NEUTRON SCATTERING STUDIES ON CORRELATED TRANSITIONMETAL OXIDES
By
Mengze Zhu

A DISSERTATION
Submitted to
Michigan State University
in partial fulfillment of the requirements
for the degree of
Physics – Doctor of Philosophy
2018

ABSTRACT
NEUTRON SCATTERING STUDIES ON CORRELATED TRANSITION-METAL OXIDES
By
Mengze Zhu
We have explored the collective phenomena of correlated electrons in two different transitionmetal oxides, Ruddlesden-Popper type ruthenates (Sr,Ca)n+1RunO3n+1 and inverse-trirutile
chromates Cr2MO6 (M = Te, Mo and W), using neutron scattering in combination with various
material characterization methods.
(Sr,Ca)n+1RunO3n+1 are 4d transition-metal oxides exhibiting competing magnetic and
electronic tendencies. The delicate balance among the competing states can be readily tuned by
perturbations, such as chemical doping and magnetic field, which gives rise to emergent
phenomena. We have investigated the effects of 3d transition-metal doping on the magnetic and
electronic properties of layered ruthenates. For instance, the single-layer (n = 1) Sr2RuO4 is an
unconventional superconductor possessing an incommensurate spin density wave instability with
a wave vector 
 = (0.3 0.3 L) driven by Fermi surface nesting. Upon Fe substitution, we have

unveiled an unexpected commensurate spin density wave order with a propagation vector 
 =
(0.25 0.25 0) in Sr2Ru1-xFexO4 (x = 0.03 and 0.05), despite the magnetic fluctuations persisting at

 . The latter feature is corroborated by the first principles calculations, which show that Fe
doping barely changes the nesting vector of the Fermi surface. These results suggest that in

addition to the known incommensurate magnetic instability, Sr2RuO4 is also in proximity to a
commensurate magnetic tendency that can be stabilized via Fe doping. We have also studied the
effects of a magnetic field. For example, the bilayer (n = 2) Ca3(Ru1-xTix)2O7 (x = 0.03) is a G-type
antiferromagnetic Mott insulator. We have revealed that a modest magnetic field can lead to

colossal magnetoresistance arising from an anomalous collapse of the Mott insulating state. Such
an insulator-to-metal transition is accompanied by magnetic and structural transitions. These
findings call for deeper theoretical studies to reexamine the magnetic field tuning of Mott systems
with magnetic and electronic instabilities, as a magnetic field usually stabilizes the insulating
ground state in Mott-Hubbard systems.
Cr2MO6 (M = Te, W and Mo) are spin dimer systems with the magnetic ions Cr3+ structurally
dimerized favoring a singlet ground state. However, all three compounds investigated exhibit longrange antiferromagnetic orders at low temperature owing to the inter-dimer interactions. We have
shown that the inter-dimer exchange coupling can be tuned from antiferromagnetic in Cr2TeO6 to
ferromagnetic in Cr2WO6 and Cr2MoO6, by altering the degree of d-p orbital hybridization
between W(Mo) and O atoms. The tunability of the inter-dimer interactions without introducing
additional complexities such as structural distortions and carrier doping offers a rare opportunity
to drive the system toward the quantum critical point (QCP) separating the dimer-based quantum
disordered state and the classical long-range antiferromagnetic order. Moreover, we have
unraveled Higgs amplitude modes in the magnetic excitation spectra of Cr2TeO6 and Cr2WO6,
which are generally believed to survive only in systems close to the QCP where the ordered
moment is suppressed significantly from its fully saturated value by quantum fluctuations.
However, these two compounds are away from the QCP with the ordered moment reduced only
by ~24%. This study suggests that Higgs amplitude modes are not the privilege of ordered systems
in the vicinity of the QCP, but may be common excitation modes in ordered spin dimer systems.

ACKNOWLEDGEMENTS

I would like to thank everyone for the help and support during the past five years.
All the work would not have been possible without the guidance from my Ph. D advisor, Dr.
Xianglin Ke. I am deeply indebted to him for the support and helpful advice throughout my
graduate studies. I also appreciate all the former and present members of our group, particularly,
Dr. Tao Zou, Dr. Tung-Wu Hsieh and Heda Zhang, for all the help on the experiments and for the
time spent together in the windowless office and laboratory in the basement of the BPS building.
The main portions of my research have been carried out using national facilities, and they
would not be possible without the efforts of various scientists and technicians. I am grateful for all
the help and fruitful discussions with the instrument scientists, and the timely assistance from the
scientific associates and the sample environment teams. Especially, I would like to thank Dr. Wei
Tian, Dr. Tao Hong, Dr. Huibo Cao, Dr. Clarina dela Cruz, Dr. Matthias Frontzek, Dr. Songxue
Chi, Dr. Masaaki Matsuda, Dr. Andrey Podlesnyak, Dr. V. Ovidiu Garlea, Dr. Matthew Stone, Dr.
Doug Abernathy and Dr. Andrew Christianson in Oak Ridge National Laboratory, Dr. Yiming Qiu,
Dr. Yang Zhao, Dr. J. W. Lynn, Dr. Leland Harriger and Dr. Brian Kirby in NIST Center For
Neutron Research, Dr. Yuelin Li, Dr. Donald Walko, Dr. David Keavney, Dr. John Freeland, Dr.
Philip Ryan and Dr. Jong-woo Kim in Argonne National Laboratory, and Dr. Eun Sang Choi in
National High Magnetic Field Laboratory.
I have been lucky to collaborate with a number of excellent scientists and colleagues during
my research. I really appreciate Dr. Zhiling Dun and Dr. Haidong Zhou in University of Tennessee,
Knoxville, Dr. Zhiqiang Mao, Yu Wang, Dr. Peigang Li, Dr. Jianjian Ge and Dr. Weifeng Sun in
Tulane University, and Dr. Jin Peng in Nanjing University, for synthesizing high-quality materials
for the experiments. I also want to acknowledge the theorists, Dr. S. D. Mahanti and Dr. D. Do in
iv

Michigan State University, Dr. D. J. Singh and Dr. K. V. Shanavas in University of Missouri, Dr.
G.-Q. Liu in Ningbo Institute of Material Technology and Engineering, and Dr. M. Matsumoto in
Shizuoka University. Their collaborations are very important to the understanding and
interpretation of our experimental results.
I would like to especially extend my gratitude to Dr. Reza Loloee for always being helpful
with all the problems we have encountered in our labs.
I would like to thank all the friends met in Michigan State University, in particular, Dr. Jie
Guan, Faran Zhou, Nan Du, Xukun Xiang, Dr. Yanhao Tang, Dr. Yaxing Zhang, Dr. Zhen Zhu,
Chaoyue Liu, Bo Xiao, Dr. Bin Hwang, Yanlian Xin, Dan Liu, Chuanpeng Jiang, Didi Luo and
Xueyin Huyan, etc. I could not imagine how life would have been without them over the five years.
I wish to give a special thank you to Dr. Xu Lu, Dr. Zhensheng Tao and Dr. Fangqin Li for sharing
experiences and giving advice.
Finally, I would like to thank my parents for the everlasting support and understanding.

v

TABLE OF CONTENTS

LIST OF TABLES ......................................................................................................................... ix
LIST OF FIGURES ........................................................................................................................ x
Chapter 1
Introduction to correlated transition-metal compounds ............................................ 1
1.1
Introduction and motivation ............................................................................................. 1
1.2
High-Tc superconductivity in cuprates and iron-based superconductors ......................... 3
1.3
Colossal magnetoresistance in manganites ...................................................................... 6
1.4
Spin-orbit physics in iridates ............................................................................................ 9
1.5
Scope of this thesis ......................................................................................................... 13
Chapter 2
Introduction to neutron scattering ........................................................................... 14
2.1
Basic properties of neutrons ........................................................................................... 14
2.2
Nuclear scattering ........................................................................................................... 16
2.3
Magnetic scattering ........................................................................................................ 17
2.4
Neutron scattering instrumentation ................................................................................ 20
2.4.1
Triple-axis spectrometer ......................................................................................... 20
2.4.2
Time-of-flight spectrometer .................................................................................... 21
2.4.3
Four-circle diffractometer ....................................................................................... 22
2.4.4
Powder diffractometer ............................................................................................ 24
Chapter 3
Neutron scattering studies on Ruddlesden-Popper type ruthenates ........................ 26
3.1
Introduction and motivation ........................................................................................... 26
3.2
Non-Fermi surface nesting driven spin density wave order in Sr2(Ru,Fe)O4 ................ 28
3.2.1
Materials and methods ............................................................................................ 30
3.2.2
Magnetic susceptibility, specific heat and resistivity ............................................. 31
3.2.3
Neutron diffraction.................................................................................................. 33
3.2.4
Density functional theory calculations and x-ray absorption .................................. 37
3.2.5
Inelastic neutron scattering ..................................................................................... 39
3.2.6
Summary ................................................................................................................. 40
3.3
Magnetic order and metal-insulator transition in Sr3(Ru,Fe)2O7. .................................. 41
3.3.1
Materials and methods ............................................................................................ 42
3.3.2
Magnetic susceptibility, specific heat and resistivity ............................................. 42
3.3.3
Neutron diffraction.................................................................................................. 46
3.3.4
Discussions ............................................................................................................. 49
3.3.5
Summary ................................................................................................................. 51
3.4
Field-induced insulator-metal transition and colossal magnetoresistance in Ca3(Ru,Ti)2O7
........................................................................................................................................ 52
3.4.1
Materials and methods ............................................................................................ 54
3.4.2
Magnetoresistance................................................................................................... 54
3.4.3
Neutron diffraction.................................................................................................. 55
3.4.4
Density functional theory calculations.................................................................... 59
3.4.5
Comparison with early theories .............................................................................. 61
vi

3.4.6
Comparison with CMR manganites ........................................................................ 62
3.4.7
Comparison with the parent compound .................................................................. 63
3.4.8
Discussions ............................................................................................................. 63
3.4.9
Summary ................................................................................................................. 65
3.5
Field-induced incommensurate-commensurate magnetic transitions in Ca3(Ru,Fe)2O7 66
3.5.1
Materials and methods ............................................................................................ 67
3.5.2
Neutron diffraction for B ∼ b axis ........................................................................... 68
3.5.3
Magnetoresistance and neutron diffraction for B ∼ a axis ...................................... 75
3.5.4
Discussions ............................................................................................................. 82
3.5.5
Summary ................................................................................................................. 86
3.6
Tuning the competing states in Ca3Ru2O7 by Mn doping .............................................. 88
3.6.1
Materials and methods ............................................................................................ 88
3.6.2
Magnetic properties of Ca3Ru2O7 upon Mn doping at zero field ............................ 89
3.6.3
Magnetic and transport properties of Ca3(Ru1-xMnx)2O7 (x = 0.04) in a magnetic field
................................................................................................................................. 92
3.6.4
Discussions ............................................................................................................. 97
3.6.5
Summary ................................................................................................................. 98
3.7
Temperature- and field-driven spin reorientations in triple-layer ruthenate Sr4Ru3O10100
3.7.1
Materials and methods .......................................................................................... 103
3.7.2
Temperature-driven spin reorientations at zero field ............................................ 103
3.7.3
Field-driven spin reorientations ............................................................................ 107
3.7.4
Discussions ........................................................................................................... 109
3.7.5
Summary ............................................................................................................... 111
Chapter 4
Neutron scattering studies on inverse-trirutile chromates .................................... 112
4.1
Introduction and motivation ......................................................................................... 112
4.2
Tuning the exchange interaction in Cr2(Te,W)O6 via orbital hybridization ................ 115
4.2.1
Materials and methods .......................................................................................... 115
4.2.2
Magnetic susceptibility, specific heat and powder neutron diffraction ................ 116
4.2.3
Density functional theory calculations.................................................................. 120
4.2.4
Summary ............................................................................................................... 126
4.3
Ferromagnetic inter-dimer interaction in Cr2MoO6 ..................................................... 127
4.3.1
Materials and methods .......................................................................................... 127
4.3.2
Magnetic susceptibility, specific heat and powder neutron diffraction ................ 128
4.3.3
Density functional theory calculations.................................................................. 132
4.3.4
Summary ............................................................................................................... 137
4.4
Higgs amplitude modes in the magnetic excitation spectra of Cr2TeO6 and Cr2WO6 . 138
4.4.1
Materials and methods .......................................................................................... 138
4.4.2
Inelastic neutron scattering ................................................................................... 138
4.4.3. Linear spin wave theory calculations .................................................................... 141
4.4.4
Extended spin wave theory calculations ............................................................... 143
4.4.5
Discussions ........................................................................................................... 149
4.4.6
Summary ............................................................................................................... 150
Chapter 5

Summary and perspectives ................................................................................... 152

vii

APPENDICES ............................................................................................................................ 156
APPENDIX A: Crystal symmetry of Sr2Ru1-xFexO4 (x = 0.03 and 0.05) ............................... 157
APPENDIX B: Determination of magnetic structures by neutron diffraction........................ 158
BIBLIOGRAPHY ....................................................................................................................... 161

viii

LIST OF TABLES

Table 4.1: (top) Total energy (eV/unit cell) of different [intra]-[inter] dimer magnetic
configurations. (bottom) Calculated and experimental values of intra- ( ) and inter- ( ) dimer
exchange parameters as in the model proposed by Drillon et al [189] using GGA.................... 121
Table 4.2: Structural parameters of Cr2TeO6, Cr2WO6 and Cr2MoO6 at T = 4 K, including lattice
parameters, bond lengths and bond angles ................................................................................. 129
Table 4.3: Total energy (in eV) per magnetic unit cell containing four Cr3+ ions of different
magnetic configurations .............................................................................................................. 133
Table 4.4: Exchange parameters  and  obtained from the simple isotropic Heisenberg model
 ∙ 
 ;  =  for intra-dimer and  =  for inter-dimer exchange. Values
 = −2 ∑〈,〉  
with superscript * are obtained using GGA based on a model proposed in Ref. [189] .............. 136
Table 4.5: Exchange parameters for Cr2WO6 and Cr2TeO6 used for the LSW calculations ....... 142
Table 4.6: Exchange parameters and magnetic moment for Cr2WO6 and Cr2TeO6 obtained by the
ESW calculations. The magnetic moment is normalized to the saturated value (full moment) . 145

ix

LIST OF FIGURES

Figure 1.1: d-electron orbitals in an octahedral oxygen environment [8]....................................... 2
Figure 1.2: Schematics of the crystal structures of (a) cuprate La2CuO4 [16], (b) iron pnictide
BaFe2As2 and (c) iron chalcogenide FeTe1-xSex [3] ........................................................................ 4
Figure 1.3: Phase diagrams of electron- and hole-doped high-Tc cuprates [17] ............................. 5
Figure 1.4: The structural and magnetic phase diagrams of electron- and hole-doped iron pnictide
BaFe2As2 [3] ................................................................................................................................... 6
Figure 1.5: Structural and magnetic phase diagram of the layered manganite La2-2xSr1+2xMn2O7 as
a function of temperature and chemical composition [20] ............................................................. 7
Figure 1.6: Spin-orbital orders in (a) LaMnO3 and (b) BiMnO3 [8]............................................... 8
Figure 1.7: Magnetoresistance in (a) ferromagnetic metallic Lr2/3(Pb,Ca)1/3MnO3 [21] and (b)
antiferromagnetic charge-ordered Nd0.5Sr0.5MnO3 [22] ................................................................. 9
Figure 1.8: Generic phase diagram as a function of correlation strength ⁄ and SOC ⁄ . t is the
hopping amplitude [23] ................................................................................................................. 10
Figure 1.9: d-electron orbital splitting in an octahedral crystal field in the presence of strong SOC
[24] ................................................................................................................................................ 11
Figure 1.10: Phase diagram of the Heisenberg-Kitaev model [24] .............................................. 12
Figure 2.1: Typical geometry of a neutron scattering experiment [33] ........................................ 15
Figure 2.2: Magnetic structure of Au2Mn [33] ............................................................................. 19
Figure 2.3: Schematic of the CG-4C cold neutron triple-axis spectrometer (CTAX) in HFIR,
ORNL............................................................................................................................................ 21
Figure 2.4: Schematic of the SEQUOIA time-of-flight spectrometer in SNS, ORNL [34] ......... 22
Figure 2.5: HB-3A four-circle diffractometer in HFIR, ORNL [35] ............................................ 23
Figure 2.6: Schematic representation of the angles in four-circle diffractometry [36] ................ 23
Figure 2.7: HB-2A powder diffractometer in HFIR, ORNL [37] ................................................ 25
Figure 2.8: Schematic of the HB-2A powder diffractometer in HFIR, ORNL [37]..................... 25

x

Figure 3.1: Crystal structure of Ruddlesden-Popper type ruthenates Srn+1RunO3n+1 (n = 1, 2 and ∞)
....................................................................................................................................................... 27
Figure 3.2: (a) Temperature dependence of out-of-plane DC susceptibility  of Sr2Ru0.95Fe0.05O4
after ZFC and FC, respectively. Inset shows the isothermal magnetization as a function of field at
T = 2 and 20 K after ZFC. (b) Temperature dependence of AC susceptibility measured with h = 10
Oe. (c) Temperature dependence of specific heat at zero field. Inset shows the expanded view of
the low-temperature region with the data measured at 9 T included for comparison. The solid red
line is a linear fit for 16 K < T < 30 K. (d) In-plane and out-of-plane resistivity as a function of
temperature ................................................................................................................................... 32
Figure 3.3: (a) Temperature dependence of in-plane DC susceptibility  of Sr2Ru0.95Fe0.05O4
after ZFC and FC, respectively. (b) Magnetization as a function of magnetic field at T = 2 K and
20 K............................................................................................................................................... 32
Figure 3.4: (a) Scans across 
 = (0.25 0.25 0) along the [1 1 0] direction at T = 4, 50 and 100 K
measured on Sr2Ru0.95Fe0.05O4. (b) Scans across 
 = (0.25 0.25 0) along the [0 0 1] direction at
selected temperatures. (c) The intensity of magnetic Bragg peak 
 = (0.25 0.25 0) as a function
of temperature. Note that the sample measured for (b) is smaller than that for (a,c). (d) Contour
map of elastic magnetic scattering intensity of Sr2Ru0.97Fe0.03O4 at T = 1.6 K after subtracting the
background measured at 80 K. Spurious peaks are denoted by red circles. The residue intensity
near the nuclear peaks (Âą1 Âą1 0) is presumably due to the thermal shift in the lattice parameters
....................................................................................................................................................... 35
Figure 3.5: Schematics of the SDW order of two magnetic twin domains in Sr2Ru1-xFexO4. (a),(b)
SDW with 
 = (0.25 0.25 0). (c),(d) SDW with 
 = (0.25 -0.25 0) .......................................... 36
Figure 3.6: Electronic structure for a 3 × 3 × 1 supercell of Sr2RuO4 containing one Fe substitution.
(a) Density of states and projections showing majority spin as positive and minority spin as
negative. (b) Fat band plot of the band structure showing Ru character for the unsubstituted
supercell (heavier symbols represent higher Ru character), in comparison with the Fe substituted
cell, emphasized by heavier symbols. (c) Ru character from Ru neighboring Fe and (d) Ru not
neighboring Fe .............................................................................................................................. 38
Figure 3.7: (a) x-ray absorption spectra of Sr2Ru0.97Fe0.03O4 near the Fe L edge in comparison with
FeO and Fe2O3. (b) Lower panel: contour map of inelastic neutron scattering intensity as a function
of E and K, H integrated from 0.2 to 0.4. Upper panel: cuts along [0 1 0] with E integrated from 3
to 6 meV (black) and from -0.5 to 0.5 meV (red), respectively. H is integrated from 0.2 to 0.4.
Note that the intensities of these two curves are scaled. Data were measured on Sr2Ru0.97Fe0.03O4
....................................................................................................................................................... 39
Figure 3.8: (a),(b) Temperature dependence of magnetic susceptibility  of Sr3(Ru1-xFex)2O7 (x =
0.01 and 0.03) after ZFC and FC. Insets show the in-plane magnetic susceptibility  as a
function of temperature after ZFC and FC. (c),(d) Isothermal magnetization as a function of
magnetic field of Sr3(Ru1-xFex)2O7 (x = 0.01 and 0.03), respectively. The field is applied along the
c axis. Inset shows the data of the x = 0.01 compound at 2 K up to 9 T ...................................... 44
xi

Figure 3.9: (a) Specific heat of Sr3(Ru1-xFex)2O7 (x = 0.01 and 0.03) as a function of temperature.
The blue arrow denotes TN obtained from magnetic susceptibility measurements. (b) In-plane
resistivity ρ of Sr3(Ru1-xFex)2O7 as a function of temperature. Inset shows ρ vs T2 in the lowtemperature regime for x = 0.01. The red line is a fit using linear function ................................. 44
Figure 3.10: Neutron diffraction data of Sr3(Ru1-xFex)2O7 (x = 0.03). (a) Scans around 
 = (0.25
0.25 0) along the [1 1 0] direction at representative temperatures. (b) Temperature dependence of
the peak intensity of 
 = (0.25 0.25 0). (c),(d) Scans around 
 = (0.25 0.25 0) and (0.75 0.75 0)
along the [0 0 1] direction, respectively........................................................................................ 47
Figure 3.11: (a) Schematic of the E-type antiferromagnetic order of one bilayer in
Sr3(Ru0.97Fe0.03)2O7. (b) Synchrotron XAS measurements on Sr3(Ru0.97Fe0.03)2O7 and the reference
samples FeO (Fe2+) and Fe2O3 (Fe3+) ........................................................................................... 48
Figure 3.12: (a) Magnetic and electronic phase diagram of Ca3(Ru1-xTix)2O7. A-G-Inc: coexistence
of AFM-a, G-AFM and incommensurate magnetic structures. Pink region represents Mott
insulating state, while dark green stands for weakly localized state (Modified from Ref. [100]) 53
Figure 3.13: (a),(b) Temperature dependence of resistivity  and  measured at B = 0 and 9 T
applied along the b axis. The magnetic field was applied at T = 100 K and the measurements were
taken while cooling. (c),(d) Magnetic field dependence of  and  at T = 10 K..................... 55
Figure 3.14: (a) Field dependence of lattice parameters. Statistical errors from the fitting procedure
are smaller than the symbol size. (b) Field dependence of the integrated intensity of magnetic
Bragg peaks (1 0 2) and (0 0 1), respectively, measured at T = 10 K. The magnetic field was
increased for both nuclear and magnetic Bragg peak scans. (c),(d) Rocking curves of (1 0 2) and
(0 0 1) measured at B = 0 T and 10 T applied along the b axis, T = 10 K .................................... 56
Figure 3.15: (a),(b) Neutron intensity of the (1 0 2) and (0 0 1) magnetic Bragg peaks as a function
of magnetic field measured at different temperatures. The solid curve is for increasing field and
the dashed one is for decreasing field ........................................................................................... 58
Figure 3.16: (a) T−B phase diagram of Ca3(Ru0.97Ti0.03)2O7 in a magnetic field along the b axis.
The solid squares, circles and diamonds are phase boundaries determined by neutron diffraction
measurements: solid squares denote the points where (0 0 1) shows up, and solid circles represent
where (1 0 2) disappears. The solid diamonds stand for the points where (0 0 1) disappears
completely. The open diamonds are the phase boundaries determined by the resistivity
measurements. (b) Schematics of spin structures of AFM-a, G-AFM and CAFM phases .......... 59
Figure 3.17: Projected density of states (PDOS) of the Ru  ! and  " ⁄!" orbitals. (a) PDOS
calculated using the low-temperature crystal structure and G-AFM magnetic structure with the onsite Coulomb interaction U = 0. (b) PDOS calculated using the low-temperature crystal structure
and G-AFM magnetic structure with U = 2 eV. (c) PDOS calculated using the high-temperature
crystal structure and AFM-a magnetic structure with U = 2 eV ................................................... 61

xii

Figure 3.18: (a) Temperature and (b) magnetic field dependence of the in-plane resistivity of
Ca3(Ru0.95Ti0.05)2O7 ....................................................................................................................... 64
Figure 3.19: Schematic diagrams of the magnetic structures in Ca3(Ru0.95Fe0.05)2O7. (a) AFM-a
and AFM-b: collinear antiferromagnetic structures with the magnetic moments along the a and b
axis, respectively. (b) CAFM-b (CAFM-a): canted antiferromagnetic structures consisting of an
AFM-a (AFM-b) component and a ferromagnetic one along the b (a) axis. (c) In-plane view of
one of the bilayers of the incommensurate magnetic soliton lattice. The dashed square in magenta
represents one unit cell. The orange rectangle with rounded corners encloses the domain wall. The
blue and orange arrows represent the direction of the magnetic field to be along either the a or b
axis ................................................................................................................................................ 67
Figure 3.20: (a),(b) Contour maps of the intensity of scans along the [1 0 0] direction across 
 =
(0 0 1) in a magnetic field applied along the b axis at T = 15 K. The light gray grids denote the
measurement fields. (c),(d) Scans along the [1 0 0] direction across 
 = (0 0 1) in the 0↑ % and 0↓ %
phases at T = 1.5 K, respectively. The dark green and orange arrows mark the first-order
incommensurate magnetic Bragg peaks and their third-order harmonics, respectively, while the
purple ones mark the commensurate reflections. Inset shows the zoom-in view of the third-order
harmonics at T = 1.5 K and 80 K, B = 0↑ T. The data were taken at CTAX ................................ 70
Figure 3.21: Phase diagram of Ca3(Ru0.95Fe0.05)2O7 in a magnetic field along the b axis for field
ramping up. Note that the error bars of the data points at 15 K and 30 K are smaller than the size
of the symbol................................................................................................................................. 71
Figure 3.22: (a)-(c) Scans along the [1 0 0] direction across 
 = (0 0 1) in the 0↑ % (black) and 0↓ %
(red) phases at representative temperatures. The magnetic Bragg peaks are fitted using Gaussian
functions (solid curves). The gray horizontal line indicates the instrument resolution. (d)
Incommensurability δ of the first-order incommensurate magnetic wave vector 
' = (¹δ 0 1) in
the 0↑ % and 0↓ % phases at different temperatures. The error bars obtained from Gaussian fitting
are smaller than the symbol size. The solid curves are guides to the eye. The data were taken at
CTAX............................................................................................................................................ 72
Figure 3.23: (a) Contour map of the intensity of scans along the [1 0 0] direction over 
 = (0 0 1)
while warming up in the metastable 0↓ % phase. The measurement temperatures are denoted by the
light gray grids. (b) Scans along the [1 0 0] direction over 
 = (0 0 1) at representative
temperatures. The third-order harmonics of the incommensurate magnetic reflections are denoted
by orange arrows. Inset shows the temperature dependence of the intensity ratio () ⁄( in the 0↓ %
phase. The data were taken at CTAX ........................................................................................... 74
Figure 3.24: (a) Normalized out-of-plane resistivity  of Ca3(Ru0.95Fe0.05)2O7 as a function of
magnetic field at (a) T = 15 K and (b) T = 55 K. The magnetic field is applied along the a and b
axis, respectively ........................................................................................................................... 76
Figure 3.25: Normalized out-of-plane resistivity  of Ca3(Ru0.95Fe0.05)2O7 as a function of
magnetic field at representative temperatures. The upper and lower critical fields are denoted by

xiii

red and green arrows, respectively. Inset shows the lower critical field *↓ as a function of
temperature. The red solid line is a fit using power law. The magnetic field is along the a axis . 77
Figure 3.26: Contour maps of the neutron intensity of scans across 
 = (0 0 1) along the [1 0 0]
direction at T = 15 K as (a) the magnetic field increases from 0 to 4 T and (b) the magnetic field
decreases from 4 to 0 T. (c),(d) Cuts of the scans taken at B = 0↑ , 4↓ and 0↓ T. The solid and dashed
lines are fitted curves using Gaussian functions. The gray line denotes the instrument resolution.
The magnetic field is applied along the a axis. Inset shows the scans at B = 0↑ T (blue) and 0↓ T
(red) when the magnetic field is along the b axis at T = 15 K ...................................................... 79
Figure 3.27. Contour maps of the neutron intensity of scans across 
 = (0 0 1) along the [1 0 0]
direction at T = 34 K as (a) the magnetic field increases from 0 to 4 T and (b) the magnetic field
decreases from 4 to 0 T. (c),(d) Cuts of the scans taken at B = 0↑ , 4↑ T and 1.5↓ , 0↓ T. The solid
lines are fitted curves using Gaussian functions. The gray line denotes the instrument resolution.
The magnetic field is applied along the a axis .............................................................................. 81
Figure 3.28: (a) Contour map of the neutron intensity of scans across 
 = (0 0 1) along the [1 0 0]
direction in the 0↓ % phase at various temperatures. (b) Cuts of the scans taken at T = 15, 34 and 37
K in the 0↓ % phase. The solid lines are fitted curves using Gaussian functions. The gray line
represents the instrument resolution ............................................................................................. 82
Figure 3.29: Scan along the [1 0 0] direction across the nuclear Bragg peak (0 0 4) at T = 1.5 K in
the 0↑ % phase ................................................................................................................................ 86
Figure 3.30: Temperature dependence of the intensity of representative magnetic Bragg peaks (1
0 2), (0 0 1) and (δ 0 1) of Ca3(Ru1-xMnx)2O7 (x = 0.03, 0.04 and 0.05), δ = 0.023 and 0.026 for x
= 0.03 and 0.04, respectively ........................................................................................................ 89
Figure 3.31: (a),(b) Scans along the [1 0 0] direction across the magnetic wave vector (0 0 5) at
representative temperatures. (c),(d) Scans across the magnetic wave vector (0 0 1) in the
incommensurate magnetic phase. (e),(f) Rocking curve scans across the magnetic wave vector (1
0 2) at representative temperatures ............................................................................................... 91
Figure 3.32: Lattice parameters of Ca3(Ru1-xMnx)2O7 (x = 0.03 and 0.04) as a function of
temperature ................................................................................................................................... 92
Figure 3.33: (a) Temperature dependence of the in-pane resistivity ρ at B = 0 T and 9 T, B ∼ b
axis. (b),(c) Field dependence of the in-plane resistivity ρ and magnetization M at T = 10 K and
34 K, B ∼ b axis ............................................................................................................................. 93
Figure 3.34: Rocking curve scans on Ca3(Ru0.96Mn0.04)2O7 across (1 0 2) and (0 0 1) at B = 0 T and
8 T, respectively, T = 10 K. (c),(d) The intensity of (1 0 2) and (0 0 1) as a function of magnetic
field, T = 10 K ............................................................................................................................... 95

xiv

Figure 3.35: (a),(b) Scans on Ca3(Ru0.96Mn0.04)2O7 along the [1 0 0] direction across (0 0 1) at B =
0 T and 5 T, T = 34 K. (c),(d) The intensity of (0 0 1) and (0.0245 0 1) magnetic Bragg peaks as a
function of magnetic field, T = 34 K............................................................................................. 96
Figure 3.36: Lattice constants a and c of Ca3(Ru0.96Mn0.04)2O7 as a function of magnetic field at (a)
T = 10 K and (b) T = 32 K ............................................................................................................ 97
Figure 3.37: (a) The crystal structure of Sr4Ru3O10. (b) The magnetic structure of Sr4Ru3O10. Note
that both moment size and spin direction cannot be uniquely determined due to the lack of enough
Bragg peaks with reasonably good magnetic intensity ............................................................... 101
Figure 3.38: (a),(b) Rocking curve scans across (0 0 2) and (1 1 1) Bragg reflections at T = 8, 50
and 100 K, respectively. (c),(d) Temperature dependence of the peak intensity of (0 0 2) and (1 1
1) at B = 0 T. The red and green dashed lines denote Tc and T*, respectively ............................ 105
Figure 3.39: (a),(b) Field dependence of the peak intensity of (0 0 2) and (1 1 1) at T = 1.5 K. (c),(d)
θ-2θ scans across (0 0 2) and (1 1 1) at B = 0 and 3.5 T, T = 1.5 K. Inset shows the θ-2θ scan over
(0 0 16) at B = 0 and 3.5 T, T = 1.5 K ......................................................................................... 109
Figure 3.40: θ-2θ scans across (0 0 2) and (1 1 1) at B = 0 and 3.5 T, T = 50 K........................ 109
Figure 4.1: Schematics of the crystal and magnetic structures of (a) Cr2TeO6 and (b) Cr2WO6. (c)
Dominant exchange interactions between Cr3+ ions. The black line represents a unit cell. There are
two dimer sites (A and B) in a unit cell. Each dimer consists of two spins at the top and bottom
sides strongly coupled by an antiferromagnetic exchange interaction ( > 0). The dimers interact
with each other by inter-dimer interactions ( and ) );  connects different dimers (A-B) and )
connects the same dimer sites (A-A; B-B). The magnetic structure in the ordered phase indicates
that ) < 0 (ferromagnetic) in both Cr2WO6 and Cr2TeO6, while  is ferromagnetic ( < 0) in
Cr2WO6 and antiferromagnetic ( > 0) in Cr2TeO6, respectively. We find that  ≫ | |, |) | . 113
Figure 4.2: (a) x dependence of lattice parameters measured at T = 4 K. (b),(c) Temperature
dependence of magnetic susceptibility and magnetic specific heat, respectively. (d) Normalized
neutron intensity of order parameters, (1 0 1) for x = 0, 0.5 and (0 0 1) for x = 0.8, 1, as a function
of temperature. Solid lines are guides to the eye ........................................................................ 116
Figure 4.3: Neutron powder diffraction measurements with x = 0, 0.5, 0.8 and 1.0 at T = 4 K. Black
symbols are the experimental data, red curves for fits and the difference between these two are
represented by the blue curves. For x = 0.5, there is a minor impurity phase Cr2O3 (< 1%) as
denoted by #. Symbols of + and * denote the magnetic Bragg peaks for AFM-I and AFM-II,
respectively. Insets show the expanded view of the low-angle magnetic Bragg peaks at T = 4 K
(black) and 150 K (dark green) ................................................................................................... 118
Figure 4.4: (a) TN − x phase diagram of Cr2(Te1-xWx)O6. PM represents the paramagnetic phase. (b)
Sublattice magnetization as a function of x obtained from neutron powder diffraction
measurements .............................................................................................................................. 119

xv

Figure 4.5: (a),(b) Expanded view of low-angle neutron powder diffraction data of Cr2(Te1-xWx)O6
with x = 0.6 and 0.7. Both (0 0 1), (0 0 2) and (1 0 1) magnetic Bragg peaks are observed at T = 4
K for the x = 0.7 sample indicating the coexistence of AFM-I and AFM-II, while only (0 0 2) and
(1 0 1) appear for the x = 0.6 sample corresponding to AFM-I. (c) Temperature dependence of the
(1 0 1) magnetic Bragg peak of these two samples .................................................................... 120
Figure 4.6: Spin densities on the (1 1 0) plane of Cr2TeO6 (left) and Cr2WO6 (right) where red
represents spin up and blue stands for spin down ....................................................................... 123
Figure 4.7: Projected density of states of Cr2TeO6 (left) and Cr2WO6 (right). Cr1 and Cr2 are two
inter-dimer Cr ions. O2 mediates the exchange coupling between Cr1 and Cr2 and hybridizes with
Te and W ..................................................................................................................................... 124
Figure 4.8: Energy difference between antiferromagnetic and ferromagnetic inter-dimer magnetic
configurations with antiferromagnetic configuration fixed for the intra-dimer interaction ....... 125
Figure 4.9: Schematic of the crystal and magnetic structure of Cr2MoO6. The intra-dimer
interaction is antiferromagnetic, whereas the inter-dimer exchange is ferromagnetic ............... 128
Figure 4.10: (a) Temperature dependence of magnetic susceptibility. Inset shows the inverse
susceptibility and the Curie-Weiss fit. (b) Temperature dependence of magnetic heat capacity.
Inset shows the subtraction of the phonon contribution using the scaled heat capacity of Ga2TeO6.
(c) Intensity of (0 0 1) magnetic Bragg peak as a function of temperature. The solid line is a guide
to the eye. (d) Temperature dependence of the integrated magnetic entropy. The horizontal red line
denotes the theoretical value for S = 3⁄2 ................................................................................... 130
Figure 4.11: Neutron powder diffraction data of Cr2MoO6 at T = 4 K. The black symbols are
experimental data and the red curve is the Rietveld refinement fit. The difference is represented by
the blue. The positions of the nuclear and magnetic peaks are marked by the green and magenta
lines, respectively. Inset shows the difference between the low- and high-temperature data with
the positions of magnetic Bragg peaks denoted by the symbol “*” ........................................... 131
Figure 4.12: Density of states (DOS) in the unit of number of states / (eV u.c.) and projected density
of states (PDOS) in the unit of number of states / (eV atom) of Cr2MoO6 calculated by GGA+U
..................................................................................................................................................... 133
Figure 4.13: A cartoon illustrating the ferromagnetic coupling between two Cr 3d moments via the
orbital hybridization between O 2p and Mo empty 4d orbitals leading to a virtual electron transfer
from O 2p to Mo 4d, leaving O 2p partially occupied which induces a ferromagnetic coupling
between two Cr moments via Cr 3d-O 2p hybridization ............................................................ 135
Figure 4.14: Spin density projected on the (1 1 0) plane of Cr2MoO6 calculated using GGA+U
where red and blue indicate spin up and spin down, respectively. Since the local cubic axes of the
CrO6 octahedra are rotated from the chosen tetragonal x-y-z coordinates (Fig. 4.9), in the latter
coordinate system Cr d-t2g and d-eg states are mixed ................................................................. 135

xvi

Figure 4.15: (a),(b) Powder-averaged magnetic excitation spectra as a function of energy transfer
E and momentum transfer 2 of Cr2TeO6 and Cr2WO6 at T = 4 K, respectively. The background
measured using an empty sample can has been subtracted. (c),(d) Powder-averaged magnetic
excitation spectra calculated by the LSW theory. (e),(f) Powder-averaged magnetic excitation
spectra calculated by the ESW theory ........................................................................................ 139
Figure 4.16: (a),(b) Inelastic neutron scattering intensity as a function of E at T = 4 K collected at
SEQUOIA, |2| integrated from 1.2 to 1.5 Å-1 for Cr2WO6 and 1.3 to 1.6 Å-1 for Cr2TeO6. Insets
display the zoom-in view of E between 15 meV and 28 meV for Cr2WO6, and between 20.5 meV
and 28 meV for Cr2TeO6. (c), (d) Inelastic neutron scattering intensity as a function of E calculated
by the ESW theory at T = 0 K, |2| integrated from 1.2 to 1.5 Å-1 for Cr2WO6 and 1.3 to 1.6 Å-1 for
Cr2TeO6 ....................................................................................................................................... 140
Figure 4.17: Powder-averaged magnetic excitation spectra above TN in (a) Cr2WO6 at T = 60 K,
and (b) Cr2TeO6 at T = 120 K, respectively, measured with the incident neutron energy Ei = 65
meV ............................................................................................................................................. 141
Figure 4.18: Spin wave dispersions and intensity of inelastic neutron scattering of (a) Cr2WO6 and
(b) Cr2TeO6 calculated by SpinW code using the LSW theory .................................................. 142
Figure 4.19: The spin wave dispersion and intensity of inelastic neutron scattering of the (a)
transverse and (b) longitudinal modes in Cr2WO6, respectively, calculated using the ESW theory.
(c) and (d) are the powder-averaged magnetic excitation spectra for the transverse and longitudinal
modes, respectively. The broadening factor is chosen as Γ = 1.0 meV that is close to the
instrumental energy resolution .................................................................................................... 146
Figure 4.20: The spin wave dispersion and intensity of inelastic neutron scattering of the (a)
transverse and (b) longitudinal modes in Cr2TeO6, respectively, calculated using the ESW theory.
(c) and (d) are the powder-averaged magnetic excitation spectra for the transverse and longitudinal
modes, respectively. The broadening factor is chosen as Γ = 1.0 meV that is close to the
instrumental energy resolution .................................................................................................... 147
Figure 4.21: Powder-averaged magnetic excitation spectrum of Cr2TeO6 at T = 4 K measured with
the incident neutron energy Ei = 90 meV at T = 4 K .................................................................. 147
Figure 4.22: (a),(b) | |⁄ dependence of the ordered moment 3 ⁄34 and the excitation gaps. The
black line represents the excitation gap in the disordered state. The excitation gaps of the
longitudinal modes (L-mode) and the transverse modes (T-mode) in the ordered state are plotted
by red and blue solid lines, respectively. The exchange parameters of Cr2TeO6 and Cr2WO6 are
denoted by green dashed vertical lines. The ordered moments are represented by magenta lines
..................................................................................................................................................... 149
Figure B1: (a) Schematics of possible magnetic structures. (b) Rocking curve scan across (0 0 5)
magnetic Bragg peak at T = 10 K and B = 10 T, B ∼ b axis........................................................ 159

xvii

Figure B2: Magnetization of Ca3(Ru0.97Ti0.03)2O7 as a function of magnetic field at selected
temperatures ................................................................................................................................ 160

xviii

Chapter 1
Introduction to correlated transition-metal compounds
1.1

Introduction and motivation
Electron-electron correlations in condensed matter can give rise to fascinating physical

phenomena that are completely unexpected based on non-interacting electron pictures. A
prototypical example is the correlation-driven metal-insulator transition (MIT), also known as the
Mott transition [1]. Conventional band theory suggests that whether a material is a metal or
insulator depends on whether the valence band is partially filled or fully occupied. However, many
transition-metal oxides with partially filled d orbitals are insulating rather than metallic, which has
been ascribed to the Coulomb repulsion between electrons occupying the same orbital. Other
exciting discoveries in correlated electron systems include high-Tc superconductivity, colossal
magnetoresistance (CMR) effect, multiferroicity and a diversity of charge, spin and orbital orders
[2-7]. These findings are generally referred to as “emergent phenomena”, since they are very
difficult to predict starting from the properties of their constituents [8]. Theoretical and
experimental efforts in the past few decades have made dramatic progress in the understanding of
the behaviors of correlated electrons. Nevertheless, many fundamental questions are still under
intense investigations and remain challenging to the scientific community.
Transition-metal compounds offer an ideal playground for exploring the physics of emergent
phenomena, owing to the strong correlations among the transition-metal d electrons. The
complexity in the electronic properties arises from the fact that multiple degrees of freedom, such
as charge, spin, orbital and lattice, are simultaneously active. For instance, here we consider a
transition-metal oxide with a perovskite structure as shown in Fig. 1.1, the five-fold degenerate

1

atomic d orbitals split into the triply-degenerate t2g orbitals and the doubly-degenerate eg orbitals
due to the crystal field produced by the octahedral coordination of oxygens. The electronic
configuration is determined by the interplay of the crystal field splitting and the Hund’s rule
coupling. For example, while both Mn3+ (3d4) and Ru4+ (4d4) have three spin-up electrons
occupying the t2g orbitals, the fourth electron occupies one of the eg orbitals in Mn3+ and gives rise
to spin S = 2, whereas it occupies one of the t2g orbitals in Ru4+ and leads to spin S = 1. In addition,
the degeneracy of these electron orbitals can be further lifted by the coupling to the lattice. In an
oxygen octahedron compressed along the z axis, the eg electron in Mn3+ favors the occupation of
the in-plane 

5 6! 5

orbital. This effect, a distortion of the lattice dependent on the electronic state

of the system, is called Jahn-Teller distortion and is prevalent in transition-metal compounds.

Figure 1.1: d-electron orbitals in an octahedral oxygen environment [8].

It is generally believed that the emergent states in correlated electron systems originate from
the competition among various types of interactions, which establishes a delicate balance among
the competing phases of comparable energy. Consequently, very small perturbations, such as
variations in the chemical composition, magnetic field, electric field and structural change by
applying pressure or strain, can dramatically alter the electronic properties of the entire system.
2

This is because the energy scale required to tip the balance is associated with the relative energy
of the competing states and the intervening energy barriers, rather than that of melting the ordered
state itself. The fundamental parameters (interactions) that are relevant in transition-metal
compounds include the hopping amplitude (one-electron bandwidth), crystal field splitting,
Hund’s rule coupling, on-site Coulomb repulsion, exchange interaction, spin-orbit coupling and
Jahn-Teller effect, etc.
In this chapter, we highlight some representative emergent phenomena that have been intensely
studied in correlated transition-metal compounds, including high-Tc superconductivity in cuprates
and iron-based superconductors, CMR in manganites and spin-orbit physics in iridates. As we shall
see, the key ingredients in the physics of correlated electrons, such as phase competition, interplay
among charge, spin, orbital and lattice degrees of freedom, and giant responses to perturbations,
are ubiquitous in transition-metal compounds.

1.2

High-Tc superconductivity in cuprates and iron-based superconductors
High-Tc superconductivity is an archetype of the emergent phenomena in transition-metal

compounds. It was first discovered in a hole-doped cuprate La2-xBaxCuO4 in 1986 [9] and has
attracted tremendous interest since then. The studies on high-Tc cuprates have not only unveiled
many key aspects of the unconventional superconducting state, but also accelerated the
development of condensed matter theories, which have drastically improved our understanding of
complex correlated electron systems.
The nature of the high-Tc superconductivity in cuprates is fundamentally distinct from that of
the conventional superconducting state. Conventional superconductivity was first observed by
Onnes in Mercury in 1911 [10], which has be well described by the phenomenological GinsbergLandau theory [11] and the microscopic Bardeen-Cooper-Schrieffer (BCS) theory [12] developed

3

in the 1950s. The core idea of the BCS theory is that electrons pair into a spin-singlet state
(“Cooper pairs”) at low temperature, and superconductivity arises from the Bose-Einstein
condensation of the Cooper pairs. The attractive force that binds two electrons into a Cooper pair
is the electron-phonon coupling, which limits the critical temperature Tc of conventional
superconductors to be below 30 K. Nevertheless, the much higher Tc found in cuprates, such as Tc
= 90 K in YBa2Cu3O7-δ [13] and Tc = 133 K in HgBa2Ca2Cu3O8+δ [14], cannot be explained by this
scenario, as the electron-lattice coupling in real materials is not sufficiently strong. Thus far, the
pairing mechanism in high-Tc cuprates remains an open question. In addition, the superconducting
state is suggested to be a d-wave spin-singlet state, rather than an isotropic s-wave spin-singlet
state as in conventional superconductors [15].

Figure 1.2: Schematics of the crystal structures of (a) cuprate La2CuO4 [16], (b) iron pnictide
BaFe2As2 and (c) iron chalcogenide FeTe1-xSex [3].

Despite the various types of high-Tc cuprate compounds reported, there are many common
features. First, all the cuprates have layered structures composed of CuO2 planes where
superconductivity emerges. For instance, the crystal structure of the parent compound of the first
high-Tc superconductor La2CuO4 is sketched schematically in Fig. 1.2(a). It has a quasi-two4

dimensional layered perovskite structure where each Cu ion is centered in an oxygen octahedron.
Second, the parent compounds of high-Tc cuprates are antiferromagnetic Mott insulators, and the
superconducting state emerges as the antiferromagnetic order is suppressed by doping charge
carriers, as shown in the phase diagrams in Fig. 1.3 [17]. Interestingly, on the hole-doped side,
there exists a pseudogap phase in close proximity to the superconducting state, the origin of which
is still unknown. Third, it is generally believed that understanding the normal state properties of
high-Tc superconductors is essential to the theory of high-Tc superconductivity. However, the
normal state of high-Tc cuprates exhibits a variety of unusual behaviors, which cannot be described
in the framework of existing solid-state theories. Particularly, there has been growing experimental
evidence showing that nanoscale inhomogeneity is responsible for the anomalous normal state
properties [18]. To date, the mechanism of high-Tc superconductivity in cuprates is still a big
mystery.

Figure 1.3: Phase diagrams of electron- and hole-doped high-Tc cuprates [17].

Intense experimental efforts have been devoted to searching for new high-Tc superconductors
in other transition-metal compounds without copper. Great progress has been made in 2008, when
high-Tc superconductivity was discovered in iron pnictides LaFeAsO1-xFx, A1-xKxFe2As2 (A = Ca,
5

Sr and Ba), AFe2-xBxAs2 (B = Co, Ni and Rh) and iron chalcogenides FeTe1-xSex [3]. Intriguingly,
these iron-based superconductors share similar structures and properties and in many aspects
resemble that of the high-Tc cuprates. For instance, iron-based superconductors also possess
layered crystal structures consisting of Fe-As or Fe-Te/Se planes, as shown schematically in Fig.
1.2(b) and (c). The superconducting state, which appears as a dome in the phase diagram depicted
in Fig. 1.4, emerges as charge carriers are doped into the antiferromagnetic parent compound, in
analogy to the cuprates shown in Fig. 1.3. Nevertheless, the parent compounds of iron-based
superconductors are often semimetallic rather than Mott insulating as in high-Tc cuprates, which
suggests that itinerancy may be important.
In summary, the studies on the high-Tc cuprates and iron-based superconductors are still under
way. They are believed to help improve our understanding of high-Tc superconductivity and
eventually lead to technological applications.

Figure 1.4: The structural and magnetic phase diagrams of electron- and hole-doped iron pnictide
BaFe2As2 [3].

1.3

Colossal magnetoresistance in manganites
The perovskite manganites (RE,AE)n+1MnnO3n+1 (RE = rare earth, AE = alkali earth elements,

n = 1, 2 and ∞) are another typical correlated transition-metal oxides. Early studies on the
6

ferromagnetic metallic compounds have led to the development of the theory of double exchange
interaction [19], which succeeds to account for the properties of manganites qualitatively.
However, quantitative analysis suggests that the underlying physics is far more complicated. A
rich variety of ordered phases can emerge in manganites upon varying the chemical composition.
For instance, the structural and magnetic phase diagrams of the layered La2-2xSr1+2xMn2O7 (n = 2)
is shown in Fig. 1.5. Many complex phases, such as ferromagnetic metallic state (FM), A-type, Ctype and G-type antiferromagnetic insulating (AFI) phase, canted antiferromagnetic state (CAF)
and charge-ordered (CO) state with different degrees of structural distortions are all possible in the
same material system. This suggests that these distinct states are energetically close and can readily
transform from one to another, which poses a great challenge to condensed matter theories.
Moreover, multiple electronic degrees of freedom are intimately correlated, giving rise to complex
charge-spin-orbital ordered states. The schematics of the spin-orbital orders in LaMnO3 and
BiMnO3 are illustrated in Fig. 1.6 as examples. These observations suggest that phase competition
is important in determining the physical properties of manganites.

Figure 1.5: Structural and magnetic phase diagram of the layered manganite La2-2xSr1+2xMn2O7 as
a function of temperature and chemical composition [20].
7

Figure 1.6: Spin-orbital orders in (a) LaMnO3 and (b) BiMnO3 [8].

Another striking emergent phenomenon in perovskite manganites is the CMR effects. In
ordinary metals, the magnetoresistance is usually only a few percent, while in some artificial
ferromagnetic multilayers it can be a few tens of percent due to spin-dependent scattering. Notably,
in hole-doped manganites, the change in resistivity in a magnetic field can be as high as several
orders of magnitude, which renders it promising for industrial applications. There are two main
types of CMR effects observed in manganites. In ferromagnetic metallic compounds, the CMR
effect is most remarkable near the Curie temperature Tc, as shown in Fig. 1.7(a). In contrast, in
charge-ordered manganites, CMR occurs at low temperature via a melting of the charge order
induced by a magnetic field, as shown in Fig. 1.7(b). In addition, electron-lattice coupling has been
found to play an essential role. As we have discussed in Sec. 1.1, in Mn3+ the d electron occupying
the eg orbital is Jahn-Teller active, and a splitting of the in-plane 

5 6! 5

and the out-of-plane

)" 5 67 5 orbitals can occur provided the presence of structural distortions. Such electron-lattice
coupling can give rise to novel physical properties. For example, the appearance of an insulating

behavior near Tc of a ferromagnetic metal, as shown in Fig. 1.7(a), is quite unusual. Close
examinations have revealed that a local structural distortion is induced as the conduction eg
8

electron hops from one site to another, which leads to polaron formation and dominates the
electrical transport in this temperature regime.

Figure 1.7: Magnetoresistance in (a) ferromagnetic metallic Lr2/3(Pb,Ca)1/3MnO3 [21] and (b)
antiferromagnetic charge-ordered Nd0.5Sr0.5MnO3 [22].

Despite the extensive investigations that have been made on the physics of manganites, the
story is still far from complete. Though the significance of the competition between two distinct
ground states, i.e., the ferromagnetic metallic state and the antiferromagnetic insulating state, has
been recognized, there is accumulating experimental evidence suggesting that a single-phase
picture is not sufficient. Nanoscale phase separation is found be to prevalent in manganites, which
is believed to be crucial to the realization of the CMR effect. Similar spatial inhomogeneity has
been observed in many other transition-metal compounds, suggesting that it might be a universal
element in the strongly correlated electronic systems [18].

1.4

Spin-orbit physics in iridates
Iridates are 5d transition-metal oxides characteristic of strong spin-orbit coupling (SOC),

distinct from 3d and some 4d systems where SOC is usually considered as a perturbation. The
9

interplay of SOC and electron correlation has been proposed to lead to various types of novel
phases, depending on their relative strength to the electron hopping amplitude t. A generic phase
diagram established by theories is sketched in Fig. 1.8 [23]. In the weak correlation regime,
topological insulators or semimetals have been discovered and intensely studied [23]. More
complex phases arise in the strong correlation regime, such as Axion insulators, Weyl semimetals,
Topological Mott insulators and spin liquids, which remain as a less explored territory in
condensed matter physics.

Figure 1.8: Generic phase diagram as a function of correlation strength ⁄ and SOC ⁄ . t is the
hopping amplitude [23].

In contrast to 3d transition-metal elements [Fig. 1.1], the strong SOC in 5d transition-metal
ions results in a different multiplet structure of the electronic orbitals. For instance, Ir4+ possesses
five electrons in the 5d orbitals, occupying the t2g levels due to large crystal field splitting Δ9:
generated by the octahedral oxygen coordination. SOC further lifts the degeneracy of the t2g levels,
leading to a ; = 3⁄2 level and a ; = 1⁄2 state, as shown in Fig. 1.9. The magnetism is
10

dominated by the singly occupied ; = 1⁄2 electron. Thus, 5d iridates are excellent material
systems to explore the physics dominated by both electron correlation and SOC. In the rest of this
section, we outline a few examples.

Figure 1.9: d-electron orbital splitting in an octahedral crystal field in the presence of strong SOC
[24].

A natural extension of the studies on the perovskite high-Tc cuprates and CMR manganites is
the Ruddlesden-Popper type perovskite iridates Srn+1IrnO3n+1. Sr2IrO4 and Sr3Ir2O7 are both
antiferromagnetic spin-orbit Mott insulators, though Sr3Ir2O7 is much weaker than Sr2IrO4 [25,26].
They serve as model compounds to study the exchange interaction and magnetic anisotropy in 5d
magnetic systems. Furthermore, since Sr2IrO4 is isostructural to La2CuO4, a very alluring idea is
whether a superconducting state can be achieved by carrier doping. Such a superconducting state,
if existing, is believed to be highly unconventional as pointed out by some theoretical proposals
[27-29], which will to a great extent improve our understanding of unconventional
superconductivity. Nevertheless, experimentally no superconductivity has been observed in this
material so far.
11

Figure 1.10: Phase diagram of the Heisenberg-Kitaev model [24].

In addition to the perovskite iridates, honeycomb iridates, such as Na2IrO3 and Li2IrO3, have
attracted great attention. It is suggested that a bond-dependent exchange interaction is symmetryallowed, which provides a rare experimental realization of the celebrated Kitaev physics [30].
Figure 1.10 shows the phase diagram obtained by solving the Heisenberg-Kitaev model. Notably,
quantum spin liquid states can stabilize in proximity to different types of magnetic ordered phases.
Neutron and resonant x-ray scattering measurements have revealed that Na2IrO3 exhibits a zigzag
magnetic order, which is likely to be close to the hypothetical quantum spin liquid phase [31,32].
To date, the studies on 5d transition-metal compounds in recent years have only uncovered a
tip on the iceberg of the physics dominated by both electron correlations and SOC. Many
opportunities and new frontiers are still waiting for further exploration.

12

1.5

Scope of this thesis
This thesis presents neutron scattering studies on two correlated transition-metal oxides,

namely, Ruddlesden-Popper type ruthenates (Sr,Ca)n+1RunO3n+1 and inverse-trirutile chromates
Cr2MO6 (M = Te, W and Mo). It is organized as follows:
Chapter 2 introduces the basics of neutron scattering, including nuclear and magnetic scattering,
elastic and inelastic scattering, respectively, as well as different types of instruments used in the
work presented in this thesis, such as triple-axis spectrometers, time-of-flight spectrometers, fourcircle diffractometers and powder diffractometers.
Chapter 3 presents neutron scattering studies on the emergent states in Ruddlesden-Popper
type ruthenates (Sr,Ca)n+1RunO3n+1, which are induced upon chemical doping or magnetic field.
Chapter 4 discusses neutron scattering studies on the inverse-trirutile chromates Cr2TeO6,
Cr2WO6 and Cr2MoO6, where the magnetic properties are predominated by S = 3⁄2 spin dimers.
Chapter 5 summarizes all the work and the perspectives on future studies are given.

13

Chapter 2
Introduction to neutron scattering
2.1

Basic properties of neutrons
Neutron scattering is the most powerful technique to investigate both structural and magnetic

properties of condensed matter. Thermal neutrons have the energy of 5 ~ 100 meV and have been
widely used in experiments. The de Broglie wavelength of thermal neutrons is of the same order
as the interatomic distance, therefore interference effects are strong which provide valuable
information on the microscopic structure of the system, in analogy to electron and x-ray scattering.
Nevertheless, as neutrons have no charge, they can penetrate into the system deeply and be
scattered by the nuclei through short-range nuclear forces. In addition, neutrons have spin < =

1⁄2 and can interact with the spin and orbital motion of the unpaired electrons, thus serve as ideal

probes to the magnetic properties of the scattering system. Other than static structures, the
dynamical properties of the scattering systems can also be studied as neutrons are scattered
inelastically. Since the energy of thermal neutrons is of the same order as some of the elementary
excitations in condensed matter, such as phonons and magnons, the energy of these excitations can
be measured more accurately.
A typical geometry of neutron scattering experiments is sketched in Figure 2.1 [33]. A target
is placed into the incident neutron beam and the number of neutrons scattered in the direction (=,
>) in a small solid angle Ω is measured by a neutron detector. The concept “cross section” is
commonly used to analyze the neutron scattering data, as in many other scattering experiments.
The “partial differential cross section” is defined by the equation

14

@
= (CDEFGH IJ CGDHICK KLMGHG NGH KGLIC OCI M KIPO MCQPG R
Ί A
OC OHGLOIC (=, >) TOℎ JOCMP GCGHQV FGTGGC A MC A + A) / Φ Ω A

where ÎŚ is the flux of the incident neutrons, i.e. the number through unit area per second. The
dimension of “cross section” is [area], as implied by its name.

Figure 2.1: Typical geometry of a neutron scattering experiment [33].
Other useful quantities such as “differential cross section” @⁄Ω and “total scattering cross

section” @Z[Z can be defined by the following equations
]

@
@
= \(
)A
Ί
ΊA
^

@Z[Z = \(

@
)Ί
Ί

Neutron scattering experiments measure the cross sections of difference scattering processes.
In the following sections, we give the formulae of the partial differential cross section derived in
Ref. [33] for representative scattering processes concerned in this thesis, including nuclear and
magnetic scattering. The nuclear scattering is a process where neutrons interact with the nuclei via

15

nuclear forces, whereas magnetic scattering is where neutrons couple to the magnetic moments of
unpaired electrons, respectively.

2.2

Nuclear scattering
The basic expression of the partial differential cross section for nuclear scattering is
m]

@
_` 1
=
c F F d \ 〈exph−Oi ∙ jd (0)k exphOi ∙ j ()k〉 exp(−Ol) 
Ω A _ 2aℏ d  


6]

where _ and _ ` are the wave vectors of the incident and scattered neutrons, F and Fd the
scattering lengths of the nuclei at sites ; and ; ` , i = n − no the change in the wave vectors,

jd (0) and j () the positions of the nuclei at sites ; and ; ` at time 0 and , ℏω = A − Ao the

change in the neutron energy, 〈⋯ 〉 denoting thermal average.

The scattering process can be classified into elastic scattering and inelastic scattering,
depending on whether neutrons gain or lose energy. In the case of elastic nuclear scattering, the
coherent differential cross section is
(2a))
@
r s
=w
c y(i − z)|{| (i)|
Ί [t uv
x^
z

where
{| (i) = c F}~ exp(Oi ∙ ) exp(−€~ )
~

w is the number of unit cells in the scattering system, x^ the volume of the unit cell, z a vector in
the reciprocal lattice, {| (i) the structure factor, F}~ the average scattering length of the d-th atom

in the unit cell,  the equilibrium position of the d-th atom in the unit cell, exp(−€~ ) the Debye-

Waller factor. Thus, the scattering occurs only when
i = n − n` = z
16

In the case of inelastic nuclear scattering, the coherent partial differential cross section for
creating one phonon is


_ ` (2a))
1
F}~
@
)[tm =
c c c
exp(−€~ ) exp(Oi ∙ )(i ∙ ƒ~4 )
(
Ί A
_ 2x^
l4
‚3~
4

z

~

× 〈C4 + 1〉y(l − l4 )y(i − 
 − z)

where l4 is the angular frequency of a phonon for the s-th normal mode, 3~ the mass of the d-th
atom in the unit cell, ƒ~4 the polarization vector, C4 the quantum number of the s-th mode, 
 the
wave vector of the normal mode s. The energy and momentum conservation laws must be satisfied
for the scattering to occur
A − A ` = ℏl…

n − n` = z + 

The coherent partial differential cross section for annihilating one phonon is


_ ` (2a)
1
F}~
@
(
)[t6 =
c c c
exp(−€~ ) exp(Oi ∙ )(i ∙ ƒ~4 )
_ 2x^
l4
Ί A
‚34
4

z

~

× 〈C4 〉y(l + l4 )y(i + 
 − z)

Correspondingly, the momentum and energy conservation laws require
A − A ` = −ℏl…
n − n` = z − 


2.3

Magnetic scattering
For a Bravais lattice with localized electrons, the differential cross section for elastic magnetic

scattering is
(

@
1
Š
‹‰ i
‹Š ) c exp(Oi ∙ Œ)〈‰^ 〉〈v 〉
)uv = (†H^ ) w{ Q{(i)} exp(−2€) c(y‰Š − i
Ί
2
‰Š

17

Œ

where † = 1.913 is a constant, H^ =

Ž u 5

‘ ’“

the classical radius of electrons, w the number of unit

cells, Q the Landé g-factor, {(i) the magnetic form factor, exp(−2€) the Debye-Waller factor,
‹‰ and i
‹Š the unit vectors of i , v
y‰Š the Kronecker delta, and α, β representing –, V, — , i

Š

standing for the β-component of the spin angular momentum of atom at site P. Starting from this
general formula, the expressions for different types of magnetic structures such as ferromagnets,
antiferromagnets and noncollinear spin structures can be derived.
For ferromagnets, in a single domain the electron spins align in the same direction. The elastic
differential cross section is
(

@
(2a)) 1
)uv = (†H^ ) w
{ Q{(i)} exp(−2€)(1 − ˜̂ " )〈 " 〉 c y(i − z)
Ί
x^ 2
z

where z is a reciprocal lattice vector. Thus, the magnetic Bragg peaks occur at the same wave
vectors in the reciprocal space as the nuclear Bragg peaks.
For simple antiferromagnets, the magnetic structure can be considered as two interpenetrating
sublattices, where the spins on sublattice A are antiparallel to those on sublattice B. The expression
for the differential cross section is
(

@
(2a))
‹) }y(i − z’ )
)uv = (†H^ ) w’
c |{š (z’ )| exp(−2€) {1 − (z›’ ∙ œ
Ί
x^’
z

where
{š (z’ ) =

1
Q〈ž 〉{(z’ ) c @~ exp(Oz’ ∙ )
2
~

w’ is the number of magnetic unit cells, x^’ the volume of the magnetic unit cell, z’ a vector in

the magnetic reciprocal lattice, @~ being +1 for an A ion and -1 for a B ion, respectively.

18

For noncollinear magnetic structures, for example, a helical structure in Au2Mn as shown in
Fig. 2.2, the elastic differential cross section is
(

w (2a))  1
@
〈〉 { Q{(i)} exp(−2€)(1 + i
‹" )
)uv = (†H^ )
4 x^
2
Ί
× c{y(i + 2 − z) + y(i − 2 − z)}
z

The magnetic Bragg peaks occur at
i=zÂą2
shown as a pair of satellite peaks around the nuclear Bragg peaks.

Figure 2.2: Magnetic structure of Au2Mn [33].

In analogy to the coherent one-phonon scattering, inelastic magnetic scattering can probe the
spin dynamics by the creation or annihilation of magnons. The partial differential cross section of
coherent one-magnon scattering for a single domain is
@
_ ` (2a)) 1
1

(†H
)
‹" ){ Q{(i)} exp(−2€)
=
<(1 + i
^
2
Ί A
_ x^ 2

19

× c{y(i − 
 − z)y(ℏl
 − ℏl)〈C
 + 1〉
z,


The thermal average of C
 is

+y(i + 
 − z)y(ℏl
 + ℏl)〈C
 〉}
〈C
 〉 = {exp ℏl
 ¡¢ − 1}6

where ¡ = 1⁄_£ %. The first term represents the creation of one magnon, whereas the second one
stands for the annihilation of one magnon. The energy and momentum conservation laws are
reflected in the delta functions
n − n` = z ± 


ℏ 
(_ − _ ` ) = ±ℏl¤
2E

2.4

Neutron scattering instrumentation
The neutron scattering experiments presented in this thesis are mainly carried out using the

national facilities in High Flux Isotope Reactor (HFIR) and Spallation Neutron Source (SNS) in
Oak Ridge National Laboratory (ORNL).
HFIR is a reactor-based neutron source, where neutrons are produced by the spontaneous
fission of

235

U. In contrast, SNS is built with an accelerator, and neutrons are produced by

bombarding a heavy target (e.g., mercury) with high-energy protons. The difference is that the
neutron beam is continuous at HFIR, whereas it is pulsed at SNS. Several different types of
instruments have been used for neutron scattering experiments, including triple-axis spectrometers,
time-of-flight spectrometers, four-circle diffractometers and powder diffractometers.
2.4.1

Triple-axis spectrometer

Triple-axis spectrometers are one of the most versatile instruments designed for both elastic
and inelastic neutron scattering experiments. In the work presented in this thesis, the HB-1A fixed20

incident-energy triple-axis spectrometer and CG-4C cold neutron triple-axis spectrometer (CTAX)
at HFIR are two triple-axis spectrometers used. The schematic of the CTAX spectrometer is
sketched in Fig. 2.3.

Figure 2.3: Schematic of the CG-4C cold neutron triple-axis spectrometer (CTAX) in HFIR,
ORNL.

The name “triple-axis” refers to the monochromator axis, the sample axis and the analyzer axis,
respectively. The monochromator is a single crystal which selects the energy of the incident
neutrons by the Bragg reflection. The analyzer is also a single crystal that selects the energy of the
scattered neutrons counted by the detector. Using a triple-axis spectrometer, the scans can be
conveniently made either along a certain direction in the momentum transfer axis at a fixed energy
transfer E (constant-E scan), or along the energy transfer axis at a fixed 2 vector (constant-2 scan),

which enables the measurements of the scattering function <(2, l) in a controlled fashion.
2.4.2

Time-of-flight spectrometer

Time-of-flight spectrometers are another type of widely used neutron scattering instruments
that are usually built with a spallation neutron source. For instance, the schematic of the SEQUOIA
time-of-flight spectrometer at SNS in ORNL is shown in Figure 2.4. In sharp contrast to the triple21

axis spectrometers, time-of-flight spectrometers are equipped with arrays of detectors covering a
large area of the momentum transfer space. The energy of the incident neutrons is selected by a
set of Fermi choppers (e.g., “T0 choppers”, “Fermi choppers”), and that of the scattered neutrons
is determined by the time it takes for the neutrons to arrive at the detectors. Time-of-flight
spectrometers have the advantage of measuring large regions of energy transfer ℏω and

momentum transfer 2 space simultaneously, thus are particularly efficient for inelastic neutron

scattering measurements. In the work presented in this thesis, the SEQUOIA and HYSPEC timeof-flight spectrometers at SNS are used.

Figure 2.4: Schematic of the SEQUOIA time-of-flight spectrometer in SNS, ORNL [34].

2.4.3

Four-circle diffractometer

Four-circle diffractometers are designed for the determination of nuclear and magnetic
structures of single crystals. For example, the picture of the HB-3A four-circle diffractometer at
HFIR is shown in Fig. 2.5 [35] and the schematic representation of the angles defining the
scattering geometry is shown in Fig. 2.6 [36].

22

Figure 2.5: HB-3A four-circle diffractometer in HFIR, ORNL [35].

Figure 2.6: Schematic representation of the angles in four-circle diffractometry [36].

The scattering plane defined by the incident (primary) and diffracted beams is horizontal, and
the “instrument axis” is vertical passing through the center of the instrument where the samples
are mounted. The orientation of the sample is defined by a set of angles =, l,  and >. The
scattering angle 2= is the angle between the incident and diffracted beams. The angle = denotes a
23

rotation of the sample about the instrument axis when the detector rotates by 2=. The sample can
also rotate by an additional angle l about the same axis. The -axis is in the scattering plane,

perpendicular to the  -circle (vertical circle containing the instrument axis). And the angle

between the -axis and the incident beam is = + l. The angle  is then defined as a rotation about
the χ-axis. The >-axis is along the radial direction of the χ-circle forming an angle χ with respect

to the “instrument axis”. The angle > defines a rotation about the >-axis [36].

Compared with triple-axis spectrometers where only momentum transfer 2 in a given

scattering plane can be reached, four-circle diffractometers provide much more degrees of freedom
for sample rotations. Thus, a larger portion of the momentum transfer space can be measured
without remounting the sample.
2.4.4

Powder diffractometer

Powder diffractometers are designed for investigating the structural and magnetic properties
of polycrystalline samples. The picture and schematic of the HB-2A powder diffractometer at
HFIR is shown in Fig. 2.7 and 2.8, respectively [37]. The energy of the incident neutron beam is
selected by the monochromator crystal utilizing Bragg conditions. The elastic scattering cross
section is measured for a broad range of momentum transfer 2 by a detector bank consisting of a
variety of 3He counting tubes.

24

Figure 2.7: HB-2A powder diffractometer in HFIR, ORNL [37].

Figure 2.8: Schematic of the HB-2A powder diffractometer in HFIR, ORNL [37].

25

Chapter 3
Neutron scattering studies on Ruddlesden-Popper type
ruthenates
3.1

Introduction and motivation
The 4d transition-metal oxides, particularly the Ruddlesden-Popper type ruthenates

(Sr,Ca)n+1RunO3n+1, provide excellent playgrounds for exploring the emergent phenomena in
correlated electron systems. Owing to the more extended 4d electron orbitals, the correlation effect
is weaker than that in 3d transition-metal oxides, whereas the SOC is not as strong as that in 5d
materials. In analogy to the high-Tc cuprates and CMR manganites, the electronic charge, spin,
orbital and lattice degrees of freedom are all active. Thus, the magnetic and electronic ground
states are very susceptible to perturbations, such as chemical doping, magnetic field and pressure,
due to the competition of various interactions of comparable energy scale.
(Sr,Ca)n+1RunO3n+1 crystallizes in a Ruddlesden-Popper type perovskite structure consisting of
RuO2 planes separated by (Sr,Ca)O layers. The schematics of the crystal structures of Sr2RuO4 (n
= 1), Sr3Ru2O7 (n = 2) and SrRuO3 (n = ∞) are depicted in Fig. 3.1. The Ru4+ (4d4) magnetic ions
are centered in the oxygen octahedra, with four electrons occupying the t2g levels (S = 1). While
the Sr-based compounds tend to be ferromagnetic and metallic as n increases, substituting Ca for
Sr leads to additional structural distortions and drives the system toward antiferromagnetic and
nonmetallic phases. Moreover, a diversity of exotic emergent properties have been discovered in
(Sr,Ca)n+1RunO3n+1, including unconventional superconductivity [38-40], Mott transitions [41],
metamagnetic quantum critical point (QCP) [42] and orbital order [43,44], etc.

26

In this chapter, using neutron scattering technique together with magnetic susceptibility,
specific heat and electrical transport measurements, we show that the magnetic and electronic
properties of Ruddlesden-Popper type ruthenates can be readily tuned by 3d transition-metal
doping and magnetic field. In Section 3.2 and 3.3, we present the studies on the effects of Fe doping
on Sr2RuO4 (n = 1) and Sr3Ru2O7 (n = 2). In Section 3.4 and 3.5, we discuss the studies on the
effects of magnetic field on Ca3(Ru0.97Ti0.03)2O7 and Ca3(Ru0.95Fe0.05)2O7 (n = 2). The doping
effects of Mn on Ca3Ru2O7 and the magnetic-field-induced phase transitions are described in
Section 3.6. In Section 3.7, we investigate the magnetic transitions of Sr4Ru3O10 (n = 3) driven by
temperature and magnetic field.

Figure 3.1: Crystal structure of Ruddlesden-Popper type ruthenates Srn+1RunO3n+1 (n = 1, 2 and ∞).

27

3.2

Non-Fermi surface nesting driven spin density wave order in

Sr2(Ru,Fe)O4
The single-layer Sr2RuO4 (n = 1) is isostructural to the parent compound of the high-Tc cuprate
La2CuO4. Unconventional superconductivity has been discovered in this material with Tc = 1.5 K
[45], which is proposed to be chiral p-wave different from the s-wave superconductivity in
conventional superconductors or the d-wave spin-singlet one in high-Tc cuprates [46]. Although a
variety of experiments have substantiated the unconventional character of the superconducting
state and examined the symmetry of the order parameter as well as the structure of the
superconducting gap [38-40], the pairing mechanism and the nature of the superconductivity in
Sr2RuO4 are still open questions. For instance, the absence of topologically protected edge currents
[47] is not in line with the time-reversal symmetry breaking p-wave superconductivity [48,49].
Recently it is argued that the superconducting Cooper pairs in Sr2RuO4 cannot be described in
terms of pure singlets or triplets, but are spin-orbit entangled states due to spin-orbit coupling
[50,51].
Furthermore, as in other unconventional superconductors, the correlation between
superconductivity and magnetism in Sr2RuO4 is of particular interest. That is, the superconducting
state is close to magnetic instabilities, and spin fluctuations may be responsible for the
superconducting pairing mechanism [52]. While the normal state of Sr2RuO4 shows Fermi liquid
behavior below T = 25 K [53], the system exhibits strong magnetic instabilities with ferromagnetic
and antiferromagnetic fluctuations coexisting and competing [54,55]: The Fermi surface nesting
of the quasi-one-dimensional ι/β bands leads to antiferromagnetic fluctuations, whereas the close
proximity of the Fermi level to a Van Hove singularity of the quasi-two-dimensional Îł band gives
rise to ferromagnetic fluctuations [56,57]. Ferromagnetic correlations have been corroborated by
28

nuclear magnetic resonance measurements [58], and are suggested to be responsible for the pwave superconductivity [59]. However, neutron scattering experiments found prominent
incommensurate antiferromagnetic fluctuations at 
' = (0.3 0.3 L), arising from Fermi surface
nesting of the ι/β bands [54]. Such incommensurate antiferromagnetic fluctuations along with
strong anisotropy have been proposed to account for the unconventional superconductivity in
Sr2RuO4 [60]. Additionally, recent theoretical and experimental studies have also suggested that
the superconductivity in Sr2RuO4 may be generated by the Cooper pairs on the ι/β bands but not
on the Îł band [61,62].
A fundamental challenge to the understanding of unconventional superconductivity is how the
tendency towards magnetic ordering is suppressed while strong magnetic fluctuations are
maintained that may lead to superconductivity. Intriguingly, for Sr2RuO4, at the bare density
functional level the incommensurate magnetic instability at 
' is sufficiently strong so that
ordering would be expected [63]. This ordering is presumably suppressed by spin fluctuations,
possibly associated with competing orders [55], which is a characteristic common to
unconventional superconductors. A powerful means of exploring the competing magnetic
tendencies in Sr2RuO4 is chemical doping. For instance, moderate substitutions of Ca for Sr sites,
and Ti or Mn for Ru sites have been shown to give rise to static spin density wave (SDW) ordering
with the propagation vector associated with the nesting Fermi surface [64-66]. In contrast, carrier
doping via La substitution for Sr sites enhances ferromagnetic fluctuations by elevating the Fermi
level closer to the Van Hove singularity of the Îł band [67]. These studies attest that the magnetic
ground state of Sr2RuO4 is in the vicinity of the antiferromagnetic and ferromagnetic instabilities.
In this section, we present a commensurate, quasi-two-dimensional SDW ordering in Sr2RuO4
induced by Fe doping for Ru. This magnetic ordered state is characterized by a wave vector 
 =
29

(0.25 0.25 0), in contrast to the incommensurate ones in Ti- and Mn-doped compounds [65,66].
Intriguingly, the incommensurate magnetic excitations at 
' = (0.3 0.3 0) associated with Fermi
surface nesting in the pristine Sr2RuO4 persist in the Fe-doped compounds. This suggests that the
induced static ordered state is not driven by Fermi surface nesting, which has been corroborated
by ab initio electronic structure calculations. These results imply that, in addition to the known
incommensurate magnetic instability, Sr2RuO4 is also in proximity to a commensurate magnetic
tendency which may facilitate the suppression of static magnetic order and give rise to
unconventional superconductivity.
3.2.1

Materials and methods

Single crystals of Sr2Ru1-xFexO4 (x = 0.03 and 0.05) samples were grown using the Floating
Zone method [66]. Magnetization, specific heat and electrical resistivity were measured using the
Physical Property Measurement System (PPMS, Quantum Design). Neutron diffraction
experiments were performed using the HB-1A thermal neutron triple-axis spectrometer at HFIR
in ORNL, with fixed incident neutron energy Ei = 14.6 meV and the collimation setting of 40′-40′sample-40′-80′. The samples were oriented in the (H K 0) and (H H L) scattering planes,
respectively, where (H K L) are in reciprocal lattice units 2π/a, 2π/b and 2π/c (a = b = 3.868 Å and
c = 12.684 Å at 5 K in the tetragonal space group I4/mmm). Magnetic excitations were measured
using the HB-1 triple-axis spectrometer at HFIR and the HYSPEC, CNCS and SEQUOIA timeof-flight spectrometers [68] at SNS in ORNL with samples co-aligned in the (H K 0) plane. The
incident neutron energy was set as Ei = 13 meV at HYSPEC and 3.32 meV at CNCS, respectively.
x-ray absorption spectroscopy (XAS) measurements were carried out on the beamline 4-ID-C at
the Advanced Photon Source (APS) in Argonne National Laboratory (ANL).

30

3.2.2

Magnetic susceptibility, specific heat and resistivity

The main panel of Fig. 3.2(a) shows the temperature dependence of the DC magnetic
susceptibility  of Sr2Ru1-xFexO4 (x = 0.05) measured with B = 1 T applied along the c axis. There
are three remarkable features. (i) Compared to the weak temperature dependence associated with
the Pauli paramagnetism in the parent compound [53], the Fe-doped compound exhibits enhanced
Curie-Weiss susceptibility, which implies the formation of localized moments induced by Fe
doping. The Curie-Weiss fit on the susceptibility at elevated temperatures gives rise to an effective
magnetic moment Îźeff ~ 1.8 ÎźB per Ru. (ii) A paramagnetic-antiferromagnetic phase transition is
observed at TN ~ 64 K, as evidenced by the peak in the magnetic susceptibility data. (iii) Upon
further cooling, a bifurcation between the zero-field-cooled (ZFC) and field-cooled (FC) data
emerges below Tg ~ 16 K, characteristic of a spin-glass-like state.
The inset of Fig. 3.2(a) shows the isothermal magnetization measurements performed at T = 2
K and 20 K. Hysteresis is observed at 2 K which is consistent with the fact that ferromagnetic
correlations develop in the spin-glass-like state. The spin-glass-like state below Tg is also
supported by the frequency dependence of the AC magnetic susceptibility data plotted in Fig.
3.2(b), where one can see that the peak around 16 K weakly shifts to higher temperatures with
increasing measurement frequency. Note that such a bifurcation between FC and ZFC data and the
hysteretic behavior in magnetization are absent for the in-plane magnetic susceptibility
measurements where the antiferromagnetic phase transition is also observed, as shown in Fig. 3.3(a)
and (b), indicating that the spin-glass-like state presumably arises from the development of shortrange ferromagnetic correlations between RuO2 layers. Furthermore, the magnetic moments
induced by Fe doping exhibit magnetic anisotropy with the ordered moment along the c axis.
Similar features have been observed in the Ti- and Mn-doped Sr2RuO4 [69,66].

31

Figure 3.2: (a) Temperature dependence of out-of-plane DC susceptibility  of Sr2Ru0.95Fe0.05O4
after ZFC and FC, respectively. Inset shows the isothermal magnetization as a function of field at
T = 2 and 20 K after ZFC. (b) Temperature dependence of AC susceptibility measured with h = 10
Oe. (c) Temperature dependence of specific heat at zero field. Inset shows the expanded view of
the low-temperature region with the data measured at 9 T included for comparison. The solid red
line is a linear fit for 16 K < T < 30 K. (d) In-plane and out-of-plane resistivity as a function of
temperature.

Figure 3.3: (a) Temperature dependence of in-plane DC susceptibility  of Sr2Ru0.95Fe0.05O4
after ZFC and FC, respectively. (b) Magnetization as a function of magnetic field at T = 2 K and
20 K.

32

Figure 3.2(c) presents the temperature dependence of the specific heat measured at zero field.
An anomaly is observed around TN, corresponding to the onset of antiferromagnetic ordering. The
small change in specific heat at TN might be due to the small magnetic moment size associated
with this spin ordered state. It is worth noting that a specific heat anomaly is not convincingly
observed in the Ti- and Mn-doped compounds, even though a static magnetic order develops at
low temperature in both systems [66,70]. The inset of Fig. 3.2(c) shows the plot of ¦§ ⁄% vs T2 and
the extracted Sommerfeld coefficient is in the range of 27 ~ 35 mJ mol-1 K-2, depending on the
temperature fitting regime, which is slightly smaller than the one obtained from the parent
compound [53], presumably due to the reduced carrier density upon the formation of the SDW
order [66,70]. Interestingly, as seen in the inset, the specific heat at low temperature is enhanced
and can be suppressed upon applying a 9-T magnetic field, which is most probably ascribable to
the magnetic contribution associated with the spin-glass-like state. Temperature dependence of the
out-of-plane and in-plane resistivity  and  are shown in Fig. 3.2(d). Both  and  exhibit

anomalies at TN and close to Tg. Particularly, the increase in  below TN implies a partial gap
opening on the Fermi surface arising from the formation of the antiferromagnetic order.
3.2.3

Neutron diffraction

In order to determine the magnetic structure of Sr2RuO4 induced by Fe doping, we performed
neutron diffraction measurements. Fig. 3.4(a) shows the scans along the [1 0 0] direction over 

= (0.25 0.25 0) at T = 4, 50 and 100 K measured on Sr2Ru0.95Fe0.05O4. A Gaussian-shaped Bragg
peak is clearly observed at 4 and 50 K but vanishes at 100 K, indicating the magnetic origin of this
peak. In addition, the full width at half maximum (FWHM) is found to be determined by the
instrument resolution, which implies the formation of a long-range commensurate magnetic order
in the ab plane. On the contrary, the scans around 
' = (0.3 0.3 0) and (0.3 0.3 1) do not give
33

discernible magnetic intensity. Fig. 3.4(b) shows the scans along the [0 0 1] direction across the
magnetic Bragg peak 
 = (0.25 0.25 0) measured at various temperatures. Distinct from the scans
along the [1 1 0] direction, these curves can be fitted using a Lorentzian function implying a
correlation length of about 20 Å along the c axis at T = 4 K. This suggests that the magnetic
ordering induced by Fe doping is nearly two-dimensional, with very short correlation length along
the c axis. Additionally, the strongest magnetic Bragg peak is observed at 
 = (0.25 0.25 L) with
L = 0 instead of L = 1, indicating the absence of the phase shift between neighboring RuO2 layers
[64]. These results are in sharp contrast to the earlier studies on Ti- and Mn-doped Sr2RuO4, where
short-range incommensurate SDW order with the propagation vector 
' = (0.3 0.3 1) originating
from the nesting Fermi surface are reported [65,66]. This suggests that the mechanism for the
emergence of the commensurate magnetic order in Fe-doped Sr2RuO4 is different.
The temperature dependence of the magnetic intensity at 
 , which is proportional to the
square of the staggered magnetization of the antiferromagnetic order, is shown in Fig. 3.4(c). A
well-defined phase transition is readily seen at TN = 64 K, consistent with the magnetic
susceptibility and specific heat measurements. It is worth noting that for the 3% Fe-doped
compound (x = 0.03), the magnetic Bragg peaks are also observed at 
 and other equivalent wave

vectors, but not at 
' , as presented in the contour map in Fig. 3.4(d). The observation of both

magnetic reflections associated with the magnetic propagation vectors 
 = (0.25 0.25 0) and (0.25
-0.25 0) implies the existence of magnetic twin domains due to the tetragonal symmetry of the
crystal structure. The intensity of the corresponding magnetic reflections is comparable, indicating
that the population of these two magnetic twin domains are nearly equal.

34

Figure 3.4: (a) Scans across 
 = (0.25 0.25 0) along the [1 1 0] direction at T = 4, 50 and 100 K
measured on Sr2Ru0.95Fe0.05O4. (b) Scans across 
 = (0.25 0.25 0) along the [0 0 1] direction at
selected temperatures. (c) The intensity of magnetic Bragg peak 
 = (0.25 0.25 0) as a function
of temperature. Note that the sample measured for (b) is smaller than that for (a,c). (d) Contour
map of elastic magnetic scattering intensity of Sr2Ru0.97Fe0.03O4 at T = 1.6 K after subtracting the
background measured at 80 K. Spurious peaks are denoted by red circles. The residue intensity
near the nuclear peaks (Âą1 Âą1 0) is presumably due to the thermal shift in the lattice parameters.

Possible models of the magnetic structure have been explored by the magnetic representational
analysis using the program BASIREPS in the FULLPROF suite [71] and by the magnetic
symmetry approach using the tools at the Bilbao Crystallographic Server [72]. The crystal
symmetry of Sr2Ru1-xFexO4 (x = 0.03 and 0.05) is assumed to be the same as that of the parent
compound (No. 139, I4/mmm), as discussed in Appendix A. The maximal magnetic space groups
compatible with the space group of the crystal structure and the propagation vector 
 = (0.25 0.25
0) require the magnetic moments to be oriented either along the c axis or in the ab plane. We find
that our data are best described by the SDW models with the moments parallel to the c axis, in

35

agreement with the magnetic susceptibility measurements. Since the moment distribution of the
SDW order can be described as a cosine modulation ¨v = <cos(
jv + y), there are two possible

spin configurations that depend on the choice of the initial phase δ: (i) S (+, 0, –, 0) when y = 0

(magnetic group Ccmcm), or (ii) 1/√2< (+, +, –, –) when y = (2C + 1)a/4, in which n is an

integer (magnetic group Ccmca). The parameter S represents the amplitude of the magnetic
moment which has been estimated from the diffraction data to be about 0.4 ÎźB. The schematics of
these two magnetic structure models are illustrated in Fig. 3.5(a) and (b), respectively. Note that
these two models give rise to identical neutron diffraction patterns and are different only in the
size of the local moment by a factor of √2. The corresponding magnetic twin domain with 
 =
(0.25 -0.25 0) is shown in Fig. 3.5(c) and (d).

Figure 3.5: Schematics of the SDW order of two magnetic twin domains in Sr2Ru1-xFexO4. (a),(b)
SDW with 
 = (0.25 0.25 0). (c),(d) SDW with 
 = (0.25 -0.25 0).

36

3.2.4

Density functional theory calculations and x-ray absorption

The fact that the commensurate magnetic order with a propagation vector 
 = (0.25 0.25 0)
emerges in Fe-doped Sr2RuO4 is very intriguing, considering that both the strong magnetic
fluctuations in the pristine compound and the static incommensurate magnetic order in the Ti- and
Mn-doped compounds occur at the same wave vector 
' = (0.3 0.3 L), which are ascribed to the
Fermi surface nesting of the quasi-one-dimensional ι/β bands [54,65,66]. This raises an important
question: Does the commensurate magnetic order originate from the change in the nesting vector
of the Fermi surface upon Fe substitution?
To address this question, in collaboration with Dr. D. J. Singh and Dr. K. V. Shanavas in
University of Missouri, we performed density functional theory (DFT) calculations for both the
pristine and Fe-doped Sr2RuO4, using the generalized gradient approximation of Perdew, Burke
and Ernzerhof (PBE-GGA) [73] and the projector augmented plane wave method [74] as
implemented in the VASP code [75]. We used an energy cut-off of 450 eV and a k-point sampling
of 22 × 22 × 13 for the single unit cell of Sr2RuO4 and 7 × 7 × 13 for a 3 × 3 × 1 supercell to obtain
a similar density of k-points. The structures were fully optimized. We also checked by performing
calculations for the 3 × 3 × 1 supercell containing one Fe using the all electron general potential
linearized augmented plane wave method [74] as implemented in the WIEN2k code [76] and
relaxing only the internal atomic coordinates. The calculation yielded essentially the same results
as the VASP calculations.

37

Figure 3.6: Electronic structure for a 3 × 3 × 1 supercell of Sr2RuO4 containing one Fe substitution.
(a) Density of states and projections showing majority spin as positive and minority spin as
negative. (b) Fat band plot of the band structure showing Ru character for the unsubstituted
supercell (heavier symbols represent higher Ru character), in comparison with the Fe substituted
cell, emphasized by heavier symbols. (c) Ru character from Ru neighboring Fe and (d) Ru not
neighboring Fe.

The Fermi surfaces and other properties of bulk Sr2RuO4 were similar to the prior reports [59].
All calculations with Fe spin polarized were performed. The density of states (DOS) and
projections of a 3 × 3 × 1 supercell, which contains one Fe atom replacing the Ru in Sr2RuO4, is
shown in Fig. 3.6 along with a band structure plot for the folded zone. The majority spin of the Fe
d orbitals is fully occupied, as shown in Fig. 3.6(a), suggesting that Fe enters as the high-spin
configuration Fe3+ which is in agreement with the XAS measurements presented in Fig. 3.7(a).
The calculated multiplet splitting of the Fe 3s core level in our DFT calculation is 4.45 eV,
consistent with this high-spin state. Thus, the introduction of Fe results in an electron deficiency
of 1 e / Fe for the host lattice. It is important to note that the Fermi surfaces are large and by
Luttinger’s theorem, a change of 0.03 e ~ 0.05 e per cell leads to a change in the Fermi surface
volume of 0.015 ~ 0.025 of the Brillouin zone volume, consistent with the small shifts (~0.1 eV
near EF) along with distortions in the band structure for 11% Fe doping [Fig. 3.6(b)-(d)]. These
small changes resulting from 3% and 5% Fe doping cannot explain the large shift in the magnetic
38

ordering vector, thus a simple itinerant electron explanation in terms of band filling is not operative.
However, in addition to the Fe moments, we find a strong back polarization of the Ru neighboring
Fe amounting to more than 1 ÎźB / Ru neighbor (1.08 ÎźB as obtained by integrating the spin density
over a sphere of radius 2 Bohr around the Ru). We infer that this strong local magnetic coupling
of Fe and Ru frustrates the incommensurate nesting and leads to the commensurate order observed
in our experiments.

Figure 3.7: (a) x-ray absorption spectra of Sr2Ru0.97Fe0.03O4 near the Fe L edge in comparison with
FeO and Fe2O3. (b) Lower panel: contour map of inelastic neutron scattering intensity as a function
of E and K, H integrated from 0.2 to 0.4. Upper panel: cuts along [0 1 0] with E integrated from 3
to 6 meV (black) and from -0.5 to 0.5 meV (red), respectively. H is integrated from 0.2 to 0.4.
Note that the intensities of these two curves are scaled. Data were measured on Sr2Ru0.97Fe0.03O4.

3.2.5

Inelastic neutron scattering

The robustness of the nesting vector of the ι/β bands on the Fermi surface with respect to Fe
doping is corroborated experimentally by the magnetic excitation spectra measured using the timeof-flight inelastic neutron scattering technique. The lower panel of Fig 3.7(b) shows the contour
map of the scattering intensity of Sr2Ru1-xFexO4 (x = 0.03) as a function of E (i.e., energy transfer)
and K. Surprisingly, the dominant magnetic excitations above E = 3 meV are well centered at the
incommensurate wave vectors 
' = (0.3 0.3 0) and (0.3 0.7 0) [black curve in the upper panel of
Fig. 3.7(b)], which is different from the elastic magnetic reflections (red curve) centered at the

39

commensurate wave vectors 
 = (0.25 0.25 0) and (0.25 0.75 0). In addition, the magnetic
fluctuations barely show any energy dependence, similar to that in both the pristine and the Tidoped compounds [54,77]. While the magnetic excitations related to this ordered state warrant
further investigation, the coexistence of the commensurate magnetic order at 
 = (0.25 0.25 0)

and the dynamic spin fluctuations at 
' = (0.3 0.3 0) in Fe-doped Sr2RuO4 implies that the
magnetic order is not driven by the Fermi surface nesting as in the Ti- and Mn-doped compounds
[65,66]. Thus, Fe doping reveals a previously unanticipated commensurate magnetic instability in
Sr2RuO4 at 
 = (0.25 0.25 0), which competes with the known incommensurate one. These results
suggest that the tendency toward magnetic ordering in Sr2RuO4 is suppressed by quantum
fluctuations associated with competing magnetic instabilities, while strong spin fluctuations are
maintained and may give rise to the unconventional superconducting state.
3.2.6

Summary

We have unraveled a commensurate SDW order with a propagation vector 
 = (0.25 0.25 0)

in Sr2RuO4 upon Fe doping for Ru, whereas the incommensurate magnetic fluctuations at 
' =
(0.3 0.3 L) in the pristine compound persist. This suggests that this commensurate ordered state
does not arise from Fermi surface nesting, in contrast to the previous studies on Ti-, Mn- and Cadoped Sr2RuO4 [64-66]. This study indicates that the unconventional superconducting state in
Sr2RuO4 is not only adjacent to the known incommensurate magnetic order, but also to a
commensurate one.

40

3.3

Magnetic order and metal-insulator transition in Sr3(Ru,Fe)2O7.
The bilayer Sr3Ru2O7 (n = 2) exhibits a Fermi liquid ground state that is close to a

ferromagnetic instability [78]. More intriguingly, it possesses a magnetic-field-tuned QCP, where
non-Fermi liquid behavior [42] and highly anisotropic magnetoresistance [79] emerge. Recently
neutron diffraction measurements have identified two different magnetically ordered phases close
to the QCP [80]. Furthermore, it is revealed that ferromagnetic and antiferromagnetic spin
fluctuations coexist in this system [81], and the latter is suggested to be dominant near the critical
field, which is unexpected for a metamagnetic transition [82].
Chemical doping is an effective approach to tailor the materials’ properties by stabilizing one
of the competing phases while suppressing others. In contrast to Sr2RuO4, Ti and Mn dopants have
different effects on the physical properties of Sr3Ru2O7. While both Ti and Mn doping lead to
insulating electronic transport behaviors at low temperature, Ti doping gives rise to an
incommensurate SDW order with a propagation vector 
' = (0.24 0.24 0) [83,84], but Mn-doped

Sr3Ru2O7 exhibits a commensurate E-type antiferromagnetic order characterized by 
 = (0.25

0.25 0) [85,86]. In both cases, the static magnetic orders do not reflect the dominant dynamical
magnetic correlations at 
 = (0.09 0 0) and (0.25 0 0) in the pristine Sr3Ru2O7 [81], distinct from
those in Ti- and Mn-doped Sr2RuO4 [65,66]. In addition, synchrotron XAS measurements have
found that Mn dopants in Sr3Ru2O7 display a valence value of Mn3+, and show an inversion of the
conventional crystal-field level hierarchy, which is suggested to be due to the hybridization
between the Ru-O 4d-2p bands and the Mn 3d orbitals [87]. Inspired by the discovery of the
commensurate SDW order in Fe-doped Sr2RuO4, it would be interesting to study the effects of Fe
doping on the ground state properties of Sr3Ru2O7.

41

In this section, we present the magnetic and electronic properties of Fe-doped Sr3Ru2O7. In
contrast to the paramagnetic metallic state in the parent compound, Sr3(Ru1-xFex)2O7 (x = 0.01)
shows a metallic spin-glass-like ground state, whereas for x = 0.03 an insulating phase with a
quasi-two-dimensional E-type antiferromagnetic order with the propagation vector 
 = (0.25 0.25
0) is observed below TN ~ 40 K. These features are similar to that in Mn-doped Sr3Ru2O7 [88],
which suggests that the magnetically ordered state induced upon Fe and Mn doping originates
from the intrinsic instability of Sr3Ru2O7.
3.3.1

Materials and methods

Single crystals of Sr3(Ru1-xFex)2O7 (x = 0.01 and 0.03) were grown using the Floating Zone
method. Magnetization, specific heat and resistivity measurements were performed using PPMS.
Neutron diffraction experiments were carried out using the HB-1A thermal neutron triple-axis
spectrometer at HFIR in ORNL. The energy of the incident neutrons is fixed as Ei = 14.6 meV.
The single crystals were oriented in the (H K 0) and (H H L) scattering planes, and were mounted
in an aluminum sample can and cooled using a closed-cycle helium refrigerator down to 4 K. For
convenience, we describe our neutron diffraction data using the tetragonal lattice symmetry
I4/mmm with a = b = 3.874 Å and c = 20.69 Å. The neutron intensity was presented in the unit of
counts per monitor count unit (mcu), with 1 mcu corresponding to approximately 1 second.
Synchrotron XAS experiments were performed using the beamline 4-ID-C at APS in ANL to
measure the valence state of the Fe dopants.
3.3.2

Magnetic susceptibility, specific heat and resistivity

The main panels of Fig. 3.8(a) and (b) show the temperature dependence of the magnetic
susceptibility  of Sr3(Ru1-xFex)2O7 (x = 0.01 and 0.03) after ZFC and FC, respectively. In the
pristine Sr3Ru2O7, the magnetic susceptibility is nearly isotropic with a pronounced peak at T = 16

42

K in both  and  [78], which is ascribable to the crossover of the nature of the dominant
magnetic fluctuations from ferromagnetic to antiferromagnetic upon cooling [81]. No hysteresis
effect is observed between the ZFC and FC data [78]. In contrast, for x = 0.01, the peak at 16 K is
completely suppressed in both  and  . Instead,  (T) shows a maximum at Tg ~ 4 K, below
which a bifurcation between the ZFC and FC data is clearly seen, characteristic of a spin-glasslike state. Figure 3.8(c) displays the magnetization as a function of the magnetic field applied along
the c axis at T = 2 and 6 K, respectively. The hysteretic behavior observed at 2 K indicates that
short-range ferromagnetic correlations develop between the neighboring RuO2 layers along the c
axis below Tg. On the contrary, as shown in the inset of Fig. 3.8(a), the in-plane magnetic
susceptibility  shows a paramagnetic behavior down to 2 K, and there is no bifurcation between
the ZFC and FC curves. These results suggest that Fe doping in Sr3Ru2O7 induces strong magnetic
anisotropy with the easy axis along the c direction, similar to Fe-doped Sr2RuO4 [89]. Furthermore,
the metamagnetic transition observed in the parent compound has been completely smeared out,
as shown in the inset of Fig. 3.8(c), which implies that Fe doping drives the system away from
metamagnetism, as reported in Ti-doped Sr3Ru2O7 [83].
Interestingly, for x = 0.03, a sharp peak in magnetic susceptibility develops in both  and 
at TN ~ 40 K, suggestive of the onset of a paramagnetic-antiferromagnetic phase transition. Neither
the bifurcation in the magnetic susceptibility curves M (T) measured with ZFC and FC histories
nor the hysteresis in the isothermal magnetization M (H) is observed. In addition, it is noteworthy
that at 9 T, the magnetization at 2 K is much smaller than that at 50 K. These results suggest that
the nature of these peaks in the magnetic susceptibility data is fundamentally different from that
in the x = 0.01 compound, which implies that 3% Fe dopants lead to the formation of a long-range
antiferromagnetic ordering.

43

Figure 3.8: (a),(b) Temperature dependence of magnetic susceptibility  of Sr3(Ru1-xFex)2O7 (x =
0.01 and 0.03) after ZFC and FC. Insets show the in-plane magnetic susceptibility  as a
function of temperature after ZFC and FC. (c),(d) Isothermal magnetization as a function of
magnetic field of Sr3(Ru1-xFex)2O7 (x = 0.01 and 0.03), respectively. The field is applied along the
c axis. Inset shows the data of the x = 0.01 compound at 2 K up to 9 T.

Figure 3.9: (a) Specific heat of Sr3(Ru1-xFex)2O7 (x = 0.01 and 0.03) as a function of temperature.
The blue arrow denotes TN obtained from magnetic susceptibility measurements. (b) In-plane
resistivity ρ of Sr3(Ru1-xFex)2O7 as a function of temperature. Inset shows ρ vs T2 in the lowtemperature regime for x = 0.01. The red line is a fit using linear function.

44

Figure 3.9(a) shows the temperature dependence of the specific heat for Sr3(Ru1-xFex)2O7 (x =
0.01 and 0.03). It is worth pointing out two distinctive features. First, for the x = 0.03 compound,
an anomaly is clearly observed close to TN ~ 40 K, which is indicative of the formation of a longrange antiferromagnetic order. In contrast, no anomaly in specific heat has been observed in the x
= 0.01 compound in this temperature range. Second, the x = 0.01 compound exhibits an upturn
below T ~ 10 K, which is presumably of magnetic origin. Note that ¦§ ⁄% measured at the lowest
temperature (~0.21 J / K2 mol) is much larger than that for the x = 0.03 compound (~0.06 J / K2
mol). Similar behaviors have been observed in Ti-doped Sr3Ru2O7 [84], which has been suggested
to be associated with the spin fluctuations. The increase of the specific heat in the low-temperature
limit in the disordered state (e.g., x = 0.01) can be ascribed to the softening of the magnetic
fluctuations when being closer to the magnetically ordered state induced upon doping, whereas the
suppression of the value of ¦§ ⁄% in the antiferromagnetically ordered phase for the x = 0.03
compound is due to the opening of the gap in the spin excitation spectrum [84].
The temperature dependence of the in-plane resistivity ρ of Fe-doped Sr3Ru2O7 is presented
in Fig. 3.9(b). Similar to the pristine Sr3Ru2O7, the x = 0.01 compound exhibits metallic behavior
down to 2 K. At low temperature, ρ (T) shows T2 dependence as shown in the inset, which is
characteristic of the Fermi liquid behavior. However, the field-induced anisotropic and highly
resistive state in the parent compound [79] is completely suppressed in the x = 0.01 compound,
consistent with the absence of the metamagnetic transition in the magnetic susceptibility data
shown in the inset of Fig. 3.8(c). In contrast, a metal-insulator transition (MIT) is observed in the
x = 0.03 compound at TMIT ~ 44 K, similar behaviors have been observed in Ti- and Mn-doped
Sr3Ru2O7 [84,86]. It has been suggested that the MIT in the Mn-doped compound is Mott type,
induced by electron correlations instead of Slater type due to the formation of the antiferromagnetic

45

order [85]. As TMIT of the x = 0.03 compound of Fe-doped Sr3Ru2O7 is very close to the
antiferromagnetic transition temperature TN ~ 40 K, it might be helpful to study this material
system with higher Fe doping concentrations to resolve the nature of this MIT.
3.3.3

Neutron diffraction

In order to determine the spin structure of the antiferromagnetically ordered phase in
Sr3(Ru0.97Fe0.03)2O7, we have carried out neutron diffraction measurements. The magnetic Bragg
peaks were observed at 
 = (0.25 0.25 0) and the equivalent wave vectors in the reciprocal space,
such as (0.75 0.75 0) and (1.25 1.25 0), etc. Figure 3.10(a) shows the scans along the [1 1 0]
direction across 
 at representative temperatures. The data are well fitted by Gaussian functions
and the peak width is resolution limited, indicating a long-range ordering in the ab plane.
Temperature dependence of the peak intensity of 
 is shown in Fig. 3.10(b). The magnetic
intensity starts to increase at T ~ 40 K on cooling, consistent with TN obtained in the magnetic
susceptibility measurements. Figure 3.10(c) shows the scans along the [0 0 1] direction across 

at selected temperatures. These curves are best fitted by Lorentzian functions, in contrast to
Gaussian for the scans along the [1 1 0] direction, which indicates short-range magnetic
correlations (1 / FWHM = 1.2c, c = 20.69 Å at T = 4 K) along the c axis. These results suggest that
the magnetic ordering is quasi-two-dimensional, similar to that in Ti- and Mn-doped Sr3Ru2O7
[84,88]. Figure 3.10(d) displays the same scan across 
 = (0.25 0.25 0) and (0.75 0.75 0) for a
wider L range at T = 4 K. It is worth noting that the magnetic intensities are only observed at even
values of L. Furthermore, while the intensity of (0.25 0.25 L) with L = 0 and 2 is much greater than
that of (0.75 0.75 L), at L = 4 and 6 the intensities of these two types of magnetic Bragg peaks are
comparable.

46

Figure 3.10: Neutron diffraction data of Sr3(Ru1-xFex)2O7 (x = 0.03). (a) Scans around 
 = (0.25
0.25 0) along the [1 1 0] direction at representative temperatures. (b) Temperature dependence of
the peak intensity of 
 = (0.25 0.25 0). (c),(d) Scans around 
 = (0.25 0.25 0) and (0.75 0.75 0)
along the [0 0 1] direction, respectively.

Possible magnetic structure models have been explored by the magnetic representational
analysis using FULLPROF [71] and the magnetic symmetry analysis using the programs at Bilbao
Crystallographic Server [72]. The results obtained are in agreement with each other. Due to the
orthorhombic crystal symmetry and the propagation vector 
 = (0.25 0.25 0) (i.e., (0.5 0 0) in
orthorhombic symmetry notation [88]), our data are best described by the E-type antiferromagnetic
structure with zigzag spin chains in the ab plane, as shown in Fig. 3.11(a). The magnetic moments
in a bilayer are ferromagnetically aligned, similar to that in Mn-doped Sr3Ru2O7 [88]. For this
magnetic structure model, the squared structure factors of the magnetic reflections 
 = (0.25 0.25
L) and (0.75 0.75 L) are nonzero only for even values of L:
|<(0.25 0.25 ­)| = |<(0.75 0.75 ­)| ~LIK  (2a∆­)
47

where 2∆ ~ 0.20 is the separation between neighboring RuO2 planes. This is in line with the data
shown in Fig. 3.10(d). In addition, the comparison of the intensity of (0.25 0.25 L) and (0.75 0.75
L) also provides clues on the spin orientation. The cross section of the magnetic neutron diffraction
is given by
‹ ∙ ³́) ]|<(
)|
@(
) ∝ |{(
)| [1 − (


‹ , ³́ the unit
where {(
) is the magnetic form factor, <(
) the magnetic structure factor, and 


vectors of the wave vector 
 and the magnetic moment M. The fact that the intensity of (0.25 0.25
L) and (0.75 0.75 L) for L = 4 is comparable suggests that the magnetic moments are along the c
‹ ∙ ³́) ] is greater for the latter, since the structure
axis, such that the polarization factor [1 − (


factors <(
) are equal for these two reflections and the magnetic form factors {(
) decrease

rapidly with the increasing modulus of 
. Similar argument holds for the L = 6 case. The ordered

moment is estimated to be ~0.5 ÎźB for the x = 0.03 compound, which is much smaller than 2 ÎźB
expected for fully localized Ru4+ magnetic moments.

Figure 3.11: (a) Schematic of the E-type antiferromagnetic order of one bilayer in
Sr3(Ru0.97Fe0.03)2O7. (b) Synchrotron XAS measurements on Sr3(Ru0.97Fe0.03)2O7 and the reference
samples FeO (Fe2+) and Fe2O3 (Fe3+).

48

3.3.4

Discussions

There are several remarkable features observed in Fe-doped Sr3Ru2O7 that may help to
elucidate the effects of 3d transition-metal dopants on this system.
First of all, the magnetic structure of Sr3(Ru0.97Fe0.03)2O7 is very similar to that in the Mndoped compound, which is independent on the Mn doping concentration [86]. This suggests that
the magnetic ordering is an inherent instability of the system and the role of Fe and Mn dopants is
to tip the balance between the competing magnetic tendencies.
Second, the propagation vectors of the magnetic orders induced by 3d transition-metal doping,
e.g., Ti [84], Mn [86] and Fe in this study, do not correspond to the dominant spin fluctuations in
the pristine Sr3Ru2O7, which are centered on the principal axes of the tetragonal Brillouin zone at

 = (0.09 0 0) and (0.25 0 0) [81]. It has been reported that these incommensurate
antiferromagnetic fluctuations originate from the nesting of the Fermi surface [81]. Therefore, the
observation of a commensurate E-type antiferromagnetic order with the propagation vector distinct
from the wave vectors of the spin fluctuations in the pristine Sr3Ru2O7 suggests that the impurityinduced magnetic order in this bilayer ruthenate system is not due to Fermi surface nesting, a
feature different from that in Ti- and Mn-doped Sr2RuO4 [65,66].
Third, it is well-known that the magnetism in layered ruthenates is strongly correlated with
structural transitions. In Mn-doped Sr3Ru2O7, a change in the lattice constants of ~0.1% has been
reported at TN by x-ray diffraction measurements [86]. However, no change in the lattice constants
at either TN or TMIT was convincingly observed in Fe-doped Sr3Ru2O7 via single-crystal neutron
diffraction measurements, although we cannot exclude the possibility that the change is too small
to be seen within the instrumental resolution. Since the ionic radius of the Fe dopants is important
to the structural distortions of the RuO6 octahedra, we have performed synchrotron XAS

49

measurements on the x = 0.03 compound at the Fe L2,3 edge at room temperature to determine the
valence value of Fe. Two reference samples, FeO and Fe2O3, with well-defined valence values
were measured simultaneously. As shown in Fig. 3.11(b), the peak position and the line shape of
the data suggest that the Fe dopants are in the Fe3+ valence state, the same as Mn3+ in Mn-doped
Sr3Ru2O7 [87]. Since the ionic radii of Fe3+ and Mn3+ are the same and are slightly larger than that
of Ru4+ [90], we expect that Fe dopants give rise to similar structural effects as Mn. It has been
reported that Mn doping reduces the rotation of the RuO6 octahedron and enhances the octahedral
flattening via a reduction in the apical Ru-O bond length in the magnetically ordered state [91].
Theoretical studies on the correlation between structure and magnetism have been performed on
Ca2-xSrxRuO4 and are valuable references to the Ruddlesden-Popper type perovskite ruthenates. It
is revealed that the rotation of the RuO6 octahedron favors ferromagnetism and the subsequent
tilting leads to antiferromagnetism, while the flattening of the octahedron is essential to stabilize
the ferromagnetic or antiferromagnetic state [92]. However, the observed E-type antiferromagnetic
order in Fe- and Mn-doped Sr3Ru2O7 is not in line with this picture that the magnetic order is due
to the doping-induced changes in the structural distortions discussed above.
Finally, the interplay between the 3d orbitals of the dopants and the Ru 4d orbitals might be
essential to the impurity-induced magnetic ordering and MIT. For instance, it has been reported
that the hybridization between the Mn 3d and Ru-O orbitals can reverse the conventional hierarchy
of the crystal-field levels of the half-filled Mn eg orbital in Mn-doped Sr3Ru2O7 [87]. Recent
studies on Fe-doped Sr2RuO4 has found a commensurate magnetic ordering characterized by the
propagation wave vector 
 = (0.25 0.25 0). DFT calculations show that Fe doping barely changes
the Fermi surface but induces strong back polarization on the Ru neighboring Fe, which may lead
to the formation of the commensurate magnetic order [89]. Considering the similarity between the

50

electronic structures of Sr2RuO4 and Sr3Ru2O7 [87], it is very likely that a similar scenario holds
true for the Fe-doped Sr3Ru2O7, though first principles calculations are highly desired to examine
this conjecture.
3.3.5

Summary

We have studied the magnetic and electronic properties of the bilayer ruthenate Sr3Ru2O7
doped by Fe. We find that in contrast to the Fermi liquid ground state in the pristine compound,
1% Fe substitution leads to a metallic spin-glass-like state, whereas an insulating E-type
antiferromagnetic ordered phase is induced below TN ~ 40 K in the 3% Fe-doped compound. This
magnetic ordering is quasi-two-dimensional with short correlation length along the c direction,
similar to that in Mn-doped Sr3Ru2O7. These results suggest that this doping-induced ordered state
is associated with the intrinsic magnetic instability of the pristine compound, which can be readily
tuned via the local exchange coupling between the magnetic dopants and the Ru hosts.

51

3.4

Field-induced insulator-metal transition and colossal magnetoresistance

in Ca3(Ru,Ti)2O7
The bilayer Ca3Ru2O7 (n = 2) crystallizes in an orthorhombic structure with the space group
Bb21m (No. 36) [93]. Upon cooling, it undergoes an antiferromagnetic transition at TN = 56 K and
a MIT at TMIT = 48 K [94]. Below 30 K, the material exhibits quasi-two-dimensional metallic
behavior [95] due to small un-nested Fermi surface pockets surviving through the MIT [96].
Neutron diffraction measurements have revealed that the magnetic moments in the bilayer are
aligned ferromagnetically, and the bilayers are stacked antiferromagnetically along the c axis. The
magnetic moments are along the a axis (AFM-a) at TMIT < T < TN but are along the b axis (AFMb) below TMIT [97].
Previous studies have shown that 3d transition-metal doping for Ru can dramatically change
the magnetic and electronic properties of ruthenates. In Ca3Ru2O7, a Mott insulating state can be
achieved by doping as low as ~3% Ti into the Ru sites. The Mott character of the charge gap has
been confirmed by the recent hard x-ray photoemission measurements [98]. Concomitantly, the
ground state magnetic structure transforms from AFM-b to G-type with the nearest-neighbor spins
coupled antiferromagnetically (G-AFM). The major effect of Ti substitution is to disrupt carrier
hopping and narrow the bandwidth [99]. The phase diagram of Ca3(Ru1-xTix)2O7 as a function of
temperature T and Ti doping concentration x is shown in Fig. 3.12(a).

52

Figure 3.12: (a) Magnetic and electronic phase diagram of Ca3(Ru1-xTix)2O7. A-G-Inc: coexistence
of AFM-a, G-AFM and incommensurate magnetic structures. Pink region represents Mott
insulating state, while dark green stands for weakly localized state (Modified from Ref. [100]).

Ca3(Ru0.97Ti0.03)2O7 shows a metallic state with AFM-a type antiferromagnetic order at TMIT
(~46 K) < T < TN (~62 K), and a Mott insulating state with G-AFM order at T < TMIT. According
to the T − x phase diagram of Ca3(Ru1-xTix)2O7, Ca3(Ru0.97Ti0.03)2O7 sits right at the boundary of
the correlated metallic phase and the Mott insulating phase [100], where the physical properties
and the responses to external stimuli are to a great extent determined by the competition between
these two distinct magnetic and electronic instabilities. Thus, it serves as a model system to explore
the exotic properties of strongly correlated electrons in the vicinity of the MIT.
In this section, we present the CMR phenomenon observed in Ca3(Ru0.97Ti0.03)2O7 which
occurs concurrently with a magnetic phase transition and a drastic change in the lattice structure.
We argue that the underlying mechanism of the CMR effect in Ti-doped Ca3Ru2O7 is
fundamentally different from that in manganites. Instead, we ascribe it to a collapse of the Mott
insulating ground state driven by a magnetic field.

53

3.4.1

Materials and methods

Ca3(Ru0.97Ti0.03)2O7 single crystals were grown using the Floating Zone method. The Ti doping
concentration was verified using the energy dispersive spectroscopy (EDS). Magnetoresistance
and magnetic susceptibility measurements were done using PPMS. Neutron diffraction
experiments were performed at Helmholtz Zentrum Berlin (HZB) using the E4 two-axis
diffractometer equipped with a 20 × 20 cm2 position-sensitive detector. The neutron incident
wavelength was λ = 2.434 Å. The sample of ~100 mg was aligned in the (H 0 L) scattering plane
and cooled by a helium cryostat. The magnetic field was applied along the b axis, while the Bragg
reflections were measured up to 14.5 T.
3.4.2

Magnetoresistance

Figure 3.13(a) and (b) show the temperature dependence of the out-of-plane resistivity  and

the in-plane resistivity  of Ca3(Ru0.97Ti0.03)2O7 measured with a magnetic field of 0 T and 9 T

applied along the b axis. At zero field, Ca3(Ru0.97Ti0.03)2O7 undergoes a MIT on cooling, with a
sharp increase in  and  by ~5 and ~3 orders of magnitude, respectively, at TMIT. In contrast,

at B = 9 T, both  and  gradually increase at low temperature by only a smaller factor,
attributable to disorder scattering. Such a dramatic magnetoresistive effect is also observed in the
isothermal resistivity measurements at T = 10 K shown in Fig. 3.13(c) and (d). With increasing
magnetic field applied along the b axis, both  and  decreases sharply at a critical field Bc ~
8.5 T and show a hysteresis with decreasing field, indicating a first-order insulator-to-metal
transition (IMT). On the other hand, both resistivity values decrease slightly with the field applied
along the a axis, which is close to the magnetic hard axis.

54

Figure 3.13: (a),(b) Temperature dependence of resistivity  and  measured at B = 0 and 9 T
applied along the b axis. The magnetic field was applied at T = 100 K and the measurements were
taken while cooling. (c),(d) Magnetic field dependence of  and  at T = 10 K.
3.4.3

Neutron diffraction

To obtain a deeper insight into the magnetic-field-induced IMT in Ca3(Ru0.97Ti0.03)2O7, we
performed single-crystal neutron diffraction measurements at T = 10 K with the field applied along
the b axis. Interestingly, the lattice parameters of Ca3(Ru0.97Ti0.03)2O7 change drastically when the
magnetic field approaches the critical value where the resistivity drops significantly. As shown in
Fig. 3.14(a), the lattice constant c increases by ~0.78%. Similar behavior has been observed by
heating the sample at zero magnetic field from low temperature to above TMIT [99]. These
observations demonstrate that the Mott insulating ground state of Ca3(Ru0.97Ti0.03)2O7 is stabilized
by a strong coupling to the lattice distortions, in analogy to the bandwidth-controlled Mott system
Ca2-xSrxRuO4 [41,101]. In both cases, the emergence of the Mott insulating state is accompanied
55

by a structural transition from a long c axis to a short one [99,102]. The enhanced RuO6 octahedral
flattening and tilting below TMIT reduce the bandwidth [103] and change the occupancy of the t2g
orbitals [92,104], leading to a MIT [103,105,106].

Figure 3.14: (a) Field dependence of lattice parameters. Statistical errors from the fitting procedure
are smaller than the symbol size. (b) Field dependence of the integrated intensity of magnetic
Bragg peaks (1 0 2) and (0 0 1), respectively, measured at T = 10 K. The magnetic field was
increased for both nuclear and magnetic Bragg peak scans. (c),(d) Rocking curves of (1 0 2) and
(0 0 1) measured at B = 0 T and 10 T applied along the b axis, T = 10 K.

The remarkable lattice structure and resistivity changes in Ca3(Ru0.97Ti0.03)2O7 induced by a
magnetic field are accompanied by a magnetic structure transition. Fig. 3.14(c) and (d) show the
rocking curves of (1 0 2) and (0 0 1) magnetic Bragg peaks, respectively, measured at T = 10 K, B
= 0 T and 10 T after the sample was cooled down in zero magnetic field. Note that the (1 0 2)
magnetic Bragg peak refers to G-AFM magnetic structure as in Ti-doped Ca3Ru2O7 [99], while (0
0 1) corresponds to AFM-b or AFM-a type magnetic structures as in the pristine Ca3Ru2O7 [97].
The disappearance of (1 0 2) and the emergence of (0 0 1) at B = 10 T indicate a field-induced

56

magnetic structure transition from G-AFM to one similar to AFM-a (or AFM-b). However, the
field-induced state is not a collinear magnetic structure but a canted antiferromagnetic phase
(denoted as CAFM), i.e., a vector superposition of AFM-a and a ferromagnetic component along
the b axis, as discussed in Appendix B. The strong coupling between the field-induced structural
transition and the magnetic transition can be readily seen. The integrated intensity of (1 0 2) and
(0 0 1) magnetic Bragg peaks measured at T = 10 K as a function of increasing magnetic field is
shown in Fig. 3.14(b). The intensity of (0 0 1) increases significantly at Bc where that of (1 0 2)
has a sharp drop, indicative of a magnetic phase transition. Clearly, the critical field of the magnetic
transition coincides with that of the structural transition. These observations are similar to those
observed in the temperature-dependent studies on the same compound but in the absence of a
magnetic field, reinforcing the presence of strong spin-lattice coupling in this system [99]. The
peak intensities of (1 0 2) and (0 0 1) as a function of magnetic field at representative temperatures
are shown in Fig. 3.15. The sample was initially cooled down to 10 K at zero field and the peak
intensities were measured with the field increasing first, followed by the field decreasing, prior to
warming up to the next measurement temperature. The hysteresis observed at low temperature
indicates the first-order nature of the field-induced transition between G-AFM and CAFM. The
CMR effects are observed only accompanying the field-driven G-AFM to CAFM transition below
TMIT. On the contrary, the resistivity shows little change at TMIT < T < TN, where (0 0 1) is
suppressed as the magnetic field increases until the magnetic moments are fully polarized by the
magnetic field.

57

Figure 3.15: (a),(b) Neutron intensity of the (1 0 2) and (0 0 1) magnetic Bragg peaks as a function
of magnetic field measured at different temperatures. The solid curve is for increasing field and
the dashed one is for decreasing field.

Figure 3.16 presents the T − B phase diagram which summarizes our electronic transport and
neutron diffraction measurements. At TMIT < T < TN, upon applying the magnetic field along the b
axis, Ca3(Ru0.97Ti0.03)2O7 undergoes a magnetic phase transition from AFM-a to CAFM, and
finally to a fully spin polarized paramagnetic state (PM), a feature similar to that in the parent
compound [97]. In contrast, below TMIT, Ca3(Ru0.97Ti0.03)2O7 exhibits a magnetic phase transition
from G-AFM to CAFM, which occurs simultaneously with the electronic structure changing from
an insulating state to a weakly localized state, implying the strong correlation between the fieldinduced IMT and magnetic phase transition.

58

Figure 3.16: (a) T−B phase diagram of Ca3(Ru0.97Ti0.03)2O7 in a magnetic field along the b axis.
The solid squares, circles and diamonds are phase boundaries determined by neutron diffraction
measurements: solid squares denote the points where (0 0 1) shows up, and solid circles represent
where (1 0 2) disappears. The solid diamonds stand for the points where (0 0 1) disappears
completely. The open diamonds are the phase boundaries determined by the resistivity
measurements. (b) Schematics of spin structures of AFM-a, G-AFM and CAFM phases.

3.4.4

Density functional theory calculations

To understand the CMR phenomena we observed in Ti-doped Ca3Ru2O7, we have carried out
DFT calculations in collaboration with Dr. G.-Q. Liu in Ningbo Institute of Material Technology
and Engineering. The calculations were performed with the full-potential linear augmented plane
wave (FLAPW) within the local spin density approximation (LSDA) which is implemented in
WIEN2K package [76]. The irreducible Brillouin zone was sampled with 100 k-points. To
investigate the interplay of structural, magnetic and electronic properties, we assumed that the Ti
induced change in electron count is very small. Thus, the calculations were done for the parent
Ca3Ru2O7. Below are three major findings from our calculations.
(i) The role of electron correlation: In agreement with Ref. [99], at the bare density functional
level the low-temperature crystal structure and G-AFM magnetic structure give rise to a metallic
59

state shown in Fig. 3.17(a). This is inconsistent with the experimental observations, implying the
essential role of electron correlations to stabilize an insulating ground state. Taking into account
an on-site Coulomb repulsion U = 2 eV [104,105] leads to the opening of an electronic band gap
of ~0.2 eV, as shown in Fig. 3.17(b).
(ii) The spin-charge-lattice interplay: In contrast to the insulating behavior [Fig. 3.17(b)] in the
low-temperature G-AFM state with a larger RuO6 octahedral flattening, Fig. 3.17(c) shows a
metallic character for the field-induced AFM-a type magnetic structure accompanied by a
suppressed octahedral flattening, which agrees with our experimental observation. This drastic
change in the nature of the ground state suggests strong interplay among spin, charge and lattice
degrees of freedom.
(iii) The importance of electronic orbital polarization: In the insulating state, the electronic
orbitals are polarized with the spin-down channel of the 

!

orbital almost fully occupied while

the  " ⁄!" orbitals are nearly empty, as shown in Fig. 3.17(b). This arises from a larger
octahedral flattening that lifts the degeneracy of the t2g orbitals (

!

and  " ⁄!" ). In contrast, in

the metallic state [Fig. 3.17(c)], the reduced structural distortion gives rise to a suppression of the


!

orbital occupancy and a drastic reduction of the orbital polarization. These results suggest that

the field-induced orbital depolarization in this multiband system may play an important role in the
collapse of the Mott insulating ground state.

60

Figure 3.17: Projected density of states (PDOS) of the Ru  ! and  " ⁄!" orbitals. (a) PDOS
calculated using the low-temperature crystal structure and G-AFM magnetic structure with the onsite Coulomb interaction U = 0. (b) PDOS calculated using the low-temperature crystal structure
and G-AFM magnetic structure with U = 2 eV. (c) PDOS calculated using the high-temperature
crystal structure and AFM-a magnetic structure with U = 2 eV.

3.4.5

Comparison with early theories

In Mott insulators, the insulating ground state is stabilized by Coulomb repulsion. Generally,
there are two fundamental tuning parameters in Mott systems, i.e., bandwidth and band filling [1],
which can be controlled experimentally by varying pressure, carrier concentration and electric
field, etc [107-110]. However, a collapse of the Mott gap induced by a magnetic field is rarely
observed. Theoretical studies based on a half-filled single-band Hubbard model predicted a
magnetic-field-driven first-order metamagnetic localization transition from a strongly correlated
metallic paramagnetic state to an insulating state via a competition between the local
antiferromagnetic exchange and the Zeeman energy [111-113]. This prediction has been suggested
to account for the field-induced Mott transition in a quasi-two-dimensional organic conductor Îş(BEDT-TTF)2Cu[N(CN)2]Cl [114]. However, the observation of a collapse of the Mott gap in Ti-

61

doped Ca3Ru2O7 is not in line with this simple picture. Instead, the Mott insulating state in
Ca3(Ru0.97Ti0.03)2O7, as in various other material systems, is much more complicated. It is often
antiferromagnetic and the MIT is accompanied by lattice distortions [99]. Hence, the observation
of a collapse of the Mott insulating state in Ti-doped Ca3Ru2O7 induced by a magnetic field implies
that the coupling among lattice, spin and orbital degrees of freedom is crucial to the understanding
of the Mott insulating states and the associated phase transitions, which has not been explored in
great depth theoretically.
3.4.6

Comparison with CMR manganites

We would like to point out that the physical origin of the CMR effect observed in Ti-doped
Ca3Ru2O7 is fundamentally distinct from that in the hole-doped manganites. The canonical CMR
effect is observed in manganites that become ferromagnetic at low temperature via the double
exchange mechanism [115], such as La1-xSrxMnO3 (x = 0.175) [4]. The application of a magnetic
field leads to large magnetoresistance near the transition temperature, which has been ascribed to
the reduction in spin scattering and the polaron effects due to strong electron-lattice interactions.
The occurrence of CMR in manganites can also originate from the field-induced melting of the
charge-ordered state, for example, in Pr0.5Ca0.5MnO3 and Nd0.5Sr0.5MnO3 with the ratio of Mn3+ to
Mn4+ being commensurate with the crystalline lattice [116,22]. However, neither of these two
mechanisms can be applied to account for the observations in Ca3(Ru0.97Ti0.03)2O7, considering the
fact that the CMR effect emerges only at low temperature below TMIT and the single valence state
of Ru ions (Ru4+). Furthermore, the CMR effect in manganites can be explained using the one-eg
orbital model [117], although a later study using the two-eg orbital model explains the first-order
CMR transition for some manganites [118]. In contrast, it is essential to take into account the

62

multiband (t2g orbitals) structure of Ti-doped Ca3Ru2O7 in order to understand the field-induced
IMT, as discussed in Section 3.4.4.
3.4.7

Comparison with the parent compound

The magnetoresistive effect in Ca3(Ru0.97Ti0.03)2O7 is also distinct from that in the parent
compound.
(i) The pristine Ca3Ru2O7 exhibits a metallic ground state. Thus, no magnetic-field-induced
transition in the electronic structure occurs as in Ca3(Ru0.97Ti0.03)2O7.
(ii) When the magnetic field is applied along the magnetic easy axis (b axis), an enhancement
in the conductivity by ~1 order of magnitude has been observed at B = 6 T (4 K), which is
accompanied by a first-order metamagnetic transition. Based on neutron diffraction measurements,
this magnetoresistive effect has been attributed to a bulk spin-valve effect associated with spin
scattering [97].
(iii) A reduction in the resistivity in the metallic ground state (AFM-b) has been observed with
a field of 15 T applied along the magnetic hard axis (a axis), which is argued to originate from the
suppression of orbital ordering at high fields [119]. However, no orbital ordering in Ca3Ru2O7 at
zero field has been convincingly detected in resonant x-ray scattering experiments [120].
In contrast to the parent compound, the CMR effect observed in Ca3(Ru0.97Ti0.03)2O7 cannot be
interpreted in the framework of a spin-valve mechanism. Instead, it suggests a magnetic-fieldinduced change of the electronic ground state from a Mott insulator to a weakly localized state,
i.e., a collapse of the Mott state.
3.4.8

Discussions

In addition to the spin-lattice-orbital coupling and the field-induced orbital depolarization
discussed above, we would like to underline the essential role of phase competition on the collapse

63

of the Mott gap in a magnetic field in Ca3(Ru0.97Ti0.03)2O7. It is noteworthy that the closure of the
Mott gap in Ca3(Ru0.97Ti0.03)2O7 by a modest magnetic field is quite unexpected, considering that
the upper limit of the associated Zeeman energy per Ru4+ spins is about 1.4 meV, much smaller
than the Mott gap of ~120 meV estimated from the optical conductivity measurements [121].
However, due to the proximity to the phase boundary [Fig. 3.12], two dramatically different states,
one a weakly localized state with AFM-a or AFM-b spin structure and the other a Mott insulating
state with G-AFM magnetic structure, are close in energy. Owing to the phase competition, the
energy required to trigger the transition is associated with the difference between the energies of
these two states and the height of the intervening energy barriers, thus is much smaller than that
needed to melt the order completely. This idea is in analogy to the CMR effects in manganites
[122], though the microscopic mechanisms are very different as discussed above. The significance
of the delicate competition is also manifested by the fact that for materials slightly away from the
phase boundary, for example x = 0.05, no magnetic-field-induced IMT is observed up to 9 T [Fig.
3.18].

Figure 3.18: (a) Temperature and (b) magnetic field dependence of the in-plane resistivity of
Ca3(Ru0.95Ti0.05)2O7.

64

3.4.9

Summary

The presence of electronic and magnetic instabilities with strong spin-lattice-charge couplings
and potential changes in the orbital occupancy, enables a drastic change in the electronic structure
of Ca3(Ru0.97Ti0.03)2O7 by a modest magnetic field, leading to a collapse of the Mott insulating
ground state. Thus, Ca3(Ru0.97Ti0.03)2O7 serves as a rare example among the antiferromagnetic
Mott insulators whose electronic ground state can be tuned using the magnetic field as a controlling
parameter. This study would stimulate future theoretical studies on the field-induced IMT in Mott
systems with multiband electronic structure, strong magnetic and electronic instabilities, as well
as subtle interplay of various degrees of freedom.

65

3.5

Field-induced incommensurate-commensurate magnetic transitions in

Ca3(Ru,Fe)2O7
Upon Fe doing for Ru, Ca3(Ru1-xFex)2O7 (x = 0.05) exhibits the coexistence of the
commensurate AFM-b and an incommensurate magnetic soliton lattice below TMIT ~ 43 K, in
addition to the metallic state with AFM-a type magnetic structure at TMIT < T < TN, TN ~ 83 K. The
incommensurate magnetic soliton lattice is a distorted cycloidal structure propagating along the a
axis [Fig. 3.19(c)], with a nearly temperature-independent propagation wave vector 
' = (δ 0 1),
δ ~ ¹0.017, as evidenced by the observation of the third-order harmonics of the magnetic Bragg
reflections 
' in the neutron diffraction measurements [123]. Owing to the non-centrosymmetric
crystal symmetry, Dzyaloshinskii-Moriya (DM) interaction is expected to play an essential role in
the formation of the incommensurate magnetic soliton lattice [124]. The incommensurability δ of
Ca3(Ru0.95Fe0.05)2O7 is rather small, indicating a long period of the incommensurate phase of ~58
unit cells or ~315 Å along the a axis, which is typical for incommensurate magnetic structures
stabilized by DM interactions.
The evolution of the magnetic soliton lattice phase in the presence of magnetic field is an
intriguing problem to investigate, as it essentially reflects the interplay among exchange
interactions, magnetic anisotropy, Zeeman interaction and DM interaction in this system. In this
section, we study the field-induced spin structure transitions of Ca3(Ru0.95Fe0.05)2O7 upon applying
the magnetic field both parallel and perpendicular to the spin propagation direction (i.e. a axis).
First-order incommensurate-to-commensurate magnetic transitions have been observed for both B
∼ a and B ∼ b axes. Intriguingly, the field-induced magnetic phase transitions display irreversible
behaviors, where metastable states with different ordering wave vectors have been observed upon
decreasing the magnetic field. The metastable states persist below a characteristic temperature Tg
66

~ 37 K, above which the equilibrium ground state is restored, which implies that thermal
fluctuations play a crucial role.

Figure 3.19: Schematic diagrams of the magnetic structures in Ca3(Ru0.95Fe0.05)2O7. (a) AFM-a
and AFM-b: collinear antiferromagnetic structures with the magnetic moments along the a and b
axis, respectively. (b) CAFM-b (CAFM-a): canted antiferromagnetic structures consisting of an
AFM-a (AFM-b) component and a ferromagnetic one along the b (a) axis. (c) In-plane view of
one of the bilayers of the incommensurate magnetic soliton lattice. The dashed square in magenta
represents one unit cell. The orange rectangle with rounded corners encloses the domain wall. The
blue and orange arrows represent the direction of the magnetic field to be along either the a or b
axis.

3.5.1

Materials and methods

Single crystals of Ca3(Ru0.95Fe0.05)2O7 were synthesized by the Floating Zone method.
Magnetotransport measurements were performed using PPMS. Neutron diffraction experiments
were carried out using the E4 two-axis diffractometer in HZB and the CG-4C cold neutron tripleaxis spectrometer (CTAX) at HFIR in ORNL. The wavelength of the incident and scattered
67

neutrons is λ = 2.434 Å at E4 and is chosen as 4.045 Å at CTAX to have better resolution in the
momentum transfer. The single-crystal sample was aligned in the horizontal (H 0 L) scattering
plane, where H and L are in the reciprocal lattice units (r.l.u.). At E4, the sample was mounted
inside a vertical-field cryomagnet with the magnetic field applied along the b axis, while at CTAX
it was loaded into a similar vertical-field cryomagnet as well as a horizontal-field one where the
magnetic field was applied along the a axis up to 4 T, respectively. The horizontal-field
cryomagnet contains two pairs of windows for the incident and outgoing neutron beams.
Constraint by the scattering geometry and the size of the window, only magnetic reflections around

 = (0 0 1) are accessible in the neutron diffraction study for B ∼ a axis. The neutron intensity is
presented in the unit of counts per monitor count unit (mcu), where 1 mcu corresponds to ~1 second.
3.5.2

Neutron diffraction for B ∼ b axis

In this section, we present the neutron diffraction study as the magnetic field is applied along
the b axis, perpendicular to the spin propagation axis as indicated by the orange arrow in Fig.
3.19(a) and (c). Figure 3.20(a) and (b) show the contour maps of the neutron intensity of scans
along the [1 0 0] direction across the magnetic Bragg peak 
 = (0 0 1) at T = 15 K after ZFC the

sample. At zero field, both commensurate and incommensurate magnetic peaks are observed at 


= (0 0 1) and 
' = (δ 0 1), δ ~ ¹0.017, respectively. There is sizable diffuse scattering intensity
centered at the commensurate peak 
 , in agreement with the previous report [123]. As the

magnetic field increases, the intensity of the incommensurate magnetic Bragg peaks at 
' is
suppressed and vanishes at the critical field *↑ = 4.7 ± 0.1 T. Concomitantly, the commensurate

peak centered at 
 gets enhanced and the intensity due to diffuse scattering disappears. This
observation clearly suggests a first-order magnetic-field-driven incommensurate-to-commensurate

phase transition with an abrupt change in the magnetic wave vector. Note that the width of these

68

magnetic Bragg peaks is not resolution limited, indicating a correlation length of ~370a for the
commensurate state and ~340a for the incommensurate phase at zero field, and ~580a for the
commensurate phase at B = 9 T, T = 1.5 K, where a is the lattice constant. Since the reflection
condition due to the symmetry of the crystal structure requires that either both H and L are even,
or the sum of H and L is even, the enhancement in the intensity of (0 0 1) suggests the formation
of an antiferromagnetic phase at B > *↑ . We have observed a series of magnetic Bragg peaks in
the reciprocal space at (0 0 3), (0 0 5) and (0 0 7), etc. The analysis of the spin structure of the
field-induced commensurate phase at B > *↑ in Ca3(Ru0.95Fe0.05)2O7 is very similar to that of the
pristine and Ti-doped Ca3Ru2O7 as described in Appendix B. It is determined to be a canted
antiferromagnetic structure (CAFM-b), as shown in Fig. 3.19(b). Such a field-induced
incommensurate-to-commensurate transition may arise from the competition among Zeeman
energy, exchange energy and DM interaction [125], a mechanism similar to that reported in a DMinduced spin spiral Ba2CuGe2O7 [126].
At TMIT < T < TN, in a magnetic field along the b axis, the system transforms from AFM-a to
CAFM-b, and finally to a fully spin polarized state (PM). The magnetic phase diagram established
by the neutron diffraction data collected at E4 and CTAX is shown in Fig. 3.21.

69

Figure 3.20: (a),(b) Contour maps of the intensity of scans along the [1 0 0] direction across 
 =
(0 0 1) in a magnetic field applied along the b axis at T = 15 K. The light gray grids denote the
measurement fields. (c),(d) Scans along the [1 0 0] direction across 
 = (0 0 1) in the 0↑ % and 0↓ %
phases at T = 1.5 K, respectively. The dark green and orange arrows mark the first-order
incommensurate magnetic Bragg peaks and their third-order harmonics, respectively, while the
purple ones mark the commensurate reflections. Inset shows the zoom-in view of the third-order
harmonics at T = 1.5 K and 80 K, B = 0↑ T. The data were taken at CTAX.
Very intriguingly, as the magnetic field decreases, the system cannot restore to its original
zero-field state. As shown in Fig. 3.20(b), the commensurate CAFM-b spin structure transforms
into the coexistence of commensurate and incommensurate ones below the critical field *↓ = 4.3
Âą 0.5 T, corroborating the first-order character of the field-induced magnetic phase transition.
While the commensurate magnetic peak is still located at 
 = (0 0 1), the incommensurate ones

are centered at 
' = (Âą0.005 0 1), in contrast to (Âą0.017 0 1) prior to applying the magnetic field.
As the magnetic field decreases further down to 0 T, this new incommensurability remains
unchanged and does not restore to the initial values, i.e., 
' = (Âą0.017 0 1).

70

Figure 3.21: Phase diagram of Ca3(Ru0.95Fe0.05)2O7 in a magnetic field along the b axis for field
ramping up. Note that the error bars of the data points at 15 K and 30 K are smaller than the size
of the symbol.

Figure 3.20(c) and (d) present the scans measured at T = 1.5 K and B = 0 T before (denoted as
0↑ %) and after (0↓ %) ramping the magnetic field up to 9 T. We have observed several remarkable
features. (i) In addition to the first-order incommensurate magnetic reflections (green arrows)
discussed above, the third-order harmonic peaks (orange arrows) are also observed for both 0↑ %

and 0↓ % states. This feature suggests that the magnetic soliton lattice reemerges in the 0↓ % state,

but with a much longer periodicity (~200 unit cells or ~1100 Å along the a axis). (ii) The intensity
ratio of the third- to the first-order harmonic peaks () /( is much larger at B = 0↓ T, suggesting
that the incommensurate magnetic structure is more distorted from the uniform cycloidal structure
with narrower domain wall width. (iii) While the FWHM of the commensurate peak remains nearly
unchanged compared to the one in the 0↑ % state, the FWHM of the incommensurate magnetic

peaks in the 0↓ % state becomes broader, implying a shorter correlation length of the magnetic

soliton lattice in the 0↓ % state. It is worth noting that such a 0↓ % phase is rather persistent and a
subsequent magnetic field cycling between 0 and 9 T shows a reversible behavior with the
71

magnetic wave vector transforming between the incommensurate 
' = (Âą0.005 0 1) and the

commensurate 
 = (0 0 1), unless the temperature is raised high enough. These results suggest

that the system could not restore to its equilibrium ground sate after the field-induced
incommensurate-to-commensurate phase transition. Instead, it forms a persistent metastable state
at low temperature, leading to the irreversible behavior of the modulation wave vector in the fieldinduced magnetic phase transition.

Figure 3.22: (a)-(c) Scans along the [1 0 0] direction across 
 = (0 0 1) in the 0↑ % (black) and 0↓ %
(red) phases at representative temperatures. The magnetic Bragg peaks are fitted using Gaussian
functions (solid curves). The gray horizontal line indicates the instrument resolution. (d)
Incommensurability δ of the first-order incommensurate magnetic wave vector 
' = (¹δ 0 1) in
the 0↑ % and 0↓ % phases at different temperatures. The error bars obtained from Gaussian fitting
are smaller than the symbol size. The solid curves are guides to the eye. The data were taken at
CTAX.

Figure 3.22(a)-(c) present the irreversibility of the field-driven incommensuratecommensurate magnetic structure transition at representative temperature. The scans along the [1

72

0 0] direction were performed at zero field under different histories following the aforementioned
procedure, one right after ZFC to the designated measurement temperature (0↑ %) and the other
after ZFC and applying the magnetic field up to 9 T, then removing the magnetic field (0↓ %). One
can see that the wave vector 
' of the incommensurate magnetic reflection in the 0↑ % state stays

almost constant below TMIT, consistent with the previous report [123]. However, the 0↓ % state

behaves in a completely different way where 
' of the incommensurate peaks is dependent on the
measurement temperature. A summary of the zero-field incommensurability δ measured in both
0↑ % and 0↓ % states is presented in Fig. 3.22(d). In contrast to the constant δ = 0.017 in the 0↑ %

state, δ in the 0↓ % state is relatively smaller and remains unchanged below 18 K, above which δ
increases monotonically and gradually reaches the value of the equilibrium state. This indicates
that a complete recovery of the equilibrium ground state at B = 0↓ T takes place at a characteristic
temperature Tg ~ 37 K that is slightly lower than TMIT. In addition, it is noteworthy that the
incommensurate peaks in the 0↓ % state, for example at 20 K, are much broader than that in the

equilibrium 0↑ % state, implying that the 0↓ % state has a shorter correlation length. These features

suggest that the emergence of the field-induced metastable 0↓ % state at low temperature is due to

the fact that the system gets trapped at a local minimum in the free energy landscape. As a result,
the transformation to the equilibrium ground state is kinetically prohibited by the energy barriers
unless thermal fluctuations are strong enough to overcome this barrier.

73

Figure 3.23: (a) Contour map of the intensity of scans along the [1 0 0] direction over 
 = (0 0 1)
while warming up in the metastable 0↓ % phase. The measurement temperatures are denoted by the
light gray grids. (b) Scans along the [1 0 0] direction over 
 = (0 0 1) at representative
temperatures. The third-order harmonics of the incommensurate magnetic reflections are denoted
by orange arrows. Inset shows the temperature dependence of the intensity ratio () ⁄( in the 0↓ %
phase. The data were taken at CTAX.

To further understand the temperature evolution of the field-induced metastable state, we
prepared the initial 0↓ % state at T = 1.5 K, then performed the measurements at various
temperatures while warming up. The data are shown in Fig. 3.23(a) (2 K / step). The
incommensurate wave vector 
' = (Âą0.005 0 1) remains almost unchanged until T ~ 31 K, then

starts to evolve gradually toward the equilibrium state with 
' = (Âą0.017 0 1), which is eventually

reached at Tg ~ 37 K. In the meantime, the intensity of the commensurate peak 
 = (0 0 1) gets

much weaker and the peak width becomes narrower, due to the phase transformation from the
metastable state toward the equilibrium phase. Notably, the temperature where 
' starts to deviate
from that of the low-temperature 0↓ % state is sensitively dependent on the measurement

procedures, as shown in Fig. 3.22(d) and 3.23(a), i.e., the former at T ~ 18 K and the latter at T ~
31 K, indicating that the incommensurability of the 0↓ % state obtained using different
measurement protocols can be quite different for 18 K < T < 31 K. This implies that a mechanism
which is sensitively dependent on both temperature and magnetic field history has to be invoked
in order to explain this irreversible behavior. Fig. 3.23(b) shows the evolution of the third-order
74

harmonics measured in the 0↓ % state at representative temperatures. The temperature dependence

of the intensity ratio () ⁄( is shown in the inset. As the temperature increases, () ⁄( decreases

indicating that the incommensurate magnetic soliton lattice evolves toward the uniform cycloidal
spin structure with broader domain wall width at elevated temperatures, similar to the zero-field
study reported previously [123].
3.5.3

Magnetoresistance and neutron diffraction for B ∼ a axis

It is well-known that the magnetic phase transitions of DM helimagnets are rather complicated,
depending on the orientation of the external magnetic field. In this section, we present the studies
on the field-induced spin structure transitions of Ca3(Ru0.95Fe0.05)2O7 upon applying the magnetic
field along the a axis, parallel to the propagation direction of the incommensurate magnetic soliton
lattice, as denoted by the blue arrows in Fig. 3.19(a) and (c).
Figure 3.24(a) shows the normalized out-of-plane resistivity  as a function of magnetic field
at T = 15 K, as the magnetic field is applied along the a and b axis, respectively. The measurements
were performed after the sample was initially cooled in zero magnetic field for each case. For B ∼

b, the magnetoresistance displays a minimum at *↑ = 5 ± 0.25 T as the magnetic field increases,

corresponding to the incommensurate-to-commensurate spin structure transition to CAFM-b as
revealed by neutron diffraction measurements [127]. The transition is of first order, as indicated
by the appearance of hysteresis as the magnetic field increases and decreases (*↓ = 4.5 ± 0.25 T).
Intriguingly, when the magnetic field is applied along the a axis, a first-order transition is also
observed at *↑ = 2.25 ± 0.25 T, in sharp contrast to the parent compound where no anomaly is
observed in this field range [128,129]. This transition is likely to be associated with a field-induced
spin structure transition as well. For comparison, Figure 3.24(b) shows the data taken at T = 55 K,
where the material displays a metallic state with the AFM-a type magnetic structure at zero field

75

[123,127]. One can see that the features are completely different from that observed at 15 K. The
magnetoresistance increases continuously as the magnetic field is along the b axis, whereas it
exhibits a sudden jump for B ∼ a. In the AFM-a phase, one would expect a first-order spin-flop or
spin-flip transition when the magnetic field is applied along the easy axis (a axis), but a continuous
transformation into the fully polarized state (PM) if the magnetic field is applied along the hard
axis (b axis) [97]. The observations at 55 K seem to be well explained by this scenario.

Figure 3.24: (a) Normalized out-of-plane resistivity  of Ca3(Ru0.95Fe0.05)2O7 as a function of
magnetic field at (a) T = 15 K and (b) T = 55 K. The magnetic field is applied along the a and b
axis, respectively.
Interestingly, the magnetoresistance data [Fig. 3.24(a)] for both B ∼ a and B ∼ b axes, which
are essentially dominated by the spin scattering process of different magnetic structures, also
exhibit irreversible behaviors. The zero-field resistance in the 0↓ % state is much greater than that
measured in the 0↑ % phase (0↑ % and 0↓ % are defined in Sec. 3.5.2). The emergence of such

hysteresis implies the formation of metastable magnetic phases when the magnetic field decreases,
as discovered by the neutron diffraction study described above [127]. Similar to the one induced
with the magnetic field applied along the b axis, the metastable phase induced for B ∼ a is also
quite persistent. The change in the resistance is found to be less than ~0.1% in the time scale up to
~104 seconds, and further cycling the magnetic field between 0 and 9 T gives rise to reversible
76

phase transitions between the low-field metastable phases and the high-field states. Such a
magnetic memory effect can be erased only by heating the material up to a high enough
temperature, which suggests that thermal fluctuations play an essential role.

Figure 3.25: Normalized out-of-plane resistivity  of Ca3(Ru0.95Fe0.05)2O7 as a function of
magnetic field at representative temperatures. The upper and lower critical fields are denoted by
red and green arrows, respectively. Inset shows the lower critical field *↓ as a function of
temperature. The red solid line is a fit using power law. The magnetic field is along the a axis.
Figure 3.25(a)-(d) shows the irreversible field-induced phase transitions for B ∼ a at
representative temperatures. Because of the history effect, each measurement was done after
heating the sample to a temperature above TN then ZFC to the measurement temperature to clean
up the magnetic memory. Below 26 K, as the magnetic field decreases from 9 T, the
magnetoresistance monotonically increases and becomes saturated, giving rise to a large open
hysteresis. On the contrary, at T > 26 K, the magnetoresistance [Fig. 3.25(b)-(d)] shows a decrease
at the critical field *↓ (marked by green arrows) as the magnetic field decreases, suggesting that
77

the phase transformation starts to occur. In addition, the remnant magnetoresistance at zero field
(0↓ %) decreases with increasing temperature, and finally disappears at T ~ 34 K [Fig. 3.25(d)],
which may suggest that the equilibrium phase (coexistence of commensurate and incommensurate
spin structures with δ ~ ¹0.017) has been recovered.
In order to determine the magnetic structure of the field-induced phase below TMIT and the
metastable phase at B < *↓ , we have carried out neutron diffraction measurements with the
magnetic field applied along the a axis. Due to the limited access to the reciprocal space with the
horizontal-field cryomagnet, we only focus on the magnetic reflections around 
 = (0 0 1). Figure

3.26(a) and (b) show the contour maps of the neutron intensity of scans across 
 = (0 0 1) along

the [1 0 0] direction at T = 15 K (every 0.5 T from 0 to 4 T). At zero field, the system exhibits
coexistence of the commensurate magnetic peak 
 = (0 0 1) and incommensurate ones 
' = (δ 0
1), δ = ¹0.017, in agreement with the previous study [123]. As the magnetic field increases, the
incommensurate peaks are suppressed at a critical field *↑ = 2.25 ± 0.25 T (defined as the point

where the incommensurate peaks are gone completely), much smaller than that when B ∼ b, and
the incommensurability δ keeps nearly constant during the phase transition. In the meantime, the
intensity of the commensurate one is enhanced, and the peak width becomes broader. As the
magnetic field increases further, the (0 0 1) peak intensity gets stronger and the peak width
becomes narrower. However, the integrated intensity decreases slightly, implying that the
antiferromagnetic staggered magnetization is gradually suppressed by the magnetic field. Note that
the reflection condition of the crystal symmetry of Ca3(Ru0.95Fe0.05)2O7 (Bb21m, No. 36) requires
that both H and L are even, or the sum H + L is even [93]. The emergence of Bragg reflections at
(0 0 1) suggests that the system is in an antiferromagnetic phase, similar to the field-induced states
in the pristine and other doped Ca3Ru2O7 [97,130,131]. Although only one magnetic reflection is
78

available in this experiment, considering that in all other related studies on Ca3Ru2O7 this magnetic
reflection corresponds to an AFM-a or AFM-b type spin structure [97,127,130], we propose that
the field-induced magnetic structure is CAFM-a, a superposition of AFM-b and a ferromagnetic
component along the a axis, as shown in Fig. 3.19(b).

Figure 3.26: Contour maps of the neutron intensity of scans across 
 = (0 0 1) along the [1 0 0]
direction at T = 15 K as (a) the magnetic field increases from 0 to 4 T and (b) the magnetic field
decreases from 4 to 0 T. (c),(d) Cuts of the scans taken at B = 0↑ , 4↓ and 0↓ T. The solid and dashed
lines are fitted curves using Gaussian functions. The gray line denotes the instrument resolution.
The magnetic field is applied along the a axis. Inset shows the scans at B = 0↑ T (blue) and 0↓ T
(red) when the magnetic field is along the b axis at T = 15 K.

The irreversibility of the field-induced magnetic transition is clearly observed in the neutron
diffraction measurements. As the magnetic field decreases, surprisingly, both the peak width and
peak intensity of the commensurate (0 0 1) magnetic reflection stay nearly unchanged and the
incommensurate ones do not reemerge down to 0 T at T = 15 K. This agrees well with the
magnetoresistance measurements, where the critical field *↓ , i.e. the phase transformation into the
equilibrium phase (coexistence of commensurate and incommensurate spin structures with δ ~
79

¹0.017), is absent [Fig. 3.24(a)]. This observation is different from that for B ∼ b, where the system
transforms into a metastable state with much smaller incommensurability δ ~ ¹0.004 (T = 15 K)
[127]. Figure 3.26(c) and (d) show the scans at B = 0↑ , 4↓ and 0↓ T. Inset shows the scans at B =

0↑ T and 0↓ T as the magnetic field is applied along the b axis for comparison. All the
incommensurate and commensurate magnetic peaks can be well fitted by Gaussian functions (a
broad peak is added in Fig. 3.26(c) to account for the broad diffuse scattering intensity, as discussed
in Ref. [127]). The widths of all the peaks are not resolution limited (the resolution is denoted by
the short gray line). Particularly the width of (0 0 1) at B = 4 T is much broader than that in the
0↑ % phase, which suggests a much shorter correlation length ~210a in the field-induced CAFM-a

phase and the metastable state at B = 0↓ T.

We also performed the same neutron diffraction measurements at a higher temperature T = 34
K, where a critical field *↓ is seen in the magnetoresistance measurements [Fig. 25(b)-(d)], as
shown in Fig. 3.27(a) and (b). Similarly, the incommensurate-to-commensurate magnetic
transition occurs at *↑ = 2.75 ± 0.25 T, close to that shown in the magnetoresistance data [Fig.

3.25(d)]. However, as the magnetic field decreases, at *↓ = 2 T the commensurate (0 0 1) peak

becomes weaker and two incommensurate peaks appear at 
' = (δ 0 1), δ ~ ¹0.01. Upon further
reducing the magnetic field, the incommensurability δ gradually evolves toward that of the
equilibrium state, which indicates a decrease in the period of the incommensurate magnetic
structure. Nevertheless, the equilibrium value of the incommensurability δ ~ ¹0.017 cannot be
reached down to 0 T at this temperature. A complete recovery of the initial incommensurability
occurs in the measurements done at Tg = 37 K, the same as that when the magnetic field is along
the b axis [127]. The small discrepancy in Tg where the equilibrium state is recovered determined
by neutron diffraction and magnetotransport measurements might be due to the fact that the
80

incommensurability can be better revolved by neutron diffraction. Figure 3.27(c) and (d) show the
representative scans at B = 0↑ , 4↑ T and 1.5↓ , 0↓ T respectively. The peak width of (0 0 1) at B >

*↑ (CAFM-a) is not resolution limited, corresponding to a correlation length of ~450a. In the

metastable state at B < *↓ , the correlation length of the commensurate phase is comparable to that
of the CAFM-a phase, but the incommensurate peaks are broader and become slightly narrower as
the magnetic field decreases.

Figure 3.27: Contour maps of the neutron intensity of scans across 
 = (0 0 1) along the [1 0 0]
direction at T = 34 K as (a) the magnetic field increases from 0 to 4 T and (b) the magnetic field
decreases from 4 to 0 T. (c),(d) Cuts of the scans taken at B = 0↑ , 4↑ T and 1.5↓ , 0↓ T. The solid
lines are fitted curves using Gaussian functions. The gray line denotes the instrument resolution.
The magnetic field is applied along the a axis.
To make a further comparison with the B ∥ b case, we also studied the evolution of the 0↓ %
metastable phase as a function of temperature. The metastable state was obtained using the same
procedure as the 0↓ % phase shown in Fig. 3.26(b). Namely, we first ZFC the sample to 15 K and
applied 4 T magnetic field along the a axis. The magnetic field was then reduced to zero. Figure
81

3.28(a) shows the contour map of the neutron intensity obtained by scanning across 
 = (0 0 1)
along the [1 0 0] direction at various temperatures (every 2 K from 15 to 50 K). Upon warming
the magnetic reflection (0 0 1) almost doesn’t change until T = 31 K, where the incommensurate
peaks start to appear and move towards the equilibrium state with 
' = (δ 0 1), δ ~ ¹0.017. As
soon as the incommensurate peaks show up, the intensity of the commensurate (0 0 1) peak
becomes much weaker, indicating that the commensurate magnetic structure (CAFM-a) transforms
into the incommensurate ones. Fig. 3.28(b) shows the scans at selected temperatures T = 15, 34
and 37 K. One clearly see that the equilibrium states is reached at Tg = 37 K, similar to the study
for B ∼ b [127]. In addition, the peak width of (0 0 1) at T = 34 K is much narrower than that at 15
K, which implies that the correlation length becomes much longer (~440a). In contrast, the peak
width of the incommensurate peaks is broader and corresponds to a correlation length of ~230a.
For T > 37 K, the behavior of the magnetic Bragg peaks is the same as that of the equilibrium 0↑ %
phase [123].

Figure 3.28: (a) Contour map of the neutron intensity of scans across 
 = (0 0 1) along the [1 0 0]
direction in the 0↓ % phase at various temperatures. (b) Cuts of the scans taken at T = 15, 34 and 37
K in the 0↓ % phase. The solid lines are fitted curves using Gaussian functions. The gray line
represents the instrument resolution.

3.5.4

Discussions

In Ca3(Ru0.95Fe0.05)2O7, due to the non-centrosymmetric crystal symmetry, it is convenient to
82

ascribe the formation of incommensurate magnetic structures to the DM interaction [124]. In fact,
incommensurate magnetic structures have been observed in Ti- and Mn-doped Ca3Ru2O7, but only
in a small temperature window near the MIT [100,131]. These results suggest that the
incommensurate magnetic structures are close in free energy to AFM-b or AFM-a, thus can be
readily induced by light chemical doping. Owing to the noncollinear, long-period incommensurate
magnetic structures, first-order spin structure transitions have been observed as the magnetic field
is applied along both the a and b axis, which is in sharp contrast to the parent compound [132,129].
In the pure Ca3Ru2O7, below TMIT, the material displays a collinear AFM-b type magnetic structure,
with the staggered magnetization along the b axis [97]. The magnetic field leads to a first-order
spin-flop transition at Bc = 6 T (T = 4 K) only when the magnetic field is applied along the b axis
[129]. On the contrary, as the magnetic field is applied along the a axis, the system continuously
evolves toward the fully spin polarized state (PM) and no first-order spin structure transition is
observed [97]. Note that although a field-induced transition is seen at 15 T, it is attributed to a
change in the orbital occupancy [129]. As a DM helimagnet, the stabilization of the
incommensurate magnetic soliton lattice in Ca3(Ru0.95Fe0.05)2O7 and its evolution as a function of
temperature and magnetic field are governed by the competition among Zeeman interaction,
exchange interaction, DM interaction and magnetocrystalline anisotropy [133], a mechanism that
accounts for different types of spin structure transitions in various other DM magnets such as
Ba2CuGeO7 [134,135] and FeGe [136-138], etc. However, a more detailed theory is required to
interpret these observations quantitatively.
The irreversibility in the incommensurate-to-commensurate magnetic phase transitions driven
by a magnetic field in Ca3(Ru0.95Fe0.05)2O7 is characteristic of a metastability phenomenon. The
supercooling and superheating phases associated with first-order transitions are prototypical

83

metastable states [139]. Metastability often emerges when the phase transformation to the
thermodynamic equilibrium state is kinetically prohibited, particularly at low temperature where
thermal fluctuations are not sufficiently strong to overcome the intervening energy barriers in the
free energy landscape. It has been observed in different systems regardless of the microscopic
details, such as the field-induced magnetic transitions in phase-separated manganites [22,140-142]
and the field-induced metastable vortex lattices in superconducting MgB2 [143,144]. A common
feature among these metastable states is that the history effect can be erased by heating up the
system to a high enough temperature, which suggests that thermal fluctuations are essential in the
emergence of metastability.
Metastability concerning incommensurate magnetic structures, especially on the modulation
wave vectors, have been observed in other magnetic systems such as TbMnO3 [145], DyMn2O5
[146], UNiAl [147], where a different incommensurability or even an alternative modulation wave
vector emerges as the magnetic field decreases. In some other cases, the systems maintain the
pressure- or field-induced state upon releasing the pressure or removing the magnetic field, for
example, Cr in a pressure-induced transition [148], Ba0.5Sr1.5Zn2(Fe1-xAlx)12O22 [149] and
NaFe(WO4)2 [150] in the magnetic-field-driven phase transitions. A plausible qualitative
explanation for the field-induced irreversibility in the magnetic modulation vector in
Ca3(Ru0.95Fe0.05)2O7 may be given from the perspective of the free energy landscape as a function
of thermodynamic variables. Upon decreasing the magnetic field, due to the first-order nature of
the phase transition, the nucleation of the new phase requires an activation energy in order to
overcome the intervening energy barriers between the local minima. However, thermal fluctuations
at low temperature are not strong enough, which prevents the formation of large enough nucleus
of the stable phase [139]. The critical field *↓ signifies the magnetic field at which the strength of
84

the thermal energy ~kBT and the height of the energy barrier U between CAFM and the metastable
phase become comparable. Such a mechanism has been applied to account for the irreversibility
observed in phase-separated manganites Nd0.5Sr0.5MnO3 and Pr1-xCaxMnO3, where large hysteresis
is seen in the phase transition between a ferromagnetic metallic state and an antiferromagnetic
charge-ordered phase driven by a magnetic field [22,140]. Following the analysis in Ref. [22], the
temperature dependence of *↓ obtained by magnetoresistance measurements as summarized in the
(^)

Š

inset of Fig. 3.25(a), can be approximately scaled using an empirical function % ∝ ¶* − * ¶ ,
(^)

where * is the critical field when U becomes zero [22]. The fact that *↓ becomes greater as the

temperature increases suggests that the height of the energy barrier U decreases as the magnetic
field is swept down, in analogy to that in the manganite systems. The value of the critical exponent
(^)

β is sensitively dependent on the choice of * . In the inset of Fig. 3.25(a), the solid line represents
the fit with β = 1.5.
The microscopic origin of the metastability in the modulation wave vector of the
incommensurate magnetic structures in Ca3(Ru0.95Fe0.05)2O7 remains elusive. One may be tempted
to attribute the irreversibility to the coupling between the commensurate AFM-b and the
incommensurate magnetic structures, since they coexist in this material. However, such a scenario
might not be valid since no convincing evidence that suggests any correlation between the intensity,
correlation length, or the diffuse scattering intensity of the commensurate peak and the modulation
wave vector 
' is seen in the neutron diffraction experiment, as discussed in Ref. [127]. In spite
of the absence of a unifying microscopic theory to address this irreversibility problem, the
phenomenological theories proposed to account for the metastability in Cr might shed light on the
underlying mechanism [151,152]. First, we discuss spin-lattice coupling. It has been suggested
that the lattice distortions induced by the SDW ordering in Cr have negative free energy, which
85

results in the persistence of the domain walls when ramping down the pressure and leads to the
metastability in the modulation wave vector [151]. Nevertheless, this is not likely the origin of the
metastability observed in Ca3(Ru0.95Fe0.05)2O7, since no lattice modulation peaks at 2
 displaced
from the nuclear peak (0 0 4) are convincingly observed, as shown in Fig. 3.29, in contrast to those
in Cr indicating the presence of strong exchange striction. Second, we discuss domain wall pinning.
It has been proposed that the metastability in Cr can also stem from the pinning of domain walls
by nonmagnetic impurities [152]. In Fe-doped Ca3Ru2O7, we speculate that the interaction between
the magnetic Fe dopants and the domain walls might be responsible for the metastability. Although
these randomly doped magnetic impurities can enhance the heterogeneous nucleation process of
the magnetic domain walls during the commensurate-to-incommensurate transition when the
magnetic field ramps down, they also pin the domain walls thus create energy barriers and prevent
the domain wall density from reaching the equilibrium value, leading to the observed metastability
[152]. However, a more comprehensive theoretical study is desired to account for this metastability
phenomenon.

Figure 3.29: Scan along the [1 0 0] direction across the nuclear Bragg peak (0 0 4) at T = 1.5 K in
the 0↑ % phase.
3.5.5

Summary

We have investigated the magnetic-field-induced phase transitions of Ca3(Ru1-xFex)2O7 (x =
86

0.05) with the field applied both along and perpendicular to the propagation direction of the
incommensurate magnetic soliton lattice. First-order incommensurate-to-commensurate magnetic
transitions into canted antiferromagnetic structures (CAFM-a or CAFM-b) have been observed. In
addition, the field-induced transitions display distinct irreversible behaviors below a characteristic
temperature Tg and give rise to persistent metastable states, where the system either maintains the
high-field phase or transforms into an intermediate incommensurate state with smaller
incommensurability. These observations can be qualitatively described in the framework of frozen
kinetics at low temperature, which prevents the system from overcoming the energy barriers and
reaching the equilibrium state. Therefore, Fe-doped Ca3Ru2O7 provides an ideal material system
to study the metastability problem of the magnetic soliton lattice, which deserves further
theoretical investigations.

87

3.6

Tuning the competing states in Ca3Ru2O7 by Mn doping
In the previous two sections, we have shown that nonmagnetic Ti and magnetic Fe impurities

substituted for Ru in Ca3Ru2O7 give rise to significantly different doping effects. An interesting
question arises: What is the role of the 3d transition-metal dopants in determining the magnetic
and electronic states? In this section, we discuss the doping effects of another magnetic impurity
Mn on the spin structure of Ca3Ru2O7. We find that Mn induces an incommensurate cycloidal spin
structure in a narrow temperature range close to TMIT for x ≤ 0.03, while the ground state magnetic
structure is the same as that of the parent compound. However, the system exhibits a G-AFM Mott
insulating state with a simultaneous change in the lattice parameters at TMIT for x ≥ 0.04.
Furthermore, in Ca3(Ru0.96Mn0.04)2O7 that is close to the phase boundary [153], a magnetic field
applied along the b axis drives a spin structure transition from G-AFM to CAFM below TMIT,
which is accompanied by an IMT and a drastic change in the lattice constants. In contrast, at TMIT
< T < TICM, an incommensurate-to-commensurate magnetic transition has been observed without
any detectable structural change in the neutron diffraction measurements. Our results show that
the effects of the magnetic Mn doping is very similar to the nonmagnetic Ti but is different from
the magnetic Fe.
3.6.1

Materials and methods

The single-crystal samples of Ca3(Ru1-xMnx)2O7 with doping concentrations ranging from x =
0 to 0.1 were grown by the Floating Zone technique. The magnetization and resistivity
measurements were performed using PPMS. Neutron diffraction experiments were carried out
using the HB-1A and CG-4C triple-axis spectrometer at HFIR in ORNL. The energy of the incident
neutrons of HB-1A and CG-4C was fixed as Ei = 14.6 meV and 5 meV, respectively. During the
measurements on the materials of different doping concentrations at zero field, the samples were

88

oriented in the (H 0 L) and (0 K L) scattering planes and mounted in an aluminum can which is
cooled using a closed-cycle helium refrigerator down to 4 K. To study the magnetic-field-induced
transitions, the Ca3(Ru0.96Mn0.04)2O7 sample was aligned in the horizontal (H 0 L) scattering plane
and loaded into a vertical-field cryomagnet such that the magnetic field was applied along the b
axis up to 8 T.

Figure 3.30: Temperature dependence of the intensity of representative magnetic Bragg peaks (1
0 2), (0 0 1) and (δ 0 1) of Ca3(Ru1-xMnx)2O7 (x = 0.03, 0.04 and 0.05), δ = 0.023 and 0.026 for x
= 0.03 and 0.04, respectively.

3.6.2

Magnetic properties of Ca3Ru2O7 upon Mn doping at zero field

Figure 3.30 shows the temperature dependence of the intensity of the representative magnetic
peaks for Ca3(Ru1-xMnx)2O7 with different doping concentration x. For x = 0.03, at TN ~ 61 K the
magnetic peaks show up at the nuclear-forbidden wave vectors 
 = (0 0 1) and other equivalent
89

positions in the reciprocal space, for example (0 0 5) and (0 0 7), etc. The intensity keeps increasing
until TICM ~ 42 K, where the incommensurate magnetic Bragg peak 
' = (δ 0 1) emerges while
the commensurate ones get suppressed. This incommensurate magnetic structure exists only in a
very narrow temperature range TCM (~34 K) < T < TICM, below which the intensity of the
commensurate magnetic peaks is enhanced and persists down to the lowest temperature measured.
For x = 0.04, the high-temperature magnetic phases are similar to those in the x = 0.03 compound.
However, at TMIT ~ 30 K the intensity of both (0 0 1) and (δ 0 1) disappears completely, and new
magnetic peaks show up at (1 0 2), (1 0 4) and (1 0 6), etc., which suggests that the ground state
spin structure is different from that of the parent compound and the x = 0.03 one. For x = 0.05,
only one first-order magnetic transition is observed at TN ~ 60 K, and the low-temperature
magnetic structure is characterized by the magnetic peak 
 = (1 0 2).
To determine the spin structures of different magnetic phases in Mn-doped Ca3Ru2O7, we have
collected a series of nuclear and magnetic Bragg peaks in the (H 0 L) and (0 K L) scattering planes
for each Mn doping concentration. Figures 3.31(a)-(d) show the scans along the [1 0 0] direction
across the magnetic reflections (0 0 5) and (0 0 1) for x = 0.03 and 0.04 at representative
temperatures. The rocking curve scans over (1 0 2) for x = 0.04 and 0.05 are shown in Fig. 3.31(e)
and (f), respectively. We have performed magnetic representational analysis to explore the
possible magnetic structures using FULLPROF [71]. We find that the strongest (0 0 1) peak
corresponds to AFM-a or AFM-b where the magnetic moments are aligned parallel to each other
within the bilayer but are antiparallel between adjacent bilayers [97]. The strongest (1 0 2) peak
represents G-AFM with all nearest-neighbor magnetic moments aligned antiparallel to each other
[99]. The incommensurate magnetic phase characterized by 
' = (δ 0 1) is a cycloidal magnetic
structure propagating along the a axis, with the period determined by the incommensurability δ. It

90

is worth noting that the magnetic wave vectors of the incommensurate magnetic structures are
strongly temperature-dependent, as shown in Fig. 3.31(c) and (d). No higher-order harmonics are
observed, suggesting that the incommensurate magnetic phase is uniformly modulated, in contrast
to the magnetic soliton lattice in Fe-doped Ca3Ru2O7 [123]. For x = 0.03, similar to the parent
compound, the (2 0 1) magnetic peak is present at T = 4 K but is absent at 50 K, which suggests
that at T < TCM, the magnetic moments are along the b axis (AFM-b). In contrast, in the hightemperature phase at TICM < T < TN, the moments are along the a axis (AFM-a) [97], which is also
the case for the x = 0.04 compound.

Figure 3.31: (a),(b) Scans along the [1 0 0] direction across the magnetic wave vector (0 0 5) at
representative temperatures. (c),(d) Scans across the magnetic wave vector (0 0 1) in the
incommensurate magnetic phase. (e),(f) Rocking curve scans across the magnetic wave vector (1
0 2) at representative temperatures.

Very intriguingly, the magnetic transition to the low-temperature G-AFM state is accompanied
by a dramatic change in the lattice constants. Figure 3.32 shows the lattice constants a, b and c as
a function of temperature. For x = 0.03, there is no observable anomaly in all the lattice constants
throughout the temperature range measured. In contrast, for x = 0.04 upon cooling the lattice
constant b increases discontinuously by ~0.98%, while c decreases by ~0.71% at TMIT ~ 30 K,

91

which is much more pronounced compared with that in the parent compound (~0.1%) [93]. The
shortening along the c axis and the expansion in the in-plane b axis imply a flattening of the RuO6
octahedron, similar to that in Ca2RuO4, which may lead to a change in the orbital occupancy of
the Ru t2g electrons [92,102]. For x = 0.05, a structural change of comparable magnitude is also
observed at the magnetic transition TN.

Figure 3.32. Lattice parameters of Ca3(Ru1-xMnx)2O7 (x = 0.03 and 0.04) as a function of
temperature.

3.6.3

Magnetic and transport properties of Ca3(Ru1-xMnx)2O7 (x = 0.04) in a magnetic field

We have further investigated the effects of the magnetic field on the magnetic and electronic
properties of Mn-doped Ca3Ru2O7. Note that since the ground state magnetic structure of the x =
0.03 compound is similar to the parent compound (AFM-b), applying a magnetic field along the b
axis is expected to lead to a transition to CAFM [97]. In contrast, no field-induced transition (up
to 9 T) is observed in the resistivity measurements on the x = 0.05 sample. Notably, the x = 0.04
compound is close to the phase boundary [153], bridging the correlated AFM-b phase and the Mott
insulating G-AFM state. Therefore, we have performed single-crystal neutron diffraction,
magnetization and resistivity measurements on Mn-doped Ca3Ru2O7 of this doping concentration
(x = 0.04).

92

Figure 3.33: (a) Temperature dependence of the in-pane resistivity ρ at B = 0 T and 9 T, B ∼ b
axis. (b),(c) Field dependence of the in-plane resistivity ρ and magnetization M at T = 10 K and
34 K, B ∼ b axis.
Figure 3.33(a) shows the temperature dependence of the in-plane resistivity ρ of
Ca3(Ru0.96Mn0.04)2O7 at B = 0 and 9 T applied along the b axis, respectively. In contrast to the zerofield data where a series of transitions have been observed, in a magnetic field of 9 T, the system
displays a localized behavior throughout the entire temperature range measured, suggesting a
dramatic change in the electronic structure. In addition, the field-induced IMT is accompanied by
the magnetic transitions. Figure 3.33(b) and (c) show the isothermal magnetoresistance and
magnetization as a function of magnetic field at T = 10 and 34 K, respectively. At T = 10 K, the
Mott insulating ground state is suppressed by a magnetic field at *↑ = 7.2 T with a change in the
resistivity by ~3 orders of magnitude. Concurrently, a large change in the magnetization is
93

observed, indicating a field-induced spin structure transition. This field-induced phase transition
is of first order, which is manifested by the large hysteresis loops as the magnetic field increases
and decreases. At T = 34 K where the zero-field state shows the coexistence of commensurate and
incommensurate magnetic structures, the magnetic field can also drive a first-order magnetic
transition, as shown in both magnetoresistance and magnetization measurements. However, a
much smaller change in resistivity is observed upon applying the magnetic field, presumably
related to the spin scattering associated with a magnetic structure transition, which is distinct from
the 10 K data that arise from the change in the electronic structure.
In order to determine the spin structure of the field-induced state in Ca3(Ru0.96Mn0.04)2O7, we
performed neutron diffraction measurements. Figure 3.34(a) and (b) show the rocking curve scans
across the magnetic wave vectors 
 = (1 0 2) and (0 0 1) at T = 10 K, B = 0 T and 8 T, respectively.
One clearly sees that (1 0 2) is suppressed completely at 8 T, while an alternative magnetic Bragg
peak (0 0 1) appears. Figure 3.34(c) and (d) show the intensity of (1 0 2) and (0 0 1) magnetic
peaks as a function of magnetic field at T = 10 K. A field-induced magnetic structure transition is
observed at *↑ = 7.3 ± 0.1 T. The large hysteresis loop observed when the magnetic field increases
and decreases indicates the first-order nature of the field-induced spin structure transition, in
agreement with both magnetoresistance and magnetization measurements. We have collected a
series of magnetic Bragg peaks such as (0 0 3), (0 0 5) and (2 0 1), etc. at B = 8 T, and performed
representational analysis using FULLPROF [71]. The antiferromagnetic structure that best
describes the neutron diffraction data is of AFM-a type, where the staggered magnetic moments
are along the a axis, perpendicular to the applied magnetic field. In combination to the
ferromagnetic component along the b axis revealed in the magnetization measurements, the
resultant field-induced magnetic state is CAFM, which is a vector sum of AFM-a and a

94

ferromagnetic component (~1.4 ÎźB at B = 8 T, 10 K) along the b axis, similar to the field-induced
state in the parent and Ti-doped compounds discussed in Appendix B. It is worth noting that above
*↑ , the intensity of (0 0 1) starts to decrease as the magnetic field increases further, which can be
ascribed to the fact that CAFM transforms toward the fully spin polarized state (PM) with a
decrease in the staggered antiferromagnetic moment.

Figure 3.34: Rocking curve scans on Ca3(Ru0.96Mn0.04)2O7 across (1 0 2) and (0 0 1) at B = 0 T and
8 T, respectively, T = 10 K. (c),(d) The intensity of (1 0 2) and (0 0 1) as a function of magnetic
field, T = 10 K.

We have also studied the field-induced magnetic transition of the incommensurate magnetic
structures at TMIT < T < TICM. Figure 3.35(a) and (b) show the scans along the [1 0 0] direction
across the magnetic wave vector 
 = (0 0 1) at T = 34 K. At zero field, the magnetic structure
exhibits the coexistence of commensurate and incommensurate phases. However, at B = 5 T, the
incommensurate peaks are suppressed, and the intensity of the commensurate peak becomes much
stronger,

indicating an

incommensurate-to-commensurate magnetic transition.
95

Similar

enhancements in the magnetic intensity are also observed at the wave vectors (0 0 3), (0 0 5) and
(0 0 7), etc. Figure 3.35(c) and (d) show the field dependence of the magnetic intensity of (0 0 1)
and (0.0245 0 1) at T = 34 K. The system undergoes an incommensurate-to-commensurate
↑
= 4.4 Âą 0.1 T, where the (0.0245 0 1) peak is suppressed completely
magnetic transition at *

while the (0 0 1) peak increases then stays almost constant. This field-induced incommensuratecommensurate magnetic transition is also of first order in nature, as the hysteresis loops are clearly
seen when the magnetic field increases and decreases. By performing an analysis similar to that at
↑
T = 10 K, we conclude that the field-induced magnetic state at *
is CAFM as well. In addition,

as the magnetic field increases further, the intensity of the (0 0 1) peak starts to decrease and finally
↑
disappears at *
= 6.6 T, which indicates that the AFM-a component of the CAFM state is

completely suppressed and the system is in a fully spin polarized state (PM).

Figure 3.35: (a),(b) Scans on Ca3(Ru0.96Mn0.04)2O7 along the [1 0 0] direction across (0 0 1) at B =
0 T and 5 T, T = 34 K. (c),(d) The intensity of (0 0 1) and (0.0245 0 1) magnetic Bragg peaks as a
function of magnetic field, T = 34 K.
96

The field-induced IMT of the G-AFM Mott insulating state is accompanied by a simultaneous
change in the lattice parameters. As shown in Fig. 3.36(a), at T = 10 K the lattice parameter c
increases at *↑ by nearly ~1%, while the lattice parameter a remains unchanged across the
transition. The magnetoelastic coupling is similar to that observed at zero field, where the Mott
insulating state is accompanied by a flattening of the RuO6 octahedra. On the contrary, no
structural change is observed for the field-induced magnetic transition at T = 34 K, as shown in
Fig. 3.36(b).

Figure 3.36: Lattice constants a and c of Ca3(Ru0.96Mn0.04)2O7 as a function of magnetic field at (a)
T = 10 K and (b) T = 32 K.

3.6.4

Discussions

Very surprisingly, the effects of the magnetic Mn doping on Ca3Ru2O7 resemble that of the
nonmagnetic Ti substitution [99,100], but are in contrast to the magnetic Fe doping [123]. This
suggests that these emergent states are related to the intrinsic instabilities of the Ru-O network and
the role of 3d transition-metal dopants is to tip the balance between competing tendencies that
already exist in this bilayer ruthenate system.
The scenarios proposed in the studies on Ti-doped Ca3Ru2O7 are likely to be applicable to the
Mn-doped compounds. The electron correlation, the strong coupling between spin and lattice

97

degrees of freedom, the flattening of the RuO6 octahedra which is expected to change the orbital
occupancy of the Ru t2g levels, and the phase competition are the key ingredients to understand the
emergence of the G-AFM Mott insulating state and the field-induced IMT in Ca3(Ru1-xMnx)2O7.
The mechanism underlying the similarity in the doping effects of nonmagnetic Ti and magnetic
Mn on Ca3Ru2O7, which are significantly different from that of the other magnetic dopants Fe, is
an intriguing but challenging question to investigate. In a recent work, J. Peng et al. [153] find that
the MIT temperatures TMIT of Ca3Ru2O7 with different 3d transition-metal dopants are
predominated by the structural parameter c⁄√MF not only in the low-temperature ordered phase,
but also in the high-temperature paramagnetic state. It is proposed that the magnetic and electronic
states of 3d transition-metal doped Ca3Ru2O7 are determined by lattice-orbital coupling, i.e. the GAFM Mott insulating state can stabilize only below a critical value of this structural control
parameter, where orbital polarization is expected based on first principles calculations. Therefore,
due to the larger structural distortions induced by Ti and Mn dopants compared to that of the Fe
dopant, Ti- and Mn-doped Ca3Ru2O7 exhibit G-AFM Mott insulating state, whereas Fe-doped
compound shows incommensurate magnetic structures resulting from the competition among DM
interaction, exchange interactions and magnetic anisotropy. Furthermore, it has been suggested
that the structural distortions caused by 3d transition-metal impurity is dominated by the electronic
scattering rather than the difference in the ionic radius or the magnetic moments [153].
3.6.5

Summary

We have investigated the magnetic properties of the emergent phases in the bilayer ruthenate
Ca3Ru2O7 induced by Mn doping. The system shows an incommensurate magnetic structure for x
≤ 0.03, where the ground state magnetic structure is the same as the parent compound. For x ≥
0.04, the ground state changes to a G-AFM Mott insulator. We have also investigated the magnetic-

98

field-induced phase transitions in Ca3(Ru0.96Mn0.04)2O7 that is positioned at the phase boundary of
the T − x phase diagram. Upon applying a magnetic field along the b axis, the G-AFM ground state
transforms into CAFM through a first-order transition, together with a collapse of the Mott
insulating state and drastic changes in the lattice constants. Our results unambiguously demonstrate
the presence of competition among various states in this bilayer ruthenate system, which can be
tuned by 3d transition-metal doping and external magnetic field.

99

3.7

Temperature- and field-driven spin reorientations in triple-layer

ruthenate Sr4Ru3O10
To date, compared to the intense efforts invested on the single-, double-layer and threedimensional ruthenates (n = 1, 2 and ∞), there have been much fewer studies on the triple-layer (n
= 3) compounds. Sr4Ru3O10 crystallizes in an orthorhombic space group Pbam [154], as shown in
Fig. 3.37(a), which displays interesting but perplexing magnetic properties. From the magnetic
susceptibility measurements, it has been reported that Sr4Ru3O10 undergoes a paramagneticferromagnetic transition at Tc = 105 K, below which the easy axis is along the c direction [154].
Intriguingly, while an additional transition has been found in the magnetic susceptibility at T* =
50 K [154,155], no anomaly is revealed in the specific heat measurements [156]. Although several
distinct scenarios have been proposed to account for the feature at T*, its intrinsic character remains
an open question. On the one hand, below T* a canted antiferromagnetic structure with a
ferromagnetic component along the c axis and an antiferromagnetic component in the ab plane has
been proposed [155,157]. Nevertheless, no evidence of antiferromagnetism has been observed via
neutron diffraction experiments [158,159]. On the other hand, previous neutron diffraction
measurements have yielded contradictory results regarding the direction of the ferromagnetic
moments in Sr4Ru3O10 [158,159]. It is initially argued that the spins are aligned in the ab plane
based on the observation of (0 0 L)-type magnetic Bragg peaks and no anomaly has been observed
at T* [158]. As a result, the feature at T* in the magnetic susceptibility measurements is ascribed
to a magnetic domain process [158]. In contrast, a distinct magnetic easy axis has been proposed
more recently, where the magnetic moments are determined to be along the c axis since the (0 0
L)-type magnetic Bragg peaks are found to be absent. In addition, a kink-like feature has been
reported in the temperature dependence of the Bragg reflection (2 0 0) at T*, which is also claimed

100

to be observed in all other reflections [159]. Therefore, the easy axis of the ferromagnetic moments
and the nature of the anomaly at T* of Sr4Ru3O10 remain elusive.

Figure 3.37: (a) The crystal structure of Sr4Ru3O10. (b) The magnetic structure of Sr4Ru3O10. Note
that both moment size and spin direction cannot be uniquely determined due to the lack of enough
Bragg peaks with reasonably good magnetic intensity.

Another intriguing property of Sr4Ru3O10 is the field-induced metamagnetic transition. While
a typical magnetic hysteresis loop of ferromagnets is observed as the magnetic field is applied
along the c axis, a first-order metamagnetic transition is seen below T* for the field applied in the
ab plane [154,155]. Experimental signatures of the electronic phase separation during the phase
transition have been observed [160]. Nevertheless, previous neutron diffraction studies reveal
controversial features regarding this metamagnetic transition. Across the critical field, while no
anomaly has been observed in the field dependence of the magnetic reflection (0 0 8) in Ref. [158],
a field-induced transition at the same wave vector is reported in Ref. [159]. In addition, it has been
101

revealed that the metamagnetic transition is accompanied by a change in the lattice, as evidenced
in the Raman scattering [157], neutron diffraction [159] and magnetostriction studies [161], which
suggests the presence of strong spin-lattice coupling in this system. To date, the nature and the
origin of this metamagnetic transition are yet to be resolved.
In this section, we present a single-crystal neutron diffraction study on the magnetic order and
the metamagnetic transition in Sr4Ru3O10. We find that the material orders ferromagnetically at Tc
~ 100 K without any signature of antiferromagnetic components, in agreement with the previous
studies [158,159]. However, the magnetic moments are found to possess both in-plane and out-ofplane components below Tc, in contrast to the previous neutron diffraction studies where the spins
are proposed to align either in the ab plane [158] or along the c axis [159]. In addition, we have
observed spin reorientations below T* ~ 50 K, where the magnetic moments incline toward the
out-of-plane direction. Furthermore, we show that while the magnetic moments are nearly along
the c axis at T = 1.5 K, a magnetic field applied along the in-plane [1 -1 0] direction gives rise to
a spin reorientation at a critical field Bc, above which there exists a spin component perpendicular
to the plane defined by the field direction and the c axis, in accordance to the metamagnetic
transition. This observation is distinct from the previous studies, where the metamagnetic
transition is ascribed to the magnetic domain processes [158] or the coexistence of zero-field caxis component and the field-induced in-plane spin polarization [159]. Both temperature- and
magnetic-field-driven spin reorientations are presumably ascribable to the change in the
magnetocrystalline anisotropy that is strongly correlated with the RuO6 octahedral distortion. Our
results reconcile the inconsistency among previous neutron diffraction studies on the magnetic
easy axis of Sr4Ru3O10, resolve the puzzling character of the transition at T*, and elucidate the
nature of the field-induced metamagnetic transition.

102

3.7.1

Materials and methods

The single-crystal Sr4Ru3O10 was grown by the Floating Zone method [162]. The lattice
constants are a = 5.5280 Å, b = 5.5260 Å and c = 28.651 Å. The phase and quality of the crystal
have been verified by x-ray diffraction measurements. The magnetization as a function of both
temperature and magnetic field has been measured using SQUID, and the results are in agreement
with previous studies [154,155]. A very small amount of SrRuO3 intergrowth has been revealed in
the magnetic susceptibility measurements, which is easy to be separated in the neutron diffraction
measurements due to its very different lattice parameter c (c = 7.8446 Å in the orthorhombic unit
cell). Zero-field and field-dependent neutron diffraction measurements were performed using the
HB-3A four-circle diffractometer (λ = 1.5426 Å) and the HB-1A triple-axis spectrometer (λ = 2.36
Å), respectively, at HFIR in ORNL. At HB-3A, the sample was mounted on an aluminum stick
and loaded into a closed-cycle Helium displex. At HB-1A, the sample was oriented in the
horizontal (H H L) scattering plane and loaded into a vertical-field cryomagnet such that the
magnetic field was applied along the [1 -1 0] direction. The Bragg reflections (H K L) were in the
reciprocal lattice units 2π/a, 2π/b and 2π/c of the orthorhombic space group Pbam (No. 55) [154].
3.7.2

Temperature-driven spin reorientations at zero field

We start with the zero-field neutron diffraction data collected at the HB-3A four-circle
diffractometer. Figure 3.38(a) and (b) present the rocking curves of the nuclear Bragg reflections
(0 0 2) and (1 1 1) at T = 8, 50 and 100 K, respectively. The intensity of (0 0 2) shows a nonmonotonic behavior, which is significantly enhanced at 50 K compared to that measured at 8 K
and 100 K. In contrast, the (1 1 1) intensity becomes stronger with decreasing temperature. These
observations suggest that there is magnetic intensity superimposed on the nuclear Bragg peaks. To
further elucidate the non-monotonic temperature dependence of the magnetic intensity, Figure

103

3.38(c) and (d) present the peak intensity of (0 0 2) and (1 1 1) Bragg reflections as a function of
temperature, respectively. Upon cooling, additional reflection intensity in both peaks emerges
below 100 K. Similar enhancement has been observed in other nuclear Bragg peaks including (0
0 6) and (0 0 8), indicating the development of a ferromagnetic ordering below Tc ~ 100 K in line
with the magnetic susceptibility measurements [154]. Consistent with Fig. 3.38(a), the intensity of
the (0 0 L)-type magnetic reflections with L = 2, 6 and 8, exhibits non-monotonic temperature
dependence, with a maximum at T* ~ 50 K that is reminiscent of the anomaly at T* in the magnetic
susceptibility data [154,155]. In accordance to Fig. 3.38(b), the magnetic intensity of (1 1 1)
emerges at Tc and continues to increase below T*. On the one hand, the observation of the (0 0 L)type magnetic reflections in our study is in agreement with the previous neutron diffraction
experiments reported in Ref. [158], but is different from that in Ref. [159] where they are found to
be absent. On the other hand, we have revealed distinct temperature-dependent behaviors in the
intensity of (0 0 L) and (1 1 1) in our experiment, which is in sharp contrast to the previous studies
where either no anomaly was observed [158], or a kink feature was claimed to exist in the
temperature dependence of all the reflections [159]. Note that the HB-3A diffractometer is
equipped with a two-dimensional neutron detector, from which the change in the scattering angle
2θ of the diffracted neutrons at different temperatures is found too small to be resolved due to the
instrumental resolution limitation. This is in agreement with a change in the lattice constants of
only ~0.04% upon cooling observed in the previous studies [159]. Therefore, the variation in the
peak intensity of these Bragg peaks cannot be ascribed to the structural change, but is magnetic in
origin.

104

Figure 3.38: (a),(b) Rocking curve scans across (0 0 2) and (1 1 1) Bragg reflections at T = 8, 50
and 100 K, respectively. (c),(d) Temperature dependence of the peak intensity of (0 0 2) and (1 1
1) at B = 0 T. The red and green dashed lines denote Tc and T*, respectively.

In order to explore the possible magnetic configurations in Sr4Ru3O10, we carried out the
magnetic representational analysis using the program SARAH [163]. The space group of the
crystal structure is No. 55 Pbam and the propagation vector is n = (0 0 0). There are four
inequivalent Ru atoms in a chemical unit cell which are located at Ru1 (0 0 0), Ru2 (0 0 0.1402),
Ru3 (0.5 0 0.3598) and Ru4 (0.5 0 0.5). The analysis shows that an in-plane antiferromagnetic
component would give rise to strong magnetic reflections at (1 0 0), (1 0 1) or (0 1 1), which,
nevertheless, are all absent in our measurements. Therefore, the feature at T* in the magnetic
susceptibility cannot be ascribed to the formation of antiferromagnetic components in the ab plane
that was claimed previously [155,157]. As a result, the only symmetry-allowed magnetic
configuration in Sr4Ru3O10 is the ferromagnetic structure with the magnetic moments either along
the c axis or in the ab plane. Since neutrons couple to the magnetic moment perpendicular to the
105

momentum transfer 
, the observation of magnetic intensities at (0 0 L) with L = 2, 6 and 8 at T*
< T < Tc suggests the development of a magnetic order with the spins aligned in the ab plane. This,
combined with the magnetic susceptibility measurements which show an easy axis closer to the c
axis, suggests that the ferromagnetic moments have both in-plane and out-of-plane components in
this temperature regime. Note that at T* < T < Tc, the intensity of (0 0 2) increases more rapidly
than that of (1 1 1), implying that the magnetic moments incline toward the ab plane, though the
easy axis is always closer to the c axis. Below T*, the decrease in the intensity of the (0 0 L)-type
magnetic peaks suggests that the in-plane ferromagnetic moment reduces, and the spins incline
continuously toward the out-of-plane direction upon cooling. This speculation is supported by the
observation of an increase in the intensity of the (1 1 1) Bragg peak below T*, as the direction of
the (1 1 1) wave vector is almost perpendicular to the c axis (note that a* ≈ b* ≈ 1.137 Å-1, c* ≈
0.219 Å-1). We have collected the nuclear and magnetic Bragg peaks at 8 K, 50 K and 100 K, and
attempted to perform Rietveld analysis using the program FULLPROF [71]. It is worth noting that
the magnetic moments of the outer-layer Ru ions (Ru1 and Ru4) and the central-layer ones (Ru2
and Ru3) can be different, owing to the difference in the direction and magnitude of the RuO6
octahedral rotation [154]. At T = 8 K, due to the very weak intensity of the (0 0 L) peaks, the spins
are nearly aligned along the c axis. Since only the (1 1 1) peak is associated with this spin
component, the magnitude of the magnetic moments of the outer- and central-layer Ru ions cannot
be exclusively determined at this temperature. Similarly, more magnetic reflections are required
in order to determine the moment sizes and the canting angles quantitatively at T*. The schematic
of the magnetic structure of Sr4Ru3O10 at zero field is plotted in Fig. 3.37(b).

106

3.7.3

Field-driven spin reorientations

Next, we discuss the neutron diffraction measurements using the HB-1A triple-axis
spectrometer with the magnetic field applied along the in-plane [1 -1 0] direction, where a firstorder metamagnetic transition is observed in the magnetic susceptibility measurements [154].
Figure 3.39(a) and (b) display the peak intensity of the (0 0 2) and (1 1 1) Bragg peaks as a function
of the magnetic field at T = 1.5 K. It is remarkable that the intensity of (0 0 2) increases significantly
at Bc = 1.5 T, whereas that of (1 1 1) decreases. These observations are distinct from the previous
study where no anomaly has been observed in the intensity of the magnetic reflections as a function
of magnetic field [158]. It is worth noting that in another neutron diffraction study, a similar trend
to (0 0 2) shown in Fig. 3.39(a) has been observed in the field dependence of the intensity of (0 0
8) [159]. However, a slight reduction in the scattering intensity of the (H K 0) reflections was
claimed and suggested to be associated with the ferromagnetic moment along the c axis, which led
the authors to argue that the field-induced transition is due to the coexistence of the initial zerofield ferromagnetic order with a field-induced in-plane spin polarization [159]. If this were the
case, one would expect a slight increase in the magnetic intensity of the (1 1 1) reflection above
the critical field Bc, as (1 1 1) is perpendicular to the [1 -1 0] direction but is about 82.2° with
respect to the [0 0 1] direction. This is not in line with our observation of a field-induced decrease
in the intensity of the (1 1 1) reflection, which suggests that the applied magnetic field gives rise
to a spin component perpendicular to the plane defined by the field direction and the c axis. This
is supported by a recent study which, by measuring the magnetic moment vectors, reports the
existence of a magnetic moment component that is perpendicular to the field rotation plane [164].
Note that the slight difference in the critical field compared with that in the magnetic susceptibility
data may arise from the different applied field directions in the ab plane. To further support the

107

variation of the magnetic intensity below and above Bc, Figure 3.39(c) and (d) plot the θ-2θ scans
over the (0 0 2) and (1 1 1) Bragg peaks measured at T = 1.5 K with B = 0 and 3.5 T, respectively.
We can clearly see that above Bc the intensity of the (0 0 2) peak becomes enhanced while that of
the (1 1 1) peak is suppressed. Furthermore, there is no noticeable change in the 2θ values of both
peak positions across the field-induced magnetic transition, which is consistent with the previous
study that the lattice constants a and c change only by ~0.04% at the critical field that is beyond
the instrumental resolution in our study [159]. For comparison, in the inset of Fig. 3.39(d), we
present the θ-2θ scan of a high-|2| nuclear peak (0 0 16), where the magnetic intensity is expected
to be very weak due to the small magnetic form factor. The very little difference in the intensity
of (0 0 16) at B = 0 and 3.5 T further substantiates that the field-induced intensity variation of the
low-|2| peaks (0 0 2) and (1 1 1) is magnetic in origin, rather than due to a structural change.
Figure 3.40(a) and (b) show the θ-2θ scans of the (0 0 2) and (1 1 1) Bragg peaks measured at
T = 50 K with B = 0 and 3.5 T, respectively. The intensity of both (1 1 1) and (0 0 2) displays
negligible changes upon applying the magnetic field. These results, combined with the features
observed at low temperature shown in Figure 3.39, are in agreement with the magnetic
susceptibility measurements that the metamagnetic transition only occurs below T* ~ 50 K.

108

Figure 3.39: (a),(b) Field dependence of the peak intensity of (0 0 2) and (1 1 1) at T = 1.5 K. (c),(d)
θ-2θ scans across (0 0 2) and (1 1 1) at B = 0 and 3.5 T, T = 1.5 K. Inset shows the θ-2θ scan over
(0 0 16) at B = 0 and 3.5 T, T = 1.5 K.

Figure 3.40: θ-2θ scans across (0 0 2) and (1 1 1) at B = 0 and 3.5 T, T = 50 K.

3.7.4

Discussions

The observation of a spin reorientation at T* in Sr4Ru3O10 at zero field has elucidated the longstanding puzzle on the anomaly in the magnetic susceptibility measurements [154]. This spin

109

reorientation suggests a change in the magnetic easy axis as a function of temperature. This feature
is reminiscent of the widely studied spin reorientation transitions in rare earth magnets and
orthoferrites [165], which are ascribed to the change in the magnetic anisotropy constants as a
function of external parameters, such as temperature, magnetic field and pressure, etc [166]. Our
finding raises an intriguing question: What is the underlying mechanism responsible for the change
in the magnetic easy axis in Sr4Ru3O10? It is known that in Ruddlesden-Popper type ruthenates,
the magnetic anisotropy is strongly coupled to the structural distortions of the RuO6 octahedra. For
instance, the magnetocrystalline anisotropy has been extensively investigated in SrRuO3 thin films,
where the magnetic anisotropy and the saturated magnetic moments can be readily tuned by
controlling the RuO6 octahedral distortion via the epitaxial strain imposed by substrates [167].
Considering the fact that the ferromagnetic state with Tc ~ 100 K in the triple-layer (n = 3)
Sr4Ru3O10 bridges the physics between the double-layer (n = 2) Sr3Ru2O7 (Fermi liquid,
ferromagnetic instability) [78] and the three-dimensional (n = ∞) SrRuO3 (ferromagnetic, Tc = 160
K) [168], the change in the magnetocrystalline anisotropy in Sr4Ru3O10 across T* may arise from
the corresponding changes in the lattice structures. Indeed, anomalies in the lattice constants with
c expanding while a contracting below T* at zero field have been observed previously [159,161].
These changes in lattice parameters are expected to alter the occupancy of the Ru t2g orbitals in
this multiband system, which consequently affects the magnetic anisotropy through spin-orbit
coupling [169].
The nature of the in-plane field-induced metamagnetic transition in Sr4Ru3O10 below T*, which
has been an unresolved puzzle, can now be readily understood. It is also due to a spin reorientation
where the magnetic moments reorient from the nearly c axis toward the ab plane. Intriguingly, we
find that there exists a magnetic moment component perpendicular to the plane defined by the field

110

direction and the c axis above the critical field, which is consistent with the observation of a recent
study where it has been ascribed to the Dzyaloshinskii-Moriya interaction assuming that the easy
axis is along the c direction [164]. On the other hand, it is worth noting that across the field-induced
metamagnetic transition below T* the lattice parameter a expands while c shrinks [159], a trend
which is opposite to the lattice change observed upon cooling through T* at zero field discussed
above [159,161]. This suggests that the magnetic anisotropy is altered above the critical field due
to strong spin-lattice coupling, which needs to be taken into account in the future theoretical
modeling.
3.7.5

Summary

The magnetic structure and the metamagnetic transition in the triple-layer ruthentate Sr4Ru3O10
(n = 3) are revisited by neutron diffraction measurements. The magnetic order below Tc ~ 100 K
is found to be ferromagnetic with both in-plane and out-of-plane spin components, and the
ferromagnetic moments evolve toward the c axis below T* ~ 50 K. These findings elucidate the
long-standing puzzle on the nature of the magnetic transition at T* in Sr4Ru3O10. Below T*, in a
magnetic field applied along the in-plane [1 -1 0] direction, the ferromagnetic moments undergo a
spin reorientation from the c axis toward the ab plane with a spin component perpendicular to the
plane defined by the magnetic field direction and the c axis. Both temperature- and field-induced
spin reorientations can be attributed to a change in the magnetocrystalline anisotropy due to the
change in the RuO6 octahedral distortion.

111

Chapter 4
Neutron scattering studies on inverse-trirutile chromates
4.1

Introduction and motivation
¸m
Inverse-trirutile compounds ¡)m
 * š¸ (A = Ga, V, Cr, Mn and Fe; B = Te, W) crystallize in

a tetragonal structure characterized by a tripling of the rutile unit cell along the c axis. The crystal
structure is composed of chains of edge-sharing AO6 octahedra separated by a BO6 octahedron.
The materials of interest in this chapter are the chromates Cr2MO6 (M = Te, W and Mo). The
crystal and magnetic structures of Cr2TeO6 and Cr2WO6 were first reported in the late 1960s, as
sketched in Fig. 4.1(a) and (b) [170]. The nearest-neighbor Cr3+ ions with S = 3⁄2, separated by a
distance of ~3 Å, are coupled antiferromagnetically ( > 0) which is much stronger compared to

the next-nearest-neighbor interaction  (~3.8 Å). Thus, the two nearest-neighbor Cr3+ spins are

expected to form a S = 3⁄2 spin dimer, with the dominant exchange interactions between Cr3+
spins illustrated in Fig. 4.1(c). Both compounds exhibit long-range antiferromagnetic orders with
the Neel temperature TN = 93 K for Cr2TeO6 and TN = 45 K for Cr2WO6, respectively [171].
Interestingly, even though the lattice parameters (a = b = 4.545 Å and c = 8.995 Å for Cr2TeO6;
a = b = 4.583 Å and c = 8.853 Å for Cr2WO6), bond lengths and bond angles are very close in
these two materials, their ground state magnetic structure are very distinct [170]. The inter-dimer
interaction in Cr2TeO6 is antiferromagnetic ( > 0), whereas that in Cr2WO6 is ferromagnetic

(  < 0 ). It is important to emphasize that the difference in the inter-dimer superexchange

interaction  , which is mediated by the intervening oxygen ions, cannot be explained by the

Goodenough-Kanamori rules based on the symmetry consideration of the occupied electron
orbitals [172,173]. Thus, understanding the nature of the inter-dimer exchange interactions in these

112

isostructural inverse-trirutile compounds is of fundamental importance to elucidate the origin of
different types of magnetic orderings observed.

Figure 4.1: Schematics of the crystal and magnetic structures of (a) Cr2TeO6 and (b) Cr2WO6. (c)
Dominant exchange interactions between Cr3+ ions. The black line represents a unit cell. There are
two dimer sites (A and B) in a unit cell. Each dimer consists of two spins at the top and bottom
sides strongly coupled by an antiferromagnetic exchange interaction ( > 0). The dimers interact
with each other by inter-dimer interactions ( and ) );  connects different dimers (A-B) and )
connects the same dimer sites (A-A; B-B). The magnetic structure in the ordered phase indicates
that ) < 0 (ferromagnetic) in both Cr2WO6 and Cr2TeO6, while  is ferromagnetic ( < 0) in
Cr2WO6 and antiferromagnetic ( > 0) in Cr2TeO6, respectively. We find that  ≫ | |, |) |.
The magnetic excitations of the inverse-trirutile compounds are also of particular interest. For
classical long-range ordered spin systems, the magnetic excitations are generally well described
by the linear spin wave (LSW) theory, which gives rise to transverse Nambu-Goldstone modes
arising from the phase fluctuations of the order parameter. Nevertheless, for spin dimer systems,
the LSW theory does not always work well. With a dominant antiferromagnetic intra-dimer
exchange interaction, the system forms a singlet ground state (<Z[Z = 0) with a gapped triplet first

excited state (<Z[Z = 1). As the inter-dimer interactions become stronger, it undergoes a phase

transition from the quantum disordered state to a long-range ordered state. Such a long-range
magnetic order can be induced via applying pressure to enhance the inter-dimer interactions, or
via applying magnetic field to split the spin-triplet state by the Zeeman term in the Heisenberg
113

Hamiltonian. Intriguingly, in the ordered phase close to the quantum critical point (QCP), in
addition to the gapless transverse Goldstone modes, the longitudinal fluctuations in the magnitude
of the sublattice magnetization are expected to give rise to Higgs amplitude modes, which are in
analogy to Higgs bosons in high energy physics [174,175]. However, to date, the experimental
realizations and observations of Higgs amplitude modes in quantum spin systems are rather rare,
particularly limited to quasi-one-dimensional S = 1 spin chains [176-178], two-dimensional
coupled S = 1⁄2 two-leg spin ladders [179] and spin dimer systems with S = 1⁄2 [180-182]. This
is in part because it is rare to find spin systems with the ground state close to a QCP and excitations
within an appropriate energy range that is approachable using spectroscopic probes, and in part
because the Higgs amplitude modes have a finite life time, decaying into a pair of transverse
Goldstone modes which makes the observation difficult [183-185].
In this chapter, we explore the nature of the exchange interactions as well as the magnetic
excitations of the inverse-trirutile chromates. In Sec. 4.2, we show that the inter-dimer interactions
in Cr2(Te1-xWx)O6 can be tuned from antiferromagnetic to ferromagnetic via mixing the
nonmagnetic Te and W ions. Our density functional theory (DFT) calculations suggest that the
hybridization between the W 5d and O 2p orbitals plays an essential role in the observed
ferromagnetic inter-dimer interactions in Cr2WO6. To substantiate this idea, in Sec. 4.3 we
investigate the magnetic and electronic structures of another isostructural inverse-trirutile
Cr2MoO6, and we find that the orbital hybridization between Mo 4d and O 2p indeed leads to a
ferromagnetic inter-dimer coupling. Finally, in Sec. 4.4, we present the observation and
explanation of Higgs amplitude modes in the magnetic excitation spectra of Cr2TeO6 and Cr2WO6
via inelastic neutron scattering and extended spin wave (ESW) theory calculations.

114

4.2

Tuning

the

exchange

interaction

in

Cr2(Te,W)O6

via

orbital

hybridization
As soon as the magnetic structures of Cr2TeO6 and Cr2WO6 are determined, it is pointed out
that the magnetic coupling in these materials is strongly influenced by the nonmagnetic Te and W
ions [173]. The implications of these observations are remarkable: one may tune the inter-dimer
exchange interactions in a spin dimer system without introducing severe structural distortions or
adding additional charge carriers. In this section, we present the magnetic phase diagram and the
electronic structure of Cr2(Te1-xWx)O6 system (0 < x < 1) obtained via neutron powder diffraction
measurements and first principles calculations, respectively. We observe a crossover in the sign of
the inter-dimer exchange interaction at xc ~ 0.7, where both the transition temperature TN and the
sublattice magnetic moment 34 reach the minimum values. Our ab initio electronic structure
calculations show that the presence of the low-lying W 5d states and their hybridization with the
O p states play an important role in the exchange interaction between Cr 3d moments. However, a
quantum phase transition to the quantum disordered spin-singlet state cannot be achieved by the
chemical substitution of Te and W.
4.2.1

Materials and methods

Polycrystalline samples of Cr2(Te1-xWx)O6 with various x values were synthesized using the
solid-state chemistry reaction method. The mixtures of Cr2O3, TeO2 and WO3 were grounded
together and annealed in air around 700 ~ 1000 ℃ for 40 hours. The magnetization was measured
using the Superconducting Quantum Interference Device (SQUID) magnetometer and the specific
heat was measured using PPMS. Neutron powder diffraction measurements on each sample of ~4
grams were performed using the HB-2A neutron powder diffractometer at HFIR in ORNL. The

115

data were taken with the incident neutron wavelength λ = 2.41 Å using the collimation of 12′open-6′.
(a)

(b) 0.0036

4.58

Cr2Te1-xWxO6

9.00

c (Å)

a, b (Å)

8.95

4.56

8.90

M (ÂľB / Cr)

0.0032
4.57

0.0028

x=0
0.1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

0.0024

4.55
8.85
0.2

0.4

(c)
C mag (J/K mol Cr)

12

x

0.6

Cr2Te1-xWxO6

8

0.8

0.0020
0

1.0

x=0
0.1
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1.0

4

0
0

100

200

300

T (K)

(d)

1.0
I (T) / I (4K)

4.54
0.0

x=0
0.5
0.8
1.0

0.8
0.6
0.4
0.2
0.0

60

120
T (K)

180

20

40

60
T (K)

80

100

Figure 4.2: (a) x dependence of lattice parameters measured at T = 4 K. (b),(c) Temperature
dependence of magnetic susceptibility and magnetic specific heat, respectively. (d) Normalized
neutron intensity of order parameters, (1 0 1) for x = 0, 0.5 and (0 0 1) for x = 0.8, 1, as a function
of temperature. Solid lines are guides to the eye.

4.2.2

Magnetic susceptibility, specific heat and powder neutron diffraction

Figure 4.2(a) presents the lattice parameters at T = 4 K for Cr2(Te1-xWx)O6, which are obtained
by refining the neutron powder diffraction data using FULLPROF [71]. The in-plane lattice
expands while the c axis contracts slightly with increasing W concentration. The linear relationship
of a and c as a function of x suggests a homogenous solid solution of the Te and W mixture. The
temperature dependence of the magnetic susceptibility measured with an applied field H = 3000
Oe is shown in Fig. 4.2(b). There are two noteworthy features: (i) All the samples display a broad
peak with the peak position Tp evolving non-monotonically with x. (ii) As to be shown later, the
antiferromagnetic transition temperature TN of these compounds is smaller than the corresponding
116

Tp, implying the development of strong low-dimensional correlations prior to forming the threedimensional long-range magnetic order. The increase in the magnetic susceptibility at low
temperature presumably originates from the paramagnetic impurities.
The non-monotonic dependence of TN on x is corroborated by the temperature dependence of
the magnetic specific heat shown in Fig. 4.2(c). The magnetic specific heat is obtained by
subtracting the phonon contribution from the total heat capacity, using the scaled specific heat of
an isostructural nonmagnetic compound Ga2TeO6. Compared with the relatively sharp drop in the
heat capacity at T > TN in both end members, the magnetic heat capacity of Cr2(Te1-xWx)O6 with
nonzero x, particularly for 0.3 ≤ x ≤ 0.8, exhibits broad peaks reminiscent of a spin-glass-like
transition. However, no difference is found in the temperature dependence of the DC magnetic
susceptibility measured under ZFC and FC conditions. In addition, the AC magnetic susceptibility
measurements reveal nearly frequency-independent (f = 10 ~ 10kHz) signal even for x = 0.5 at
which the compound shows the strongest chemical site disorder. This excludes the occurrence of
a spin-glass transition. Instead, all the compounds display long-range antiferromagnetic orders as
revealed by the neutron powder diffraction measurements discussed next. On the other hand, the
integrated magnetic entropy is in the range of 4.43 ~ 9.53 J / (K mol Cr), much smaller than the
theoretical value (11.5 J / (K mol Cr)) for Cr3+ ions with S = 3⁄2. Thus, the broadening of the
magnetic heat capacity suggests that in Cr2(Te1-xWx)O6 the low-temperature long-range ordered
antiferromagnetic state coexists with strong magnetic fluctuations.

117

Figure 4.3: Neutron powder diffraction measurements with x = 0, 0.5, 0.8 and 1.0 at T = 4 K. Black
symbols are the experimental data, red curves for fits and the difference between these two are
represented by the blue curves. For x = 0.5, there is a minor impurity phase Cr2O3 (< 1%) as
denoted by #. Symbols of + and * denote the magnetic Bragg peaks for AFM-I and AFM-II,
respectively. Insets show the expanded view of the low-angle magnetic Bragg peaks at T = 4 K
(black) and 150 K (dark green).

Some representative neutron powder diffraction data for x = 0, 0.5, 0.8 and 1.0 measured at T
= 4 K (and 150 K shown in the insets) are shown in Fig. 4.3. For x = 0 and 0.5, the magnetic Bragg
peaks show up at 
 = (0 0 2), and the refined spin structure is with antiferromagnetic inter-dimer
exchange interaction (AFM-I), as shown in Fig. 4.1(a); For x = 0.8 and 1, the magnetic Bragg
peaks occur at 
 = (0 0 1), and the corresponding magnetic structure is with ferromagnetic interdimer interaction (AFM-II), as shown in Fig. 4.1(b). It is noteworthy that the FWHM of the
magnetic Bragg peaks of all the samples is determined by the instrumental resolution according to
the Rietveld refinement, confirming the existence of a long-range magnetic order at low
temperature. Temperature dependence of the order parameters of these four samples is plotted in

118

Fig. 4.2(d), which shows a non-monotonic dependence of TN on x, as discussed previously, with
TN for x = 0.8 smaller than the others.

Figure 4.4: (a) TN − x phase diagram of Cr2(Te1-xWx)O6. PM represents the paramagnetic phase. (b)
Sublattice magnetization as a function of x obtained from neutron powder diffraction
measurements.

The TN − x phase diagram of Cr2(Te1-xWx)O6 is shown in Fig. 4.4(a). Interestingly, the system
displays a crossover of the magnetic state at xc ~ 0.7. The compounds with x < 0.7 exhibits an
AFM-I type magnetic structure, whereas those with x > 0.7 show an AFM-II type magnetic
structure. Accordingly, TN varies non-monotonically as a function of x and reaches the minimum
value (TN ~ 29.6 K) at xc. At the crossover point (xc ~ 0.7), both AFM-I and AFM-II types of
magnetic structures coexist, as shown in Fig. 4.5. Intriguingly, the sublattice magnetization 34 of
Cr2(Te1-xWx)O6 obtained from the data refinement at T = 4 K also displays a non-monotonic
dependence on x and reaches a minimum at xc, as shown in Fig. 4.4(b). All 34 values are smaller
than the expected value 3 ÎźB for fully localized Cr3+ ions (with negligible SOC).

119

Figure 4.5: (a),(b) Expanded view of low-angle neutron powder diffraction data of Cr2(Te1-xWx)O6
with x = 0.6 and 0.7. Both (0 0 1), (0 0 2) and (1 0 1) magnetic Bragg peaks are observed at T = 4
K for the x = 0.7 sample indicating the coexistence of AFM-I and AFM-II, while only (0 0 2) and
(1 0 1) appear for the x = 0.6 sample corresponding to AFM-I. (c) Temperature dependence of the
(1 0 1) magnetic Bragg peak of these two samples.

These experimental observations bring out several interesting questions: (i) What are the
underlying mechanisms that determine the magnetic ground states of the end members? Because
of very similar crystal structures and lattice parameters of Cr2TeO6 and Cr2WO6, the difference in
their magnetic structures cannot be explained simply by the Goodenough-Kanamori rules based
on superexchange interactions. Thus, one has to understand the differences in the electronic
structures of these materials. (ii) Why do both TN and 34 depend non-monotonically on x and
display the minimum values at xc ~ 0.7?
4.2.3

Density functional theory calculations

In order to understand the ground state magnetic structures and the nature of the intra- and
inter-dimer exchange interactions, we have carried out DFT calculations within the generalized
gradient approximation (GGA) and GGA+U [186] as implemented in the Vienna ab initio
simulation package (VASP) [186,187,75], using the projector-augmented wave method [188,74]
and Perdew-Burke-Ernzerhof exchange-correlation functional [73] in collaboration with Dr. S. D.
Mahanti and Dr. D. Do in Michigan State University. We have chosen four different long-range
ordered magnetic states denoted as A-B (AFM-AFM = AFM-I, AFM-FM = AFM-II, FM-AFM
120

and FM-FM), where A refers to intra-dimer and B refers to inter-dimer magnetic coupling. In all
the calculations, the structural parameters and ionic positions are allowed to relax. The plane-wave
energy cutoff and the total energy accuracy are set as 400 eV and 10-3 eV, respectively. To sample
the Brillouin zone, we use the Monkhorst-Pack schemes with the k-mesh of 14 × 14 × 7 for selfconsistent calculations and 20 × 20 × 10 to get the density of states (DOS).
Table 4.1: (top) Total energy (eV/unit cell) of different [intra]-[inter] dimer magnetic
configurations. (bottom) Calculated and experimental values of intra- ( ) and inter- ( ) dimer
exchange parameters as in the model proposed by Drillon et al [189] using GGA.
Configuration
AFM-AFM AFM-FM FM-AFM FM-FM
Cr2TeO6 -131.711
-131.547 -131.634 -131.411
GGA
Cr2WO6 -159.930
-160.002 -159.854 -159.815
Cr2TeO6 -122.603
-122.572 -122.583 -122.561
GGA+U, U=4
Cr2WO6 -150.714
-150.768 -150.701 -150.750

Compound
Cr2TeO6
Cr2WO6
Parameter
 (meV)  (meV)  (meV)  (meV)
GGA
-4.3
-2.3
-10.4
1.0
Theo.
GGA+U, U=4 eV
-1.12
-0.46
-1.0
0.75
Ref. [189]
-2.9
-0.4
-3.8
0.12

In Table 4.1 (top), we give the GGA energies (per magnetic unit cell containing four Cr atoms).
The ground state is AFM-AFM (AFM-I) for Cr2TeO6 and AFM-FM (AFM-II) for Cr2WO6, in
agreement with the neutron diffraction results. The magnetic moments are nearly the same for all
four Cr atoms and lie in the range of 2.6 ~ 2.8 ÎźB, lower than 3.0 ÎźB for the Cr3+ spin, indicating
the hybridization between Cr d and O p, Te s and W d states. Since GGA generally does not
adequately describe the d electrons in transition-metal atoms, we have also done GGA+U
calculations [190,191]. In the same table, we give the energies for U = 4 eV (this incorporates both
intra-site Coulomb repulsion and exchange through a single parameter [186]). The lowest energy
states are consistent with those obtained in the GGA calculations. The major effect of U is to
121

reduce the splitting between the ground state and the excited states, indicating a reduction in the
strength of the effective exchange coupling between the Cr moments. At the same time, the
magnetic moments of Cr ions increase to ~3.0 ÎźB.
To understand the nature of the exchange couplings between different Cr moments, we look at
the geometry and local coordination of the intra- and inter-dimer Cr3+ pairs. The distance between
the intra-dimer Cr atoms (Cr1 and Cr3 in Fig. 4.6) is ~3 Å, whereas that between the inter-dimer
Cr atoms (Cr1 and Cr2, or Cr3 and Cr4 in Fig. 4.6) is ~3.6 Å. The exchange interaction between
Cr1 and Cr3 is dominated by the Anderson antiferromagnetic (A-AFM) kinetic exchange [192]
(~2  ⁄ in the Hubbard model representation where t is the hopping between the d orbitals of Cr
and U is an effective intra-atomic Coulomb repulsion). The direct A-AFM exchange between the
inter-dimer Cr atoms is likely to be negligible because the distance is ~3.60 Å and t falls off
exponentially. On the other hand, for both intra- and inter-dimer, one has to consider the
superexchange via the O atom which is bonded to either a Te or W atom. Since the Cr1-O-Cr3
angle is close to 90°, we expect the strength of the Cr1-Cr3 superexchange to be weak. Thus, AAFM exchange dominates leading to an antiferromagnetic alignment. In contrast, the Cr1-O-Cr2
angle is ~130° thus superexchange should be appreciable. In Cr2TeO6, this superexchange is
antiferromagnetic, consistent with the filled oxygen states providing the superexchange path. On
the contrary, in Cr2WO6 the Cr1-O-Cr2 coupling is ferromagnetic whose origin may be attributed
to the low-lying unoccupied W d state that hybridizes with the O p and mediates this ferromagnetic
exchange, as to be discussed next.

122

Figure 4.6: Spin densities on the (1 1 0) plane of Cr2TeO6 (left) and Cr2WO6 (right) where red
represents spin up and blue stands for spin down.

In Fig. 4.6, we give the ground state spin densities for the two compounds. The spin densities
associated with Cr1 and Cr3 (intra-dimer coupling) and the nearest oxygen (O1) are very similar
in the two compounds. The superexchange through these oxygen atoms is very weak due to the
nearly 90° exchange path (as one can see in Fig. 4.6), where different O p orbitals hybridize with
Cr1 and Cr3. In contrast, the spin densities associated with the inter-dimer exchange between Cr1
and Cr2 differ dramatically. In Cr2TeO6 this coupling is dominated by the O2 induced
superexchange (~130° path) with very little Te s or p state mixing, whereas in Cr2WO6 the O2
charge and spin distributions are strongly altered by the W d states. This hybridization (and the
basic difference between Cr2TeO6 and Cr2WO6) is also seen in the projected density of states
(PDOS) given in Fig. 4.7. One can see that both Te p and W d hybridize with O2 p, but the latter
is more strongly. This is due to the fact that the W d states are closer in energy to the O2 p band.
The difference in the strength of this hybridization also shows up clearly in the O2 p partial DOS,
which consequently affects the hybridization between the Cr1, Cr2 d states and the O2 p states.
Why the difference in the Cr d and O2 p hybridization gives a ferromagnetic inter-dimer coupling
for Cr2WO6 is more subtle. A simple pictorial (and perturbative) way of thinking about this is as
123

follows: O2 p and W d (or Te p) hybridization creates a virtual hole in the O2 p band (which can
be either spin up or spin down). This virtual hole can mediate a ferromagnetic double exchange
between Cr1 and Cr2. For W d this hybridization is much stronger, and the resultant ferromagnetic
coupling strength is larger than the usual antiferromagnetic superexchange. Therefore, the net
effect is a weak ferromagnetic exchange. In Cr2TeO6 the ferromagnetic coupling is much weaker
and the antiferromagnetic superexchange wins. This underlying physics is somewhat similar to
that proposed by Kasuya to explain the ferromagnetic coupling between the rare-earth moments in
EuS [193].

Figure 4.7: Projected density of states of Cr2TeO6 (left) and Cr2WO6 (right). Cr1 and Cr2 are two
inter-dimer Cr ions. O2 mediates the exchange coupling between Cr1 and Cr2 and hybridizes with
Te and W.
In Table 4.1 (bottom), we give the values of the intra-dimer ( ) and inter-dimer ( ) exchange
obtained by fitting the energies of different spin configurations obtained within GGA to an S =
3⁄2 Heisenberg model. Clearly, the intra-dimer exchange is antiferromagnetic in both compounds,
while the inter-dimer one is antiferromagnetic in Cr2TeO6 but ferromagnetic in Cr2WO6.
124

Introduction of the intra-site Coulomb repulsion U reduces the strength of the exchange,
particularly the antiferromagnetic exchange. These results are in qualitative agreement with the
values extracted from the high-temperature susceptibility measurements by Drillon et al [189],
although there are quantitative differences.

Figure 4.8: Energy difference between antiferromagnetic and ferromagnetic inter-dimer magnetic
configurations with antiferromagnetic configuration fixed for the intra-dimer interaction.

Finally, to understand the magnetic structures of Cr2(Te1-xWx)O6 with different x, we have
used the virtual crystal approximation (VCA) and GGA to calculate the energies of the AFM-I and
AFM-II magnetic structures. The energy difference between these two magnetic configurations is
plotted in Fig. 4.8. We find that the ground state switches from AFM-I to AFM-II when x ~ 0.7,
consistent with the experimental results. Although VCA addresses the problem in an average way
(it does not probe the effects of local fluctuations caused by disorder, clustering, etc.), the results
suggest that by controlling the effective coupling between Cr 3d and W 5d states (indirectly
through intervening O) through the substitution of W into Te sites, we can indeed tune the
competition of the magnetic interactions between the inter-dimer Cr spins, i.e., antiferromagnetic
superexchange and ferromagnetic exchange induced by the orbital hybridization. At x ~ 0.7, these

125

two magnetic interactions are comparable in strength, leading to the strongest spin fluctuations
thus the minimum values in both 34 and TN.
4.2.4

Summary

We have discovered a crossover of the magnetic ground states in isostructural Cr2(Te1-xWx)O6
compounds at xc ~ 0.7 where both the Neel temperature TN and the sublattice magnetization 34
reach the minimum. These phenomena have been attributed to the competition between the
antiferromagnetic superexchange and an unusual ferromagnetic exchange that arises from the
orbital hybridization between the unoccupied W 5d orbitals and the O 2p states which provide the
exchange path between two Cr moments. This work highlights a new approach to tune the
magnetic exchange via chemical doping without introducing additional charge carriers or
structural distortions.

126

4.3

Ferromagnetic inter-dimer interaction in Cr2MoO6
To further substantiate the proposed orbital hybridization effect of the nonmagnetic ions with

empty d orbitals in determining the nature of the exchange interaction, we have studied the
magnetic properties and the electronic structure of another inverse-trirutile compound Cr2MoO6.
In this system, the presence of the low-lying unoccupied Mo 4d orbitals hybridizing with the
occupied 2p orbitals of the oxygen atom in the Cr-O-Cr superexchange path would give rise to a
ferromagnetic coupling between the inter-dimer Cr moments, as in Cr2WO6. This is exactly what
we have observed.
4.3.1

Materials and methods

Polycrystalline samples of Cr2MoO6 were synthesized [194] under high-pressure and hightemperature conditions using a cubic anvil system. The mixtures of Cr2O3 and MoO3 in the
stoichiometric ratio were grounded thoroughly and reacted at 1000 ℃ for 30 min under 4 GPa.
Phase purity of the obtained product was first examined at room temperature using the powder xray diffraction. The magnetic susceptibility was measured using the SQUID magnetometer and the
specific heat was characterized using PPMS. Neutron powder diffraction measurements on the
sample of ~1 gram were performed on the HB-2A neutron powder diffractometer at HFIR in
ORNL. The data were taken with the neutron wavelength λ = 2.41 Å and a collimation of 12′open-6′, and were analyzed using FULLPROF [71].

127

Figure 4.9: Schematic of the crystal and magnetic structure of Cr2MoO6. The intra-dimer
interaction is antiferromagnetic, whereas the inter-dimer exchange is ferromagnetic.

4.3.2

Magnetic susceptibility, specific heat and powder neutron diffraction

The schematics of both crystal structure and the obtained magnetic structure of Cr2MoO6 are
shown in Fig. 4.9. It has an inverse-trirutile structure with the tetragonal space group P42/mnm,
and the lattice parameters are a = b = 4.58717(9) Å and c = 8.81138(22) Å at T = 4 K. As seen in
Table 4.2, the Cr-O-Cr bond angles and bond lengths are close to those of Cr2WO6 and Cr2TeO6
due to the similar ionic radii of Mo6+, Te6+ and W6+. As to be shown later, Cr2MoO6 undergoes a
paramagnetic-antiferromagnetic transition at TN = 93 K with the same spin order as Cr2WO6. The
Cr3+ spins are aligned in parallel within the ab plane but are antiparallel within the dimer (Cr1-Cr3
and Cr2-Cr4). The inter-dimer coupling (Cr1-Cr2 and Cr3-Cr4) is ferromagnetic, the same as that
in Cr2WO6 but is in contrast to the antiferromagnetic coupling in Cr2TeO6 [170,195]. The close
similarity in the crystal structures of these compounds with dramatically distinct magnetic
structures underscores the limitations of the Goodenough-Kanamori rules [172,173] in predicting
the nature of the exchange interaction in theses complex systems with competing exchange
128

mechanisms. The role of other nonmagnetic ions (e.g., Mo, Te and W) in the superexchange
mechanism has to be reexamined.
Table 4.2: Structural parameters of Cr2TeO6, Cr2WO6 and Cr2MoO6 at T = 4 K, including lattice
parameters, bond lengths and bond angles.
Cr2TeO6
Cr2WO6
Cr2MoO6
a
4.54472(8) 4.58346(5) 4.58717(9)
Lattice Parameters (Å)
b
4.54487(8) 4.58346(5) 4.58717(9)
c
8.99539(21) 8.85319(13) 8.81138(22)
Cr3-Cr1
2.968(14)
2.930(13)
2.78(3)
Bond Length (Å)
Cr3-Cr4
3.559(7)
3.558(6)
3.603(8)
Cr3-Cr3
4.54487(8) 4.58346(5) 4.58717(9)
Cr3-O1-Cr1
98.3(3)
97.6(3)
96.65(16)
Bond angle (degree)
Cr3-O2-Cr4
127.0(4)
128.6(3)
130.3(3)

Figure 4.10(a) shows the temperature dependence of the magnetic susceptibility measured in
an external field H = 1000 Oe. A broad peak is observed at Tp ~ 98 K, implying the presence of
short-range antiferromagnetic correlations near and above this temperature. Inset displays the
inverse susceptibility and a Curie-Weiss fit that is performed in the temperature range 200 ~ 300
K. The fitting gives a Weiss temperature Θ½ ~ -254.4 K and an effective magnetic moment Οeff ~
2.92 ΟB / Cr. The negative Θ½ is consistent with the dominance of antiferromagnetic correlations
in Cr1-Cr3 and Cr2-Cr4 dimers [189] in this system. The increase in the susceptibility at low
temperature is presumably due to paramagnetic impurities [171,195]. The magnetic heat capacity
as a function of temperature ¦’¾ (T) is shown in Fig. 4.10(b) with the phonon contribution
subtracted by the scaled heat capacity of an isostructural nonmagnetic compound Ga2TeO6 from
the total heat capacity measured [inset of Fig. 4.10(b)]. The sharp peak at TN ~ 93 K in ¦’¾ (T)
indicates a transition into a long-range ordered antiferromagnetic state. It is noteworthy that the
transition temperature TN is slightly lower than Tp determined from the broad peak in the magnetic
susceptibility, which seems to be a common feature in the inverse-trirutile compounds such as
129

Cr2TeO6, Cr2WO6 and Fe2TeO6 [171,195]. Such a feature is associated with the presence of shortrange low-dimensional magnetic fluctuations prior to entering a three-dimensional long-range
antiferromagnetic state.

Figure 4.10: (a) Temperature dependence of magnetic susceptibility. Inset shows the inverse
susceptibility and the Curie-Weiss fit. (b) Temperature dependence of magnetic heat capacity.
Inset shows the subtraction of the phonon contribution using the scaled heat capacity of Ga2TeO6.
(c) Intensity of (0 0 1) magnetic Bragg peak as a function of temperature. The solid line is a guide
to the eye. (d) Temperature dependence of the integrated magnetic entropy. The horizontal red line
denotes the theoretical value for S = 3⁄2.
The integrated magnetic entropy is shown in Fig. 4.10(d) with a saturated value of 5.30 J / (K
mol Cr), which is about ~50% of the theoretical value (11.5 J / (K mol Cr)) for the S = 3⁄2 Cr3+
ions. The presence of the resultant residual magnetic entropy implies the existence of magnetic
correlations in the system above TN. Note that the residual magnetic entropy (6.20 J / (K mol Cr))
is nearly half of the maximum value which has been commonly observed in low-dimensional or
frustrated magnetic systems with competing interactions. It has also been seen in the isostructural
compounds Cr2(Te1-xWx)O6 [195].

130

Figure 4.11: Neutron powder diffraction data of Cr2MoO6 at T = 4 K. The black symbols are
experimental data and the red curve is the Rietveld refinement fit. The difference is represented by
the blue. The positions of the nuclear and magnetic peaks are marked by the green and magenta
lines, respectively. Inset shows the difference between the low- and high-temperature data with
the positions of magnetic Bragg peaks denoted by the symbol “*”.

The neutron powder diffraction data measured at T = 4 K are shown in Fig. 4.11. The magnetic
propagation vector is determined to be (0 0 0). Representational analysis using the BasIreps
program in FULLPROF [71] suggests that the magnetic structure shown in Fig. 4.9 is symmetry
compatible and the most plausible fit to the data. This magnetic structure is identical to that of
Cr2WO6, that is, the inter-dimer coupling is ferromagnetic. Note that the in-plane spin direction
could not be unambiguously determined because of the tetragonal structure of the system and the
nature of the neutron powder diffraction data. The FWHM of the magnetic Bragg peaks is
determined by the instrumental resolution according to the refinement, confirming the long-range
character of the magnetic order. Inset shows the zoom-in view of the magnetic Bragg peaks after
subtracting the 150 K neutron diffraction data from the 4 K one. The temperature dependence of
the intensity of the (0 0 1) magnetic Bragg peak is presented in Fig. 4.10(c) with the solid curve as
a guide to the eye. The magnetic signal disappears at the antiferromagnetic transition temperature
TN ~ 93 K, consistent with the specific heat measurement. Note that the magnitude of the static

131

magnetic moment (~2.47 ÎźB / Cr) extracted from the neutron powder diffraction measurements is
smaller than the theoretical value of 3.0 ÎźB / Cr, which is attributed to quantum spin fluctuations
as well as the covalency [196-198].
4.3.3

Density functional theory calculations

The observed ferromagnetic inter-dimer coupling in Cr2MoO6 is consistent with a mechanism
we proposed earlier for Cr2WO6 [195]. The nearby unoccupied Mo 4d orbitals (W 5d) play an
essential role in determining the sign of the Cr-Cr exchange by hybridizing with the intervening
filled O p orbitals. That is, the d - p hybridization induced ferromagnetic exchange dominates the
usual antiferromagnetic superexchange interaction in Cr2MoO6 and Cr2WO6. In contrast, in
Cr2TeO6, a compound with a similar crystal structure but without the empty Te d orbitals, the
superexchange interaction between two inter-dimer Cr moments is antiferromagnetic. However,
in spite of the similarity, i.e. the presence of the empty d orbitals, the Neel temperature of Cr2MoO6
is considerably higher than that of Cr2WO6 (TN ~ 45 K).
To understand the electronic structure and the magnetic properties of Cr2MoO6, we have
carried out first principles calculations for four different intra-dimer and inter-dimer magnetic
configurations, i.e., AFM-AFM, AFM-FM, FM-AFM and FM-FM, where AFM and FM denote
antiferromagnetic and ferromagnetic alignments, respectively. DFT calculations were done in a
similar way to that described in Sec. 4.2.3. According to the total energies of different magnetic
configurations listed in Table 4.3, we can see that GGA and GGA+U calculations give the same
magnetic ground state AFM-FM (AFM-II), consistent with the experimental results. However, in
contrast to Cr2WO6 and Cr2TeO6 [195], GGA gives a small negative band gap in Cr2MoO6, while
the experiment shows an insulating behavior. Note that GGA usually underestimates the band gap

132

in semiconductors [199] and the problem is severe in systems containing 3d electrons such as Cr.
This problem can be partially corrected through the GGA+U approximation [190,191].
Table 4.3: Total energy (in eV) per magnetic unit cell containing four Cr3+ ions of different
magnetic configurations.
Configuration
GGA
GGA+U

AFMAFM
-153.325
-143.649

AFM-FM

FM-AFM

FM-FM

-153.520
-143.671

-153.457
-143.562

-153.422
-143.635

Figure 4.12: Density of states (DOS) in the unit of number of states / (eV u.c.) and projected density
of states (PDOS) in the unit of number of states / (eV atom) of Cr2MoO6 calculated by GGA+U.

Figure 4.12 presents the density of states (DOS) and projected density of states (PDOS) of
Cr2MoO6 calculated using GGA+U with U = 4 eV. GGA+U indeed opens up a band gap and
increases the Cr3+ magnetic moment from ~2.5 ÎźB for GGA to ~2.9 ÎźB for GGA+U. The reduction
from 3.0 ÎźB to 2.9 ÎźB is due to the covalency, and a comparison with the experimental value 2.47
ÎźB suggests that the effect of quantum spin fluctuations is important in this system [196-198]. Note
133

that in our electronic structure calculations, the principal axis is along the tetragonal axes, which
are not aligned with the cubic axis of the local crystal field of the CrO6 octahedra, giving rise to
the mixture of both t2g and eg states in the former coordinate system for Cr 3d orbitals. Also note
that the Mo 3d orbitals were treated as core states and were not a part of the manifold of active
orbitals because of the large energy difference (~235.45eV) between O 2p and Mo 3d. Therefore,
the notation of d orbitals of Mo hereafter refers to its 4d orbitals. The projected DOS shows that
there is strong hybridization between Mo d and O p near the bottom of the O p band (-4 to -6.5
eV), and the hybridization also shows up in the lower part of the conduction band [200]. This
strong hybridization is due to the small distance between Mo and O atoms (~1.9 Å). The Cr d
states, which are mainly in the energy range 0 to -4 eV, do not hybridize strongly with the Mo d
states due to the large Cr-Mo distance (~3.0 and ~3.5 Å). However, the perturbed (by mixing with
Mo d) O p bands hybridizing with the Cr d states (Cr-O distance is ~1.9 Å) lead to a ferromagnetic
interaction between the inter-dimer Cr spins. This physics is very similar to what we found in
Cr2WO6 [195]. As elaborated in the cartoon shown in Fig. 4.13, a simple perturbative way of
understanding this ferromagnetic coupling is that the hybridization of O p with the empty Mo d
orbitals creates a virtual hole in the O p band. This hole leads to a ferromagnetic coupling between
two Cr3+ d spins, which flank this oxygen atom. In this sense, the empty 4d orbitals of Mo are
responsible for the observed ferromagnetic coupling between inter-dimer Cr spins, similar to what
happens in the Cr2WO6 compound.
To visualize the magnetic ordering and the orbital hybridization effect, we show the spin
densities projected onto the (1 1 0) plane in Fig. 4.14. We can see that the inter-dimer (Cr1-Cr2,
Cr3-Cr4) Cr spin densities are the same (ferromagnetic, blue-blue or red-red) and the intra-dimer
(Cr1-Cr3, Cr2-Cr4) spin densities are opposite (antiferromagnetic, red-blue), which is very similar

134

to that of Cr2WO6 but in contrast to Cr2TeO6 [195]. The remarkable difference is seen in the spin
density associated with the oxygen atom (O2), which mediates the inter-dimer exchange. In
Cr2WO6 and Cr2MoO6 both lobes are of the same polarization, whereas in Cr2TeO6 they have
opposite polarizations [195], characteristic of the ferromagnetic and antiferromagnetic coupling
between Cr1 and Cr2, respectively.

Figure 4.13: A cartoon illustrating the ferromagnetic coupling between two Cr 3d moments via the
orbital hybridization between O 2p and Mo empty 4d orbitals leading to a virtual electron transfer
from O 2p to Mo 4d, leaving O 2p partially occupied which induces a ferromagnetic coupling
between two Cr moments via Cr 3d-O 2p hybridization.

Figure 4.14: Spin density projected on the (1 1 0) plane of Cr2MoO6 calculated using GGA+U
where red and blue indicate spin up and spin down, respectively. Since the local cubic axes of the
CrO6 octahedra are rotated from the chosen tetragonal x-y-z coordinates (Fig. 4.9), in the latter
coordinate system Cr d-t2g and d-eg states are mixed.
We have calculated the sign and strength of the intra-dimer ( , Cr1-Cr3 distance ~2.780 Å)
and inter-dimer ( , Cr3-Cr4 distance ~3.604 Å) exchange interactions between Cr moments using
135

 for Cr2WO6 and Cr2MoO6
a simple nearest-neighbor Heisenberg model  = −2 ∑〈,〉   ∙ 

within GGA and GGA+U. The respective values are given in Table 4.4. Note that compared to 

and  , further-neighbor interactions () , Cr3-Cr3 distance within the ab plane ~4.587 Å) are much
weaker thus are neglected here. A similar approximation was made previously in Ref. [189]. As
seen in the table, the introduction of U reduces the strengths of the exchange couplings in general.
However, because of the competition between ferromagnetic and antiferromagnetic contributions
to a given exchange parameter, the introduction of U affects  and  differently, not only for a
given system but also between different systems. The precise quantitative values should not be
taken seriously but the qualitative trends are consistent with the experiments. The smaller  value
of Cr2MoO6 than that of Cr2WO6 given by GGA+U is consistent with the scenario that the Mo 4d
orbital is less extended than the W 5d one, which is expected to contribute a weaker ferromagnetic
interaction induced via a less extent of d - p orbital hybridization. More careful calculations of the
total energy using improved approximations to the exchange correlation potential within DFT are
needed to pin down the parameters of the exchange Hamiltonian. Also, magnon dispersion
measurements and theoretical calculations will help us understand the nature of the magnetic
exchange in these interesting systems.
Table 4.4: Exchange parameters  and  obtained from the simple isotropic Heisenberg model
 ∙ 
 ;  =  for intra-dimer and  =  for inter-dimer exchange. Values
 = −2 ∑〈,〉  
with superscript * are obtained using GGA based on a model proposed in Ref. [189].
Compound
Cr2MoO6
Cr2WO6
Parameter
 (meV)  (meV)  (meV)  (meV)
Theo. (GGA)
-5.4
2.7
-10.4
1.0
Theo.(GGA+U)
-1.9
0.15
-1.0
0.75
Ref. [189]
-3.8*
0.12*

136

4.3.4

Summary

We have studied the magnetic and electronic properties of an inverse-trirutile compound
Cr2MoO6 to examine the idea that the low-lying empty d bands associated with Mo can induce a
ferromagnetic coupling between two Cr moments by perturbing the p orbitals of the O atom that
mediates the exchange interaction. The underlying physics is similar to the one we have suggested
for Cr2WO6 with low-lying empty W 5d bands. GGA+U appears to give a better description of
this system. However, for a quantitative understanding of the exchange parameters more
theoretical work is necessary. Also, inelastic neutron scattering measurements on the magnon
spectra will be helpful in pinning down the exchange parameters to validate the theoretical
predictions.

137

4.4

Higgs amplitude modes in the magnetic excitation spectra of Cr2TeO6 and

Cr2WO6
In this section, we present the magnetic excitations of Cr2TeO6 and Cr2WO6 measured by
inelastic neutron scattering on polycrystalline samples. Longitudinal Higgs (amplitude) modes
have been observed, together with transverse Goldstone (phase) modes, which can be well
described using the extended spin wave (ESW) theory. Although these two compounds are not
close to a QCP, the large spin quantum number (S = 3⁄2 ) in combination with the narrow
bandwidth of the amplitude modes, makes the detection of the Higgs amplitude modes feasible in
the polycrystalline samples. This work provides new insights into the search for the amplitude
fluctuations in condensed matter systems, particularly in spin dimers beyond S = 1⁄2.
4.4.1

Materials and methods

The growth of the polycrystalline samples of Cr2TeO6 and Cr2WO6 has been described in Sec.
4.2.1. The quality of the samples has been verified by the powder x-ray and powder neutron
diffraction measurements. Inelastic neutron scattering measurements were performed using the
SEQUOIA time-of-flight spectrometer at SNS in ORNL. The incident neutron energy was fixed
as Ei = 65 meV in order to cover a large enough area in the energy and momentum transfer (E |2|) space. Samples of ~4 grams each were loaded in a standard aluminum sample can with helium
exchange gas and mounted onto a closed-cycle refrigerator. At each temperature, an identical
empty sample can was used for background subtraction.
4.4.2

Inelastic neutron scattering

Figure 4.15(a) and (b) show the powder-averaged magnetic excitation spectra as a function of
energy transfer E (in meV) and momentum transfer |2| (in Å-1) at T = 4 K. In Cr2WO6, for |2| =

0.72 Å-1 corresponding to the magnetic Bragg reflection (0 0 1) and the equivalent |2| positions in
138

the reciprocal space, gapless excitations have been observed in the energy range of 0 < E < 10
meV. The intensity of these excitations is strongest around |2| ~ 1.27 Å-1 and E ~ 9 meV. The
magnetic dispersions below 10 meV energy transfer can be well described by the LSW theory
taking into account three exchange interactions  ,  and ) presented in Fig. 4.1(c), where the
powder-average of two branches of the magnetic excitations accounts for the observed spectra.
Thus, the excitations in the energy range 0 < E < 10 meV are transverse Goldstone modes
originating from the spontaneously broken rotational symmetry in the magnetically ordered state.
Very strikingly, an additional gapped mode centered at E ~ 12 meV is also observed. This mode
is less dispersive than the transverse modes, and the dependence of its intensity on |2| indicates
that it is also associated with the long-range magnetic order. Furthermore, there is another
excitation mode around E ~ 21 meV, although it is not obvious in the intensity map [Fig. 4.15(a)].

Figure 4.15: (a),(b) Powder-averaged magnetic excitation spectra as a function of energy transfer
E and momentum transfer |2| of Cr2TeO6 and Cr2WO6 at T = 4 K, respectively. The background
measured using an empty sample can has been subtracted. (c),(d) Powder-averaged magnetic
excitation spectra calculated by the LSW theory. (e),(f) Powder-averaged magnetic excitation
spectra calculated by the ESW theory.

139

To better visualize these two additional modes discussed above, Fig. 4.16(a) shows the cuts of
the neutron intensity as a function of E, with |2| integrated from 1.2 to 1.5 Å-1. At T = 4 K, three
distinct modes are observed at E ~ 9, 12 and 21 meV, respectively. In contrast, at T = 60 K which
is slightly above TN = 45 K, while a broad peak associated with the transverse modes is still present
due to the short-range correlations of the magnetic order surviving above TN (also see Fig. 4.17(a)),
the latter two modes at E ~ 12 meV and 21 meV are completely suppressed, affirming the magnetic
character of these two modes. It is important to point out that these two additional modes cannot
be captured by the LSW theory calculation, as shown in Fig. 4.15(c).

Figure 4.16: (a),(b) Inelastic neutron scattering intensity as a function of E at T = 4 K collected at
SEQUOIA, |2| integrated from 1.2 to 1.5 Å-1 for Cr2WO6 and 1.3 to 1.6 Å-1 for Cr2TeO6. Insets
display the zoom-in view of E between 15 meV and 28 meV for Cr2WO6, and between 20.5 meV
and 28 meV for Cr2TeO6. (c),(d) Inelastic neutron scattering intensity as a function of E calculated
by the ESW theory at T = 0 K, |2| integrated from 1.2 to 1.5 Å-1 for Cr2WO6 and 1.3 to 1.6 Å-1 for
Cr2TeO6.

140

Similarly, in Cr2TeO6, there are transverse modes in the magnetic excitation spectra but in the
energy range 0 < E < 19 meV, higher compared with that in Cr2WO6, which can also be adequately
described by the LSW calculations. Intriguingly, in spite of the difference in the sign of  in these
two compounds, a gapped mode is also observed at E ~ 23 meV in Cr2TeO6, which disappears
above TN as shown in the E-cut plots at T = 4 and 120 K presented in Fig. 4.16(b). As in the case
of Cr2WO6, this additional mode cannot be explained using the LSW theory [Fig. 4.15(d)].

Figure 4.17: Powder-averaged magnetic excitation spectra above TN in (a) Cr2WO6 at T = 60 K,
and (b) Cr2TeO6 at T = 120 K, respectively, measured with the incident neutron energy Ei = 65
meV.

4.4.3. Linear spin wave theory calculations
The spin wave dispersions of Cr2WO6 and Cr2TeO6 have been calculated using the LSW theory
in collaboration with Dr. S. D. Mahanti in Michigan State University. The dispersion relation for
Cr2TeO6 is
∗
l = ±< (4 + ¿)À(1 + †)n ) − Á†n ± †n
Á



and that for Cr2WO6 is


l = ±< (4 + ¿)À(1 + †)n ) + |†n | − |†n | ± à 
141

where
†n =


4
G ÄÅ (6Æ) cos(_ M⁄2)cos(_! M⁄2)
4+Âż

†n =

†)n

Âż
G ÄÅ Æ
4+Âż

4Âż`
1
=
[1 − (cos(_ M) + cos(_! M))]
2
4+Âż

∗
∗
à = [4(1 + †)n ) − 2|†n | ]|†n | + (†n  †n  + †n
†n
)




¿ and ¿` are defined as ¿ =  ⁄| | and ¿` = |) |⁄| |, respectively. y is the separation between

two Cr3+ spins in a dimer: y = L ⁄3. The calculated spin wave dispersions are shown in Fig. 4.18,
and the associated powder-averaged spectra are shown in Fig. 4.15(c) and (d). The exchange
parameters used are tabulated in Table 4.5.

Figure 4.18: Spin wave dispersions and intensity of inelastic neutron scattering of (a) Cr2WO6 and
(b) Cr2TeO6 calculated by SpinW code using the LSW theory.

Table 4.5: Exchange parameters for Cr2WO6 and Cr2TeO6 used for the LSW calculations.

Cr2WO6
Cr2TeO6

 [meV]  [meV] ) [meV]
5.80
-0.38
-0.10
11.5
0.87
-0.10

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4.4.4

Extended spin wave theory calculations

Since the weakly dispersing high-energy modes (gapped modes) in the magnetic excitation
spectra of both Cr2WO6 and Cr2TeO6 cannot be explained within the LSW approximation where
only phase fluctuations are included, these gapped modes can potentially be attributed to the
amplitude modes associated with the fluctuations in the magnitude of the magnetic moments. In
order to better understand the nature of these magnetic excitations, we have extended the LSW
calculations for interacting spin dimer systems in collaboration with Dr. M. Matsumoto in
Shizuoka University [201]. Extended spin wave (ESW) theory was introduced to study the
multipole dynamics of the quadrupole ordered phases in CeB6 [201] and PrOs4Sb12 [202], and was
also applied to interacting spin dimer [203], spin trimer [204] and spin tetramer systems [205,206].
It is equivalent to the bond operator formulation introduced to study the excited states in bilayer
spin systems [207] and interacting spin dimer systems [208,209]. The present formulation has an
advantage for investigating the magnetic excitations in complicated systems such as S = 3⁄2 spin
dimers, where there are 16 local states of a dimer.
Starting from the crystal structure of Cr2WO6 and Cr2TeO6 shown in Fig. 4.1(c), the
Hamiltonian for the interacting S = 3⁄2 spin dimer systems is given by the following general form:
 = c c ÇZ7 (O, ¨) +


Ž

c

È,Ž,,Ž d É

ÇZu7 (O, ¨, ;, ¨ ` )

(1)

The first term ÇZ7 (O, ¨) represents the Hamiltonian of a dimer on the ¨(= ·, *) sublattice (see
Fig. 4.1(c)) in the ith unit cell. It can be written as
ÇZ7 =  Žv ∙ Ž7

(2)

ŽÊ represents the S = 3⁄2 spin operator at the γ (= l, r which are equivalent to the top and bottom
spins of the dimer in Fig. 4.1(c)) site in a dimer on the Âľ (= A, B) sublattice in the ith unit cell. The

143

second term ÇZu7 (O, ¨, ;, ¨ ` ) in Eq. (1) is the Hamiltonian for the inter-dimer interaction between
dimers on the µ and µ′ sublattices in the ith and jth unit cell, respectively. It is expressed as
ÇZu7 (O, ¨, ;, ¨ ` ) = c ŽÊ,Žd Êd ŽÊ ∙ Žd Êd
Ê,Êd Ìv,7

(3)

where ŽÊ,Žd Êd represents the exchange coupling constant. In Eq. (1), the summation ∑ÈŽŽd É is
taken over the pairs of interacting spins by the  and ) interaction shown in Fig. 4.1(c).

First, we solve the Hamiltonian under a mean-field approximation on the basis of the dimer
states. For weak inter-dimer interactions, we obtain a singlet (<ÍÎÍ = 0) local ground state with no
magnetic moment. When the inter-dimer interactions are increased, the magnetic (<ÍÎÍ ≠ 0) states
of a dimer participate in the ground state and finite magnetic moments appear. The ordered phase
is realized for
20(| | + |) |) ≥ 

(4)

20(| | + |) |) = 

(5)

and a QCP is obtained when

The finite Neel temperatures of Cr2WO6 and Cr2TeO6 indicate that these two compounds satisfy
Eq. (4). On the basis of a dimer lattice, the ordered moment is staggered in Cr2WO6. The ordering
wave vector is expressed by 2 = (0 0 1) in the reciprocal lattice unit. In Cr2TeO6, the ordered
moment is uniform on the dimer lattice and the ordering wave vector is expressed by 2 = (0 0 0).

The detailed ESW calculations can be found in the Supplemental Materials of Ref. [210].
Figure 4.15(e) and (f) show the calculated powder-averaged magnetic spectra based on the ESW
theory for Cr2WO6 and Cr2TeO6, respectively. They are in very good agreement with the
experimental results presented in Fig. 4.15(a) and (b). The exchange parameters used to obtain the
best fit between the experimental and calculated spectra are tabulated in Table 4.6, together with

144

the calculated magnetic moments that are consistent with the experimental values. As shown in
Fig. 4.19 and 4.20, the observed gapless magnetic excitations in both compounds are indeed the
transverse Goldstone modes, which can be well described by both LSW and ESW calculations.
The additional gapped modes observed in both compounds are however captured only in the ESW
theory, as illustrated in both the calculated contour map [Fig. 4.15(e) and (f)] and the E-cut plots
[Fig. 4.16(c) and (d)]. Specially, we would like to stress that the features at E ~ 12 meV in Cr2WO6
and E ~ 23 meV in Cr2TeO6 with narrow bandwidth are ascribed to the amplitude modes, which
stem from the fluctuations in the magnitude of the magnetic moment in the ordered state (with the
ground state a mixture of both <ÍÎÍ = 0 spin-singlet state and <ÍÎÍ = 1 spin-triplet state) − the Higgs
amplitude modes. Note that the intensity of the longitudinal modes relative to the transverse modes
is weaker in the experimental data compared to the calculated value. The overestimated intensity
of the longitudinal modes is due to the fact that the ESW calculations are done on the basis of a
harmonic theory where the magnon-magnon interaction is not taken into account. In reality, the
presence of this interaction would lead to the decay of the longitudinal mode into a pair of
transverse modes, resulting in broadened linewidth and reduced intensity of the longitudinal modes.
Table 4.6: Exchange parameters and magnetic moment for Cr2WO6 and Cr2TeO6 obtained by the
ESW calculations. The magnetic moment is normalized to the saturated value (full moment).

Cr2WO6
Cr2TeO6

 [meV]  [meV] ) [meV] 3 ⁄34
5.25
-0.475
-0.10
0.794
9.4
1.035
-0.10
0.820

Another interesting point is that there are additional higher energy modes around 21 meV and
40 meV in Cr2WO6 and Cr2TeO6, respectively, predicted in the calculations. These are attributed
to the <ÍÎÍ = 2 excited state, although their intensities are weak in the inelastic neutron scattering

measurements. In spite of this, the <ÍÎÍ = 2 excitation mode is convincingly observed in Cr2WO6,
145

as shown in Fig. 4.16(a), which in turn unambiguously supports the scenario of an S = 3⁄2 spin
dimer and the resultant observation of the amplitude modes in this system. We also tried to look
for the <ÍÎÍ = 2 excitation mode in Cr2TeO6 using an incident neutron energy Ei = 90 meV, but the
phonons around 40 meV mask this weak magnetic signal, as shown in Fig. 4.21.

Figure 4.19: The spin wave dispersion and intensity of inelastic neutron scattering of the (a)
transverse and (b) longitudinal modes in Cr2WO6, respectively, calculated using the ESW theory.
(c) and (d) are the powder-averaged magnetic excitation spectra for the transverse and longitudinal
modes, respectively. The broadening factor is chosen as Γ = 1.0 meV that is close to the
instrumental energy resolution.

146

Figure 4.20: The spin wave dispersion and intensity of inelastic neutron scattering of the (a)
transverse and (b) longitudinal modes in Cr2TeO6, respectively, calculated using the ESW theory.
(c) and (d) are the powder-averaged magnetic excitation spectra for the transverse and longitudinal
modes, respectively. The broadening factor is chosen as Γ = 1.0 meV that is close to the
instrumental energy resolution.

Figure 4.21: Powder-averaged magnetic excitation spectrum of Cr2TeO6 at T = 4 K measured with
the incident neutron energy Ei = 90 meV at T = 4 K.

147

It is known that in quantum antiferromagnets the amplitude excitations may emerge when the
ground state of the system is in the vicinity of a QCP where the ordered moment is reduced
significantly by quantum fluctuations. The observation of Higgs amplitude modes in the magnetic
excitation spectra of Cr2WO6 and Cr2TeO6 even in the polycrystalline samples is quite striking,
considering that their ordered moments are reduced by only ~24%. A natural question arises: Are
the ground states of both compounds close to the QCP? To quantitatively evaluate the exchange
parameters of Cr2WO6 and Cr2TeO6 with respect to the QCP, in Fig. 4.22(a) and (b) we show the
| |⁄ dependence of the excitation gap and the magnetic moment with the values of  and )

fixed as listed in Table 4.6. The magnetic moment increases with | |⁄ in the ordered phase, as
illustrated by the magenta curves in Fig. 4.22. The black line represents the excitation gap in the
disordered phase, which is threefold degenerate. The excitation gap is located at 2 = (0 0 1) and
(0 0 0) for Cr2WO6 and Cr2TeO6, respectively, and it is analytically given by
A¾§ = À − 20 (| | + |) |)
in the disordered phase. In the ordered phase, the excitation modes split into two transverse modes
and one longitudinal mode. The transverse mode (T-mode) stays gapless, reflecting the global
rotational symmetry of the spin Hamiltonian. The gap of the Higgs amplitude mode (L-mode)
scales with | |/ . This excitation gap corresponds to the Higgs mass in high-energy physics.

According to Eq. (5), the QCP is located at | |⁄ = 0.031 and 0.040 for Cr2WO6 and Cr2TeO6,

respectively. The estimated values of this ratio are | |⁄ = 0.0905 and 0.101 for Cr2WO6 and
Cr2TeO6, respectively, as indicated by the vertical green dashed lines. Correspondingly, the
calculated ordered moments normalized to the fully saturated value for Cr3+ are consistent with
the observed reduction in the ordered moment of ~24%.

148

Figure 4.22: (a),(b) | |⁄ dependence of the ordered moment 3 ⁄34 and the excitation gaps. The
black line represents the excitation gap in the disordered state. The excitation gaps of the
longitudinal modes (L-mode) and the transverse modes (T-mode) in the ordered state are plotted
by red and blue solid lines, respectively. The exchange parameters of Cr2TeO6 and Cr2WO6 are
denoted by green dashed vertical lines. The ordered moments are represented by magenta lines.

4.4.5

Discussions

As the system goes away from the QCP, the energy of the amplitude mode increases and the
neutron scattering intensity weakens. This, together with the tendency of decaying into transverse
modes, makes it difficult to detect Higgs amplitude modes. Thus, our observation brings about
another intriguing question: What makes it feasible to observe the longitudinal excitation modes
in these two compounds? There are several contributing factors. i) These compounds are spin
dimers with S = 3⁄2. Similar to the S = 1⁄2 spin dimer systems, the low-lying states here are <ÍÎÍ

= 0 singlet and <ÍÎÍ = 1 triplet, suggesting a similar low-energy structure. However, the matrix
elements of the spin operators (between the singlet and triplet states) are different between S =
3⁄2 and S = 1⁄2 dimer systems, with the matrix element enhanced by √5 for the S = 3⁄2 dimer.
As a result, the intensity of the amplitude mode is enhanced relative to the transverse mode in the
case of the S = 3⁄2 dimer. ii) Interestingly, as shown in Fig. 4.19 and 4.20, the dispersion of the
amplitude modes in both Cr2WO6 and Cr2TeO6 is weak and the inelastic neutron scattering
intensity stays in a narrow energy region separated from the transverse modes, which enables us

149

to distinguish these two modes even with polycrystalline samples. Our results suggest that Higgs
amplitude modes are not the privilege of the ordered quantum spin dimers in the vicinity of a QCP,
but may be common excitation modes that can survive even away from it.
Finally, we would like to point out that the inverse-trirutile systems composed of dimerized
edge-sharing octahedra, such as Cr2WO6 and Cr2TeO6 in this study, provide substantial
opportunities to search for and tune Higgs amplitude modes in quantum antiferromagnets. First,
as both the intra-dimer interaction  and inter-dimer interaction  can be effectively tuned by
varying the concentration of the nonmagnetic atoms in Cr2(Te1-xWx)O6, one can anticipate the
presence of the amplitude modes in the whole series of compounds. In particular,  changes its
sign at x ~ 0.7, giving rise to minimum values of both the Neel temperature TN and the ordered
moment 3 ⁄34 ~ 15%, as shown in Fig. 4.4 [195]. The small ordered moment suggests that the
ordered phase is very close to the QCP at x ~ 0.7 that would yield stronger intensity of the
amplitude modes. Second, the availability of various isostructural inverse-trirutile compounds
consisting of edge-sharing (M2O10)14- (M = Cr, Fe, Mn and V, etc.) entities with reduced ordered
moments and low Neel temperatures, enables the exploration of a wider range in the exchange
parameter space that may lead to the observation of Higgs amplitude modes in other interacting
spin dimer systems beyond S = 1⁄2.
4.4.6

Summary

We present the observation of the Higgs amplitude modes in the inelastic neutron scattering
measurements on polycrystalline Cr2TeO6 and Cr2WO6 compounds with interacting S = 3⁄2 spin
dimers that are away from a QCP. The measured magnetic excitation spectra, both the longitudinal
and transverse modes, agree well with the extended spin wave (ESW) calculations. This study
suggests that Higgs amplitude modes may be common excitation modes in quantum spin dimers

150

instead of being limited to a few exceptional systems close to a QCP, thus paving a new avenue to
search for and understand the physics of Higgs-like quasiparticles in condensed matter systems.

151

Chapter 5
Summary and perspectives
Collective behaviors of correlated electrons are one of the mysteries in condensed matter
physics even after decades of intense investigations. Studies on transition-metal oxides, where the
effects of electron correlations are essential, have contributed tremendously to the development of
this field. In this thesis, we have presented neutron scattering studies on Ruddlesden-Popper type
ruthenates (Sr,Ca)n+1RunO3n+1 and inverse-trirutile chromates Cr2MO6 (M = Te, W and Mo).
Perovskite (Sr,Ca)n+1RunO3n+1 is a prototypical 4d correlated electron system ideal for
exploring the physics of emergent phenomena. Owing to the strong coupling among charge, spin,
orbital and lattice degrees of freedom, the magnetic and electronic properties are very susceptible
to perturbations, such as chemical doping and magnetic field. For instance, upon 3% ~ 5% Fe
doping we have unraveled a commensurate SDW order with the propagation vector 
 = (0.25
0.25 0) in Sr2RuO4 (n = 1), which cannot be ascribed to Fermi surface nesting as suggested by
inelastic neutron scattering measurements and electronic structure calculations. For Sr3Ru2O7 (n =
2) we have found that Fe doping leads to an E-type antiferromagnetic insulating state distinct from
the Fermi liquid ground state in the parent compound. In the presence of a magnetic field, we have
observed a field-induced collapse of the Mott gap in Ca3(Ru0.97Ti0.03)2O7 accompanied by changes
in the spin structure and the lattice, which results in a CMR effect that is fundamentally different
in nature from that in phase-separated manganites. For Ca3(Ru0.95Fe0.05)2O7, the magnetic field
drives an incommensurate-to-commensurate magnetic transition when applied either along the
crystallographic a or b axis. The magnetic phase transitions are found to be irreversible when

152

removing the magnetic field, leading to history-dependent metastable states that transform into the
equilibrium one only upon warming up the system above a characteristic temperature Tg.
Cr2MO6 (M = Te and W) have inverse-trirutile structures with structurally dimerized Cr3+
magnetic moments. We show that the nature and strength of the inter-dimer interaction can be
effectively tuned by mixing Te and W in the solid solution Cr2(Te1-xWx)O6. The sign of the interdimer exchange interaction changes at xc ~ 0.7, where the Neel temperature TN and the sublattice
magnetization 34 reach the minimum. Our DFT calculations suggest that the hybridization
between the W 5d and O 2p orbitals in the Cr-O-Cr exchange path might play a crucial role. To
further validate this idea, we have also studied the magnetic structure of another isostructural
compound Cr2MoO6 and found that the inter-dimer interaction is indeed ferromagnetic as expected
from the Mo 4d-O 2p hybridization. The magnetic excitation spectra of Cr2TeO6 and Cr2WO6
measured by the inelastic neutron scattering also exhibit intriguing physics. In addition to the
transverse Goldstone modes, we have observed longitudinal modes, also known as Higgs
amplitude modes, that arise from the fluctuations in the magnitude of the ordered moment. It is
generally believed that Higgs amplitude modes are only observable when the system is close to a
QCP separating the dimer-based quantum disordered phase and the long-range ordered state,
where the ordered moment and the transition temperature are strongly suppressed by quantum
fluctuations. Intriguingly, our ESW calculations have shown that these two materials are away
from the QCP, which is highly unexpected according to the conventional wisdom.
A number of intriguing questions remain open and are worth further experimental and
theoretical investigations. Here we give a few examples. (i) What is the microscopic origin of the
commensurate SDW order in Sr2RuO4; and more importantly, does it have any correlation with
the p-wave unconventional superconductivity? (ii) The spin wave excitations of this commensurate

153

SDW order are not well captured in the current inelastic neutron scattering measurements. Is it
because of its low energy scale? (iii) What are the roles of the 3d transition-metal dopants? Why
is the effect of Ti and Mn similar in Sr2RuO4 but different from that of Fe, whereas in Sr3Ru2O7,
Mn and Fe give rise to similar effects which are different from Ti. (iv) Is the insulating state in Fedoped Sr3Ru2O7 a Mott type originating from electron correlations, or Slater type arising from the
magnetic order? (v) Is the collapse of the Mott insulating state in a magnetic field a universal
phenomenon? (vi) Is it possible to experimentally verify the presence of orbital order in the Mott
insulating ground state of Ti-doped Ca3Ru2O7? (vii) What’s the origin of the irreversibility in the
field-induced transitions in Fe-doped Ca3Ru2O7?
There are still ample opportunities in Ruddlesden-Popper type ruthenates and inverse-trirutile
compounds for exploring the collective behaviors of correlated electrons. Other than chemical
doping and magnetic field, applying hydrostatic or uniaxial pressure is also quite effective in
tuning the magnetic and electronic properties of ruthenates. Recent studies on Ca3(Ru0.97Ti0.03)2O7
have shown that the G-AFM Mott insulating ground state collapses with a very low hydrostatic
pressure, which is ascribed to the coupling between the lattice distortions and the orbital
polarization by DFT calculations [211]. In addition, the electric field is also a promising tuning
parameter. Researchers have found that an insulator-to-metal transition can be induced in Ca2RuO4,
which is accompanied by a structural transition [212]. Moreover, the photo-doping effects have
been studied on some Mott insulators using femtosecond lasers. Upon exciting the charge carriers
across the Mott gap, transient novel phases may emerge [109]. The relaxation process from the
excited states to the equilibrium one may provide valuable information on the electron correlations.
The idea is likely to be applicable to correlated ruthenates. Another emerging field that has been
developing rapidly in recent years is the transition-metal oxide thin films and heterostructures

154

[213]. It has been demonstrated that the physical properties can be tailored by epitaxial strains
imposed by different substrates or substrate orientations. SrRuO3 is a rare 4d ferromagnetic metal
with a Curie temperature Tc = 160 K and strong magnetocrystalline anisotropy, and it has been
widely used as the conducting layer in heterostructures [167]. Prototypical interfacial phenomena
such as exchange bias effects have been observed in SrRuO3 / La0.67Sr0.33MnO3 bilayers grown on
SrTiO3 substrates [214]. Furthermore, more exotic phenomena can occur at the interface. For
instance, highly confined two-dimensional electron gas and superconductivity have been observed
in SrTiO3 / LaAlO3 [215,216]. While a similar two-dimensional electron gas has been theoretically
predicted in SrRuO3 / SrTiO3 superlattices [217], it has not been observed experimentally.
For the inverse-trirutile compounds, the availability of isostructural materials consisting of
different magnetic ions (e.g. Fe, Mn and V) enables further studies regarding the dependence of
Higgs amplitude modes on the spin quantum numbers and the ratio of the inter- and intra-dimer
interactions | |⁄ . Further experiments, such as magnetic-field-dependent measurements and
polarized neutron scattering studies, will indeed provide more information once high-quality
single-crystal samples are synthesized successfully.
In conclusion, though understanding the electron correlations remains challenging for
condensed matter physicists, transition-metal compounds have offered an important stage where
new discoveries are always expected. Further experimental and theoretical efforts are still highly
desired.

155

APPENDICES

156

APPENDIX A: Crystal symmetry of Sr2Ru1-xFexO4 (x = 0.03 and 0.05)

The symmetry of the crystal structure is an important starting point to explore the symmetryallowed magnetic structure models. In addition, the assignment of the observed 
 = (0.25 0.25 0)
peak to a magnetic Bragg peak, rather than a nuclear one arising from a change in the crystal
symmetry, requires a determination of the space group. Here, the space group of Sr2Ru1-xFexO4 (x
= 0.03, 0.05) is assumed to be the same as that of the parent compound (No. 139 I4/mmm). The
rationales are as follows:
(i) The ionic radii of Fe3+ (0.645 Å) and Ru4+ (0.62 Å) are very close, similar to Sr2Ru1-xTixO4
where the ionic radius of Ti4+ is 0.605 Å. In fact, the space group of the end members of all three
compounds are the same, i.e. I4/mmm.
(ii) In spite of the significant difference in the ionic radii of Sr2+ (1.31 Å) and Ca2+ (1.18 Å),
the space group of Ca2-xSrxRuO4 remains the same as Sr2RuO4 for 1.5 < x ≤ 2 (i.e., within 25% of
Ca substitution for Sr) [102].
Therefore, due to the low doping concentration 3% ~ 5% and the similar ionic radii of Fe3+ and
Ru4+, the crystal symmetry of Fe-doped Sr2RuO4 is not likely to change. In fact, even assuming
that there were structural distortions such that the space group changed to I41/acd (Ca2-xSrxRuO4,
0.5 < x < 1.5 due to RuO6 rotation) or Pbca (Ca2-xSrxRuO4, x < 0.2 due to rotation, tilting and
flattening), the wave vector 
 = (0.25 0.25 0) in tetragonal notation becomes 
 = (0.5 0 0) in
orthorhombic notation, which still cannot be assigned as a nuclear Bragg peak. Therefore, the
emergence of this new Bragg peak must be associated with the formation of a magnetic order,
rather than related to the structural change that does not actually take place in Sr2Ru1-xFexO4. (Note
that all the ionic radius data are taken from Ref. [90])

157

APPENDIX B: Determination of magnetic structures by neutron diffraction

In this section, we describe the details on determining the different magnetic structures in
Ca3Ru2O7 and Ti-doped Ca3Ru2O7 based on neutron diffraction measurements. The AFM-a/AFMb and G-AFM type magnetic structures have been reported previously by W. Bao et al [97] and X.
Ke et al [99], respectively. The determination of the CAFM phase above the critical field Bc at T
< TMIT is presented as follows. The space group of Ca3(Ru1-xTix)2O7 (x = 0.03) is Bb21m (No. 36),
and the magnetic propagation vector for the CAFM phase is (0 0 1). We performed magnetic
representational analysis using FULLPROF [71]. According to this symmetry analysis, only four
types of collinear antiferromagnetic structure models are possible, as shown in Fig. B1(a). Since
the L index of the observed magnetic reflections (0 0 L) is odd, model 3 and 4 can be ruled out
immediately as the squared magnetic structure factor |{(ℎ0P)| = 0. In both model 1 and 2, the
magnetic moments in the RuO2 plane are coupled parallel to each other.
In model 1, the magnetic moments are parallel with the bilayer and antiparallel between
adjacent bilayers (AFM-a or AFM-b).
In model 2, within the bilayer the spin configuration of two RuO2 layers is antiparallel and the
bilayers are coupled antiparallel along the c axis.
The squared magnetic structure factor is |{(ℎ0P)| = 64LIK  (2aPÑ) for model 1 and

|{(ℎ0P)| = 64KOC (2aPÑ) for model 2, where 2Ñ = 0.1978. Since we have observed sizable
magnetic intensity at (0 0 5) at T = 10 K and B = 10 T [Fig. B1(b)], model 2 can be readily excluded
as |{(005)| = 64KOC (2aPÑ) ≈ 0. In addition, the magnetic reflections (0 0 1) and (2 0 1) have

the same structure factor |{(ℎ01)| = 64LIK  (2aPÑ) ≈ 0.6610. Thus, the absence of the (2 0 1)
magnetic reflection suggests that the staggered antiferromagnetic moment is aligned along the a
axis, since neutrons couple to the moments perpendicular to the momentum transfer 
.
158

Figure B1: (a) Schematics of possible magnetic structures. (b) Rocking curve scan across (0 0 5)
magnetic Bragg peak at T = 10 K and B = 10 T, B ∼ b axis.
The evidence to the presence of spin canting is the following:
(i) As shown in Fig. 3.15(b), the intensity of the field-induced (0 0 1) peak at T = 10 K is
weaker than the intensity of (0 0 1) at T = 50 K and B = 0 T.
(ii) The isothermal magnetization measurements [Fig. B2] with the magnetic field applied
along the b axis yield a metamagnetic transition and the field-induced phase has a large
ferromagnetic moment. Hence, the field-induced magnetic structure below TMIT is a canted
antiferromagnetic phase (CAFM): a superposition of AFM-a and a ferromagnetic component along
the b axis. For instance, at T = 20 K, the canted spins above the critical field (~9 T) have an
antiferromagnetic component of ~0.71 ÎźB and a ferromagnetic component of ~1.76 ÎźB along the b
axis.

159

Figure B2: Magnetization of Ca3(Ru0.97Ti0.03)2O7 as a function of magnetic field at selected
temperatures.

160

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