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Understanding the relationship between crystal chemistry and physical properties in magnetic garnets

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Content UNDERSTANDING THE RELATIONSHIP BETWEEN CRYSTAL CHEMISTRY AND
PHYSICAL PROPERTIES IN MAGNETIC GARNETS
by
Abbey J. Neer
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
(CHEMISTRY)
August 2018
Copyright 2018 Abbey J. Neer
for my family and women in science
ii
Acknowledgments
First and foremost, I want thank my advisor, Professor Brent C. Melot. Brent
introduced me to the world–or should I say rabbit-hole–of solid state chemistry.
These past five years, his endless knowledge and passion for science pushed me
to know more about magnetism and solid state physics. Most importantly, Brent
taught me the power of self-reliance and confidence in science and for that I am
so thankful.
I owe so much to all the past and present members of the Melot group. You
guys truly are family and it was a pleasure to work with you all. I deeply appre-
ciate the tremendous amount of support and encouragement you guys gave me
throughout the years. I am so proud of our group for not tearing each other
down, but building each other up. I know we will keep in contact and continue
to support each other for many, many years to come!
A huge THANK YOU to Shiliang "Nemo" Zhou. He was truly an integral part
of my success. Nemo was always there when anyone needed help with literally
anything, and always pushed me to know more and be better–I cannot thank
him enough! Erica Howard and JoAnna Milam-Guerrero, Thank you so much for
endless the support, love, genuine friendship, fruitful science discussions, fruitful
non-science discussions, laughter and sneaky shenanigans we’ve shared; Erica,
you truly are an amazing scientist and have such a big heart, I’m so proud to
call you a best friend; JoAnna, I could not have asked for a more supportive
person to share a project with, it been a joy to grow close to you and watch your
family grow. Nick Bashian, thank you for always worrying about my potassium
intake and providing the most obvious but useful advice; Laura Estergreen, even
though you aren’t officially in the Melot Group, your p-chem perspective has
iii
always served us well, and I greatly appreciate your ability to keep a conversation
anywhere (especially during planks at the gym).
Of course, I would not have accomplished nearly as much without the help of
some incredible undergraduate and high school interns–Veronika Fischer, Justin
So, Michelle Zheng, Nicole Spence, Joseph Stiles, April (Beatriz) Lopez-Burmudez,
Kyle Nolan, Sabrina Mir and Allyson Ee. Watching all of their love for science
grow over the years motivated me through the hardest of days in the lab, thank
you. I owe much of my passion for mentoring to Veronika, who continues to be
my biggest fan. Witnessing her confidence and passion flourish from her fresh-
man to senior year of undergrad has been so inspiring!
None of my this work would have been possible without the gracious help of
my collaborators and USC’s Chemistry Department. Andrew Clough was one of
my first friends in graduate school. It has truly been a joy to collaborate, study
and laugh through it all. My PhD would not have been possible without all the
impromptu beamtime at BT-1 through Craig Brown, or the condensed matter
physics insight from Professor Kate Ross. I am forever thankful for Judy Fong,
Magnolia Benitez, and Michele Dea for solving all of my non-science academic
problems! I owe so much for the support I got through the Women in Chemistry
at USC, especially Betsy, Courtney, Prof. Smaranda Marinescu and Prof. Hanna
Reisler. Empowering and supporting the women in science has always been a
passion of mine and it was a blessing to have been a part of a group who shares
the same values.
I am so incredibly grateful for my non-science friends in LA–John, Kim, Pavel,
Renée, Jenn, Jack, Simone, Tiffany, Lauren, Lily, Jessica, Anthony, Sam, and
Matt. A special thank you to John for introducing me to all of these amazing
people and being such a great friend through these years. Kim, thank you so
iv
much for always reminding me to celebrate accomplishments, and of course I’ll
always think fondly of our endless hours at coffee shops. Renée, thanks for
always reminding me that there’s a world outside of science, I’ve loved attending
all of your art events! Jess, I love you and you’ve been an angel of a homemaker,
roomie, big sister these past few years! I’m really going to miss you!! You all
truly made Los Angeles feel like home. I look forward to spending many more
destination Thanksgivings with you all!
My family may mostly be on the east coast but their love and support was
always a phone call away. Thank you all for supporting my dreams. I apologize
for not visiting as much as I should have. Mom, your encouraging words truly
helped me get through my toughest days. I’m glad I could always count on you
to answer your phone–as long as it was before 9:00 PM EST. Thank you Grammy
Jean for always reminding me of the positive outlook in all of life’s situations.
My two best friends Kelly and Ali: I appreciate every phone call you answered,
at all hours, to support me through all my science highs and lows. Ali, thank
you specifically for listening to me almost everyday. Sometimes just hearing your
voice was enough to keep me grounded!
v
Curriculum Vitae
Education
2013-2018 Ph.D., Chemistry, Department of Chemistry
University of Southern California, Los Angeles, CA
2009-2013 B.S., cum laude Chemistry, Department of Chemistry and Biochemistry
Florida State University, Tallahassee, FL
Publications
1. A.J. Neer, V.A. Fischer, M.Zheng, N.R. Spence, B.C. Melot, Ca
2
M
2
Ge
3
O
12
,
M = Cr
3+
and Fe
3+
Garnet’s Magnetic Structure’s Relationship to Magne-
todielectric Properties (in preparation)
2. A.J. Neer, M. Zheng, N.R. Spence, B.C. Melot, Physical Properties and Mag-
netic Strucutre of Mn
3
Al
2
Ge
3
O
12
(in preparation)
3. A.J. Neer, M. Zheng, N.R. Spence, J. Xue, B.C. Melot, One-dimensional
magnetic garnet Y
3
Fe
2
Ga
3
O
12
’s physical and magnetic properties (in prepa-
ration)
4. E.S. Howard, A.J. Neer, K. Nolan, B.C. Melot, Low Temperature Synthesis,
Structure and Magnetic Properties of Layered Ferromagnet Co
3
Si
2
O
5
(OH)
4
(in preparation)
5. A.J. Clough, A.J. Neer, K. Chen, S. Allen, B.C. Melot, S.C. Marinescu, Con-
ductivity and Magnetism for a Series of Metal Dithiolene Two-dimensional
Coordination Nanosheets (in preparation)
vi
6. A.J. Neer, V.A. Fischer, M. Zheng, C.M. Brown, C. Cozzan, and B.C. Melot,
Magnetodielectric Effects at Quantum Critical Fields in Cobalt-containing
Garents (in preparation)
7. A.J. Neer, J. Milam-Guerrero, J. E. So, B. C. Melot, K. A. Ross, Z. Hulvey, C.
M. Brown, A. A. Sokol, D. O. Scanlon, Ising Magnetism on the Octahedral
Sublattice of a Cobalt-containing Garnet and the Potential for Quantum
Criticality Phys. Rev. B. 95 (2017) 144419 [doi]
8. D. Russel, A.J. Neer, B. C. Melot, S. Derakhshan, Long Range Antiferro-
magnetic ordering in B-site Ordered Double Perovskite: Ca
2
ScOs
6
Inorg.
Chem 155 (2016) 2240-245 [doi]
vii
Understanding the Relationship Between Crystal Chemistry and Physical
Properties in Magnetic Garnets
by
Abbey J. Neer
viii
Contents
List of Tables xi
List of Figures xii
1 Introduction 1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Magnetdioelectricity Potential within the Garnet Structure . . . . 11
2 Quasi-One-Dimensional Ising Magnetism and Quantum Criticality on
the Octahedral Sublattice of a Cobalt-containing Garnet 15
2.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
2.1.1 Materials Preparation . . . . . . . . . . . . . . . . . . . . 16
2.1.2 Neutron Diffraction . . . . . . . . . . . . . . . . . . . . . . 17
2.1.3 Physical Property Measurements . . . . . . . . . . . . . . 17
2.1.4 Density Functional Theory Calculations . . . . . . . . . . 17
2.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 20
2.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.4 Supplemental Information . . . . . . . . . . . . . . . . . . . . . . 30
2.4.1 Table of fitted parameters from the Nuclear and Magnetic
Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.4.2 Table of fitted parameters from the Free Fermion Model . 30
2.4.3 Single Ion Calculations and Estimate of theg-Tensor . . . 31
3 Magnetodielectric Effects at Quantum Critical Fields in Cobalt-Containing
Garnets 35
3.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
3.1.1 Materials Preparation . . . . . . . . . . . . . . . . . . . . 36
3.1.2 Neutron Diffraction . . . . . . . . . . . . . . . . . . . . . . 36
3.1.3 Physical Property Measurements . . . . . . . . . . . . . . 37
3.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 39
3.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
ix
3.4 Supplemental Information . . . . . . . . . . . . . . . . . . . . . . 51
3.4.1 Table of Fitted Parameters from the Nuclear and Magnetic
Refinements . . . . . . . . . . . . . . . . . . . . . . . . . . 51
3.4.2 Figure of Magnetization-Capacitance Comparison and A.C
Magnetocapcitance . . . . . . . . . . . . . . . . . . . . . . 52
4 Low field Dependence in Magnetocapcitance of Ca
3
M
2
Ge
3
O
12
Garnets,
M= Fe
3+
, Cr
3+
54
4.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.1.1 Materials Preparation . . . . . . . . . . . . . . . . . . . . 55
4.1.2 Spark Plasma Sintering (SPS) . . . . . . . . . . . . . . . . 55
4.1.3 Physical Property Measurements . . . . . . . . . . . . . . 55
4.1.4 Capacitance Measurements . . . . . . . . . . . . . . . . . 56
4.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 56
4.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
5 Magnetodielectricity in One-Dimensional Garnet Y
3
Fe
2
Ga
3
O
12
64
5.1 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64
5.1.1 Materials Preparation . . . . . . . . . . . . . . . . . . . . 64
5.1.2 Spark Plasma Sintering (SPS) . . . . . . . . . . . . . . . . 65
5.1.3 Physical Property Measurements . . . . . . . . . . . . . . 65
5.1.4 Capacitance Measurements . . . . . . . . . . . . . . . . . 66
5.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . 66
5.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
6 Future Directions 77
6.1 Preliminary Results of Mn
3
Al
2
Ge
3
O
12
. . . . . . . . . . . . . . . . 78
6.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78
6.2.1 Materials Preparation . . . . . . . . . . . . . . . . . . . . 78
6.2.2 Spark Plasma Sintering (SPS) . . . . . . . . . . . . . . . . 79
6.2.3 Physical Property Measurements . . . . . . . . . . . . . . 79
6.2.4 Capacitance Measurements . . . . . . . . . . . . . . . . . 79
6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
Reference List 85
x
List of Tables
1.1 Reportedd andf Block Cations on Garnet Sublaatices . . . . . . 9
1.2 Reported Non-Magnetic Cations on Garnet Sublattices . . . . . . 10
2.1 Rietveld Refinement Results of CaY
2
Co
2
Ge
3
O
12
. . . . . . . . . . 30
2.2 Basis Functions of CaY
2
Co
2
Ge
3
O
12
. . . . . . . . . . . . . . . . . 31
2.3 Free-Fermion Model Fitted Parameters in CaY
2
Co
2
Ge
3
O
12
. . . . . 31
2.4 Calculated Single-Ion Energies in CaY
2
Co
2
Ge
3
O
12
. . . . . . . . . 34
3.1 Crystallographic Data of CaY
2
Co
2
Ge
3
O
12
and NaCa
2
Co
2
V
3
O
12
. . 44
3.2 Rietveld Refinement Results of NaCa
2
Co
2
V
3
O
12
. . . . . . . . . . 51
3.3 Basis Functions of NaCa
2
Co
2
V
3
O
12
. . . . . . . . . . . . . . . . . 51
5.1 Crystallographic Refinement Results of Y
3
Fe
2
Ga
3
O
12
. . . . . . . . 66
6.1 Heating Profiles of Garnets . . . . . . . . . . . . . . . . . . . . . . 77
xi
List of Figures
1.1 Atom Displacement in BaTiO
3
’s Electric Dipole . . . . . . . . . . . 2
1.2 Geometric frustration in a triangular lattice . . . . . . . . . . . . 3
1.3 Garnet Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . 6
1.4 Connecticity of the Octahedral Sublattice in the Garnet Structure 7
1.5 Frustration in a Linear Chain . . . . . . . . . . . . . . . . . . . . 11
1.6 Octahedral Chains in Garnet Structures . . . . . . . . . . . . . . . 13
2.1 Rod-packing in CaY
2
Co
2
Ge
3
O
12
Garnet . . . . . . . . . . . . . . . 16
2.2 Magnetic Structure of CaY
2
Co
2
Ge
3
O
12
. . . . . . . . . . . . . . . 19
2.3 Magnetism of CaY
2
Co
2
Ge
3
O
12
. . . . . . . . . . . . . . . . . . . . 21
2.4 Specific heat Results of CaY
2
Co
2
Ge
3
O
12
. . . . . . . . . . . . . . . 22
2.5 Magnetic Phase Diagram of CaY
2
Co
2
Ge
3
O
12
. . . . . . . . . . . . 27
2.6 Decompostion of Co
2+
Ground Wavefunctions . . . . . . . . . . . 34
3.1 Magnetism of NaCa
2
Co
2
V
3
O
12
. . . . . . . . . . . . . . . . . . . . 38
3.2 Garnet Octahedral-Sublattice Connectivity in Rods . . . . . . . . 40
3.3 Specific Heat Results of NaCa
2
Co
2
V
3
O
12
. . . . . . . . . . . . . . 41
3.4 Magnetic Phase Diagram of NaCa
2
Co
2
V
3
O
12
. . . . . . . . . . . . 43
3.5 Magnetic Structure of NaCa
2
Co
2
V
3
O
12
. . . . . . . . . . . . . . . 45
xii
3.6 Angles and Distances of Nearest Neighbors in NaCa
2
Co
2
V
3
O
12
and
CaY
2
Co
2
Ge
3
O
12
. . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.7 Calculated Density of States for CaY
2
Co
2
Ge
3
O
12
and NaCa
2
Co
2
V
3
O
12
48
3.8 Magnetization-Magnetodielectricity Comparison of CaY
2
Co
2
Ge
3
O
12
and NaCa
2
Co
2
V
3
O
12
. . . . . . . . . . . . . . . . . . . . . . . . . 50
3.9 Magnetocapcitance A.C. Frequency Dependence of NaCa
2
Co
2
V
3
O
12
and CaY
2
Co
2
Ge
3
O
12
. . . . . . . . . . . . . . . . . . . . . . . . . 52
4.1 Magnetism of Ca
3
Cr
2
Ge
3
O
12
. . . . . . . . . . . . . . . . . . . . . 57
4.2 Magnetism of Ca
3
Fe
2
Ge
3
O
12
. . . . . . . . . . . . . . . . . . . . . 58
4.3 Specific Heat Results of Ca
3
Fe
2
Ge
3
O
12
. . . . . . . . . . . . . . . . 59
4.4 Specific Heat Results of Ca
3
Cr
2
Ge
3
O
12
. . . . . . . . . . . . . . . 60
4.5 Field Dependent Magnetocapcitance of Ca
3
Fe
2
Ge
3
O
12
. . . . . . . 61
4.6 Field Dependent Magnetocapcitance of Ca
3
Fe
2
Ge
3
O
12
. . . . . . . 62
5.1 Garnet Crystal Structure, with a quarter of the unit cell (a), in
which the octahedra are shown in green, and the terahedra in
periwinkle. The cubic site is symbolized by a black sphere for sim-
plicity. (b) the connectivity between the tetrahedral site and octa-
hedral site within a rod. (c) The connectivity of the tetrahedral
site and octahedral site btween neighboring rods. . . . . . . . . . 67
5.2 Rietveld Refinement Results for Y
3
Fe
2
Ga
3
O
12
. . . . . . . . . . . . 68
5.3 Magnetism of Y
3
Fe
2
Ga
3
O
12
. . . . . . . . . . . . . . . . . . . . . . 70
5.4 A.C. Magnetic Susceptibility of Y
3
Fe
2
Ga
3
O
12
. . . . . . . . . . . . 72
5.5 Specific Heat Results of Y
3
Fe
2
Ga
3
O
12
. . . . . . . . . . . . . . . . 73
5.6 Temperature Dependent Magnetodielectric Data of Y
3
Fe
2
Ga
3
O
12
. 74
5.7 Magnetization-Magnetodielectricity Comparison of Y
3
Fe
2
Ga
3
O
12
. 75
xiii
6.1 Rietveld Refinement Results for Mn
3
Al
2
Ge
3
O
12
. . . . . . . . . . . 80
6.2 Magnetism of Mn
3
Al
2
Ge
3
O
12
. . . . . . . . . . . . . . . . . . . . . 82
6.3 Magnetic Susceptibilty of Mn
3
Al
2
Ge
3
O
12
at Various Fields . . . . . 83
6.4 Specific Heat Results of Mn
3
Al
2
Ge
3
O
12
. . . . . . . . . . . . . . . 84
xiv
Understanding the Relationship Between Crystal Chemistry and Physical
Properties in Magnetic Garnets
by
Abbey J. Neer
xv
Abstract
Multiferroic materials, which simultaneously exhibit more than one of the
ferroic properties, offer unique opportunities to study the relationship between
crystal structures and their physical properties. In particular, single phase mate-
rials that exhibit a cross-coupling between their magnetic and electrical polariza-
tions can give rise to novel functionality such as the ability to manipulate magne-
tizations through applied voltages or vice versa.[1] The fundamental challenge
described by Nicola Spaldin, that resurrected interest in the field of multiferroics
is the contraindicated origins of ferroelectricity and magnetism. That may be
circumvented by induced polarization via novel magnetic structures. [2, 3, 4, 5,
6, 7]
Magnetically frustrated materials offer a route to unconventional magnetic
properties with the capability to induce coupling of charge and spin. [5] The
work here aims to develop an understanding of magnetic frustration as a vehicle
to multiferrocity in garnet structures.
We began our search of magnetic frustration by studying garnet systems in
which Co
2+
occupies the octahedral sublattice. CaY
2
Co
2
Ge
3
O
12
and NaCa
2
Co
2
V
3
O
12
are two antiferromagnetic materials exhibit a moderate degree of frustration
with field-induced transitions. Their magnetic phase diagrams were constructed
via field-dependent specific heat measurements and revealed Quantum Criti-
cality corresponding to the field-induced transitions seen in the magnetization.
We then discuss the relationship between field-induced transitions at the critical
fields with the magnetocapacitance of these garnets. These two garnet systems
were also found to exhibit Ising-esque magnetism seen in their magnetic struc-
ture along the body diagonals. Thus we explored the correlation between the
xvi
efficiency of packing within the unit cell and their magnetic structure as well as
the suppression of Quantum Criticality at higher fields in ideal packing. Specif-
ically, a more efficiently packed rod system will suppress quantum criticality to
substantially higher fields. We also observe a correlation between the quantum
critical fields and anomalies in the magnetodielectricity measurements.
Then, we went on to study how altering the magnetic octahedral sublattice
affects the magnetism in identical diamagnetic backbones in garnets Ca
3
Cr
2
Ge
3
O
12
and Ca
3
Fe
2
Ge
3
O
12
. Both Ca
3
Cr
2
Ge
3
O
12
and Ca
3
Fe
2
Ge
3
O
12
are low temperature
antiferromagnets with a noticeably small degree of frustration. However, in
these systems their collinear magnetism occuring along the rods allows for field-
dependent magnetodielectric anomalies.
Lastly we explore the magnetodielectricity of the one-dimensional garnet
Y
3
Fe
2
Ga
3
O
12
. At first glance, Y
3
Fe
2
Ga
3
O
12
appears to be a spin-glass, however
the magnetic susceptibility fits in agreement with the one-dimensional Bonner-
Fisher model for a uniform X-Y chain. The glass-like character observed in the
specific heat was found to be attributed to the site-mixing between the gar-
net’s octahedral and tetrahedral sublattices. Two anomalies are evident in the
specific heat which manifest in the the A.C. susceptibility unveiling complex
spin-dynamics and magnetic ordering within the system. It is suscepted that
one anomaly corresponds to the one-dimensional ordering of Y
3
Fe
2
Ga
3
O
12
, and
another to a second three-dimensional ordering of the material. The temperature
at which the suspected three-dimensional ordering occurs shows an anomaly in
the temperature-dependent magnetodielectricity; however, the hysteresis found
between the cooling and heating measurements alludes to a first order transi-
tion structural change. Aside from a structural change, the material below itsT
C
shows subtle anomalies at low fields of magnetodielectric measurements.
xvii
Chapter 1
Introduction
The goal of this work is to develop an understanding of competing mag-
netic interactions of the sublattices in garnet-type structures via systematic com-
positional substitutions. These studies combine structural characterization with
physical property measurements in order to elucidate the structure-property rela-
tionships in magnetodielectric garnet structures. The purpose of this introduc-
tory chapter is to provide a foundation of the garnet crystal structure as well as
describe the relationship between magnetodielectricity and magnetocapacitance.
1.1 Motivation
Multiferroic materials, which simultaneously exhibit more than one ferroic
property, offer unique opportunities to study the relationship between crystal
structures and their physical properties. Spin-lattice coupling in magnetodi-
electrics is an attractive area of research for sensor development; however, con-
traindicated origins create challenges for new phase discovery. [8] The magne-
toelectric effect describes a multiferroic which only couples magnetic order and
ferroelectric ordering.
Single phase materials that exhibit a cross-coupling between their magnetic
and electrical polarizations can give rise to novel functionality like the ability
to manipulate magnetizations with through applied voltages or vice versa. [1]
Magnetoelectrics, with their ability to control charge and magnetism, are then
1
(a) (b)
Figure 1.1: Above the ordering temperature, T
c
, of 403 K BaTiO
3
is a dielectric
(paraelectric) phase (a) which crystallizes into the cubic space group Pm3m
(No. 221). Once cooled below the T
c
, BaTiO
3
becomes ferroelectric (b) an
the symmetry reduces to a tetragonal space group P 4mm (No. 99), as Ti
4+
displaces from the center of the octahedral creating a charge separation between
the cations and anions.
promising for technological applications including field sensors, memory storage,
signal processing, and lower-energy data storage. [9, 10, 11, 12, 13, 14]
Electric and magnetic order are often thought of separately, where the electric
charges of electrons produce ferroelectricity and electron’s spin produce mag-
netism. Ferroelectrics, in order to preserve an electric dipole, require highly
insulating behavior while magnetic materials often contain conductive transi-
tion metals. To further convolute these origins, a typical ferroelectric’s electric
polarization occurs due to the noncentrosymmetricity created by assymmetric
displacement of non-magnetic cations. This establishes necessary charge sepa-
ration for an electric dipole, as shown in Figures1.1 (a) and (b) of the classic
ferroelectric BaTiO
3
in which the octahedral coordinate Ti
4+
cation shifts up sep-
arating the cations and anions thus resulting in an electric dipole, Figure 1.1 (b).
Meanwhile, magnetic cations that possess partially filledd-orbitals tend to prefer
2
the centrosymmetric position of their octahedra based on thermodynamic stabil-
ity of the localized unpaired electrons’ Coulombic repulsion within their covalent
bonds. Thus, these competing origins are deemed contraindicated; this is also
known as thed
n
vsd
0
problem.
Magnetic frustration within transition-metal and rare-earth oxides is a
promising route for circumventing this contraindication. Magnetic frustration
can simply be described as the inability to simultaneously balance all compet-
ing magnetic interactions through a single and unique ground state. [15, 16]
Archetypal frustration is conveyed by arranging Ising spins onto a triangular lat-
tice in an antiferromagnetic configuration; however, a chain of spins can also
exhibit frustration and will be explained in detail later.
Figure 1.2: An illustration of geometric frustration as a result of a triangular
lattice geometry.
Figure 1.2 depicts frustration on the triangular lattice, when the first two
corners are chosen to be antiparallel to one another, then the third can not con-
currently satisfy both competing spins and thus is considered to be frustrated.
A frustrated system will adopt one of six possible degenerate states, where the
3
third corner will be align parallel with one of the spins. [17] In order to forgo the
frustrated state, a material will adopt a more energetically favorable spin-state
often resulting in a non-collinear magnetic structure. [18] These alluring spin-
arrangements form non-collinear magnetic structures such as helical magnetic
structures which are frequently liable mechanism to couple electrical polariza-
tion and magnetic order. [18, 19, 20, 21, 22]
Several single phase magnetodielectrics have been reported in literature. [5,
23, 24, 19, 3, 25] Unconventional mechanisms such as breaking space-inversion
symmetry within magnetic structures have been implemented in order to induce
polarization and circumvent the d
0
vs d
n
problem. [26, 27] In the cubic spinel
CoCr
2
O
4
, as well as LiCu
2
O
2
and BiFeO
3
, helical ordering of the magnetic
moments is capable of breaking spatial inversion symmetry. [28, 29, 30, 31,
32, 5] Structural geometries with strong competition between nearest- and
next-nearest neighbors have also been shown to result in highly frustrated sys-
tems that can exhibit electrical polarizations. This is exemplified in TbMnO
3
where the reversal of the Ising moments on Tb
3+
induces a reorientation of
the Mn
3+
spins and triggers an elongation along the a-axis. [5] Similarly,
commensurate-incommensurate magnetic transitions were reported to give rise
to large anomalies in the magnetic field-dependent dielectric properties of
orthorhombicRMn
2
O
5
(whereR=Tb, Dy, Ho). [23, 33]
Accurate measurements of macroscopic polarizations typically require large
single crystals in order to detect the extremely small currents produced by mag-
netoelectric materials. Considering that crystal growth is rarely straightforward,
measurements capable of screening polycrystalline samples for spin-lattice cou-
pling are highly desirable. [34] In response, magnetocapacitance measurements
4
have quickly grown to be an essential tool since the temperature- or field-
dependent capacitance of a well-sintered pellet can often reveal very minor
changes in charge localization or expansions/contractions in the unit cell. [35]
The capacitance of a sintered polycrystalline pellet measured as a function of
magnetic field can be described by Equation 1.1, where C is the capacitance, ε
is the dielectric constant, andA as well asd refer to the geometry of the sample
pellet.
C =
εA
d
(1.1)
ε = 1 + 4π
∂P
∂E
(1.2)
Specifically, the sintered pellet acts as a parallel plate capacitor when the faces
are coated with silver epoxy, allowing the capacitance to be a function of the
pellet, using the pellet’s area,A, andd, the thickness of the pellet which separates
the makeshift capacitor plates. Then from Equation 1.2, one can indirectly probe
averaged changes in the dielectric polarization from the dielectric response.
This method of measuring polycrystalline pellets has proven to agree with
single-crystal responses. [28, 29] Examination of the magnetodielectric proper-
ties of a material can also be useful in identifying symmetry-breaking changes
in the magnetic moments that would normally preclude the evolution of macro-
scopic polarizations based on the nuclear symmetry alone. [36, 7, 19]
5
1.2 Crystal Structure
(a) (b)
Figure 1.3: (a) First quadrant of the garnet structure showing the tetrahedral
sites in periwinkle, octahedral sites in maroon, and cubic sites as black spheres.
(b) The packing of the sublattices within each rod.
Garnets, with the general formula R
3
B
2
A
3
O
12
, can be considered a playground
for magnetism because their robust structural and compositional diversity is
optimal for methodical studies of magnetic interactions. This mineral crystal-
lizes in the cubic space group Ia
¯
3d (No. 230), which creates a 3D network of
interwoven-packed rods along the body diagonals of the unit cell. [37, 38] Each
rod consists of a BO
6
octahedral center surrounded by corner-sharing AO
4
, tetra-
hedra, which then creates a 8-coordinate or dodecahedra environment for the
remaining R-site, shown in Figure 1.3 (a). Within this picture, the oxygen anions
sit around each of the four three-fold axes of rotation that run along the body
diagonals of the cubic unit cell. This symmetry creates rods out of the octahedral
sites that are then joined together by vacant triangular prisms that are formed
by the faces of the cubic sites (Figure 1.3 (b)).
The connectivity of the three uniquely coordinated sublattices create multiple
exchange pathways making this crystal structure promising for novel magnetic
6
(a) (b)
Figure 1.4: (a) Within a rod, the magnetic octahedral sublattice super-exchange
pathway is through the oxygens of the cubic site. (b) While between neighboring
chains, the tetrahedral’s oxygen then facilitates the exchange.
properties, which greatly motivates this work. The study of magnetism on each
of the sublattices separately has not comprehensively been reported, and would
provide insight to understanding and tuning of related functional properties.
Much of the work revolves around magnetism on the octahedral sublattice, in
which the exchange pathways are shown in Figure 1.4. The octahedral site occu-
pies the center of the rod, and thus the major exchange pathways can then be
simplified to within a single rod Figure 1.4 (a), or between neighboring rods Fig-
ure 1.4 (b). Other exchange pathways are possible within a garnet structure, but
are not applicable to the scope of this work.
Transition-metal oxides, like garnets, are very promising for magnetoelectric
materials due to their insulating quality. When transition-metal oxides remain
stoichiometric, the energy gap between the filled valence bands and virtually
empty conduction band is quite large creating an insulating character. [39] It
should be noted that magnetic metal oxides do have partially filled bands cor-
responding to thed-orbitals, however generally these are deemed too narrow to
7
support metallic behavior, therefore not lowering the valence bands. [40] Insu-
lating magnetic materials are needed for the polarization to accurately be mea-
sured as depicted in Equation 1.1, otherwise the materials conductivity will cause
leakage and conclusions cannot be drawn from magnetoelectric measurements.
A challenge with garnet structures is the lack of preference for each sublat-
tice within the periodic table. Unlike the perovskite structure which adheres to
a tolerance factor, the garnet structure lacks rules to determine which element-
combinations from the periodic table form garnets. [41] In fact, garnets are so
compositionally diverse it can often be difficult to evade site-mixing among sub-
lattices due to the fact that few cations prefer exclusively one sublattice. Tables
1.1 and 1.2 summarizes the lack of design rules with respect to the cations
reported in single phase garnet structures and are located at the end of this
section. [42, 43, 44, 45]
8
Cation CubicR-site OctahedralB-site TetrahedralA-site
Sc × ×
Ti × ×
V × ×
Cr ×
Mn × ×
Fe × ×
Co × × ×
Ni × ×
Cu × ×
Zn × × ×
Y × × ⊗
Zr ×
Nb ×
Rh ×
Ag ×
Cd × × ⊗
La ×
Hf ×
Ta ×
Pr ×
Nd ×
Sm × ⊗
Eu × ⊗
Gd ×
Tb × ×
Dy × ⊗
Ho × ⊗
Er × × ×
Tm ⊗ ×
Yb × ×
Lu ×
Table 1.1: A summary of d and f block Cations cations reported on the indi-
vidual sublattices to showcase their lack of design rules. When a× is shown,
this refers to the particular cation’s ability to occupy the site in whole integers.
Whereas a⊗ describes when the specific cation only occupies this site in small
fractional amounts. A blank indicates the cation has not been reported on the
sublattice. [45]
9
Cation CubicR-site OctahedralB-site TetrahedralA-site
Li × ×
Na × ⊗
Mg × ×
K ×
Ca × ⊗
Sr ×
Ba ×
Al × ×
Si ×
Ga × ×
Ge × ×
As ×
In × ×
Sn ×
Sb ×
Te ×
Pb ×
Bi ×
Table 1.2: A summary of non-magnetic cations reported on the individual sub-
lattices to showcase their lack of design rules. When a× is shown, this refers
to the particular cation’s ability to occupy the site in whole integers. Whereas
a⊗ describes when the specific cation only occupies this site in small fractional
amounts. A blank indicates the cation has not been reported on the sublat-
tice. [45]
10
1.3 Magnetdioelectricity Potential within the Gar-
net Structure
It is particularly the garnet’s three unique sublattices that hold the key to
the promise of magnetocapacitance effects. The cubic lattice, which is made up
of triangular lattices, is intrinsically geometrically frustrated thus opening it to
possibilities for non-collinear magnetic routes to polarization. Cubic magnetism
in Gd
3
Ga
5
O
12
is arguably the most studied frustrated garnet phase reported, and
does not show onset of magnetic-ordering down to 25mK, it was later reported to
have spin-liquid behavior. [46, 47] Mn
3
Al
2
Ge
3
O
12
, another magnetic cubic gar-
net, adopts a non-collinear magnetic ground state resulting in triangular mag-
netic structure. [48, 49, 50] A magnetoelectric effect has yet to be published
with this composition, but measurements are shown in future chapters here.
While frustration is a often a vehicle to potential exotic magnetism, the cubic
site is not the only site within the garnet structure capable of frustration. The
rod-packing of the garnet structure creates linear chains of octahedra, and when
occupied with magnetic cations is capable of frustration through the competing
superexchange interactions. Frustration within a linear chain is cultivated from
the competing magnetic interactions from the Nearest-Neighbor (NN) and the
Next-Nearest-Neighbor (NNN), and is depicted in Figure 1.5.
J J
NN NNN
Figure 1.5: Frustration in a linear chain as a result of strong competition between
NN (nearest neighbor) and NNN (next nearest neighbor) magnetic interactions
11
Systems with linear chains are then capable of producing an electrical polar-
ization if Dzyaloshinskii-Moriya interactions induce a magnetically driven polar-
ization. [18] Specifically these system’s polarizations obey Equation 1.3, which
combine the relationships of the strength of the spin-orbit coupling, a, and the
unit vector,e
ij
, that connects the neighboring spins within a chain,S
i
andS
j
.
P =ae
ij
× [S
i
×S
j
] (1.3)
The Dzyaloshinskii-Moriya interactions clarifies a prerequisite for the coupling of
electrical polarization and magnetic order, that space-inversion symmetry must
be broken. Once broken, the oxygen anionic ligands within structures are able to
displace from their original centrosymmetric postion and thus a net macroscopic
polarization is measurable. [51, 36, 24, 19, 18, 52] When space-inversion sym-
metry is broken, oxygen ligands are able to displace in such a manner suitable
for electrical polarization, that often is referred to as exchange striction. [25, 53]
Magnetoelectric effects have been reported in various magnetic chain struc-
tures including LiCu
2
O
2
, Ca
3
CoMnO
6
and CoNb
2
O
6
, however through slightly
different magnetic mechanisms. In LiCu
2
O
2
, the linear chain adopts a non-
collinear helical magnetic structure to avoid a frustrated state, this breaks the
inversion symmetry resulting in a net polarization. However in the ferrimag-
net Ca
3
Co
2
Mn
2
O
6
, the linear chain adopts a collinear magnetic spin order and
still is able to induce polarization through exchange striction between the Mn
4
+
and Co
2+
competing exchange interactions within the magnetic structure. Later,
we will discuss how Ising magnetism induces a magnetoelectric effect in Co
2+
-
octahedra containing garnets as a result of one dimensional magnetic linear-
chains.
12
(a) (b)
Figure 1.6: (a) Illustration of the rod-packing first described by Andersson and
O’Keefe. The projection of the structure is shown such that the four rods, each
uniquely colored and isolated are most visible. (b) View of the octahedral sub-
lattice color-coded to match their corresponding rods.
The octahedral sublattice, while not made up of inherently frustrated units
like the cubic site, holds it’s own promise for fundamentally interesting mag-
netism and routes to magnetodielectricity. As discussed in the previous section,
the garnet structure is made-up of interpentetrating rods, where the octahedral
sublattice resides in the center shown in Figure 1.6. Three of these rods (colored
blue, green, and red in Figure 1.6 (b)) run at angles around a central one and
cross within the plane without intersecting the central rod(Figure 1.6 (a)). These
rods create the ideal environment for exchange-interactions described in Figure
1.5, which can occur in low dimensional magnetic systems.
The third site in the garnet is the tetrahedral sublattice, which is made up
of four edge-sharing triangles creating the archetype of three-dimensional geo-
metric frustration. While the singularly tetrahedral magnetic garnet has yet to
be reported, higher magnetic ordering temperatures have been attributed to this
13
site. With magnetic cations on the octahedral as well as the tetrahedral site as
seen in Y
5
Fe
5
O
12
(YIG) and Ca
3
Fe
3.5
V
1.5
O
12
(CaVIG) the reported ordering tem-
peratures above 500 K. The lack of compositions with magnetism on the tetrahe-
dral site is attributed to the design rules for the garnet structure, and very few
elements have been reported on the tetrahedral site. [42, 43, 44, 45]
14
Chapter 2
Quasi-One-Dimensional Ising
Magnetism and Quantum Criticality
on the Octahedral Sublattice of a
Cobalt-containing Garnet
Unique topology motivated us to investigate how magnetic ions with strong
single-ion anisotropy would behave when fully localized to these rods. Using
low-temperature powder neutron diffraction combined with magnetic suscep-
tibility and heat capacity measurements, we have studied CaY
2
Co
2
Ge
3
O
12
and
found that the highly anisotropic nature of the superexchange pathways gives
rise to quasi-one-dimensional behavior. We also find evidence for a quantum
phase transition in magnetic fields above 6 T, suggesting the system may repre-
sent a new embodiment of a linear chain of Ising spins.
2.1 Methods
Certain commercial equipment, instruments, or materials are identified in
this document. Such identification does not imply recommendation or endorse-
ment by the National Institute of Standards and Technology nor does it imply
that the products identified are necessarily the best available for the purpose.
15
2.1.1 Materials Preparation
Polycrystalline powders of CaY
2
Co
2
Ge
3
O
12
were prepared from CaCO
3
, Y
2
O
3
,
Co(C
2
O
4
)·2H
2
O and GeO
2
by heating well-ground mixtures initially at 950
◦
C
overnight followed by regrinding, pelletizing, and heating at 1300
◦
C until phase
pure. [54] Given the well-known reactivity of cobalt-based compounds with alu-
mina, platinum crucibles were favored, with phase pure powders exhibiting a
dark purple coloration while Al contamination was found to result in a lavender
hue.
Figure 2.1: (Color online) (a) First quadrant of the garnet structure showing
the tetrahedral sites in gray, octahedral sites in red, and cubic sites as black
spheres. (b) Illustration of the rod packing first described by Andersson and
O’Keefe. The projection of the structure is shown such that the four rods, each
uniquely colored and isolated in (c), are most visible.
16
2.1.2 Neutron Diffraction
The sample was loaded in to a vanadium cell and placed into a helium flow
cryostat with temperature control. Neutron powder diffraction data were col-
lected at 295 K, 25 K and 2 K using the BT-1 high resolution neutron powder
diffractometer at the NIST Center for Neutron Research. A Ge(311) monochro-
mator with a 75
◦
take-off angle, λ = 2.0787(2)Å and in-pile collimation of 60
minutes of arc were used. Data were collected over the range of 1.3
◦
-166.3
◦
in
scattering angle (2-Theta) with a step size of 0.05
◦
.
2.1.3 Physical Property Measurements
Temperature and field-dependent magnetic susceptibility as well as specific
heat measurements were collected using a Quantum Design 14 T Dynacool Phys-
ical Property Measurement System. For the specific heat measurements, pow-
ders of the title phase were mixed with equal amounts of silver to increase the
thermal coupling to the stage, with the contribution from silver being measured
separately and subtracted. The magnetic contribution to the specific heat was
estimated by using CaY
2
Mg
2
Ge
3
O
12
to estimate the lattice contribution and sub-
tracting after properly rescaling. [55]
2.1.4 Density Functional Theory Calculations
All density functional theory (DFT) calculations were performed using the
VASP code.[56, 57] The projector augmented wave (PAW) [58] method was
used to describe the interactions between the cores (Co:[Ar], Ca:[Ar], Y:[Kr],
Ge:[Ar]and O:[He]) and the valence electrons. The calculations were performed
using the PBEsol exchange-correlation functional,[59] with the inclusion of the
17
a correction for on-site Coloumb interactions (PBEsol + U). PBEsol is a revi-
sion of the PBE functional specifically tailored for solids, and yields structural
data in excellent agreement with experiment. We apply the rotationally invari-
ant approach of Dudarev,[60], with aU value of 4.4 eV applied to the Cod states,
which has previously been shown to provide a good description of CoII contain-
ing oxides.[61, 62] Relativistic spin–orbit effects were included explicitly in all
calculations. A planewave cutoff of 900 eV and ak-point sampling of Γ-centred
2× 2× 2 were utilized for the 160 atom unit cell, with the ionic forces converged
to less than 0.01 eV Å
−1
.
18
(b)
(c)
20 40 60 80 100 120 140
2 θ (deg) [ λ=2.079Å]
2000
3000
4000
5000
6000
7000
8000
counts
(a)
Figure 2.2: (Color online) (a) Results of the Rietveld refinement against the 2 K
powder neutron diffraction data. R
nuc
Bragg
=1.5%, R
mag
=8.5%. (b) Illustration of
the resulting magnetic structure viewed off one of the cubic unit cell edges. (c)
Projection illustrating the magnetic structure down the (111) axis of the unit
cell. In both illustrations the spins are color coded to match the octahedral rods
in Figure 2.1.
19
2.2 Results and Discussion
We began our study by using powder neutron diffraction to determine the
nuclear and magnetic structure. The room temperature nuclear structure was
found to be in good agreement with the cubicIa
¯
3d structure with Co ions located
on the octahedrally coordinated 16a site, with the resulting structural parameters
given in SI Table I. In order to fit the reflections associated with the onset of
magnetic order observed in the 2 K patterns at 23.8
◦
, 30.8
◦
, 36.7
◦
, 41.8
◦
, and
58.7
◦
, the method of representational analysis was used. All reflections could
be indexed with a simple wavevector ofk = 0, with the BasiReps routine within
FullProf returning four one-dimensional, two two-dimensional, and four three-
dimensional representations within the Little groupG
k
.
A reasonable fit of the diffraction data, shown in Figure 2.2 (a), could only
be obtained using the first representation, Γ
1
, which consists of the basis vectors
listed in SI Table II. This representation corresponds to the magnetic space group
Ia
¯
3d and forces the moments to point along one of the body diagonals of the
unit as illustrated in Figure 2.2 (b) and (c). The refined moment resulting from
the fit was 2.7(1)μ
B
per Co, which will be addressed in detail later.
20
-3
-2
-1
0
1
2
3
M (µ
B
Co
-1
)
10K
9K
8K
7K
6K
5K
4K
3K
2K
-12 -8 -4 0 4 8 12
µ
0
H (T)
dM/dH
0 50 100 150 200 250 300
T (K)
0.00
0.05
0.10
0.15
0.20
0.25
χ (emu mol
-1
Oe
-1
)
0.05 T zfc
0.05 T fc
0 10 20 30
0.1
0.2
0.2
0.3
(a)
(b)
(c)
Figure 2.3: (a) Temperature-dependent susceptibility of CaY
2
Co
2
Ge
3
O
12
col-
lected with a field of 0.05 T. The inset emphasizes the cusp associated with
the onset of antiferromagnetic order around 6 K. (b) Isothermal magnetization
curves collected through the magnetic ordering transition. (c) Derivative of the
magnetization with respect to the applied field after smoothing with a running
average of ten points for clarity. Note the temperature-dependence of the field
induced transition that occurs around 6 T in the 2 K data.
The magnetic structure, illustrated in Figure 2.2 (b) and (c), constrains the
spins to point along the body diagonals of the cubic unit cell and, as a result,
21
2 4 6 8 10 12 14 16
T (K)
0.0
0.5
1.0
1.5
2.0
C
mag
/T (J mol
-1
K
-2
)
0T
1T
2T
3T
4T
5T
6T
5 10 15 20 25 30 35 40 45 50
T (K)
0.0
0.2
0.4
0.6
0.8
1.0
ΔS
mag
/R ln(2)
0 5 10 15 20 25 30
T (K)
0.0
0.1
0.2
0.3
0.4
0.5
0.6
C
mag
/T (J mol
-1
K
-2
)
7T
8T
9T
10T
11T
12T
13T
14T
5 10 15 20 25 30 35 40 45 50
T (K)
0.0
0.2
0.4
0.6
0.8
1.0
ΔS
mag
/R ln(2)
(J mol
-1
K
-2
)
(a) (b)
(c) (d)
(e)
0 4 8 12 16 20
T (K)
0
0.1
0.2
0.3
0.4
C
mag
/T
10T
11T
12T
13T
14T
Figure 2.4: (Color online) Magnetic contribution to the specific heat at (a) low
and (b) high field strengths. The lattice contribution was approximated by sep-
arately measuring and subtracting the specific heat of CaY
2
Mg
2
Ge
3
O
12
in order
to estimate the change in entropy associated with the onset of magnetic order in
(c) low and (d) high fields, normalized to the maximum value ofRln(2). (e) Fits
to the heat capacity using the free fermion model discussed in the text.
orients the moments along the long axis of the octahedral chains that Anders-
son and O’Keefe described. This magnetic order is distinct from other garnets,
like Ca
3
Cr
2
Ge
3
O
12
, which has been reported to possess ferromagnetic layers of
spins that couple antiferromagnetically to each other in a more conventional
3D magnetic structure. [63, 64] In contrast, a j
eff
= 1/2 moment description is
appropriate for octahedrally coordinated Co
2+
, due to the interplay of spin-orbit
coupling and the crystal electric field (CEF). The single ion ground state is a
Kramers doublet, often separated by∼ 30 meV (∼ 300 K) from the next excited
level. [65, 66, 67] Thej
eff
= 1/2 moment formed by the single ion ground state
doublet acquires anisotropy in a distorted octahedron and must be described by
anisotropicg-tensor. The oxygen environment of Co
2+
in CaY
2
Co
2
Ge
3
O
12
is trig-
onally distorted (point group S
6
), with the octahedron being elongated along a
local [111] direction, the same directions that define the structural “rods”. For
22
Co
2+
, this type of distortion generically produces an Ising-like g-tensor, and for
CaY
2
Co
2
Ge
3
O
12
, the Ising direction is consistent with the refined magnetic struc-
ture shown in Figure 2.2. To estimate theg-tensor anisotropy in CaY
2
Co
2
Ge
3
O
12
,
a simple point charge calculation was carried out following the methods of
Hutchings [68] to obtain the CEF Hamiltonian. This was simultaneously diago-
nalized with the spin orbit coupling Hamiltonian within the full 28× 28|L
z
,S
z
i
basis of the
4
F free ion term for Co
2+
, yieldingg-tensor componentsg
z
= 5.7 and
g
xy
= 3.9. These results are in good agreement with the observed moment deter-
mined from the neutron diffraction experiment, sinceμ
z
=
1
2
g
z
= 2.85μ
B
, which
can be compared to the refined ordered moment of 2.7μ
B
atT = 2 K.
The temperature-dependent magnetic susceptibility, shown in Figure 2.3 (a),
shows the spins begin to establish this antiferromagnetic ground state starting
around 6 K. Fitting the high temperature region (200-300K) of the magnetic sus-
ceptibility to the Curie-Weiss equation,χ=C/(T-Θ
CW
), provides some insight into
the mean-field average of the magnetic interactions. The effective moment per
Co
2+
was found to be 5.17μ
B
, in close agreement with the expected value of
5.2μ
B
in a high spin octahedral coordination environment where the contribu-
tions from the spin and orbital components of the magnetization are fully decou-
pled. The Curie-Weiss theta was found to be -35K, reflecting that a moderate
antiferromagnetic coupling appears to dominate the interactions within the octa-
hedral sublattice. Care should be exercised when interpreting the Curie-Weiss
data, however, since the temperature range over which the fits are performed
is approaching the energy of the first excited single-ion level (see supplemental
material).
23
Given that neighboring octahedral sites do not share any common bonds,
these interactions will be affected by the ions on the cubic and tetrahedral posi-
tions. These superexchange pathways typically consist of Co–O–(A/R)–O–Co
linkages or super-superexchange along the edge of the polyhedra in a way simi-
lar to theA-site magnetic spinels like CoAl
2
O
4
and FeSc
2
S
4
, [69, 70, 71, 72, 73].
The preference for antiferromagnetic order along the rods suggest that the path-
way through the cubic site, which couples spins along the length of the rods,
dominates over the tetrahedral pathways that bridge the neighboring chains.
Despite a much shorter and less complex coupling between the octahedral and
tetrahedral sites when magnetic ions are present on both sites, the intrachain
pathways are strengthened because of their 12-fold degeneracy and the shorter
Co–Co distance along the rod (5Å) compared to the interchain distance (6Å).
The effects of magnetic coupling were further examined with ab initio peri-
odic Density Functional Theory using a noncollinear spin model. It is important
to note that the inclusion of spin-orbit interactions and a careful calibration of the
initial spin directions in the self-consistent field procedure is essential to build a
realistic picture of the ground and excited states. Crucially, the real spin distribu-
tion across the unit cell and atomic relaxations are included into the calculation
of the magnetic coupling constants.
The procedure was based on a delta self consistent field (SCF) method, in
which the strength of a particular interaction was estimated from energy dif-
ferences between different states of the system of interest. In the current case,
we first established that the experimentally observed one-dimensional Ising anti-
ferromagnetic arrangement along the principal diagonals of the unit cell is the
ground state of CaY
2
Co
2
Ge
3
O
12
(+-+-). We then introduced local perturbations
into the system, which would correspond to highly localized spin waves, in the
24
form of (i) one ferromagnetically ordered chain per unit cell (++++), and (ii)
a ferromagnetically ordered chain with one “defect” (+++-). The SCF itera-
tions in each case preserved the initial trial spin directions, but led to notice-
able spin density readjustment followed by atomic relaxations thus reducing the
energy penalty of the spin flip. Mapping the difference in total energies of these
three configurations onto the 1-D Ising Hamiltonian description of
3
2
~ spins, the
nearest neighbor J
1
coupling constant was calculated to be 23 K and the next-
nearest-neighbor along the chainJ
2
coupling constant as 9 K; with an estimated
error of less than 1 K. As each cobalt ion can interact with two nearest neighbors
through the available superexchange pathways, and two next-nearest neighbors
along the length of the rod, the calculated coupling constants unambiguously
support the experimentally determined ground state magnetic order. The rel-
atively high value of J
2
(∼
1
3
J
1
) also implies the ease of generating magnetic
excitations that on heating will destroy the ground state ordering.
Having thoroughly evaluated the ground state, isothermal magnetization
curves were collected at a variety of temperatures between 10 K and 2 K to assess
the stability of the magnetic order against external perturbations (Figure 2.3
(b)). For small applied fields, the usual linear response expected from a well-
ordered antiferromagnet can be clearly seen at all temperatures. Starting at 4 K,
however, a distinct field-dependent magnetic transition begins to evolve around
2.5 T, which moves to fields of 4 T and 5 T as the temperature is decreased fur-
ther. Field-dependent metamagnetic transitions such as this have been observed
in several other Co-based compounds, [74, 75, 76] and is typically attributed
to the magnetic field strength becoming strong enough to overcome the single-
ion anisotropy of the moments and drive the order towards a ferromagnetic
state. [74, 75]
25
The complex field- and temperature-dependence of the magnetism encour-
aged us to explicitly map out these transitions using specific heat measurements
as shown in Figure 2.4 (a). In the absence of a magnetic field, a sharp lambda-
like anomaly evolves at 6 K, in very good agreement with the temperature-
dependent magnetic susceptibility data. As lambda anomalies are not expected
for purely one dimensional materials due to the lack of long-range magnetic
order, [55] the system clearly undergoes a three-dimensional transition despite
the reduced dimensionality of the magnetic structure and in agreement with the
magnetic structure seen from neutron diffraction.
As increasing field strengths are applied, the peak in the heat capacity is con-
tinuously suppressed until a field of 6 T past which point the peak reverses trend
and begins increasing in temperature. In these large fields, the features in the
specific heat begin to resemble the broad peaks typically associated with either a
Schottky anomaly or the free-fermion model (Figure 2.4 (e)) that has been used
to describe the magnetically disordered state of CoNb
2
O
6
,[77, 78, 79] suggesting
the system transitions into a field polarized paramagnetic state. Between 6 T and
10 T, however, the specific heat cannot be described well using either model. Cor-
relating the specific heat with the sharp uptick in the magnetization curves sug-
gests that material is in a transitionary state from the antiferromagnetic ground
state into a new phase with a larger concentration of uncompensated moments.
26
Figure 2.5: (Color online) Magnetic phase diagram of the garnet CaY
2
Co
2
Ge
3
O
12
suggesting the presence of a quantum critical point when exposed to fields
around 6 T. The open black circles were extracted as the peak position of the
lambda-anomaly while the open red diamonds represent the energy of the mag-
netic excitations in the disordered phase, Δ/2. AFM = antiferromagnet, PM =
thermally disordered paramagnet, QPM = field polarized quantum paramagnet.
Attempts to fit the data using the free-fermion model of Pfeuty [79] were
performed by starting with the coupling constants determined from the DFT
calculations. It was found that fixing a value of J=23 K failed to produce a
27
satisfactory fit; however, good agreement was obtained with a value of 27 K as
illustrated in Figure 2.4 (e). This discrepancy may be attributed to the artificial
cation order that is necessary in the computational cell, but still represents a
good first approximation. Within the free fermion-model, there are two impor-
tant parameters: the exchange coupling,J, and Γ, a parameter used to estimate
the extent of the disorder along the chain, with a table of fitted values given in
SI Table III. [79] In our fits, J was held constant while Γ was optimized using
a least-squares fitting algorithm. Importantly, all of our fitted values of Γ are
greater than J/2, indicative of a fully disordered chain of spins, confirming the
paramagnetic state above 10 T.
Combining the specific heat and magnetization data, the field-dependence of
the transition temperature can be used to map out a first approximation for the
magnetic phase diagram as illustrated in Figure 2.5. We note that, considering
this data comes from polycrystalline samples, the effects of anisotropic superex-
change pathways may be under represented by these field lines, and represents
an average of the effects from the magnetic field along all directions of the unit
cell. Regardless, the transition to the antiferromagnetic ground state shows a
continuous suppression of T
N
that can be fit with a 1-aH
2
c
dependence that
extrapolates to the complete suppression of antiferromagnetism to 0K around
7.5 T. At high fields, the phase boundary is drawn using the energy gap, Δ/2,
associated with the magnetic excitations in the disordered phase as obtained
from the fits to the free-fermion solution discussed earlier. The dashed red line
represents a linear extrapolation of the gap to zero kelvin.
The phase diagram constructed for CaY
2
Co
2
Ge
3
O
12
strongly suggests that
the material undergoes a quantum critical phase transition with a critical field
between 6 T and 7.5 T. If this high field transition is indeed a critical point,
28
the green region between the AFM and disordered phases in Figure 2.5 would
correspond to a regime where quantum fluctuations dominate the behavior of
the system, which would explain the difficulty in fitting the specific heat data
collected in this region. Given the combination of one-dimensional chains of
Ising moments, this material is reminiscent of the transverse field Ising model,
which is normally associated with the columbite CoNb
2
O
6
or LiHoF
4
. Yet, unlike
CoNb
2
O
6
, the chains within the unit cell of CaY
2
Co
2
Ge
3
O
12
are not parallel, but
instead cross each other so that one of the chains always be subjected to some
degree of a longitudinal field. Dmitriev and coworkers have described such a
case in a recent report [80] and find, despite the introduction of a longitudinal
field component, that Ising chains should still exhibit a critical point similar to
what we see here, supporting the one-dimensional Ising character of the garnet.
2.3 Conclusions
In summary, we have investigated the observation of quasi-one-dimensional
magnetism in the garnet CaY
2
Co
2
Ge
3
O
12
, and found a complex antiferromag-
netic ground state that is strongly influenced by the presence of external mag-
netic fields. Our results suggest that despite a 3D nuclear structure, the single-ion
anisotropy of Co
2+
is sufficient to dominate the magnetism and cause the system
to behave like a chain of Ising moments, resulting in a quantum phase transition
with the application of moderate fields. Many questions remain regarding the
nature of the magnetic excitations in the quantum critical region of the phase
diagram as the effect of field orientation with respect to the individual chains.
The growth of single crystal samples and neutron spectroscopy will be critical
29
for addressing many of these questions and furthering our understanding of the
criticality in this novel cobalt-based garnet.
2.4 Supplemental Information
2.4.1 Table of fitted parameters from the Nuclear and Mag-
netic Refinements
Neutron powder diffraction data were collected using the BT-1 high resolu-
tion neutron powder diffractometer at the NIST Center for Neutron Research
using a Ge(311) monochromator with a 75
◦
take-off angle,λ = 2.0787(2)Å and
in-pile collimation of 60 minutes of arc. Data were collected over the range of
1.3
◦
-166.3
◦
in scattering angle (2-Theta) with a step size of 0.05
◦
. The instru-
ment is described at http://www.ncnr.nist.gov/
Parameter 300 K 25 K 2 K
a 12.354(4) 12.341(3) 12.341(5)
O position
-0.334(7) -0.0326(7) -0.0325(0)
0.056(5) 0.0567(6) -0.0569(8)
0.157(6) 0.1573(7) -0.1565(3)
R
Bragg
9.7% 3.1% 5.3%
R
mag
– – 8.7%
Table 2.1: Results of the Rietveld refinement of CaY
2
Co
2
Ge
3
O
12
against the pow-
der neutron diffraction data. Note that Ca and Y sit at (0, 0.25, 0.125), Co at (0,
0, 0), and Ge at (0, 0.25, 0.375) and are fixed at all temperatures.
2.4.2 Table of fitted parameters from the Free Fermion Model
Powders of the title phase were mixed with equal amounts of silver to increase
the thermal coupling to the stage, with the contribution from silver being mea-
sured separately and subtracted. The magnetic contribution to the specific heat
30
Atom coordinates
~
ψ
1
1
Atom coordinates
~
ψ
1
1
Co1 (0, 0, 0) (1 1 1) Co5 (
3
4
,
1
4
,
1
4
) (1 1 -1)
Co2 (
1
2
, 0,
1
2
) (-1 -1 1) Co6 (
3
4
,
3
4
,
3
4
) (-1 -1 -1)
Co3 (0,
1
2
,
1
2
) (-1 1 -1) Co7 (
1
4
,
1
4
,
3
4
) (1 -1 1)
Co4 (
1
2
,
1
2
, 0) (1 -1 -1) Co8 (
1
4
,
3
4
,
1
4
) (-1 1 1)
Table 2.2: The basis functions
~
ψ
λ
1
for each Co ion site in CaY
2
Co
2
Ge
3
O
12
. All
basis functions are real and correspond to the irreducible representation that fit
the magnetic reflections best.
was was estimated by using CaY
2
Mg
2
Ge
3
O
12
to estimate the lattice contribution
and subtracting after properly rescaling. Fits to the magnetic specific heat data
were performed using the free-fermion model,by fixing a value of J=27 K and
performing a least-squares fit to determine Γ.
Field (T) J
fixed
(K) Γ
fitted
(K)
10 27 14.3
11 27 23.2
12 27 25.8
13 27 28.5
14 27 31.2
Table 2.3: Results of Fitting Field Dependent Specific Heat Data to the Free-
Fermion Model
2.4.3 Single Ion Calculations and Estimate of theg-Tensor
Single ion calculations were carried out for Co
2+
in the trigonally distorted
octahedral environment relevant to CaY
2
Co
2
Ge
3
O
12
(point group S
6
). The
ground state single-ion behavior of Co
2+
in an undistorted octahedral environ-
ment is that of a fully isotropic j
eff
= 1/2 moment, due to the combination of
the cubic crystal electric field (CEF) and the spin orbit coupling (SOC), which
produce a Kramer’s doublet as the ground state [66, 65]. Any distortion away
from perfect O
h
symmetry introduces anisotropy into this Kramer’s doublet; i.e.,
31
it produces an anisotropic g-tensor. To estimate the g-tensor for Co
2+
ions in
CaY
2
Co
2
Ge
3
O
12
, the single ion Hamiltonian (H = H
SOC
+H
CEF
) was diagonal-
ized within the 28× 28 manifold of states formed by the|L
z
,S
z
i basis of the
S = 3/2 andL = 3 free ion term,
4
F.
The CEF Hamiltonian was estimated based on point charge calculations, fol-
lowing Hutchings [68]. The CEF Hamiltonian can be written in general as:
H
CEF
=
X
l,m
B
l,m
ˆ
O
l,m
where
ˆ
O
l,m
are the Stevens operator equivalents. For transition metal ions,
these operators are written in terms of the orbital angular momentum matrix
operators
ˆ
L
+
,
ˆ
L
−
and
ˆ
L
z
(as opposed to the total angular momentum opera-
tors
ˆ
J
+
,
ˆ
J
−
and
ˆ
J
z
relevant for f-electron systems) [68]. For the point group
symmetry relevant to CaY
2
Co
2
Ge
3
O
12
, (l,m) = (2, 0), (4, 0) and (4, 3) are the
only nonzero terms. The B
l,m
parameters were calculated using a point charge
model, assuming charge -2 on the oxygen sites and +2 on the cobalt site. The
oxygen positions were obtained from the structural refinement detailed in Table
2.1, with the z axis chosen to lie along the 3-fold symmetric axis of the point
group, which is a local [111] crystallographic axis for each ion. y is chosen to
lie along the projection of the [010] axis in the plane perpendicular to [111], a
choice that ensures real-valued eigenvectors and does not require the use of the
‘sine’ Steven’s operators that are needed when B
l,m
is nonzero for negative m.
These can always be made to be zero using appropriate choices of the coordinate
axes [81, 68]. With these conventions, the point charge calculation gives the
followingB
l,m
values: B
2,0
=−8.62,B
4,0
= 0.819,B
4,3
= 20.7 meV.
32
The spin orbit coupling term is given by,
H
SOC
=λ
~
S·
~
L =λ(
ˆ
S
x
ˆ
L
x
+
ˆ
S
y
ˆ
L
y
+
ˆ
S
z
ˆ
L
z
),
where for cobalt, λ =−22.32 meV [82]. The x and y spin operators are linear
combinations of
ˆ
S
+
and
ˆ
S
−
as usual:
ˆ
S
x
=
1
2
(
ˆ
S
+
+
ˆ
S
−
) and
ˆ
S
y
=
1
2i
(
ˆ
S
+
−
ˆ
S
−
) and
similarly for
ˆ
L
x
and
ˆ
L
y
.
The eigenenergies of the full single-ion Hamiltonian (H
CEF
+ H
SOC
) are
listed in Table 2.4, relative to the ground state energy. Each level retains a
double degeneracy due to Kramer’s theorem. The ground doublet is separated
from the next excited doublet by 35.6 meV; this means that the ground state
doublet is expected to dominate the single-ion properties at temperatures well
below 406 K, including the ordering temperature (T
N
= 6 K).
The amplitudes of the|L
z
,S
z
i basis states in the ground doublet eigenvectors
are shown schematically in Figure 2.6. The components of theg-tensor along the
principal axes are given by,
g
xy
=−2hψ
1
|(
ˆ
L
x
+ 2
ˆ
S
x
)|ψ
2
i = 3.97,
and
g
z
= 2hψ
2
|(
ˆ
L
z
+ 2
ˆ
S
z
)|ψ
2
i = 5.66
These values give a single-ion anisotropy ratio of g
z
/g
xy
= 1.4, i.e., the
moments are expected to be predominantly Ising-like, leading to Ising-like effec-
tive exchange in an j
eff
= 1/2 model that could be built from the ground state
Kramer’s doublet (for instance, see Ref. [67]).
33
State labels Energy (meV)
ψ
1
,ψ
2
0.0
ψ
3
,ψ
4
35.6
ψ
5
,ψ
6
42.2
ψ
7
,ψ
8
101.3
ψ
9
,ψ
10
106.3
ψ
11
,ψ
12
114.7
ψ
13
,ψ
14
508.3
ψ
15
,ψ
16
514.7
ψ
17
,ψ
18
671.1
ψ
19
,ψ
20
681.1
ψ
21
,ψ
22
691.1
ψ
23
,ψ
24
703.1
ψ
25
,ψ
26
1320.2
ψ
27
,ψ
28
1322.6
Table 2.4: Calculated energies of single-ion levels of Co
2+
in CaY
2
Co
2
Ge
3
O
12
.
|-3,-3/2⟩
|-3,-1/2⟩
|-3,1/2⟩
|-3,3/2⟩
|-2,-3/2⟩
|-2,-1/2⟩
|-2,1/2⟩
|-2,3/2⟩
|-1,-3/2⟩
|-1,-1/2⟩
|-1,1/2⟩
|-1,3/2⟩
|0,-3/2⟩
|0,-1/2⟩
|0,1/2⟩
|0,3/2⟩
|1,-3/2⟩
|1,-1/2⟩
|1,1/2⟩
|1,3/2⟩
|2,-3/2⟩
|2,-1/2⟩
|2,1/2⟩
|2,3/2⟩
|3,-3/2⟩
|3,-1/2⟩
|3,1/2⟩
|3,3/2⟩
-1
-0.5
0
0.5
1
wavefunction amplitude
ψ
1
ψ
2
Figure 2.6: Decomposition of the single-ion ground doublet wavefunctions into
the|L
z
,S
z
i basis
34
Chapter 3
Magnetodielectric Effects at
Quantum Critical Fields in
Cobalt-Containing Garnets
We recently reported the observation of a quasi-one-dimensional magnetic
order in the garnet CaY
2
Co
2
Ge
3
O
12
, where the Co
2+
ions on the octahedral site
adopt an antiferromagnetic ground state with the moments fixed along the body
diagonals of the unit cell. [83] This highly anisotropic orientation of the spins
forms discrete antiferromangetic rods that were found to undergo a magnetic-
field driven quantum critical phase transition above fields of 6 T. This motivated
us to look for analogous materials that could be used to discriminate whether
the one-dimensional magnetic structure was necessary to realize quantum criti-
cality, and ultimately led us to another Co-containing garnet, NaCa
2
Co
2
V
3
O
12
. In
both the germanate and vanadate garnets, the lattice, magnetic cation, and mag-
netic topologies are effectively identical, providing an ideal comparison where
only the diamagnetic portions of the host structure have been varied. Herein,
we report that NaCa
2
Co
2
V
3
O
12
shows similar signature of quantum criticality
at substantially higher fields, despite exhibiting a radically different magnetic
structure. We also find signatures of these quantum phase transitions appear in
the field-dependent magnetocapacitance data, suggesting a correlation between
suppression of the magnetic order and the dielectric properties of the materials.
35
3.1 Methods
Certain commercial equipment, instruments, or materials are identified in
this document. Such identification does not imply recommendation or endorse-
ment by the National Institute of Standards and Technology nor does it imply
that the products identified are necessarily the best available for the purpose.
3.1.1 Materials Preparation
Polycrystalline powders of NaCa
2
Co
2
V
3
O
12
were prepared by grinding stoi-
chiometric ratios of CaCO
3
, Na
2
CO
3
, Co(C
2
O
4
)·2H
2
O and V
2
O
5
and pressing into
pellets before firing in air through a multistage heat treatment. All pellets were
isolated from the ZrO
2
crucible using a sacrifical layer of powder with the same
stoichiometry as the target phase. Co(C
2
O
4
)·2H
2
O was freshly prepared in house
by precipitating a solution of Co(SO
4
)·7H
2
O with an excess of oxalic acid and
drying at room temperature overnight. The first calcination was performed at
500
◦
C for 24 hours, at which point the pellets were ground and mixed well
before re-pressing into pellets and heating at 850
◦
C in 24 hour increments until
phase pure, usually requiring one to two treatments. For final densification,
powders were loaded into a 9 mm carbon die and spark plasma sintered (SPS) at
600
◦
C for 5 minutes until 9 kN of pressure was achieved to produce pellets with
densities greater than 85%.
3.1.2 Neutron Diffraction
Polycrystalline powders were loaded into vanadium cans and placed in a
helium flow cryostat. Data sets were collected at 300 K, 50 K and 3 K using the BT-
1 high resolution neutron powder diffractometer at the NIST Center for Neutron
36
Research. A Ge(311) monochromator with a 75
◦
take-off angle,λ = 2.0787(2)Å
and in-pile collimation of 60 minutes of arc were used. Data were collected over
the range of 1.3
◦
-166.3
◦
in scattering angle (2Θ) with a step size of 0.05
◦
.
3.1.3 Physical Property Measurements
Temperature and field dependent magnetic susceptibility as well as specific
heat measurements were collected using a Quantum Design 14T Dynacool Phys-
ical Property Measurement System. For the specific heat measurements pow-
ders of NaCa
2
Co
2
V
3
O
12
were ground together with equal parts silver in order to
increase thermal coupling to the sample stage. The contribution of the silver was
measured separately and subtracted. [84]
Magnetocapacitance measurements were performed on pellets that were den-
sified using SPS with calculated densities between 86-98%. Dense pellets of
the compound of interest were painted with Ag-epoxy (Epotek EE129-4) and
an epoxy-coated copper wire were attached to act as a parallel-plate capacitor.
The edges of the pellet were then sanded to ensure no short circuits are created
between the sides of the electrode. Shielded stainless steel co-axial cables were
affixed to the electrode face and to the top of the custom-built measurement
probe. The capacitance was measured on a high precision capacitance bridge
Andeen-Hagerling 2700A at a frequency of 1 kHz. A Quantum Design 14 T Dyna-
cool Physical Property Measurement System (PPMS) was used for control of the
magnetic field and temperature. In order to rule out any effects due to magne-
toresistance or leakage current through the pellet, AC Impedance measurements
were performed over an array of frequencies at various fields. [85] As seen in SI
Figure 1, neither material shows any evidence for field-dependent changes in the
37
impedance, which is in agreement with the extremely low dielectric loss (tanδ≈
0.000174-0.000589) obtained for each pellet.
0 50 100 150 200 250 300
T (K)
0.00
0.05
0.10
0.15
0.20
𝜒 (emu mol
-1
Oe
-1
)
0.05 T zfc
0.05 T fc
-3
-2
-1
0
1
2
3
M ( μ
B
Co
-1
)
11 K
10 K
9 K
8 K
7 K
6 K
5 K
4 K
3 K
2 K
-12 -8 -4 0 4 8 12
H (T)
dM/dH
0 10 20
0.12
0.16
0.20
(a)
(b)
Figure 3.1: Magnetic Susceptibility: (a.) Temperature dependent suspectibility
of NaCa
2
Co
2
V
3
O
12
with an antiferromagnetic transition around 6K. (b.) Isother-
mal magnetization taken below and above the ordering temperature with the
respected derivatives below.
38
3.2 Results and Discussion
Figure 5.3 (a) shows the temperature-dependent magnetic susceptibility of
NaCa
2
Co
2
V
3
O
12
, which indicates a transition to an antiferromagnetic ground
state around 8 K. Fitting the high-temperature region (200-300 K) of the sus-
ceptibility to the Curie-Weiss equation and including a temperature independent
paramagnetic contribution, χ=C/(T-Θ
CW
)+χ
0
, yields a Θ
CW
of -44 K, an effec-
tive paramagnetic moment of 7.33μ
B
per formula unit (5.18μ
B
per Co) and
χ
0
=6.6×10
−4
emu mol
−1
Oe
−1
. This moment is in close agreement with the
expected value for Co
2+
in a high-spin octahedral coordination environment (d
7
,
S=3/2,L=3), when the orbital moment is unquenched and decoupled from that
of the spin (μ
L+S
=
q
4S(S + 1) +L(L + 1)).[86] The negative sign of Θ
CW
indi-
cates that antiferromagnetic exchange is dominant between the spins and there
is only a modest suppression of the ordering temperature (Θ/T
N
= 5.5).
The suppressed ordering temperature can be understood by considering that
garnets consist of a network of BO
6
octahedra bound together at their corners
by AO
4
tetrahedra and RO
8
dodecahedra on the edges. As O’Keefe first high-
lighted, and we later demonstrated in CaY
2
Co
2
Ge
3
O
12
, the octahedral sublattice
forms rods, as illustrated in Figure 3.2 (c), which run along the body diagonals
of the unit cell.[37] Given there is no direct connectivity between adjacent octa-
hedral sites, more complex superexchange pathways must mediate the magnetic
interactions either through the tetrahedral/cubic sites or along the edges of the
polyhedra through super-superexchange pathways exclusively involving oxygen
[see Figures 3.2 (a) and (b)]. The presence of so many competing exchange
pathways that must all be simultaneously satisfied in the ground state magnetic
structure is likely the cause of the suppressed ordering temperature.
39
(a)
(c)
(b)
Figure 3.2: Garnet Sublattice Connectivity: (a) Octahedral-Cubic site connectiv-
ity. (b.) Octahedral-tetrahedral site connectivity. (c) Connectivity of a rod within
the garnet structure. The rods consist of the octahedral site surrounded by the
cubic site through the sharing of oxygen bonds and run along the body diagonal
of the unit cell.
Below the ordering temperature of 6 K, the isothermal magnetization of
NaCa
2
Co
2
V
3
O
12
shows a deviation from the linear response expected for an anit-
ferromagnetically ordered system of spins at fields beginning around 10 T [see
Figure 5.3(b)]. A more faint deviation around 1 T can be noticed in the deriva-
tive curve of Figure. Field-induced magnetic transitions are fairly common in
40
divalent Co compounds because of its substantial anisotropic character. Mea-
surements were taken between 2 K and 11 K, to track the field-induced transition
passed the ordering temperature, with the curves flattening out toward the tra-
ditional linear response for a paramagnetic state by 11K.
0 10 20 30 40 50
T (K)
0.0
0.3
0.6
0.9
1.2
1.5
C
mag
/T
(J mol
-1
K
-2
)
0 T
1 T
3 T
5 T
7 T
9 T
11 T
13 T
0 10 20 30 40 50
T (K)
0.0
0.2
0.4
0.6
0.8
1.0
ΔS
mag
/ R
ln(2)
(a)
(b)
Figure 3.3: Specific heat results. (a.) Expected Lambda anomalies correspond-
ing to the three-dimensional ordering of NaCa
2
Co
2
V
3
O
12
with a broad anaomly
forming at fields of 11T and above. (b.) Entropy released from magnetic transi-
tion.
41
The field-dependence of the magnetization of NaCa
2
Co
2
V
3
O
12
lead us to map
out these transitions using specific heat measurements, shown in Figure 3.3 (a).
In the absence of a magnetic field, a sharp lambda anomaly is seen around
6 K, which agrees with the onset of three-dimensional order in the temperature-
dependent susceptibility. At small fields, between 0 T and 1 T, no significant
change in the peak position is noticeable, but in fields starting at 9 T the sharp
anomaly is significantly blunted and smears out to resemble a broad feature that
alludes to a disordered state.
Combining the magnetization and specific heat measurements, a magnetic
phase diagram was constructed from the observed transition temperatures as
illustrated in Figure 3.4. Fitting a quadratic response (1−aH
2
c
) to the antiferro-
magnetic transition temperature suggests a critical magnetic field near 11 T very
similar to what was fond in CaY
2
Co
2
Ge
3
O
12
albeit at much stronger fields. The
broad anomalies in the specific heat measurements are reminiscent of Schottky-
like anomalies where the system has transitioned into a field-polarized paramag-
netic state.
Low temperature neutron scattering was collected on the high resolution
powder diffractometer, BT-1, at the NIST Center for Neutron Research for
NaCa
2
Co
2
V
3
O
12
in order to further elucidate its magnetic nature. Previous neu-
tron diffraction studies on NaCa
2
Co
2
V
3
O
12
reported that the magnetic structure
consists of ferromagnetic interactions within the layers that couple antiferro-
magnetically to each other. [87] At 300 K, the nuclear structure was refined
using the FullProf suite of programa nd found to agree well with the cubicIa
¯
3d
structure with Co
2+
octahedrally coordinated on the 16a (octahedral) site. The
most reasonable fit of the 3 K diffraction data, shown in Figure 3.5 (a), could
only be obtained using the seventh representation, Γ
7
and corresponds to the
42
0 2 4 6 8 10 12 14
μ
0
H (T)
2
3
4
5
6
7
8
T
N
(K)
PM
QPM
Néel
AFM
Figure 3.4: Magnetic Phase Diagram of NaCa
2
Co
2
V
3
O
12
: Magnetic Phase dia-
gram was constructed by plotting the transitions in 3.3 (a). A quantum critical
point was determined to be around 11T in the heat capacity and magnetization
measurements.
space group Ia
¯
3d. Representational analysis was used to fit the 3 K data with
associated magnetic reflections at: 19.2
◦
, 33.6
◦
, 60.2
◦
and 67.4
◦
. All observed
magnetic reflections were indexed using a simple wave vector ofk = 0. SARAh
returned two one-dimensional, one two-dimensional and two three-dimensional
43
representations within the Little Group G
k
. The resulting topology of this rep-
resentation returns a result very similar to what was previously reported, but
indicates that the magnetic moments do not point in a perfectly parallel fashion
and instead in two different directions as seen in Figure 3.5 (b) and (c) as a
result of the single ion anisotropy of the cobalt.
Composition NaCa
2
Co
2
V
3
O
12
CaY
2
Co
2
Ge
3
O
12
Magnetic Ordering Temperature ∼8K ∼6K
Lattice Parameters (Å) 12.430(3) 12.354(4)
Nearest Neighbor Along Chains (Å) 2.97 2.76
Nearest Neighbor Between Chains (Å)
2.87 2.97
2.68 2.68
Octahedral Angle (deg)
90.17 93.8
89.83 86.2
Octahedral Distortion 0.19% 4.2%
Table 3.1: Crystallographic data for the two Co garnets discussed in the text.
All lattice parameters, atomic distances, polyhedra angles were determined from
refinements against the neutron diffraction data taken at 50 K. Distortions in
polyhedra were found by calculating the difference between a refined angle and
an perfectly symmetric octahedral coordination geometry.
Despite the similar ordering temperatures, it is interesting to note that
CaY
2
Co
2
Ge
3
O
12
and NaCa
2
Co
2
V
3
O
12
adopt very distinct magnetic structures
where CaY
2
Co
2
Ge
3
O
12
contains chains of antiparallel spins along the body diag-
onals whereas NaCa
2
Co
2
V
3
O
12
adopts a more two dimensional orientation with
ferromagnetic layers that are coupled antiferromagnetically. [83] Clearly the sin-
gle ion anisotropy of the moments in the NaCa
2
Co
2
V
3
O
12
is significantly differ-
ent from CaY
2
Co
2
Ge
3
O
12
areas the spins prefer to lay canted adding a three-
dimensionality which breaks the Ising-like character of the structure along the
rods, shown in Figure 3.5 (c).
44
2 θ (deg) [ λ=2.079 Å]
0
5000
10000
15000
20000
25000
30000
35000
counts
(b) (c)
(a)
Figure 3.5: (a) Results of the Rietveld refinement of the nuclear and magnetic
contributions to against the 3 K powder neutron diffraction data, R
nuc
Bragg
=1.5%,
R
mag
=12.6%. (b) Illustration of the resulting magnetic structure viewed off one
of the edges of the cubic unit cell edges and (c) down the unit cell’s diagonal,
along the length of the rod. In both panels the two colors coded in lavender and
teal.
45
The diamagnetic substitutions within the garnet create larger atomic dis-
tances in NaCa
2
Co
2
V
3
O
12
as well as a change in polyhedral angle distortions
listed in Table 3.1. By substituting Na
+
for Y
3+
on the cubic sublattice in
NaCa
2
Co
2
V
3
O
12
, the sublattice’s average radii is expanded, resulting in a 8%
increase in the through-space distance between neighboring Co
2+
ions com-
pared to the germanate. Conversely, the tetrahedral sublattice’s average radii in
NaCa
2
Co
2
V
3
O
12
contracts by 9% with the substitution from Ge
4+
to V
5+
. More
critically, the octahedra in the vanadate are significantly less distorted than the
germanate, with less than 0.2% deviation from 90
◦
O–Co–O bond angles Figure
3.6(b). This change in the local coordination environment likely drives the reori-
entation of the easy axis as the octahedra are no longer compressed along the
direction of the rods.
To better understand the impact that these changes in the crystal chemistry
have on the electronic structure of the material, density functional theory calcu-
lations were performed. Figure 3.7 (a) and (b) shows the calculated densities of
state for CaY
2
Co
2
Ge
3
O
12
and NaCa
2
Co
2
V
3
O
12
respectively using the experimen-
tally determined non-collinear magnetic structure In both materials the top of the
valence band consist predominantly of Cod mixed with Op states. Notably, the
valence band of CaY
2
Co
2
Ge
3
O
12
is somewhat broader than the NaCa
2
Co
2
V
3
O
12
due to a lobe of Ge states laying slightly lower in energy than those of V. It can
also be seen that, in contrast to the germanate, the unoccupied manifold of d
states in the vanadate dominate the bottom of the conduction band resulting in
substantial rehybridization, which may explain the stronger coupling between
the spins as reflected by the larger Θ
CW
(-44 K vs -32 K).
46
(a)
(b)
2.76 Å
2.97 Å
89.6°
90.4°
93.9°
86.1°
Figure 3.6: Angles and distances of nearest neighbors in CaY
2
Co
2
Ge
3
O
12
(a) and
NaCa
2
Co
2
V
3
O
12
(b) garnets along the rods. As the distance between the Co
2+
decrease the deviations from 90
◦
increase, creating a more distorted octahedra
in CaY
2
Co
2
Ge
3
O
12
, than in NaCa
2
Co
2
V
3
O
12
.
Figure 3.8 shows the magnetocapacitance of both Co-based garnets exhibits a
strong dependence on the applied magnetic field. Given the absence of any struc-
tural distortions in the low temperature neutron diffraction, the origin of this
47
Arb units -10 -8 -6 -4 -2 0 2 4 6 8
Energy (eV)
Arb. units
Total DoS
Co
V
O
(a)
(b)
Total DoS
Co
Ge
O
Figure 3.7: Calculated Densities of State for (a) CaY
2
Co
2
Ge
3
O
12
and (b)
NaCa
2
Co
2
V
3
O
12
.
dielectric coupling cannot be attributed to a proper ferroelectric coupling due to
the cubic symmetry of the unit cell. For small applied fields both NaCa
2
Co
2
V
3
O
12
and CaY
2
Co
2
Ge
3
O
12
magnetocapacitance exhibits an approximately quadratic
dependence on the magnetic field following the onset of antiferromagnetic order.
48
Comparing the square of magnetization with the magnetocapacitance measure-
ments and agrees well for the feature between 4 T in CaY
2
Co
2
Ge
3
O
12
, which
suggests a there is coupling of the form P
2
M
2
that is allowed for all materi-
als regardless of space group symmetry. [88] This type of coupling is typically
attributed to spin-phonon coupling and is reminiscent of what has previously
been observed in CoTiO
3
, [89] NiCr
2
O
4
,[90] and SeCuO
3
.[35] Interestingly, at
both
Interestingly, significant deviations from the quadratic dependence are found
in both compositions at fields correlating with the suppression of long-range
magnetic order at the quantum critical points, 7 T and 11 T for the ger-
manate and vanadate respectively. The field-dependence of the capacitance for
NaCa
2
Co
2
V
3
O
12
’s becomes less pronounced on heating above the ordering tem-
perature of 6 K as do the features in CaY
2
Co
2
Ge
3
O
12
. Well above the magnetic
ordering temperatures the parabolic response flattens entirely, resulting in very
little field-dependence. Interestingly CaY
2
Co
2
Ge
3
O
12
’s features appear to linger
above the ordering temperature to about 20 K (SEE SI), which was mentioned in
SeCuO
3
, [35] (see SI).
3.3 Conclusions
The magnetic behavior of NaCa
2
Co
2
V
3
O
12
has been studied from suscepti-
bility measurements, and the magnetic structures have been determined using
powder neutron diffraction data. We find that altering the composition of the
cubic and tetrahedral sites in the garnet structure results in substantial changes
to the local octahedral coordination environment in NaCa
2
Co
2
V
3
O
12
compared
to CaY
2
Co
2
Ge
3
O
12
and results in a reorientation of the easy axis for the magnetic
49
-12 -8 -4 0 4 8 12
0.00
0.02
0.04
0.06
0.08
0.10
dM
/dH
-12 -8 -4 0 4 8 12
-0.010
-0.005
0.000
0.005
0.010
Δ %C
-12 -8 -4 0 4 8 12
H (T)
0.00
0.10
0.20
0.30
dM
/dH
-12 -8 -4 0 4 8 12
H (T)
0.000
0.005
0.010
0.015
Δ %C
Figure 3.8: Field Dependent Capacitance Measurements overlayed onto Magneti-
zation’s derivative taken at 2 K: (a.) CaY
2
Co
2
Ge
3
O
12
’s capacitance measurements
(wine) show features at the critical fields involved in a quantum phase transition.
These fields are seen in the susceptibility’s derivative (blue) between 6 and 7T.
(b.) NaCa
2
Co
2
V
3
O
12
’s capacitance also shows features at the fields in which there
are field induced transitions. These features are below 1T, and around 11T.
moment. In CaY
2
Co
2
Ge
3
O
12
the rods were previously found to be axis in which
the spins align, however in NaCa
2
Co
2
V
3
O
12
the axis of their alignment changes to
align along the b-axis of the unit cell. Both NaCa
2
Co
2
V
3
O
12
and CaY
2
Co
2
Ge
3
O
12
were found to exhibit Quantum Critical Phase transitions that correlate strongly
with features found in the magnetocapacitance.
50
3.4 Supplemental Information
3.4.1 Table of Fitted Parameters from the Nuclear and Mag-
netic Refinements
Neutron powder diffraction data were collected using the BT-1 high resolu-
tion neutron powder diffractometer at the NIST Center for Neutron Research
using a Ge(311) monochromator with a 75
◦
take-off angle,λ = 2.0775(0)Å and
in-pile collimation of 60 minutes of arc. Data were collected over the range of
1.3
◦
-166.3
◦
in scattering angle (2-Theta) with a step size of 0.05
◦
. The instru-
ment is described at http://www.ncnr.nist.gov/
Parameter 300 K 50 K 3 K
a 12.428(8) 12.408(2) 12.410(5)
O position
0.0327(8) 0.0380(9) 0.0382(2)
0.0710(3) 0.0524(4) 0.0525(2)
0.6595(4) 0.6548(1) 0.6250(3)
R
Bragg
3.1% 2.4% 3.6%
R
mag
– – 27%
Table 3.2: Results of the Rietveld refinement of NaCa
2
Co
2
V
3
O
12
against the pow-
der neutron diffraction data. Note that Na and Ca sit at (0, 0.25, 0.125), Co at
(0, 0, 0), and V at (0, 0.25, 0.375) and are fixed at all temperatures.
Atom coordinates
~
ψ
1
1
Atom coordinates
~
ψ
1
1
Co1 (0, 0, 0) (1 1 1) Co5 (
3
4
,
1
4
,
1
4
) (-1 -1 1)
Co2 (
1
2
, 0,
1
2
) (-1 -1 1) Co6 (
3
4
,
3
4
,
3
4
) (1 1 1)
Co3 (0,
1
2
,
1
2
) (-1 1 -1) Co7 (
1
4
,
1
4
,
3
4
) (-1 1 -1)
Co4 (
1
2
,
1
2
, 0) (1 -1 -1) Co8 (
1
4
,
3
4
,
1
4
) (1 -1 -1)
Table 3.3: The basis functions
~
ψ
λ
1
for each Co ion site. All basis functions are real
and correspond to the irreducible representation that fit the magnetic reflections
best.
51
3.4.2 Figure of Magnetization-Capacitance Comparison and
A.C Magnetocapcitance
-10
-5
0
5
10
M
2
( μ
B
2
Co
-2
)
-12-8 -4 0 4 8 12
H (T)
-0.010
0.000
0.010
Δ %C
0
2
4
6
8
10
M
2
( μ
B
2
Co
-2
)
-12-8 -4 0 4 8 12
H (T)
0.000
0.005
0.010
0.015
Δ %C
0
10
20
30
C
(pF)
0 T
14 T
0
3
6
9
C
(pF)
0 T
14 T
10
2
10
3
10
4
10
5
10
6
10
7
AC Frequency (Hz)
-100
0
100
tan( δ)
10
2
10
3
10
4
10
5
10
6
10
7
AC Frequency (Hz)
-100
0
100
tan( δ)
(a.)
(b.)
(c.)
(d.)
Figure 3.9: (a) CaY
2
Co
2
Ge
3
O
12
’s magnetocapcitance plotted against the square of
the magnetization. (b) CaY
2
Co
2
Ge
3
O
12
’s isothermal capacitance taken at 2 K as
a function of A.C. frequency at 0 and 14 T. (c) NaCaY
2
Co
2
V
3
O
12
’s magnetocapc-
itance plotted against the square of the magnetization. (b) NaCaY
2
Co
2
V
3
O
12
’s
isothermal capacitance taken at 2 K as a function of A.C. frequency at 0 and 14 T.
In order to ensure that these features were not due to magnetoresistance and
leakage of the materials, AC Impedance measurements were performed over an
array of frequencies at various fields. The AC Impedance measurements in both
52
Co-V and Co-Ge garnets did not show any alterations or features at the frequency
at which the magnetocapcitance measurements were taken. Square of magne-
tization of the two garnets was compared to the magnetocapccitance measure-
ments as well to rule of spurious features due to magnetostructural changes.
53
Chapter 4
Low field Dependence in
Magnetocapcitance of Ca
3
M
2
Ge
3
O
12
Garnets, M= Fe
3+
, Cr
3+
Previous studies investigated the roles of the diamagnetic backbone of octahe-
dral magnetic garnets CaY
2
Co
2
Ge
3
O
12
and NaCa
2
Co
2
V
3
O
12
, in which we report a
correlation between ionic radius size and quantum critical fields. Here we inves-
tigate the role of the type of magnetic cation on the octahedral site with identical
diamagnetic backbones in the garnets Ca
3
Cr
2
Ge
3
O
12
and Ca
3
Fe
2
Ge
3
O
12
.
4.1 Methods
Certain commercial equipment, instruments, or materials are identified in
this document. Such identification does not imply recommendation or endorse-
ment by the National Institute of Standards and Technology nor does it imply
that the products identified are necessarily the best available for the purpose.
54
4.1.1 Materials Preparation
Polycrystalline powders of Ca
3
Fe
2
Ge
3
O
12
and Ca
3
Cr
2
Ge
3
O
12
were prepared
from stoichiometric amounts of CaCO
3
, Fe
2
O
3
or Cr
2
O
3
and Ge
2
O
3
by perform-
ing multiple heating profiles of well-ground mixtures. The initial 12 hour heat-
ing is at 900
◦
C followed by regrinding, pelletizing and heating at temperatures
exceeeding 1100
◦
C until phase pure. Specifically all heatings of Ca
3
Cr
2
Ge
3
O
12
were done under N
2
gas, the final heating was done for 48 hours consecutively
at 1300
◦
C. Whereas Ca
3
Fe
2
Ge
3
O
12
’s heatings were all done in air, the second
heating was for 36 hours at 1175
◦
C.
4.1.2 Spark Plasma Sintering (SPS)
The sample was loaded into a 9mm carbon die and placed into a vaccuum
SPS instrument in the Materials Research Laboratory at University of Califor-
nia, Santa Barabara. The sample was held at 900
◦
C for 5 minutes with 8kN of
pressure in order to ensure density above 85%.
4.1.3 Physical Property Measurements
Temperature and field dependent magnetic susceptibility as well as specific
heat measurements were collected using a Quantum Design 14T Dynacool Phys-
ical Property Measurement System. For the specific heat measurements powders
of Ca
3
Fe
2
Ge
3
O
12
and Ca
3
Cr
2
Ge
3
O
12
were ground together with equal parts silver
in order to increase thermal coupling to the sample stage. The contribution from
silver was measured separately and subtracted [84].
55
4.1.4 Capacitance Measurements
Magnetocapacitance measurements were taken using a homemade measure-
ment probe. An SPS’d pellet is painted with Ag-epoxy with insulated copper
wire attached. The sides of the pellet are sanded to ensure no shorts are cre-
ated between the sides of the electrode. The copper wire is then soldered onto
a Quantum Design waffer with electrodes attacted. Lakshore co-axial cables are
then soldered to the electrodes and to the top of the probe which connects to a
Capacitance bridge. Then the field is swept as capacitance is recorded at various
temperatures.
4.2 Results and Discussion
The magnetic structures of Ca
3
Cr
2
Ge
3
O
12
and Ca
3
Fe
2
Ge
3
O
12
were previously
reported to be dual ferromagnetic lattices opposing one another. [91, 63]
A closer look and a more recent refinement ofCa
3
Fe
2
Ge
3
O
12
suggests that the
magnetic structure may allow for polarization, with an up-up-down-down pat-
tern along the diagonals of the unit cell signature of an E-Type mangetic struc-
ture. [92] Infact, early neutron investigations on Ca
3
Cr
2
Ge
3
O
12
show that like
Ca
3
Fe
2
Ge
3
O
12
this composition exhibits an E-Type magnetic structure. [63]
The Ca-Cr garnet’s magnetic susceptibility is shown in Figure 4.1 (a) shows
that the garnet orders around 9 K (inset). The 2 K magnetization of Ca-Cr does
not fully saturate up to 14 T, but is expected to at 3 spins per Cr
3+
as seen in
Figure 4.1 (b). Curie-weiss analysis shows that the effective moment is 3.61μ
B
per Cr
3+
compared to the expected moment of 3.87 μ
B
. Additonally a Curie-
Weiss Theta was found to be -1.5 K yielding a very low frustration index of 0.17.
56
0 100 200 300
T (K)
0.0
0.1
0.2
0.3
0.4
0.5
χ (emu mol
-1
Oe
-1
)
0.05 T zfc
0.05 T fc
-12 -8 -4 0 4 8 12
H (T)
-3.0
-1.5
0.0
1.5
3.0
M
( μ
B
Cr
-1
)
2K
0 10 20
0.2
0.4
(b)
(a)
Figure 4.1: Magnetic Susceptibility of Ca
3
Cr
2
Ge
3
O
12
: (a.) Temperature depen-
dent susceptibility from 2 K to 300 K taken at 0.05 T, the inset more clearly shows
the antiferromagnetic ordering around 9 K. (b.) Magnetization of Ca-Cr taken at
2 K, showing a field induced transition at low fields.
The negative sign of the Curie-Weiss temperature indicates antiferromagnetic
interactions among the Cr
3+
cations.
While the Ca-Fe garnet’s magnetic susceptibility shows the onset of antiferro-
magnetic ordering to be around 14 K shown in Figure 4.2 (a). The magnetization
57
0 100 200 300
T (K)
0.00
0.03
0.06
0.09
0.12
χ (emu mol
-1
Oe
-1
)
0.05 T zfc
0.05 T fc
-12 -8 -4 0 4 8 12
H (T)
-3.0
-2.0
-1.0
0.0
1.0
2.0
3.0
M
( μ
B
Co
-1
)
2K
0 15 30
0.09
0.11
0.12
(a)
(b)
Figure 4.2: Magnetic Susceptibility of Ca
3
Fe
2
Ge
3
O
12
: (a.) Temperature depen-
dent suscpecitbility from 2 K to 300 K taken at 0.05 T, the inset more clearly
shows the antiferromagnetic ordering around 14 K. (b.) Magnetization of Ca-Fe
taken at 2 K.
taken at 2 K of Ca-Fe shows the expected linear response to fields for an antiferro-
magnet. Curie-Weiss analysis of the high temperature region (200-300 K) yields
an effective moment of 5.21 μ
B
per Fe
3+
compared to the expected moment of
5.92 μ
B
for a d
5
system. The Curie-Weiss temperature extrapolated from the
fit was found to be -35.2 K, the negative sign is in agreement with the garnet’s
58
magnetization. Similarly to Ca
3
Cr
2
Ge
3
O
12
, Ca
3
Fe
2
Ge
3
O
12
has a relatively weak
frustration index of 2.5, but is an order of magnitude larger than Ca
3
Cr
2
Ge
3
O
12
.
0 10 20 30 40 50
T (K)
0.0
1.0
2.0
3.0
4.0
5.0
C
mag
/T
(J mol
-1
K
-2
)
0 T
0 10 20 30 40 50
T (K)
0.0
0.2
0.4
0.6
0.8
1.0
ΔS
mag
/R
ln(6)
(a)
(b)
Figure 4.3: (a) Specific heat of Ca
2
Fe
2
Ge
3
O
12
taken at 0 T. (b) Entropy of
Ca
2
Fe
2
Ge
3
O
12
.
The specific heat and entropy of the Ca-Fe system was taken at 0 T over
the temperature range of 2 K to 50 K. A sharp lambda anomaly associated with
the antiferromagnetic ordering at 15 K is present in Figure 4.3 (a), the entropy
shows at about 90% seen in Figure 4.3 (b). A broad anomaly next to the lambda
59
anomaly is also present in the specific heat, reminescent of a Schottky anomaly
and alludes to some disorder within the system.
0 10 20 30 40 50
T (K)
0.0
0.3
0.6
0.9
1.2
C
mag
/T (J mol
-1
K
-2
)
0 T
0 10 20 30 40 50
T (K)
0.0
0.2
0.4
0.6
0.8
1.0
ΔS
/R
ln(4)
(a)
(b)
Figure 4.4: (a) Specific heat of Ca
2
Cr
2
Ge
3
O
12
taken at 0 T. (b) Entropy of
Ca
2
Cr
2
Ge
3
O
12
.
The specific heat of Ca-Cr was taken at 0 T at a range of temperatures (Fig-
ure 4.4 (a)), a broad anomaly occurs at the magnetic ordering temperature of
about 9 K. A broad shoulder is present next to the lambda anomaly associated
60
with the magnetic ordering. The entropy associated with the specific heat mea-
surement is found in Figure 4.4 (b), and saturates just shy of 90%. Field depen-
-12 -8 -4 0 4 8 12
H (T)
-0.02
0.00
0.02
0.04
% ΔC
8K
7K
6K
5K
4K
3K
2K
0.00
0.01
0.02
0.03
% ΔC
20K
14K
13K
12K
11K
10K
9K
-3 -2 -1 0 1 2 3
-0.02
-0.01
0.00
-3 -2 -1 0 1 2 3
-0.003
0.000
0.003
(b)
(a)
Figure 4.5: Isothermal Capacitance Data of Ca
2
Fe
2
Ge
3
O
12
: (a) Capacitance data
taken between 2 and 8 K, and swept from -14 T to 14 T. (b) Capacitance data
taken from 9 to 14 K, and at 20 K also swept from -14 T to 14 T.
dent capacitance data was taken from 2 K through above the ordering temper-
ature of Ca
2
Fe
2
Ge
3
O
12
, and shows features below 2 T. These features are very
well defined below the ordering temperature as shown in Figure 4.6 (a). As
61
the temperature approaches the ordering temperature of the Fe
3+
garnet, 13 K,
the feature’s intensity is suppressed until finally at 14 K the feature is entirely
gone, shown in Figure 4.6 (b). Unlike previous garnets we have studied, like
CaY
2
Co
2
Ge
3
O
12
and NaCa
2
Co
2
V
3
O
12
(reference our new paper), the magnetiza-
tion in Figure 4.2 (b) shows no metamagnetic transition associated with the fields
at which the features in capacitance appear. Field dependent capacitance data
-12 -8 -4 0 4 8 12
H (T)
-0.04
-0.02
0.00
% ΔC
8K
7K
6K
5K
4K
3K
2K
-0.03
-0.02
-0.01
0.00
% ΔC
18K
14K
13K
12K
11K
10K
9K
(b)
(a)
Figure 4.6: Isothermal Capacitance Data of Ca
2
Cr
2
Ge
3
O
12
: (a) Capacitance data
taken between 2 and 8 K, and swept from -14 T to 14 T. (b) Capacitance data
taken from 9 to 14 K, and at 20 K also swept from -14 T to 14 T.
62
was also measured on the Cr
3+
garnet, and it too has a low field feature as shown
in Figure 4.6. The feature dissipates as the temperature approaches the ordering
temperature of 9 K, and eventually becomes nearly flatlined by 18 K. However,
differing from the Fe
3+
is that the Cr
3+
garnet has a metamagnetic transition at
low fields. The feature in the capacitance data is smeared out between -4 T and
4 T, while the metamagnetic transition is below 2 T.
4.3 Conclusions
The magnetic behavior of Ca
3
Fe
2
Ge
3
O
12
and Ca
3
Cr
2
Ge
3
O
12
has been studied
from susceptibility measurements. Both Ca
3
Fe
2
Ge
3
O
12
and Ca
3
Cr
2
Ge
3
O
12
were
found to exhibit low field magnetocapcitance below their respected magnetic
ordering with collinear magnetic ordering. We find that altering the octahedral
sites in the garnet structure results in significant differences to the broadness of
anomalies in the magnetocapcitance. Our results of severalB-site magnetic gar-
nets including Ca
3
Cr
2
Ge
3
O
12
, Ca
3
Fe
2
Ge
3
O
12
, NaCa
2
Co
2
V
3
O
12
and CaY
2
Co
2
Ge
3
O
12
elucidate the role of the octahedral sublattice on coupling of charge and spin.
Specifically Ca
3
Cr
2
Ge
3
O
12
and Ca
3
Fe
2
Ge
3
O
12
show that despite different mag-
netic cations occupying theB-site, allowing for the spins to interact along chains
created by the nuclear structure thus allowing for such coupling.
63
Chapter 5
Magnetodielectricity in
One-Dimensional Garnet
Y
3
Fe
2
Ga
3
O
12
Here we report the structure and magnetic properties of the newly synthe-
sized garnet Y
3
Fe
2
Ga
3
O
12
. Our previous studies on one-dimensional garnets
CaY
2
Co
2
Ge
3
O
12
and NaCa
2
Co
2
V
3
O
12
lead us to find that Y
3
Fe
2
Ga
3
O
12
magneti-
cally orders as a one-dimensional uniform chain along the octahedral sublattice.
The presence of site-mixing creates disorder as well as some spin-glass character
within the magnetic order.
5.1 Methods
Certain commercial equipment, instruments, or materials are identified in
this document. Such identification does not imply recommendation or endorse-
ment by the National Institute of Standards and Technology nor does it imply
that the products identified are necessarily the best available for the purpose.
5.1.1 Materials Preparation
Polycrystalline powders of Y
3
Fe
2
Ga
3
O
12
were prepared from stoichiometric
amounts of Y
2
O
3
, Fe
2
O
3
and Ga
2
O
3
by performing multiple heating profiles of
64
well-ground mixtures. The initial 12 hour heating is at 900
◦
C followed by
regrinding, pelletizing and heating at temperatures exceeeding 1200
◦
C until
phase pure. All heatings were performed in air and air quenched. Zirconia cru-
cibles were favored given the well-known substitution of alumina into the garnet
structure.
5.1.2 Spark Plasma Sintering (SPS)
The sample was loaded into a 9mm carbon die and placed into a vaccuum
SPS instrument in the Materials Research Laboratory at University of Califor-
nia, Santa Barabara. The sample was held at 900
◦
C for 5 minutes with 8kN of
pressure in order to ensure density above 85%.
Neutron Diffraction The sample was loaded in to a vanadium cell and placed
into a helium flow cryostat with temperature control. Neutron powder diffrac-
tion data were collected at 300 K using the BT-1 high resolution neutron powder
diffractometer at the NIST Center for Neutron Research. A Ge(311) monochro-
mator with a 75
◦
take-off angle, λ = 2.0787(2)Å and in-pile collimation of 60
minutes of arc were used. Data were collected over the range of 1.3
◦
-166.3
◦
in
scattering angle (2-Theta) with a step size of 0.05
◦
. The instrument is described
at http://www.ncnr.nist.gov/.
5.1.3 Physical Property Measurements
Temperature and field dependent magnetic susceptibility as well as specific
heat measurements were collected using a Quantum Design 14T Dynacool Phys-
ical Property Measurement System. For the specific heat measurements powders
of Y
3
Fe
2
Ga
3
O
12
were ground together with equal parts silver in order to increase
65
thermal coupling to the sample stage. The contribution from silver was measured
separately and subtracted. [84].
5.1.4 Capacitance Measurements
Magnetocapacitance measurements were taken using a homemade measure-
ment probe. An SPS’d pellet is painted with Ag-epoxy with insulated copper
wire attached. The sides of the pellet are sanded to ensure no shorts are cre-
ated between the sides of the electrode. The copper wire is then soldered onto
a Quantum Design waffer with electrodes attached. Lakshore co-axial cables are
then soldered to the electrodes and to the top of the probe which connects to a
Capacitance bridge. Then the field is swept as capacitance is recorded at various
temperatures.
5.2 Results and Discussion
Lattice Parameter, a 12.328(9) Å
Unit Cell Volume 1874.031(2) Å
3
Oxygen Positions (0.0273(8), 0.05621(6), 0.65088(6)
Octahedral Angles 83.95(3)
◦
, 96.05(6)
◦
Tetrahedral Angles 115.04(7)
◦
, 98.83(7)
◦
% Fe
3+
on Octahedral Site 80%
% Fe
3+
on Tetrahedral Site 20%
Table 5.1: A summary of the crystallographic results from the Rietveld Refine-
ment shown in Figure 5.2. The oxygen determined oxygen positions allow for
the calculation of the polyhedral angles.
Y
3
Fe
2
Ga
3
O
12
is a a spin-frustrated garnet, where Fe
3+
occupies the B and A-
sites and exhibits some spin-glass character as a result of site-mixing. In order
to determine the site-mixing quantity between the sites, neutron diffraction data
66
(a)
(b) (c)
Figure 5.1: Garnet Crystal Structure, with a quarter of the unit cell (a), in which
the octahedra are shown in green, and the terahedra in periwinkle. The cubic
site is symbolized by a black sphere for simplicity. (b) the connectivity between
the tetrahedral site and octahedral site within a rod. (c) The connectivity of the
tetrahedral site and octahedral site btween neighboring rods.
was collected at 30 K on the high resolution powder diffractometer, BT-1, at the
NIST center for Neutron Research. The nuclear structure was determined by
67
25 50 75 100 125 150
2θ (deg)
0
5000
10000
15000
20000
Intensity (counts)
Figure 5.2: Rietveld refinement results for Y
3
Fe
2
Ga
3
O
12
, with an R
Bragg
of
3.090%. Data measured on NIST’s BT-1 Powder Diffractometer.
refining against the data with a symmetry generated pattern using the Rietveld
method implemented through the GSAS-1 Suite, with the resulting parameters
given in Table 5.1. [93, 94]
The nuclear structure was found to be in good agreement with the cubic
space groupIa
¯
3d (No.230) structure with Fe
3+
cations on the 80% on the octa-
hedrally coordinated 16a site, and 20% on the tetrahedrally coordinated 24d
site, yielding the site-specific chemical formula of Y
3
}[Fe
1.6
Ga
0.4
](Fe
0.4
Ga
2.6
O
12
).
68
The distribution of Fe
3+
between theB-site andA-site agrees with earlier studies
on similar compostions. [95] A polyhedral angle is defined as the angle between
the metallic center cation and two oxygen ligands, which returns 90
◦
for an ideal
octahedral angle, and 109.5
◦
for the tetrahedral sublattice.
Our refinement shows a 7% distortion among polyhedra and a 5-9.7% in
tetrahedra, these distortions are due to refined oxygen positions. The atomic
radii of Ga
3+
and Fe
3+
are quite similar, making Y
3
Fe
5
O
12
an appropriate point
of comparison. The polyhedral angles of Y
3
Fe
2
Ga
3
O
12
as expected were found to
be analogous to Y
3
Fe
5
O
12
’s. [96]
The garnet’s rods, are woven throughout the unit cell, and each contain a
central octahedra and surrounded by six tetrahedral as shown in Figure 5.2 (b).
The distance between octahedra through a tetrahedra estimates the length of an
exchange pathway between neighboring rods, while the distance between octa-
hedra through the cubic site reflects the superexchange pathway within a single
rod. This simplification of exchange pathway assumes that with the majority
of spin (80%), resides on the octahedra sublattice and therefore the relevant
exchange pathways are either between chains or within chains.
Figure 5.3 (a) shows the temperature-dependent magnetic susceptibility of
Y
3
Fe
2
Ga
3
O
12
, in which a deviation of paramagnetism indicated by broad transi-
tion occurs around 5 K. Fitting the high-temperature region (300-400 K) of the
DC magnetic susceptibility to the Curie-Weiss equation, χ=C/(T-Θ
CW
), yields a
Θ
CW
of -56.4 K, an effective paramagnetic moment of 6.36μ
B
per formula unit
(4.49μ
B
per Fe).
This moment is quite far from agreement for the expected value for Fe
2+
of
5.92μ
B
in a high-spin octahedral coordination environment (d
5
, S=5/2, L=0),
when the orbital moment is unquenched and decoupled from that of the spin
69
0 100 200 300
T (K)
0.00
0.20
0.40
0.60
0.80
1.00
(emu mol
-1
Oe
-1
)
0.05 T zfc
0.05 T fc
Bonner-Fisher
-12 -8 -4 0 4 8 12
H (T)
-2
-1
0
1
2
M (
μ
B
Fe
-1
)
2K
0 30 60 90
0.6
0.8
1.0
-1 0 1
-0.2
0.0
0.2
(a)
(b)
χ
Figure 5.3: Magnetic Susceptibility of Y
3
Fe
2
Ga
3
O
12
: (a.) Temperature dependent
susceptibility from 2 K to 300 K taken at 0.05 T, the inset more clearly shows the
deviation of zfc to fc, characteristic of glassy behavior. The dashed red-line is the
data’s fit to the 1D Bonner-Fisher Model. (b.) Magnetization taken at 2 K.
(μ
S
=
q
4S(S + 1)).[86] The lower than expected moment yields approximately
a 20-25% moment deficient which corresponds to the 20% of Fe
3+
present on
the tetrahedral sublattice, suggesting that the spins on the octahedral sublattice
order within this broad transition.
70
The negative sign of Θ
CW
supports a dominant antiferromagnetic exchange
between the spins that exhibit a relatively high degree of frustration, as defined
by the frustration index, f = Θ/T
N
= 10.3. It is observed in Figure 5.3 (a)
that Y
3
Fe
2
Ga
3
O
12
’s magnetic susceptibility fits well to the Bonner-Fisher model
for uniform one-dimensional antiferromagnetic chains described by the rational
function:
χ
1D
=
Ng
2
μ
2
B
k
B
T
(
0.0070745x
2
+ 0.4625x + 0.0078
1x
3
+ 0.006793x
2
+ 0.0385
) (5.1)
wherex = J/k
B
T ), andg = 2.0. [97, 98] The goodness of fit is reflected in the
coefficient of determination, R
2
, which was found to be R
2
= 0.99989. From
this fit, the exchange-interaction coefficient, J, was extrapolated to be 12.1 K
which agrees with the onset of magnetic ordering of 5 K. Our previous studies
on Co
2+
garnets with low-dimensional ordering reported a DFT calculation ofJ
of 9 K. [83]
The magnetization shown in Figure 5.3 (b) shows a small open hysteresis
below 1 T, but however does not saturate up to 14 T. A subtraction of remnant
paramagnetism yields produces a saturated moment of 0.50 μ
B
per Fe
3+
or
roughly half the spins are ordered at 2 K. The curvature and open loop confirms
some ferromagnetic character within the material’s glass transition.
To affirm the absence of a true spin-glass, susceptibility was taken at various
AC frequencies and the result is shown in Figure 5.4 (a), which displays con-
servative frequency dependence. Two features are evident in the out-of-phase
susceptibility χ
AC
”, Figure 5.4 (b), these are suspected to be contributed from
different spin transitions present in this system. The first transition below 25 K is
in agreement with the broad transition in theχ
DC
susceptibility measurements.
When the frequency exceeds 100Hz, a feature begins to form 75 K and shifts
71
0 50 100 150 200 250 300
T (K)
0.0
10
-4.3
10
-4.0
10
-3.8
10
-3.7
(emu)
0.1 Hz
1.0 Hz
10 Hz
100 Hz
1000 Hz
0 50 100 150 200 250 300
T (K)
0.00
1.00
2.00
3.00
4.00
5.00
(deg) χ'' χ'
(a)
(b)
Figure 5.4: A.C. Susceptibility of Y
3
Fe
2
Ga
3
O
12
: The Real (a) and Imaginary (b)
Susceptibilities taken at various frequencies over a range of temperatures.
to 100 K by 1000Hz, this is believed to be associated with the spin-dynamics
transition as a result of site-mixing, as reported in Er
2
Sn
2
O
7
. [99]
Specific heat of Y
3
Fe
2
Ga
3
O
12
is shown in Figure 5.5, the broad anomaly at
5 K is in agreement with the onset of the one-dimensional transition shown in
the magnetic susceptibilities. An additional, sharper feature is visible around
75 K, which corresponds to the second feature in theχ
AC
”, suggesting evidence
72
0 50 100 150 200
T (K)
0.00
0.05
0.10
0.15
0.20
0.25
ΔC
mag
/ T
(J mol
-1
K
2
)
0 50 100 150 200
T (K)
0.0
0.1
0.2
0.3
0.4
ΔS
mag
/ R
ln(6)
(a)
(b)
Figure 5.5: (a) Specific heat of Y
3
Fe
2
Ga
3
O
12
taken at 0 T and corresponding (b)
Entropy
for a second magnetic transition. The entropy in Figure 5.5 (b), was found by
integrating the specific heat and shows that roughly 20% of the spins are ordered
by 50 K which corresponds to the amount of site-mixing of the Fe
3+
between the
octahedral and tetrahedral sublattice and the magnetic moment deficient in the
magnetic susceptibility. Above 50 K, the entropy begins to steady rise until it
73
plateaus again around 200 K to a ΔS of 0.38 J/ K suggesting that some magnetic
order may extend past 200 K.
0 25 50 75 100
8.80
8.90
9.00
9.10
9.20
ε
Cool
Warm
0 25 50 75 100
T (K)
0.00
0.15
0.30
0.45
tan δ (nS)
(a)
(b)
Figure 5.6: (a)Temperature dependent magnetodielectric data of Y
3
Fe
2
Ga
3
O
12
and (b) the corresponding loss taken at 0T with a alternating current of 1000
Hz.
Temperature dependent magnetodielectric measurements were performed
and showed a prominent feature in the loss due to a sharp change in the dielec-
tric constant at 60 K. Measurements were taken upon cooling and heating, the
74
hysteresis created indicates that the transition around 75 K is a first order struc-
tural transition.
Below the onset of magnetic ordering, 2 K, the field-dependent dielectric data
shows subtle features at low fields and a sharp parabolic response as shown in
Figure 5.7. When measured above the onset at 27 K, the parabolic response
broadens and the low field feature disappears. To conclude these features are due
0.000
0.020
0.040
0.060
0.080
% Δ Ɛ
2K
27K
-12 -8 -4 0 4 8 12
H (T)
0.05
0.06
tan δ (nS)
0.0
0.5
1.0
1.5
2.0
M
2
( μ
B
Co
-2
)
2K M
2
Figure 5.7: (a)The square of low-temperature magnetization plotted against the
field-dependent magnetodielectricity of Y
3
Fe
2
Ga
3
O
12
at 2 and 27 K with its cor-
responding loss (b).
to the coupling between charge and magnetic order, the square magnetization is
75
plotted against the magnetodielectric measurements below the magnetic onset
in dashed gold lines of Figure 5.7 (a). [100]
Y
3
Fe
2
Ga
3
O
12
’s magnetization in Figure 5.3 (a) shows a small open hystere-
sis at 2 K where the magnetocapcitance features materialize. We attribute these
faint features to the competing exchange interactions of the the Fe
3+
’s on both
the octahedral and tetrahedral sublattices. Find the glass papers where they talk
about the DM interactions in glasses and how this is a thing, will be interest-
ing to tie it back here.Also how one-dimensiaonl magnetism can occur in glassy
materials.
5.3 Conclusions
The magnetic behavior of Y
3
Fe
2
Ga
3
O
12
has been studied from susceptibil-
ity measurements revealing one-dimensional magnetic order which fit to the
Bonner-Fisher model for a uniform XY chain. Site-mixing quantities were deter-
mined through using powder neutron diffraction data and correspond to the
total entropy released in system. Additional specific heat and A.C. susceptibil-
ity anomalies above the magnetic ordering temperature allude to a structural
transition that manifests in the temperature dependent magnetodielectric data.
76
Chapter 6
Future Directions
Garnets with their three uniquely coordinated sites are capable of systematic
studies of all possible super-exchange pathways, the work described here focuses
on isolating magnetism of the octahedral sublattice. However work still remains
to be done for the tetrahedral and cubic sites before understanding magnetism
on multiple sublattices. Garnets have been synthesized with compositions sug-
gesting isolated cubic-site only magnetism and are listed in Table 6.1, as well as
phases synthesized but left to be studied.
Composition Sequential Heating Profile
Mn
3
Al
2
Ge
3
O
12
1200
◦
C, air
CoY
2
Co
2
Ge
3
O
12
1250
◦
C, N
2
MnY
2
Mn
2
Ge
3
O
12
1275
◦
C, N
2
Y
3
Cr
2
Ga
3
O
12
∗
1250
◦
C, N
2
Y
3
FeGa
4
O
12
∗
1250
◦
C, air
Y
3
Fe
3
Ga
3
O
12
∗
1250
◦
C, air
Y
3
Fe
4
Ga
2
O
12
∗
1250
◦
C, air
Gd
3
Al
2
Ga
3
O
12
∗
1500
◦
C, air
Dy
3
Al
2
Ga
3
O
12
∗
1500
◦
C, air
Tb
3
Al
2
Ga
3
O
12
∗
1500
◦
C, air
Table 6.1: Compositions of phase pure garnets with their heating profiles. While
this section focuses on preliminary data from Mn
3
Al
2
Ge
3
O
12
, many more inter-
esting garnets are listed here. Unless stated, it is assumed all garnets were syn-
thesized under ambient conditions. An
∗
indicates an unreported phase.
Other studies left to be completed include the effect of the diamagnetic back-
bone size in octahedral magnetism in garnets. The work on CaY
2
Co
2
Ge
3
O
12
and
NaCa
2
Co
2
V
3
O
12
concludes that dependence between the site’s size affects the
77
interaction strength between the Co
2+
, however a systematic study of altering
the sizes of the sites would elucidate the tunability of critical fields for physical
phenomena including magnetoelectric responses.
6.1 Preliminary Results of Mn
3
Al
2
Ge
3
O
12
6.2 Methods
Certain commercial equipment, instruments, or materials are identified in
this document. Such identification does not imply recommendation or endorse-
ment by the National Institute of Standards and Technology nor does it imply
that the products identified are necessarily the best available for the purpose.
6.2.1 Materials Preparation
Polycrystalline powders of Mn
3
Al
2
Ge
3
O
12
were prepared from stoichiomet-
ric amounts of Mn(C
2
O
4
)·2H
2
O, Al
2
O
3
and Ge
2
O
3
by performing multiple heat-
ing profiles of well-ground mixtures. Mn(C
2
O
4
)·2H
2
O was freshly prepared in
house by precipitating a solution of Mn(SO
4
)·7H
2
O with an excess of oxalic
acid and drying at room temperature overnight. The initial 12 hour heating
is at 900
◦
C followed by regrinding, pelletizing and heating at temperatures
exceeeding 1200
◦
C until phase pure. All heatings were performed in air and
air quenched. Zirconia crucibles were favored given the well-known substitution
of alumina into the garnet structure.
78
6.2.2 Spark Plasma Sintering (SPS)
The sample was loaded into a 9mm carbon die and placed into a vaccuum
SPS instrument in the Materials Research Laboratory at University of Califor-
nia, Santa Barabara. The sample was held at 900
◦
C for 5 minutes with 8kN of
pressure in order to ensure density above 85 %.
6.2.3 Physical Property Measurements
Temperature and field dependent magnetic susceptibility as well as specific
heat measurements were collected using a Quantum Design 14T Dynacool Phys-
ical Property Measurement System. For the specific heat measurements pow-
ders of Mn
3
Al
2
Ge
3
O
12
were ground together with equal parts silver in order to
increase thermal coupling to the sample stage. The contribution from silver was
measured separately and subtracted. [84].
6.2.4 Capacitance Measurements
Magnetocapacitance measurements were taken using a homemade measure-
ment probe. An SPS’d pellet is painted with Ag-epoxy with insulated copper
wire attached. The sides of the pellet are sanded to ensure no shorts are cre-
ated between the sides of the electrode. The copper wire is then soldered onto
a Quantum Design waffer with electrodes attached. Lakshore co-axial cables are
then soldered to the electrodes and to the top of the probe which connects to a
Capacitance bridge. Then the field is swept as capacitance is recorded at various
temperatures.
79
6.3 Results
20 30 40 50 60 70 80 90 100 110
2 θ (deg)
0
50
100
150
200
Intensity (counts)
Figure 6.1: Results of the Rietveld refinement of X-ray diffraction on
Mn
3
Al
2
Ge
3
O
12
, R
Bragg
= 10.8%.
Preliminary measurements and characterization are shown below on
Mn
3
Al
2
Ge
3
O
12
, however magnetocapacitance measurements have yet to com-
plete the study of a solely cubic magnetic system. Polycrystalline powders of
Mn
3
Al
2
Ge
3
O
12
were prepared from Mn(C
2
O
4
)·2H
2
O, Al
2
O
3
and GeO
2
by heat-
ing well-ground mixtures initially at 950
◦
C overnight followed by regrinding,
pelletizing, and heating at 1200
◦
C until phase pure. A two hour scan on the
laboratory X-ray diffractometer confirmed a single phase with space groupIa
¯
3d
80
(No. 230) and a lattice parameter of 11.908Å, Figure 6.1 with no extraneous
peaks and a Bragg factor of 10.8%. Due to the limitations of the diffractometer
site occupancies were not refined, and such is reflected in the higher Bragg Fac-
tor. The scheduled neutron time on BT1 at NIST will not only provide further
insight to the nuclear structure’s possible site-mixing, and more accurate thermal
parameters but allow for the determination of the magnetic structure.
Figure 6.2 (a) shows the temperature dependent magnetic susceptibility of
Mn
3
Al
2
Ge
3
O
12
, which suggests the material transitions into an antiferromagnetic
ground state around 6.5 K. The high-temperature region (200-300 K) of the sus-
ceptibility was fit to the Curie-Weiss equation, χ=C/(T-Θ
CW
) yielding a Θ
CW
of -23 K, and and effective paramagnetic moment of 9.98μ
B
per formula unit
(5.76μ
B
per Fe
3+
). This moment is in close to the expected value of Fe
3+
a in
a cubic coordination environment (d
7
, S = 5/2, L = 0). [101] The negative
sign of Θ
CW
supports a dominant antiferromagnetic exchange between the spins
which exhibit a relatively low degree of frustration, as defined by the frustration
index,f = Θ/T
N
= 3.5.
Figure 6.2 (b) shows the field dependent magnetization of Mn
3
Al
2
Ge
3
O
12
,
which predominantly shows a linear response with a small curvature at fields
below 4 T. This small curvature could possibly allude to a metamagnetic transi-
tion, as we have seen the octahedral Co
2+
garnets. Magnetocapcitance measure-
ments should be taken in order to determine whether a metamagnetic transition
coincides with magnetodielectric anomalies.
Susceptibility was measured in various fields seen in Figure 6.3 (a-f), reveals
a strong field dependence on the magnetic ordering. With increasing fields, it is
expected that the magnetic ordering will be suppressed, however what it shown
81
0 100 200 300
T (K)
0.0
0.1
0.2
0.3
0.4
0.5
χ (emu mol
-1
Oe
-1
)
0.05 T zfc
0.05 T fc
0 5 10
0.36
0.40
0.44
-12 -8 -4 0 4 8 12
H (T)
-6
-4
-2
0
2
4
6
M ( μ
B
f.u.
-1
)
2 K
(a)
(b)
Figure 6.2: (a)Temperature-dependent susceptibility of Mn
3
Al
2
Ge
3
O
12
collected
with a field of 0.05 T. The inset emphasizes the cusp associated with the onset
of antiferromagnetic order around 6.5 K. (b) Isothermal magnetization curves
collected below the magnetic ordering transition.
is a plateau of suppression to about 7 T, and then an increased sharpness with an
increasing field.
This peculiar field dependence motivated the field-dependent specific heat
measurements in Figure 6.4. Low Field specific heat in Figure 6.4 shows the
82
0 5 10 15 20
T (K)
0.30
0.40
0.50
0.05 T
0.35
0.42
0.49
χ (emu mol
-1
Oe
-1
)
1 T
0.30
0.36
0.42
3 T
0 5 10 15 20
T (K)
0.32
0.36
0.40
7 T
0.30
0.40
0.50
10 T
0.30
0.33
0.36
13 T
(a)
(b)
(c)
(d)
(e)
(f)
Figure 6.3: Temperature dependent susceptibility measured under increasing
field strengths, (a) 0.05 T, (b) 1 T (c) 3 T (d) 7 T (e) 10 T and (f) 13 T.
expected suppression of the Lambda anomaly corresponding to the magnetic
order as a function of field. However at higher fields, Figure 6.4, we see a more
prevalent temperature dependence with the suppression corresponding to the
isofield susceptibility measurements in Figure 6.3.
83
0 10 20 30 40 50
T (K)
0.0
1.0
2.0
3.0
4.0
C
mag
/T
(J mol
-1
K
-2
)
0 T
1 T
2 T
3 T
4 T
5 T
6 T
7 T
0 10 20 30 40 50
T (K)
0.0
0.3
0.6
0.9
1.2
1.5
ΔS
mag
/R
ln(6)
0 10 20 30 40 50
T (K)
0.0
1.0
2.0
3.0
C
mag
/T
(J mol
-1
K
-2
)
8 T
9 T
10 T
11 T
12 T
13 T
14 T
0 10 20 30 40 50
T (K)
0.0
0.3
0.6
0.9
1.2
1.5
ΔS
mag
/R
ln(6)
(a)
(b)
(c)
(d)
Figure 6.4: Magnetic contribution to the specific heat at (a) low and (b) high field
strengths. The lattice contribution was approximated by separately measuring
and subtracting the specific heat of CaNa
2
Mg
2
V
3
O
12
in order to estimate the
change in entropy associated with the onset of magnetic order in (b) low and (d)
high fields, normalized to the maximum value ofRln(6).
In order to fully understand the cause of this dependence at higher fields,
neutron diffraction will be needed to determine the magnetic structure. Inelastic
neutron scattering will also shine light on additional transitions that may induce
such temperature dependence within a material. Thus much is left to be stud-
ied with Mn
3
Al
2
Ge
3
O
12
and other cubic-magnetic garnet systems. The study of
cubic-magnetic garnets elucidate further the strength of the B-site magnetism,
in which any magnetism here will result in a spin-chain system which allows for
magnetodielectricity. This is compared to non-magnetodielectric Mn
3
Al
2
Ge
3
O
12
in which magnetism occupies only the intrinsically frustratedR-site, and theB-
site remains diamagnetic.
84
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Abstract (if available)

Abstract Structure-property relationships in magnetic solid state materials and their applications in new technology. ❧ Rational control over a material's magnetic and physical properties through magnetic structure refinement combined with magnetic susceptibility, specific heat and dielectric measurements. ❧ Current research focuses on designing new magnetic garnet phases and characterizing their magnetocapacitive properties.

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Creator Neer, Abbey Jean (author)

Core Title Understanding the relationship between crystal chemistry and physical properties in magnetic garnets

School College of Letters, Arts and Sciences

Degree Doctor of Philosophy

Degree Program Chemistry

Publication Date 08/14/2018

Defense Date 05/22/2018

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag condensed matter,garnets,magnetism,multiferroics,OAI-PMH Harvest,solid state chemistry

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Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Melot, Brent C. (committee chair), Ravichandran, Jayakanth (committee member), Thompson, Mark (committee member)

Creator Email abbeyneer@gmail.com,neer@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-c89-69766

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