
Figure 6.3. Molar fraction of x=2/3 ordered phase, determined from Rietveld analysis of x-ray patterns
shown in Fig. 6.2
angle position. Similar effects can be seen in the (220) and (121) peaks with increasing temperature.
The presence of this shoulder suggests that the region above x=2/3 is actually biphasic at room
temperature, containing a solid solution phase as well as the x=2/3 ordered phase. This finding
confirms previous XRD results as a function of desodiation by Galceran, et al. [92]. Their study
showed an abrupt discontinuity in the b-axis at compositions above x=2/3, suggesting the onset of
ordering starts above a composition of x=2/3. Additionally, an inspection of the room temperature
XRD collected by Lu, et al. [16] reveals the (020) diffraction does not entirely disappear until x=
0.8. These observations, combined with our high temperature XRD, are evidence for an appreciable
two-phase region of the phase diagram between the x=2/3 intermediate phase and a solid solution
of higher concentration.
Reitveld analysis was performed with the software package GSAS, using psedo-Voigt peak profiles [40,41]. The fits to the spectra are shown as solid lines in Fig. 6.2. The instrumental broadening
was modeled as Gaussian, while the strain broadening was assumed to be entirely Lorentzian. For
accurate peak shapes, a Stephens model for anisotropic strain broadening was included [42]. The
data were fit with two diffraction patterns, one from a solid solution phase and the other from the ordered phase, using the superstructure proposed by Boucher, et al. [26]. The ordered structure can be
described by a 3x3x1 orthorhombic supercell with a Na-vacancy at every third site along the Pmna
b-axis. This unit cell is equivalently described by the monoclinic P21 /n system. Both unit cells are
shown in Fig. 6.1 (b). The sodium concentration of the ordered phase was fixed at x=2/3, and the
sodium concentration of the solid solution was constrained so the total sodium concentration was
81
(a)
320
mortsgna cibuc
318
Solid Solution
316
314
Ordered
312
310
300
350
400
450
500
550
ordered
(c)
monoclinic (P21/n)
(b)
solid solution
18.38
orthorhombic (pmna)
18.36
18.34
18.32
a-axis
10.36
a-axis
18.30
18.28
10.35
10.34
300
10.33
400
500
b-axis
4.965
4.960
300
6.20
400
500
mortsgna
b-axis
4.955
4.950
6.19
4.945
6.18
300
6.17
6.16
11.994
300
4.98
400
500
400
500
c-axis
11.992
11.990
c-axis
11.988
4.97
4.96
300
4.95
120.86
4.94
300
400
500
Temperature (K)
400
500
angle-β
120.84
120.82
120.80
300
400
500
Temperature (K)
Figure 6.4. Structural parameters determined from Rietveld fits in Fig. 6.2. (a) Volume of unit cell for
ordered and solid solution phases. Volume of ordered (P21 /n) cell has been normalized by a factor of three
for comparison with orthorhombic Pmna cell. (b) Lattice parameters for orthorhombic solid solution phase.
(c) Lattice parameters for monoclinic ordered phase. The P21 /n a-axis coincides with the Pmna b-axis and
the P21 /n b-axis coincides with the Pmna c-axis.
82
fixed at 73%.
Figure 6.3 shows the molar fraction of the x=2/3 ordered phase as a function of temperature,
as determined from the Reitveld refinements. Between 298 and 400 K the molar fraction is around
20%. Above 420 K, the fraction of ordered phase drops substantially and is totally gone at 520 K.
Fig 6.4(a) shows the thermal trend of unit cell volume for both phases. Here the volume of the
P21 /n cell has been normalized by a factor of three for comparison with the orthorhombic cell.
There is a clear volume collapse between 463 and 483 K in the solid solution phase. Fig. 6.4(b)
shows the lattice parameters for the orthorhombic solid solution phase and the ordered phase, using
the monoclinic crystal system. In the solid solution, between 298 and 483 K the a-axis shows a
gradual contraction, above which it starts to expand. The other two axes exhibit thermal expansion
throughout the entire temperature range, although both axes also show a discontinuity between 443
and 483 K.
6.3.2
Mössbauer spectrometry
Mössbauer spectra, collected in the same temperature range, are presented in Fig. 6.5. The
room temperature spectrum has two distinct ferrous components with quadrupole splittings of 2.6
and 1.7 mm/s. In what follows, these are called A- and B-type sites, respectively. A fit to the
298 K spectrum gives an area ratio of the two components, B/A ∼20/80. The presence of this
secondary Fe2+ component is consistent with other recent Mössbauer studies of this system [16,26].
A spectrum of the same sample, collected prior to chemical desodiation, exhibited a single doublet
with a splitting of 2.74 mm/s, shown in Fig. 6.6 (c). This “fully sodiated” sample contained a
residual ferric concentration of ∼3%. Two additional samples were prepared with concentrations of
x = 0.54 and x = 0.67. Room temperature spectra for these samples are shown in Figs. 6.6(a) and
(b), respectively. Fits to these spectra gave B/A site area ratios of ∼34/66.
The spectra were evaluated using the software package CONUSS [50, 81], which generated
the solid curves in Fig. 6.5. CONUSS allows for the calculation and refinement of spectra using
the theory of Blume and Tjon for random temporal fluctuations of the hyperfine field [48, 49].
This model uses a time average over the stochastic degrees of freedom of the fluctuating nuclear
Hamiltonian, making use of a relaxation matrix that describes the transition rates between the two
sets of hyperfine parameters associated with the Fe2+ and Fe3+ environments. Depending on the
83
553
525
499
473
447
420
394
368
295
-2
-1
(mm/s)
Figure 6.5. Temperature series of Mössbauer spectra taken between 295 K and 550 K. The fits (black curves)
overlay experimental data (points). Temperatures are listed to the left of the spectra in Kelvin.
84
(a)
Na0.54FePO4
100
98
96
94
92
90
(b)
-2
-2
-2
Na0.67FePO4
100
98
96
94
(c)
Na0.97FePO4
100
98
96
94
mm/s
Figure 6.6. 298 K Mössbauer spectra Nax FePO4 (x=0.54, 0.67, 0.97).
85
relaxation time relative to the lifetime of the excited state, the effective widths of the resonance
lines can either sharpen or broaden inhomogeneously and amalgamate. While the probability of a
transition between the excited state and the ground state with the emission of photon is Lorentzian
in form, the observed probability results from a sum over the possible ground states and a stochastic
average over the sampled excited states. For polycrystalline samples, the problem reduces to the
simplest case treated by Blume and Tjon in which the stochastic and quantum mechanical parts of
the problem are separable as there is no issue of non-commutativity of the Hamiltonian at different
times.
The fluctuations from polaron hopping require a set of hyperfine parameters for each iron site,
together with a relaxation matrix of transition rates,
Γρ3+
−Γρ3+
.
W=
Γρ2+B −Γρ2+B
(6.3)
To maintain charge balance, the elements of the relaxation matrix are weighted by the populations
of the sites, ρ. This allows for the refinement of the relative weight of each site and the polaron
hopping frequency, Γ(T ). The weight ratio of ferrous A/B-type iron sites was also introduced as a
fit parameter. For the fits shown in Fig. 6.5, the valence fluctuations were limited to charge hopping
between B-type ferrous sites and Fe3+ sites, while the A-type ferrous sites were treated as static.
This assumption was justified by the observation that the onset of valence fluctuations coincides
with a decrease in the fraction of the A-type ferrous site, suggesting the valence fluctuations are
largely limited to the B-type iron environment. The Blume-Tjon model was not used for the spectra
below 400 K. These spectra were fit with a static model.
Above 400 K the sample begins to show a minor impurity component as a result of oxidation.
A comparison of the initial 298 K spectrum and a final spectrum collected after cooling to room
temperature from 553 K, reveals a ∼16% decrease in the concentration of ferrous iron. This down
temperature spectrum is shown in Fig. 6.7. X-ray diffraction completed at the conclusion of the
measurement did not show any indication of a second crystalline phase. The oxidation is likely a
result of a reaction with O2 to produce amorphous Fe2 O3 . For temperatures above 500 K, a second
ferric site was added to the fit model to account for this oxidation. This site did not participate in the
86
100
98
96
94
92
90
88
86
-2
mm/s
Figure 6.7. Down-temperature spectrum, at 298 K, acquired after cooling from 550 K.
dynamics and the hyperfine parameters of this site were fixed as IS = 0.31mm/s and QS = 0.99
mm/s. As the temperature was further increased, the spectral contribution of this site was left as a fit
parameter and allowed to drift upward with the restriction that the total weight of the ferric contribution to the spectra could not exceed the observed ferric contribution from the down temperature
measurement.
All hyperfine parameters showed gradual linear trends as a function of temperature. The temperature dependence of the quadruple splittings of the ferrous iron sites (high spin) are expected to
follow an Ingalls-type model [96], where the degeneracy of the t2g levels is removed owing to a
Boltzmann occupation of the different crystal field split t2g ↓ electronic levels by the sixth valence
electron,
−∆
∆QS
2+
(T ) ∼ ∆QSv
1 − exp kB T
−∆
+ ∆QSl .
(6.4)
1 + 2 exp kB T
Here ∆QSv and ∆QSl refer to the ground state valence and lattice contributions to the quadrupole
splitting.
The ground state quadrupole splittings and the t2g level splitting (∆) were deter-
mined from best fits to the temperature trends, giving ∆ = 104 meV, QSv = 2.406 mm/s,
QSlA = 0.33 mm/s, and QSlB = −0.54 mm/s. At 473 K there is an abrupt jump in the Bsites quadrupole splitting of ∼0.09 mm/s. Following this discontinuity, the thermal trend in the
quadrupole splitting continues to follow the same Ingalls-type slope as lower temperatures.
Having one fewer valence electron, ferric sites should not exhibit the Ingalls-type crystal-field
87
splitting effect. Therefore, any thermal trend in the quadrupole splitting is presumably related to an
evolution in the local bonding environment. The temperature dependence of the Fe3+ quadrupole
splitting has an approximately linear temperature trend,
∆QS 3+ (T ) = 1.8 − 2 × 10−3 × T.
(6.5)
The isomer shift for all sites showed progressive reduction with temperature, following approximately linear trends:
2+
ISA
(T ) = 1.6 − 1.0 × 10−3 × T,
2+
(T ) = 1.5 − 7.0 × 10−4 × T,
ISB
(6.6)
IS 3+ (T ) = 1.0 − 1.0 × 10−3 × T.
This decrease is in part related to the temperature-dependent second-order Doppler shift, [43],
Z ΘD
9kB ΘD
x3
T 4
IS(T ) = δ0 −
dx ,
1+8
16M c
ΘD
exp(x) − 1
Θ2
3kB T
IS(T ) ∼ δ0 −
1 + D2 , T > ΘD ,
2M c
20T
(6.7)
where ΘD is the Debye temperature, M is the nuclear mass, c is the speed of light and δ0 is the
intrinsic isomer shift.
An inspection of Fig. 6.5 suggests that temperatures around 420 K, there is a onset of electronic
dynamics that give rise to spectral distortions, including line broadening and a collapse of the Fe2+
and Fe3+ doublets. Figure 6.8 shows a plot for the polaron hopping frequencies, Γ(T ), determined
from the fits to the spectra shown in Fig. 6.5. Figure 6.9 shows the evolution of the iron site
occupancies as a function of temperature, as determined from these fits. Starting at 450 K there is
a rapid conversion of A-type to B-type ferrous iron sites. The data set in Fig. 6.8 was fit with Eq.
6.1 to determine the activation energy for polaron hopping. The activation energy was found to be
505±50 meV, where the uncertainty arises from the choice a prefactor for Eq. 6.1. This value is
comparable to the result obtained for LiFePO4 of 470 meV [95].
88
80
MHz
60
40
20
400
450
500
550
Temperature (K)
Figure 6.8. (a) Polaron hopping frequencies, Γ(T ), as determined from the solid curves of Fig. 6.5. Solid
curve is an Arrhenius-type fit.
1.0
0.8
0.6
2+
Fe
0.4
0.2
Fe
2+
0.0
300
350
400
450
500
550
Temperature (K)
Figure 6.9. Relative weight of Fe2+ iron sites as a function of temperature, as determined from fits to Möss-
bauer spectra.
89
600
D2"
550
D1"
H"
Temperature (K)
500
450
S"
S+I"
H+I"
400
350
300
0.0
0.1
0.2
0.3
0.4
0.5
0.6
x in NaxFePO4
0.7
0.8
0.9
1.0
Figure 6.10. Revised phase Diagram, including two phase region above x=2/3.
6.4
Discussion
The solid solution lattice parameters (Fig. 6.4) show a gradual contraction of the pnma a-axis
up to 483 K. As the weight ratio of the solid-solution phase grows relative to the ordered phase,
there is a concomitant drop in the sodium concentration in the solid solution. This results in a
contraction of the a-axis that dominates the thermal expansion for temperatures up to the crossover
into a single-phase region, at which point there is no longer a driving force for sodium to leave
the solid solution. The other two axes show thermal expansion throughout the entire temperature
range, although both show a kink between 443 and 483 K. This discontinuity can be identified as the
crossing of a phase boundary into a region of complete sodium disordering on the lattice. Within
this temperature range, there is a rapid drop in the sodium concentration in the solid solution as
the lower concentration ordered phase is converted to a solid solution. This corresponds to the
temperature range of the loss of superstructure peaks. While a minority ordered component does
not entirely disappear until 523 K, this is probably a result of delayed kinetics.
90
Figure 6.10 shows a revised phase diagram for the Nax FePO4 system that includes a two phase
region above x=2/3. The points along the phase line separating the two phase region from the solid
solution were determined from the Rietveld analysis. The blue dashed lines in this figure are from
pervious phase diagram study by Lu et. al. [16] and the green dashed line is added to account
for some degree of solubility of vacancies in the heterosite phase. At the highest temperatures the
sample will revert to the maricite phase plus some combination of phosphate oxides depending on
the concentration.
The phase stability of the Nax FePO4 system is likely influenced by the details of the coulombic
interactions between the Na-ions, Na-vacancies, electrons, and holes. In Lix FePO4 , the two-body
interactions between Li+ , vacancies (V), Fe2+ , and Fe3+ are essential to understanding the phase
behavior. While Li+ /Li+ , and V/V interactions promote the formation of a solid solution, the
Li+ /Fe2+ and Li+ /Fe3+ interactions contribute to a tendency for phase separation. As temperature
increases, the electronic configurational entropy from the disordering of the electrons and holes on
the lattice dominates [25].
Solid solutions of Lix FePO4 (stable above 473 K) are easily preserved at room temperature
by quenching. Mössbauer spectra of these quenched samples show broadened quadrupole splitting
distributions and an overall decrease in the splitting compared to the pure end members, particularly
of the minority site [65]. This results from a sampling of a range of local environments, as expected
for a solid solution. The larger shift in the splitting of the minority component arises as a result of
a large number of sites that are dissimilar to the environment seen in the end member, i.e. an Fe3+
site surrounded by Li-ions. In contrast to the broadened distributions seen in LiFePO4 , the sodiated
samples show two discrete ferrous components, indicating the presence of two structurally distinct
local environments for ferrous iron. The A-type site is reminiscent of the Mössbauer spectra of the
fully sodiated structure which exhibits a single site with a quadruple splitting of 2.74 mm/s. The
abnormally low quadruple splitting of the B-site (1.7 mm/s) is suggestive of a local environment
that is quite different, consistent with a sodium-deficient local environment.
Samples prepared with sodium concentrations below x=2/3 all show ferrous components with
area ratio of B/A-type sites of ∼34/66, in reasonable agreement with previous results [16,26]. In the
study by Lu, et. al., samples in the concentration range Nax FePO4 (x = 0.1−2/3) exhibited an area
ratio of approximately 40/60. This range is within a two-phase regime of FePO4 and Na2/3 FePO4 .
91
Above this threshold, as additional sodium is inserted into the lattice, the concentration of the Bcomponent falls off with the concentration of ferric iron. The relative ratio of the two ferrous
components in the present sample (20/80) is consistent with these results. In another recent study by
Boucher, et al., the room temperature spectrum showed a 30/70 ferrous site ratio, but when the same
sample was cooled to liquid nitrogen temperatures the spectrum no longer contained a secondary
B-site with anomalously low splitting. Instead, the spectrum could be fit with two overlapping
Gaussians, having a 50/50 weight ratio and quadrupole splittings of 2.5 and 2.9 mm/s [26]. The
authors suggested the presence of the secondary doublet at room temperature may result from rapid
electron dynamics, similar to the dynamical effects that are observed at higher temperatures in
Lix FePO4 .
Because the present sample (x = 0.73) is biphasic at room temperature, it is tempting to attribute
the two doublets to ferrous iron in the two phases. A combination of previous XRD and Mössbauer
work show that the secondary component is present for concentrations below x = 2/3, where
all ferrous iron is within the ordered structure. This observation, combined with the consistent
area ratio of the two doublets for all concentrations below x=2/3, suggests the secondary ferrous
doublet is inherent to the ordered structure. At 298 K the ordered phase makes up only 20% of
the Na0.73 FePO4 sample. The retention of the two sharp spectral doublets suggests that the local
structure remains much the same in the solid solution phase, at least at room temperature. It appears
that the solid solution retains much of the framework of the ordered structure, shown in Fig. 6.1(b),
with additional sodium ions distributed randomly on the Pmna (220) planes.
The ∼34/66 weight ratio of the ferrous sites in the room temperature ordered phase seems to
contradict the proposed superstructure, determined from synchrotron x-ray diffraction and transmission electron microscopy [26]. In the fully sodiated structure, each iron has three pairs of symmetrically positioned sodium ions, forming a distorted triangular prism. In the proposed superstructure,
shown in Fig. 6.11, every third sodium along Pmna b-axis is vacant, giving rise to three distinct iron
sites with a 1/1/1 weight ratio. The three sites correspond to sites with one, two and three vacancies
in their six-fold sodium coordination shell. In 6.11 these sites are shown in red, green and blue,
respectively. Calculation of a sodium specific effective coordination number (ECoN) gives 3.72,
92
b.#
91
Figure
Sodium-ion
sublattice
x ordered
= 2/3phase.
ordered
phase.
Sodium
areand
shown in yellow and
Figure6.11.
6.10.
of the of
x (a)
=the
2/3
Sodium
ions are
shown
inions
yellow
Figure
6.1.Sodium-ion
(colorsublattice
online).
Olivine-type
structure
of MFePO
4 (M=Li, Na) with chains of M
three
distinct
sites
are inshown
in blue,
green
and red,tocorresponding
threestructurally
structurally distinct
ironiron
sites are
shown
blue, green
and red,
corresponding
iron sites with 3, 2to
andiron sites with three,
(blue),
of FeO
(red)coordination
and phosphate
tetrahedra
the 6-fold
sodium
coordination
shell.
6 octahedra
two1 vacancies
and planes
oneinvacancies
in the
six-fold sodium
shell. The
pyramidal (grey).
outline of(b)
thisIron-sodium
coordination sublattice of
shell
in
shown
in
black
for
red-type
iron
site.
superstructure for x=2/3. Three structurally distinct iron sites are shown in yellow, green and red. T
sion electron microscopy [26]. In the fully sodiated structure, each iron has three pairs of symmeton left is for the orthorhombic pmna cell. Oblique axes of P21 /n cell are shown in orange.
rically positioned sodium ions, forming a distorted triangular prism. In the proposed superstructure,
3.69,
and 2.990 for the red, green and blue sites. Here ECoN is defined as [97],
shown in Fig. 6.10, every third sodium along Pmna b-axis is vacant, giving rise to three distinct iron
exp
1−
one,6two and three vacancies
sites with a 1/1/1 weight ratio. The three sites correspond
P to sites with
ri
hri in red,
in their six-fold sodium coordination shell. In 6.10 these sites are shown
green and yellow,
ECoN=
respectively. A calculation a sodium specific effective coordination number (ECoN) gives 3.72,
i ri exp 1− rmin
hri = P ⇣ ⌘ r 6 ,
i expri 1− r
6as [97],
3.69 and 2.990 for the red, green and yellow sites. P
Here ECoN isdefined
ri
ECoN=
i exp
(6.8)
, min
hri
(6.8)
where rmin is the shortest bond in the set and
⇣ the ⌘sum is made over all Fe-Na bonds below a cutoff
radius of 4 Å.
hri =
rmin
⇣
⌘6
ri
i exp 1
i ri exp
min
Figure 6.12 illustrates the proposed evolution of the iron and sodium site occupancies with
where rmin is the shortest bond in the set and the sum is made over all Fe-Na bonds below a cutoff
increasing temperature. The three iron sites in the ordered structure are depicted with their Naradius of 4 Å.
coordination
shells
for four
temperatures
K sodium
and 500
The temperatures
are listed at
Figure 6.11
illustrates
the proposed
evolutionbetween
of the iron77and
siteK.occupancies
with
the top of the figure along with the B/A site ratios of the ferrous components in the Mössbauer
spectra. In this figure sodium ions are depicted in black and sodium vacancies in white. The central
93
77
(B/A
0/100)
~200
(B/A
15/85)
298
(B/A
33/67)
500
(B/A
100/0)
Na+
Na
vacancy
Fe
sites:
Fe3+
Fe2+A
Fe2+B
Temperature evolution of three iron sites in Na2/3 FePO4 structure. Fe3+ ions are shown in
yellow and Fe in purple, where dark purple and lavender depict A- and B-type ferrous sites. Sodium ions
are shown in black and sodium vacancies are shown in white. The pyramidal outline reflects the coordination
environment of the central iron site, corresponding to the colors of the iron ions shown in Fig. 6.11. Red
represents a crystallographic iron site with five sodium neighbors, green represents an iron site with four
sodium neighbors, blue represents an iron site with three sodium neighbors and brown represents the average
coordination environment for an iron site in the disordered solid solution phase. Temperature is listed at the
top of each column along with B/A site ratio.
Figure 6.12.
2+
94
iron ion is shown in yellow for Fe3+ and purple for Fe2+ , where the local environment of the
ferrous iron sites is indicated with different shades of purple. The sodium-rich and sodium-deficient
environments that give rise to the A- and B-type sites in the Mössbauer spectrum are depicted with
dark purple and lavender, respectively. The color of the pyramidal frame reflects the coordination
environment of the central iron site, where the red, green and blue outlines correspond to the three
iron sites in Fig. 6.11, having one, two and three surrounding vacancies, respectively. Upon total
disordering of the Na sublattice, the three iron sites in 6.11 are no longer distinct and the outline of
the coordination shells in Fig. 6.12 are shown in brown.
Electrostatic considerations suggest that the Fe3+ holes prefer the blue sites, surrounded by vacancies, while the Fe2+ electrons prefer the sodium rich red or green sites. Below room temperature,
we expect charge ordering on the lattice reflecting this preference. Assuming the quadrupole splitting decreases with sodium coordination, the 77 K Mössbauer spectrum can be interpreted as the
divalent cations exclusively occupying the the red- and green-type sites, shown in the first column
of Fig. 6.12. Given that the red and green sites have similar effective coordination numbers, this
charge ordering gives rise to two ferrous sites with similar quadrupole splittings in a 50/50 ratio.
While the diffraction results rule out a significant rearrangement of sodium ions between 77 K
and room temperature, this type of charge localization transition below room temperature is a possibility. As the temperature is raised, a disordering of electrons and holes on the lattice results in a
partial ferrous occupancy of the blue-type sites (with three sodium neighbors). The relatively low
effective coordination number of the blue site could account for the unusually low quadruple splitting of the B-type component in the Mössbauer spectrum. At even higher temperatures, a random
distribution of electrons on the three iron sites results in 1/3 of the ferrous iron sites, or ∼ 22% of
the total iron sites, with an Fe2+ ion in a sodium-deficient blue-type environment (third column of
Fig. 6.12). Moreover, if the bonding environment of the red and green sites gives rise to similar
values for QS, this electronically disordered state gives a B/A weight ratio of 33/67, consistent with
the experimentally-observed ∼34/66 ratio. For concentrations above x=2/3, as additional sodium is
inserted into the lattice, blue sites are converted to red sites and trivalent green sites become divalent.
The result is a fall-off in the weight fraction of the B-type ferrous sites that tracks the concentration
of trivalent iron.
Above 450 K, x-ray diffraction shows a loss of local order on the sodium sublattice, and Möss-
95
S25
S100
S130
S160
S190
S220
S250
S280
S310
295$K$
368$
394$
420$
447$
473$
499$
525$
553$
-1
3+
(mm/s)
2+A
2+ B
Figure 6.13. Mössbauer spectrometry of Fig. 6.5, inverted, stacked and normalized for comparison. Dotted
lines mark the A- and B-type ferrous absorptions at ∼0 mm/s.
bauer spectrometry shows that all three sites start to look like a single site with an average sodium
coordination. With this delocalization of sodium on the lattice, there is an overall reduction of
sodium coordination around the iron sites. Red and green-type sites that result from clustering of
sodium in the ordered structure are converted to sites with lower sodium coordination. This is seen
as an overall conversion of A-type sites to B-type sites as the temperature increases above 298 K.
This is particularly evident from an inspection of Fig. 6.13, where the Mössbauer spectra have been
inverted, normalized and overlaid for comparison. At low temperature the ferrous absorption lines
at ∼0 mm/s are distinct, but with increasing temperature the weight of the B-type absorption line
grows relative to the A-type line. By 473 K, it appears that the B-type environment becomes the
majority divalent component. For the absorption lines at ∼2 mm/s, this effect is obscured by the
spectral collapse of the Fe2+ and Fe3+ lines that results from the concurrent onset of fast charge
hopping. This evolution of the iron environments is also apparent from the discontinuity in the
Fe2+
B quadrupole splitting between 447 K and 473 K. This jump represents an abrupt local change
in the B-type iron site, suggestive of a moderate increase in sodium coordination, as expected upon
sodium disordering.
The temperature range of sodium delocalization on the lattice corresponds with the temperature
96
range where electronic dynamics begin to show large effects in the Mössbauer spectra. An analysis
of the data of Fig. 6.5 gave an activation energy of 505± 50 meV. This is comparable to the value
determined using nuclear resonant scattering in Lix FePO4 of 470 meV [95]. In the lithium system,
the onset of clear dynamical effects in the Mössbauer spectra is seen at a temperature ∼60 K below
the onset in NaFePO4 . As the ion activation barrier is expected to be higher in the sodium system,
a temperature offset for ion delocalization of this magnitude is not unexpected. The higher temperature onset of electronic valence fluctuations despite the comparable polaron activation barrier
point to the central role of the ion delocalization for the onset of electronic dynamics. It appears
that sodium disordering is required before the onset of electronic mobility.
Recent work looking at the activation barrier for polaron hopping in LiFePO4 suggested that
the mobility of the electronic carriers is correlated to the Li-ion mobility [95]. It was deduced that
strong binding and polaron-ion correlation should suppress the intrinsic electronic conductivity of
Lix FePO4 and may also affect the diffusion of Li+ ions. Similar to LiFePO4 , simulations suggest
that Na+ ions diffuse readily along a [010] channels and there is a high energy barrier to cross
between channels [15]. While the calculated activation barrier for ion mobility in NaFePO4 is
higher than in Lix FePO4 , it is still below 400 meV, implying reasonably facile ionic mobility. The
delayed onset of electron dynamics in the Mössbauer spectra compared to LiFePO4 is likely a result
of a higher activation energy for ion hopping.
6.5
Conclusions
Mössbauer spectra of Na0.73 FePO4 were measured for temperatures between 298 K and 553 K.
An analysis of the spectra using the Blume–Tjon model for a system with a fluctuating nuclear
Hamiltonan gave frequencies of Fe valence fluctuations that correspond to frequencies of polaron
hopping. This analysis allowed for the determination of an activation energy for polaron hopping
of 505 meV. Synchrotron x-ray diffraction measurements collected in the same temperature range
showed that the disordering of the sodium sublattice coincided with a marked enhancement in the
electronic valence fluctuations that give rise to distortions in Mössbauer spectra. Additionally, the
synchrotron x-ray diffraction data revealed the presence a two phase region between the solid solution and ordered phases.
97
The combination of the diffraction and Mössbauer results gave new information concerning the
temperature evolution of the iron and sodium site occupancies. At the lowest temperatures there
is a preference for both sodium ordering and electronic charge ordering. As the temperature increases, electronic disorder started to develop, giving rise to ferrous iron in sodium deficient local
environments. The site occupancies from the room temperature Mössbauer spectrum indicate total
electronic disorder. Despite this apparent loss of electronic ordering, there is no evidence of charge
dynamics in the Mössbauer spectra, implying the polaron rate hopping at 298 K is still below the
MHz range. Above 450 K, there is a complete loss of order on the sodium sublattice, giving rise to a
true solid solution phase. This is also the range where the Mössbauer spectra begin to exhibit distortions constant with fast charge hopping, suggesting a relationship between the onset of fast electron
dynamics and the redistribution of sodium in the lattice. These results clarify details related to the
sodium and electronic charge ordering in the structure and suggest that electron-ion interactions
play a role in both the phase stability and the elevated temperature charge dynamics.
98
Chapter 7
Conclusions and future work
The interacting transport properties of transition-metal intercalation compounds make for a fascinating research topic. The interplay between electron and ion mobility and the details of how these
transport properties are influenced by crystal structure gives rise to interesting fundamental physics.
Developing a local picture of how charge moves through the lattice is a perspective that has received
little attention. The focus of this thesis was activation barrier measurements for charge hopping in
the olivine-phosphate family. Nuclear resonant scattering measurements allowed for a study of the
local dynamics of thermally-activated polaron hopping in lithium and sodium-iron phosphate. This
led to new insights into the correlation between electronic carriers and mobile ions. There is a great
deal of interest in this family of materials due to their potential use in next generation battery electrodes and new details concerning the influence of polaron-ion interactions on the charge dynamics
are relevant for optimizing the electrochemical performance of these materials.
My first experiments employed synchrotron nuclear forward scattering measurements at elevated temperature and pressure to determine an activation volume for the charge hopping process.
This is the first use of nuclear resonant scattering to determine an activation volume. These results
showed the valence fluctuations in Lix FePO4 are strongly sensitive to pressure, giving an anomalously large and positive activation volume. This large, positive value is typical of ion diffusion,
pointing to a cooperative mobility of polarons and Li-ions. My second study looked at the sodium
analogue of Lix FePO4 . The sodiated material shows several interesting differences in phase stability, including an ordered intermediated phase. A combination of synchrotron XRD and conventional
Mössbauer spectrometry allowed for the study of the temperature evolution of the delocalization of
99
both Na-ions and electronic carriers on the lattice. These results revealed that the loss of sodiumordering coincides with a marked enhancement of electronic valence fluctuations. These results
show a new relationship between the ordering of the sodium and the electronic charge, and suggest
that polaron-ion interactions may play an important role in the dynamics of Nax FePO4 at elevated
temperature.
7.1
Future work
7.1.1
Pair distribution study of Lix FePO4
A pair distribution function (PDF) investigation of the local structure of Lix FePO4 as a function
of temperature and pressure would provide a structural counterpart to the nuclear resonant scattering
work presented in this thesis. Nuclear resonant scattering data provides valuable information about
the local electronic state of the iron ion, but the interpretation of our results would benefit from
a better picture of the local structure. While in-situ diffraction was collected during the nuclear
forward scattering experiments, these data gave only global structural information.
Sector 11 at the Advanced Photon Source has the capability to conduct both high pressure
and high temperature PDF measurements. While PDF studies as a function of temperature are
now relatively routine, studies at elevated pressure remain a challenge. The small sample volumes
necessary for conducting diamond anvil cell experiments result in a high ratio of background to
sample scattering. Additionally, quality PDF data require access to high scattering angles so the
restricted angular opening of a typical diamond anvil cell is problematic. Despite these challenges,
ongoing development work at sector 11 has made measurements up to 15 GPa possible. This is well
within the pressure range of interest for studies of polaron dynamics in Lix FePO4 . Combining high
pressure and temperature capabilities would likely involve additional development work, but two
independent studies of the temperature evolution and the pressure evolution of the pair distribution
function would have the potential to provide valuable information, including chemical short-range
order and thermal broadening of nearest-neighbor separations.
100
7.1.2
Mössbauer study of electron dynamics in maricite-NaFePO4
Maricite is the ground state structure of NaFePO4 . The triphylite structure is obtained only
via an ion-exchange route. Figure 1.7 illustrates the structural differences between these two polymorphs. In maricite, the site occupancies of the mobile ion are swapped with the iron cations. Consequently, the sodium ions are isolated. As the structure lacks viable diffusion channels, maricite
is electrochemically inactive. Despite the diminished ionic mobility, it is likely that polaron-ion
interactions still play a role in the electron dynamics. In the maricite structure the iron octahedra
have edge-sharing connectivity, compared to the corner sharing network in triphylite. This results
in a closer iron-iron distance. The reduction in jump distance likely lowers the activation barrier
for polaron hopping, although if polaron-ion correlations influence the dynamics it is possible that
the lack of ion mobility results in a diminished electronic mobility as well. A Mössbauer study of
the temperature dependence of the polaron hopping rate in the maricite polymorph compared to the
triphylite structure would reveal new information concerning the relevance of ion conduction pathways to the electronic charge dynamics. Furthermore, a study of the phase stability of this structure
as a function of sodium concentration is potentially interesting as well.
7.1.3
Nuclear resonant scattering study of activation barriers in Li2 FeSiO4
Li2 FeSiO4 is another interesting material to study using nuclear resonant scattering. As all its
constituent elements are earth abundant, Li2 FeSiO4 is an attractive candidate cathode material. Additionally, the Li2 FeSiO4 system presents the possibility of removing two electrons for each iron
cation, theoretically resulting in a higher capacity. Shown in Fig. 1.5 (e), the structure has twodimensional ion conduction networks. As a result, the ion mobility is likely improved compared
to LiFePO4 and the issues related to channel blocking by defects in LiFePO4 are not present. Unfortunately Li2 FeSiO4 has not proved successful as a battery electrode, in part for reasons of low
electronic conductivity. A study of the pressure and temperature dependence of the polaron hopping
rate in Li2 FeSiO4 could give insight into the relevance of the dimensionality of the ion conduction
pathways to the electronic activation barrier.
101
7.1.4
Activation volume studies
Experiments on activation volumes are a largely unexplored. The concept of an activation volume is not limited to polaron hopping and applies to a wide range of activated processes in solids.
Ongoing developments in tools for high pressure measurements are making pressure studies possible for different scattering techniques and expanding the availability of sample environments. With
these advancements, the study of activation volumes is becoming increasingly accessible. While
there are a number of additional iron-bearing polaronic systems that could be studied, in theory it
should be possible apply the same technique to determine of an activation volume for atomic diffusion as well by looking at “speedup” effects in nuclear forward scattering spectra due to incoherent
motions of a diffusing species. Iron diffusion is relevant to a number of structural and geological
materials. An understanding of how this type of diffusion is affected by pressure seems particularly
relevant to geological materials.
While nuclear forward scattering studies are limited to samples with resonant isotopes, other
techniques that measure dynamical properties could be extended to high pressures to obtain activation volumes as well. One possibility is using quasielastic neutron scattering to investigate the
activation volumes for the diffusion of light elements. The recent developments in high pressure
cells for neutron experiments open up the potential for these types of experiments. Diffusion of
light atoms in host structures has many parallels to polaron mobility, and conducting an activation
volume study could provide important physical information concerning the local dynamics. This
technique could be applied to a range of different types of hydrogen storage materials.
102
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