nuclear magnetic resonance study of ferromagnetism and
TRANSCRIPT
Nuclear Magnetic Resonance Study of
Ferromagnetism and Local Symmetry Breaking in
Double Perovskite Mott Insulator Ba2NaOsO6
by
Lu Lu
B.Sc., Nanjing University, Nanjing, China, 2009
A Dissertation Submitted in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy
in The Department of Physics at Brown University
PROVIDENCE, RHODE ISLAND
May 2016
c© Copyright 2016 by Lu Lu
This dissertation by Lu Lu is accepted in its present form
by The Department of Physics as satisfying the
dissertation requirement for the degree of Doctor of Philosophy.
Date
Vesna Mitrovic, Ph.D., Advisor
Recommended to the Graduate Council
Date
James M. Valles, Ph.D., Reader
Date
John B. Marston, Ph.D., Reader
Approved by the Graduate Council
Date
Peter M. Weber, Dean of the Graduate School
iii
Abstract of “ Nuclear Magnetic Resonance Study of Ferromagnetism and Local Sym-metry Breaking in Double Perovskite Mott Insulator Ba2NaOsO6 ” by Lu Lu, Ph.D.,Brown University, May 2016
Study of the combined effects of strong electronic correlations with spin-orbit cou-
pling (SOC) represents a central issue in quantum materials research. Predicting
emergent properties represents a huge theoretical problems as presence of SOC im-
plies that the spin is not a good quantum number. A multitude of exotic quantum
phases are predicted to emerge even in materials with simple cubic crystal structure
such as Ba2NaOsO6. Experimental tests by local probes, such as Nuclear magnetic
resonance (NMR), are highly sought for.
In this thesis, we first present the temperature evolution of the 23Na NMR spec-
trum. It clearly reveals a geometrical distortion, preceding a formation of the long-
range ordered (LRO) magnetism. We discovered from the angular dependence mea-
surement that a cubic to orthorhombic structural phase transition occurs, induced
by elongation/compression of the oxygen octahedra, and only one structurally equiv-
alent environment exists for 23Na nuclei. Furthermore, from a lattice sum simulation
we found that the low temperature LRO state is the canted two-sublattice ferromag-
netic (FM) state, which gives rise to the magnetic splitting in NMR spectrum. This
is the first direct observation of such exotic magnetic order in 5d transition-metal
systems. Such state is predicted to occur due to multipolar spin-spin interactions on
the frustrated fcc lattice. We provide evidence for the presence of complicated orbital
ordering. Lastly, we studied the magnetic order within the intermediate transition
region, and the possible existence of spin nematic order is discussed based on second
moment analysis.
This work paves the way for future NMR study of highly frustrated systems with
the strong SOC and electron correlations, such as Iridates.
Acknowledgements
I would like to express my deepest gratitude to people who have educated, helped
and supported me, directly or indirectly, during my PhD study.
First of all I would like to thank Vesna Mitrovic, my PhD advisor, for always being
available and providing numerous valuable suggestions. Her patience, willingness of
knowledge-sharing and expertise in science guided me through the mysterious world
of physics, and most importantly, made the experience of scientific research much
enjoyable. I will always cherish the abilities of creative and critical thinking I learned
from her, and remember all the exciting moments we shared of discovering physics.
I would also like to thank Myeonghun Song, the post-doc who joined the labo-
ratory right after me. He was the person who taught me how to conduct an NMR
experiment, hand by hand. We shared so many hours working together at Brown,
as well as the 24/7 shifts at the National High Magnetic Field Lab (NHMFL) in
Tallahassee. A great extent of experimental work in this thesis is done with his help.
It was a pleasure to work with him.
The majority of the NMR experiments in this thesis was conducted in NHMFL,
and with the help of many excellent scientists and technicians, especially Dr. Arneil
Reyes and Dr. Phillip Kuhns. They not just helped me with experimental aspects of
NMR, but also provided valuable suggestions regarding research directions and data
iii
interpretations. And I also thank Ian Fisher from Stanford University who provided
the beautiful single crystal sample I studied for this entire thesis.
Also I thank many people in the physics department at Brown University. My
labmate Wencong Liu, who joined the lab three years ago, has shared lots of time
with me running experiments and making interesting conversations. Being a talented
programmer, he contributed to part of the numerical calculation in my thesis. I thank
Brad Barry, who worked in our group as a undergraduate student, for helping me
with the lattice modeling. Many thanks to PhD students Wanchun Wei, Zhuolin Xie
and Jimmy Joy, who offered lots of help and suggestions regarding the cryogenics
system and general instrumentations over the years. Alex Loosley, “THE Canadian”,
shared his experience of thesis writing as well as many beers with me. And also I
thank Charlie Vichers from the machine shop for teaching me everything I know
about machinining.
Lastly I thank my parents, Shaohua Lu, Cuiyun Li, who dedicated their life to
providing me with the best education, taught me to love life, and never ceased to
encourage me to reach my goals.
iv
Contents
Acknowledgments iii
1 Introduction 1
2 BNOO: Structure, Properties and Bulk measurements 4
2.1 Crystal Structure and Basic Properties . . . . . . . . . . . . . . . . . 5
2.2 Bulk Magnetization and Heat Capacity . . . . . . . . . . . . . . . . . 6
2.3 Muon Spin Rotation (µ+SR) Measurement . . . . . . . . . . . . . . . 9
2.4 Sample for NMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3 Theoretical Background 13
3.1 Crystal Field Theory in Octahedral Complexes . . . . . . . . . . . . . 14
3.2 Strong Spin-Orbit Coupling (SOC) in Ba2NaOsO6 . . . . . . . . . . 15
3.3 Hamiltonian: SOC, Exchange Interactions, and Electric Quadrupolar
Interaction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
v
3.3.1 Eigenstates with strong SOC . . . . . . . . . . . . . . . . . . . 17
3.3.2 Exchange Interactions and Electric Quadrupolar Interaction . 19
3.3.3 Overall Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . 21
3.4 Mechanism of Mott Insulating . . . . . . . . . . . . . . . . . . . . . . 21
3.5 [110] Ferromagnetism: SOC, Frustration, or Multipolar Spin Interac-
tions? . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
4 NMR in Condensed Matter and Experimental Techniques 28
4.1 Basic Concepts and Principles of NMR . . . . . . . . . . . . . . . . . 29
4.1.1 Isolated Nuclear Spins and NMR Resonance . . . . . . . . . . 29
4.1.2 FID Spin Echo, T1 and T2 in Pulsed NMR . . . . . . . . . . . 31
4.1.3 Fourier Transform Spectrum . . . . . . . . . . . . . . . . . . . 34
4.1.4 Spin-Lattice Relaxation Time T1 . . . . . . . . . . . . . . . . 35
4.1.5 Bloch Equations . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.2 NMR in Solids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2.1 Full Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.2 Dipolar Field . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
4.2.3 Quadrupole Splitting . . . . . . . . . . . . . . . . . . . . . . . 39
4.2.4 Hyperfine Interaction . . . . . . . . . . . . . . . . . . . . . . . 43
4.2.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4.3 NMR Experimental Realization . . . . . . . . . . . . . . . . . . . . . 46
4.3.1 Building Blocks . . . . . . . . . . . . . . . . . . . . . . . . . . 46
4.3.2 NMR Probe and Resonance Circuit . . . . . . . . . . . . . . . 47
vi
4.3.3 NMR Coil . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.3.4 Other Apparatus . . . . . . . . . . . . . . . . . . . . . . . . . 52
4.4 Experimental Set-Up for Ba2NaOsO6 NMR . . . . . . . . . . . . . . . 54
5 Identification of Phase Transition: Temperature Dependence of
NMR Spectra 56
5.1 Temperature Evolution of NMR Spectrum . . . . . . . . . . . . . . . 57
5.2 Field dependence of spectrum . . . . . . . . . . . . . . . . . . . . . . 59
5.2.1 Uniform and staggered local fields. . . . . . . . . . . . . . . . 60
5.2.2 Transition temperature . . . . . . . . . . . . . . . . . . . . . . 63
5.3 Spin-Lattice Relaxation Rate . . . . . . . . . . . . . . . . . . . . . . 66
5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
6 Observation of Local Symmetry Breaking 72
6.1 Angle Dependence of Quadrupole Splitting . . . . . . . . . . . . . . . 73
6.2 Distortion in Tetragonal Structure . . . . . . . . . . . . . . . . . . . . 79
6.3 Point Charge Calculation . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
6.3.2 Method Description . . . . . . . . . . . . . . . . . . . . . . . . 83
6.3.3 Programming Procedure . . . . . . . . . . . . . . . . . . . . . 85
6.4 Structural Equivalence of Na Ions . . . . . . . . . . . . . . . . . . . . 86
6.4.1 One Structurally Distinct Na . . . . . . . . . . . . . . . . . . 87
6.4.2 Two Structurally Distinct Na . . . . . . . . . . . . . . . . . . 89
6.5 Orthorhombic Symmetry Revealed by In-Plane Rotation . . . . . . . 90
vii
6.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92
7 Numerical Simulation of Low Temperature Magnetic Order 94
7.1 PM State: Orbital Shifts and Hyperfine Tensor . . . . . . . . . . . . 96
7.2 Dipolar Interaction Tensor . . . . . . . . . . . . . . . . . . . . . . . . 98
7.2.1 Cubic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 100
7.2.2 Reduced Symmetry . . . . . . . . . . . . . . . . . . . . . . . . 101
7.3 Ordered State LRO . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
7.3.1 Magnetic Anisotropy . . . . . . . . . . . . . . . . . . . . . . . 102
7.3.2 Lattice Sum Method . . . . . . . . . . . . . . . . . . . . . . . 104
7.3.3 Microscopic Models of Magnetic Phase . . . . . . . . . . . . . 105
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
8 Transition Region: Possible Nematic Order 116
8.1 Temperature Dependence of Transition Region Spectrum . . . . . . . 117
8.2 Existence of Nematic Order during Transition Region . . . . . . . . . 120
8.3 Relaxation Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
9 Resistively Detected NMR of Topological Insulator 125
9.1 Sample Preparation and Method Description . . . . . . . . . . . . . . 126
9.1.1 Quantum Oscillation . . . . . . . . . . . . . . . . . . . . . . . 129
9.1.2 RDNMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
9.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . 131
9.2.1 Bulk NMR Spectrum . . . . . . . . . . . . . . . . . . . . . . . 131
viii
9.2.2 Quantum Oscillation . . . . . . . . . . . . . . . . . . . . . . . 132
9.2.3 RDNMR . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
9.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
10 Summary and Outlook 136
A Ion Positions in a Ba2NaOsO6 Unit Cell 140
B Lattice Sum Algorithm 143
C Hyperfine Tensor Symmetry 146
C.1 Tetragonal symmetry . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
C.2 Orthorhombic Symmetry . . . . . . . . . . . . . . . . . . . . . . . . . 151
D Method of Data Averaging for RDNMR Data 153
ix
List of Tables
2.1 Coordinates of each atoms in the unit cell of Ba2NaOsO6. . . . . . . 5
5.1 Transition temperature from PM to low temperature magnetic state
at various applied fields. . . . . . . . . . . . . . . . . . . . . . . . . . 65
5.2 Temperature dependence for different spin-relaxation processes for
magnetic insulator at low temperature. . . . . . . . . . . . . . . . . . 70
6.1 Quadrupole splitting values in different fields. . . . . . . . . . . . . . 74
6.2 Sample results of point charge calculation with two structurally dis-
tinct Na sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88
6.3 Sample results of point charge calculation with one structurally dis-
tinct Na sites. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90
7.1 Hyperfine coupling constants and orbital shifts. . . . . . . . . . . . . 97
A.1 Position, occupation ratio and charge values of 14 Na ions. . . . . . . 141
A.2 Position, occupation ratio and charge values of 8 Ba ions. . . . . . . . 141
A.3 Position, occupation ratio and charge values of 13 Os ions. . . . . . . 141
A.4 Position, occupation ratio and charge values of 54 O ions. . . . . . . . 142
x
List of Figures
2.1 Ba2NaOsO6 crystal structure of an unit cell. . . . . . . . . . . . . . . 5
2.2 XRD patterns of Ba2NaOsO6 at various temperatures. . . . . . . . . 6
2.3 Temperature dependence of magnetization and heat capacity. . . . . 7
2.4 Magnetic anisotropy revealed by magnetization measurement. . . . . 9
2.5 µ+SR data at temperatures above and below Tc. . . . . . . . . . . . 10
3.1 Energy splitting of octahedral complexes under crystal field. . . . . . 14
3.2 Orbital and magnetic ordering of A-type antiferromagnetism. . . . . . 15
3.3 Possible exchange paths of Ba2NaOsO6 . . . . . . . . . . . . . . . . . 20
3.4 Energy level evolution of Ba2NaOsO6 . . . . . . . . . . . . . . . . . . 22
3.5 Phase diagram predicted by quantum models. . . . . . . . . . . . . . 26
3.6 Possible electron spin configurations predicted by quantum models. . 27
4.1 Energy splitting of 23Na nucleus under the applied field H0. . . . . . 30
4.2 Schematic of the effect of the oscillating field H1. . . . . . . . . . . . 31
4.3 Illustration of the Free Induction Decay. . . . . . . . . . . . . . . . . 32
4.4 Illustration of the Spin Echo. . . . . . . . . . . . . . . . . . . . . . . 33
4.5 Fourier transformation of NMR signal. . . . . . . . . . . . . . . . . . 34
xi
4.6 T1 recovery profile. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
4.7 Orientation dependence of quadrupole energy. . . . . . . . . . . . . . 40
4.8 Spectrum with quadrupole splitting. . . . . . . . . . . . . . . . . . . 43
4.9 NMR shift with different magnetic orders. . . . . . . . . . . . . . . . 45
4.10 Energy evolution of nuclei. . . . . . . . . . . . . . . . . . . . . . . . . 46
4.11 Building blocks of NMR experiment. . . . . . . . . . . . . . . . . . . 47
4.12 Sketch for a typical NMR probe. . . . . . . . . . . . . . . . . . . . . . 49
4.13 Diagram of NMR resonance circuit. . . . . . . . . . . . . . . . . . . . 50
4.14 Typical designs for NMR RF coils . . . . . . . . . . . . . . . . . . . . 51
4.15 Block diagram of λ/4 wave and duplexer. . . . . . . . . . . . . . . . . 53
5.1 Temperature evolution of 23Na spectra at 9 T with H0 ‖ c . . . . . . 58
5.2 Temperature dependence of NMR spectra for Ba2NaOsO6. . . . . . . 61
5.3 Temperature dependence of shifts at various fields. . . . . . . . . . . 62
5.4 Temperature evolution of local fields. . . . . . . . . . . . . . . . . . . 64
5.5 Determination of transition temperature Tc. . . . . . . . . . . . . . . 65
5.6 Temperature evolution of local field and shit in reduced temperature. 67
5.7 Temperature evolution of 1/T1T . Inset Spin-lattice relaxation time T1
as a function of T. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.8 The magnon/phonon process of nuclear spin-lattice relaxation . . . . 70
6.1 Local cubic symmetry breaking in the ordered phase. . . . . . . . . . 75
6.2 Illustration of the rotation of the applied field. . . . . . . . . . . . . . 78
6.3 Schematic of the primary and basis vectors. . . . . . . . . . . . . . . 84
6.4 Schematic of the proposed lattice distortions. . . . . . . . . . . . . . . 87
6.5 Angular dependence of both in-plane and diagonal rotation. . . . . . 91
6.6 Three possible orientations of Vzz. . . . . . . . . . . . . . . . . . . . . 92
xii
7.1 Clogston-Jaccarino plots at 7 T. . . . . . . . . . . . . . . . . . . . . . 98
7.2 Spatial distribution of Os atoms around one Na atom. . . . . . . . . 102
7.3 Angular dependence of the uniform and staggered internal fields. . . . 103
7.4 Spin orientation in staggered layer model. . . . . . . . . . . . . . . . 107
7.5 Data fitting for two staggered layer model with fixed spin orientation. 108
7.6 Comparison of angular dependence of simulated and experimental
spectra. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
7.7 Comparison of angle dependence of magnetic and quadrupole splitting.110
7.8 Integrated intensity of the entire spectrum . . . . . . . . . . . . . . . 113
7.9 Fit to the angular dependence of internal fields with FM model. . . . 114
8.1 Temperature evolution of 23Na spectra at 15 T with H0 ‖ c . . . . . . 118
8.2 Spectrum as a function of temperature at θ = 50 and55 . . . . . . . 119
8.3 First moment shift as a function of temperature. . . . . . . . . . . . . 120
8.4 Theoretically proposed phase diagram from Ref. [31]. . . . . . . . . . 122
8.5 Temperature evolution of the second moment. . . . . . . . . . . . . . 123
8.6 Temperature evolution of the relaxation rate at 15 T and θ = 55. . . 124
9.1 QO and RDNMR experimental set-up. . . . . . . . . . . . . . . . . . 128
9.2 Parallel and diagonal configurations for resistance measurement. . . . 128
9.3 Bulk NMR spectra for three samples with different Sb doping per-
centage. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
9.4 Quantum oscillation of the pure sample. . . . . . . . . . . . . . . . . 132
9.5 Change of the resistance as a function of frequency. . . . . . . . . . . 134
B.1 Schematic for lattice construction in lattice sum method. . . . . . . . 144
C.1 Schematic of hyperfine tensors. . . . . . . . . . . . . . . . . . . . . . 148
xiii
Chapter One
Introduction
The 3d transition-metal oxides (TMO) systems are proven to be a rich field of re-
search due to the interplay between spin, charge and orbital degrees of freedom,
among which the orbital degree of freedom (ODF) is recognized to play an impor-
tant role in mediating the various types of interactions including double exchange [1],
super exchange [2], Jahn-Teller [12] effect, etc. Electrons in partially filled d shells
have the tendency of forming orbital ordering (OO), which is identified as the mi-
croscopic mechanism to induces structural and magnetic phase transitions via its
coupling to the charge and magnetic order [3–5].
Much less is investigated in the 5d systems where the involvement of strong spin-
orbital coupling can complicate the balance between different energy scales. The
Mott insulator Ba2NaOsO6 with a relatively simple 5d1 electron configuration is of
particular interests. It is the only osmate without other magnetic ions that demon-
strates ferromagnetic order [11], despite the fact that no geometrical distortion which
usually induces the orbital ordering is observed in the ordered state . Meanwhile, the
isostructural and isovalent compound Ba2LiOsO6 shows antiferromagnetic order, in-
dicating a delicate balance between orbital ordering and other couplings. Therefore,
the answers to the nature of the exotic magnetic order and structural configuration
in Ba2NaOsO6, the role of strong spin-orbital coupling in 5d system, and the driving
mechanism behind them are highly sought for.
The Nuclear Magnetic Resonance (NMR), owing to its sensitivity to both the
lattice and magnetic degrees of freedom, can provide valuable information toward
demystifying these questions, as will be presented in this thesis.
The contents of the thesis is organized in chronological order to reflect the con-
ceptual development of our research. Chapter 2 introduces the existing experimental
findings about Ba2NaOsO6, including crystal structure, bulk magnetization, specific
2
heat measurements, and muon spin measurement. Our NMR study is motivated by
the unorthodox ferromagnetism with easy-axis along [110] and the lack of observa-
tion of geometrical distortions in the ordered state. Chapter 3 walks through the
theoretical studies by first-principle energy band calculations as well as the quantum
models based on the effective Hamiltonian containing various exchange and coupling
effects. They lay out the guidelines of model selection for the NMR data analysis
and interpretation. Theoretical and experimental aspects of the NMR technique in
condensed matter physics is explained in Chapter 4. Chapter 5 presents the NMR
evidence of the structural and magnetic phase transitions in Ba2NaOsO6, identified
by NMR spectra as a function of temperature and external field. Numerical calcula-
tions are applied to different models in Chapter 6 in order to understand the specific
forms of geometrical distortions and quantitatively account for the quadrupole split-
ting. Chapter 7 discusses the local magnetic order that gives rise to the magnetic
anisotropy and NMR shift, and reveals an exotic canted ferromagnetic order. Chap-
ter 8 focuses on the transition region between PM and FM states, and discusses the
possible existence of the spin nematic state. Chapter 9 deviates from the Ba2NaOsO6,
and discusses briefly about resistively detected NMR experiments on topological in-
sulator, which is also characterized by strong SOC. Finally, chapter 10 concludes all
the NMR discoveries in conjunction to each other and suggests possible directions
for future work.
3
Chapter Two
BNOO: Structure, Properties and
Bulk measurements
(a)(b)
Figure 2.1: Schematic of Ba2NaOsO6 crystal structure in (a) single unit cell (b) octahedra view.Green, yellow, red and blue spheres correspond to Ba, Na, Os and O atoms, respectively.
2.1 Crystal Structure and Basic Properties
The Mott insulator Ba2NaOsO6 has a face-centered-cubic (fcc) crystal structure with
space group Fm3m and lattice constant a = 8.2870(3)A, measured at room temper-
ature [6]. It forms a double perovskite structure, with Na and Os ions inhabiting
alternate oxygen octahedron. The unit cell is shown in Figure 2.1, and consists of
rock salt lattice of [NaO6] and [OsO6] octahedra with 6 Ba atoms separating the
layers of Os and Na atoms [7]. The location of each atom in the unite cell is listed
in Table 2.1.
x (a) y (b) z (c)Ba(1) 1/4 1/4 1/4Na(1) 1/2 1/2 1/2Os(1) 0 0 0O(1) 0.2256(6) 0 0
Table 2.1: Coordinates of each atoms in the unit cell of Ba2NaOsO6.
Crystal structure of Ba2NaOsO6 was also measured at low temperatures well
below the critical temperature of magnetic phase transition (Figure 2.2) with X-
ray powder diffraction on Oxygen sites, and no discernible structural change was
5
observed [11].
Figure 2.2: XRD patterns of Ba2NaOsO6 at room temperature and low temperatures. Datafrom [11]
With one electron per Osmium atom, the band theory predicts that Ba2NaOsO6
should be a metal. However, strong electron-electron correlation in this material
hinders electrons from hopping to neighboring sites. Both DC resistivity and in-
frared reflectivity measurements demonstrate that Ba2NaOsO6 is indeed a Mott in-
sulator [11]. The hopping energy t and Hubbard repulsion U can be calculated by
considering the coupling between adjacent OsO6 octahedra and orbital hybridization
of O 2p and Os 5d state. The calculated t = ± 0.05 eV and U = ± 3.3 eV [11]
further confirm that this material is a Mott insulator since U t.
2.2 Bulk Magnetization and Heat Capacity
Simpler osmates typically demonstrate Pauli paramagnetism, including OsO2, SrOsO3
and BaOsO3 (parent compound of Ba2NaOsO6 ) [8], but a small local moment can
be observed in more complicated structures. Materials with similar structure to
6
(a) (b)
Figure 2.3: (a) Temperature dependence of magnetization and inverse susceptibility (insets) at2T. (b) Heat capacity as a function of temperature. The magnetic contribution is plotted to theright axis. Insets shows heat capacity measured up to 300K. Data from [11].
Ba2NaOsO6, for example La2NaOsO6, and the triple perovskite Ba3AOs2O9 (A =
Li, Na), are all found to be Mott insulators and have antiferromagnetic order at low
temperature ordered state.
Ba2NaOsO6 distinguishes itself from the the above osmates as the only ferro-
magnet with a small moment of ∼0.2 µB [6, 11] at zero field and low temperature.
Some 5d oxides containing Iridium also show weak ferromagnetism [9,10], but their
saturated moments are substantially smaller than the moment of Ba2NaOsO6. A
phase transition from high temperature paramagnetic state to low temperature fer-
romagnetic state is observed in both temperature dependence of magnetization, and
more clearly, heat capacity measurements.
Figure 2.3a shows the temperature dependence of magnetization measured in
three high symmetry crystallographic directions [11]. The upturn around 8 K visible
for all directions is evidence of a ferromagnetic transition. Susceptibility curve (insets
to Figure 2.3a) at high temperature paramagnetic state can be fit to the Curie-Weiss
law
χ =c
T − θ+ χ0 , (2.1)
7
where the material dependent Curie constant C = NAg2µ0µ
2BS(S + 1)/3kB. The fit
gives a negative Curie-Weiss temperature of θ = -10, -10, and -13 K and effective
moments µeff = 0.602, 0.596, and 0.647 µB in [100], [111], and [110] directions
respectively, consistent with the other measurement in reference [6]. Negative Curie
temperature is usually observed in anti-ferromagnets. It indicates that the exchange
coupling in Ba2NaOsO6 is antiferromagnetic and that the material is not simply
a weak ferromagnet, which arises the possibility of more exotic phases like canted
anti-ferromagnetism.
The sharp peak presented in heat capacity data in Figure 2.3b defines a transition
temperature T c = 6.8 K. The magnetic contribution to the entropy obtained by
subtracting the electronic and phonon contributions is ∼4.6 J/molK and is close to
Rln2 ∼ 5.76 J/molK. This fact indicates a doublet ground state of Ba2NaOsO6. As
discussed in the next chapter, Ba2NaOsO6 has a quartet ground state, and the Rln2
entropy implies that the degeneracy of the ground state has been partially lifted.
A remarkable observation from the magnetization is the existence of strong un-
usual magnetic anisotropy. Angle dependence was determined by measuring the
magnetization while rotating the sample in (011) plane, which contains several high
symmetry axes. Inset to Figure 2.4 shows that [110] direction is the magnetic easy-
axis, which is not possible in Landau theory. In octahedral structure, the Jahn-Teller
effect [12–14] due to the distortion of O atoms around Os can lift the energy degener-
acy. However no structural transition or distortion has been observed in XRD. This
might be due to the low scattering power of the oxygen ions, and a more conclusive
finding can be found by NMR, as we describe in Chapter 5.
8
Figure 2.4: Full hysteresis loop at 1.8 K and three high symmetry directions. [Inset] Angulardependence of net magnetization. H0 is rotated in (110) plane. Data from Ref. [11].
2.3 Muon Spin Rotation (µ+SR) Measurement
The experimental measurements reviewed so far have identified a ferromagnetic
ground state with unusual easy-axis for Ba2NaOsO6. However it is challenging to
infer the configuration of local magnetic order and the microscopic mechanism re-
sponsible for such magnetism by bulk probes. Further, local probes sensitive to the
spin and/or orbital degrees of freedom are highly sought for.
µ+SR experiment was conducted on Ba2NaOsO6 [15] by implanting spin-polarized
positive muons onto the sample. The muons usually reside near sites with high elec-
tron density and precess around local magnetic field, since muons carry magnetic
moments themselves. The frequency of precession is ν = γµB where γµ = 135.5
MHz−1 is the muon gyromangetic ratio. Muons only have a lifetime of 2.2 µS and
decay into a positron (e+) and two neutrinos (νµ). Information regarding the mag-
netism can be inferred from the asymmetry factor [A(t)] of the emitted e+ detection.
9
Temperature dependence of the relaxation rate, indicator of the strength of local
static field, clearly shows a magnetic phase transition with T c = 8 K, as presented
in the inset of Figure 2.5(a). The time decay of Ba2NaOsO6 asymmetry factor below
transition temperature is interpreted to consist of two damped oscillating components
plus a background decay, and is fitted to
A(t) = [2∑i=1
Aie−λit cos(2πνit) + A3e
−λ3t] + A0e−λ0t , (2.2)
where the two oscillating components are likely resulted from muons stopped on
oxygen sites near either Os or Na atoms. The temperature dependence of the ex-
tracted precession frequency ν is then fitted with a 3D Heisenberg model using the
phenomenological form ν(T ) = ν(0)[1− (T/Tc)α]β with fixed β = 0.367. The result-
ing transition temperature Tc = 7.2 ± 0.2 K. Furthermore, a numerical probability
analysis of dipole field shows that, the most likely magnetic ordering in Ba2NaOsO6
is low moment ferromagnetism, instead of canted structure with larger moments.
Figure 2.5: µ+SR data at temperatures above and below Tc. Solid lines are fits with a dampedoscillation model described in text. Inset shows the temperature dependence of relaxation rate.Data from [15]
10
However there are several questions remained from the µ+SR experiments. The
stopping sites of muons in µ+SR measurements are not known and therefore the
validity of two-component interpretation of the signal oscillation is in question. The
lack of knowledge of the exact site of µ+ decay makes this probe not local. The proba-
bility calculations are based upon this specific interpretation, and also the numerical
results are not deterministic. Indeed the conclusion of ∼ 0.2µB weak ferromag-
netism with collinear magnetic structures is contradictory to the effective moment
of ∼ 0.6µB. Also, the fit of the temperature dependence of the local field using a
3D Heisenberg model can not be justified from the first principal. Furthermore,
the µ+SR results can not explain neither the [110] easy-axis from magnetization
measurement nor the appearance of magnetic anisotropy.
2.4 Sample for NMR
High quality single crystal of Ba2NaOsO6 with a truncated octahedral morphology of
a volume of the order of 1 mm3 was used in our experiment. It was grown from molten
hydroxide fluxes as described in Ref. [6, 11]. The oxides form the double perovskite
structure that crystallize into stoichiometric Fm3m space group. Quality of the
crystal was tested by single crystal x-ray diffraction [11]. Furthermore, the quality
is confirmed by the sharpness of 23Na NMR spectra both in the high temperature
paramagnetic state and low temperature quadrupolar split spectra, shown later in
chapter 5.
11
2.5 Summary
Ba2NaOsO6 demonstrates an exotic ordered state of weak ferromagnetic order with
uncommon [110] easy axis, which is further complicated by the lack of discernible
structural phase transition. Strong SOC combined with strong electron-electron cor-
relation is believed to play an important role in driving the observed properties. Our
NMR investigation will focus on understanding the local properties of the material
in an effort to demystify the underlying microscopic mechanism with comparison
with theoretical studies.
12
Chapter Three
Theoretical Background
3.1 Crystal Field Theory in Octahedral Complexes
E
Single 5d electron
5d
eg
Within an octahedraJahn Teller e!ect
dz2 dxzdx2-y2 dyz dxydz2 dx2-y2
t2g
dxz dyz dxy
dz2
dx2-y2
dxz dyz
dxy
Figure 3.1: Energy splitting of electrons in 5d transition metal oxides under crystal field.
Crystal field theory (CFT) explains the lifting of energy degeneracy of electron
orbital states, as shown in Fig. 3.1. An isolated single 5d electron has 5 energy
degenerate d-orbitals, dz2 , dx2−y2 , dxz, dyz, and dxy. However if the electron is placed
in an octahedral environment, as in the Ba2NaOsO6 system, this degeneracy is lifted
due to the attraction between the positively charged central Os7+ ions and the neg-
atively charged O2− ions. The set of dz2 and dx2−y2 orbitals has higher energy and
is called eg, and the set of dxz, dyz, and dxy with lower energy is called t2g.
The Jahn-Teller theorem states that, the t2g and eg degeneracy systems are not
stable under structural distortions. For example, if the two apical O ions distort
toward the central Os ion along z-axis, as shown in Fig. 3.1, the degeneracy is lifted
in both t2g and eg sets and the system energy is lowered. The orbital configura-
tion with the lowest energy would be dxy in this situation, or dxz and dyz if the
O pairs elongate along z-axis. Therefore Jahn-Teller distortions can give rise to
long-range orbital ordering, which causes the anisotropy of the electron-transfer in-
14
teraction. For example in manganese oxide LaMnO3 [3], the orbital ordering due
to Jahn-Teller distortion is the alternating dx2 and dy2 orbitals in ab planes, called
A-type antiferromagnetic state, shown in Fig. 3.2. Due to the orbital ordering, the
magnetic interaction is ferromagnetic within each plane, and antiferromagnetic be-
tween planes. The structural, orbital and spin degrees of freedoms are therefore
intrinsically coupled, requiring experimental probes sensitive to all of them.
dx2
dy2
Figure 3.2: Orbital and magnetic ordering of A-type antiferromagnetism in LaMnO3.
3.2 Strong Spin-Orbit Coupling (SOC) in Ba2NaOsO6
One of the most intensively studied aspect in the research of quantum material is
the electron correlation. The inclusion of strongly correlated electrons in 3d tran-
sition metal oxides systems goes beyond the conventional band theory and fully or
partially explains the experimental observations of Mott insulators, materials that
were concluded to be conductors with band theory. The “simplest” model that in-
cludes electron-electron correlation can be described by the Hubbard model, which
considers both the electron hoping effect and electron-electron repulsion. In certain
materials, the strong local Hubbard electron-electron repulsion U add additional cor-
relation besides the ubiquitous Pauli repulsion, and leads to localized electrons at
the atom site. The material in this case behave like a magnetic insulator and is
therefore called “Mott Insulator”.
15
Things get more complicated when we add spin-orbital coupling (SOC) into the
picture. In a Mott insulator, a weak SOC only results in a small perturbation to
the overall Hamiltonian. However in the present of strong SOC, completely different
physics arises, due to the fact that spin is no longer a good quantum number. Strong
SOC is common in all the trivalent lanthanide ions with unpaired 4f electrons, which
are tightly bound to the nuclei. Although more rare, strong SOC can also be observed
in 5d transition-metal compounds. Electrons in 5d materials are more delocalized
compared to lanthanides, but their larger atomic weight can make up for the SCO,
as in the case of some Ir-based compounds [16,17].
The material of interest in this thesis, Ba2NaOsO6, carries localized magnetic
moments on Os with 5d1 electronic configuration. Strictly speaking, atomic weight
of Os is not large enough to form very strong SOC, but this is compensated by
Ba2NaOsO6’s high crystal symmetry, which suppresses crystal field, and the large
separation between Os atoms, which suppresses exchange effects, since both crystal
field and exchange are competing with SOC. This direct consequence of strong SOC
present in Ba2NaOsO6 is that spin and orbital are no longer good quantum numbers,
and have to be projected into the t2g triplets eigenstates, as discussed in the next
section.
3.3 Hamiltonian: SOC, Exchange Interactions, and
Electric Quadrupolar Interaction
The total Hamiltonian contains several parts. First we discuss the strong SOC,
which cannot be treated as a perturbation and enters into the Hamiltonian in the
form of HSOC = −ζl ·S, where l is orbital angular momentum and S is spin angular
16
momentum. Then there are both anti-ferromagnetic and ferromagnetic exchange
interactions between the Os 5d1 electron, mediated through the O 2p orbital and
will be discussed in details in 3.3.2. Finally we include the electric quadrupolar
interaction due to the static electric potential generated by the octahedral structure
is introduced in Section 3.3.2. The total Hamiltonian is based on the quantum model
proposed in reference [18].
3.3.1 Eigenstates with strong SOC
An isolated d -orbital electron has 5 degenerate orbitals, d z2, dx2−y2 , dxy, dyz and
dxz, that are equivalent in energy. In the octahedral environment with the single
magnetic ion Os surrounded by 6 O atoms, the degeneracy is partially lifted by
the crystal field. The three orbitals dxy, dyz, and dxz have lower energy and are
collectively referred as t2g, and the other two higher energy orbitals are called eg. In
Ba2NaOsO6, 5d1 electron of Os resides on the t2g with lower energy and three fold
degeneracy.
It is long known that the effective orbital angular momentum of t2g complex is
L = −l = 1 [19–21], therefore the SOC contribution to the Hamiltonian can be
written as
HSOC = −ζl · S , (3.1)
with l = 1 and S = 1/2.
The eigenstates of t2g can be expressed with a set of quantum numbers |l, s,ml,ms >,
with ml = 0,±1 and ms = ±1/2. These eigenstates can be written in terms of the
17
combination of t2g and spin states
|0,ms >= |dmsxy >, | ± 1,ms >=
∓|dmsyz > −i|dms
xz >√2
. (3.2)
In the presence of SOC, we define
J = L+ S . (3.3)
Then, the energy of the t2g complex would be split into two states depending on
the value of quantum number j. It can be proven that this splitting does occur by
using a new set of eigenstates with quantum numbers |l, s, j,mj >. Here j takes
two values j = 3/2 or 1/2. j = 3/2 and 1/2 states are four-fold (mj = ±3/2,±1/2)
and two-fold (mj = ±1/2) degenerate in energy, respectively. The eigenvalues of the
SOC Hamiltonian then correspond to two different values, depending solely on the
value of j :
< jmj| − ζL · S|jmj > = −ζ~2
2for j = 3/2
< jmj| − ζL · S|jmj > = ζ~2 for j = 1/2 ,
(3.4)
so j = 3/2 state is lower in energy compared with j = 1/2 state by 3ζ~2/2. The
new ground level after incorporating strong SOC is therefore the four-fold j = 3/2
state. As a direct consequence of the energy splitting, the |l, s,ml,ms > states no
longer have definite energy, and are not eigenstates in the presence of the spin-orbit
interaction. We now represent the four low-lying states in the Hilbert space in the
18
form of linear combinations of |l, s,ml,ms > eigenstates:
|dmj>=
∑ml,ms
Cmjml,ms
|ml,ms > , (3.5)
where mj = ±3/2,±1/2 and Cmjml,ms denotes the Clebsch-Gordan coefficient:
Cmjml,ms
=< ml,ms|j,mj > . (3.6)
For the reason above, all the spin and orbital operators have to be projected into the
same j -defined subspace. The projection will induce several non-trivial multipolar
terms in Hamiltonian and significantly affect magnetic interactions. Details will be
discussed in Section 3.5.
3.3.2 Exchange Interactions and Electric Quadrupolar In-
teraction
In this section we first consider the anti-ferromagnetic (AFM) and ferromagnetic
(FM) exchange interactions between nearest-neighbor (NN) electrons. These ex-
change interactions happen through the overlapping between osmium d orbitals and
the intermediate oxygen p orbitals, and are constrained by structural symmetries.
The direct exchange can be neglected due to the large separations between magnetic
ions.
The possible paths for AFM and FM exchange are strictly governed by Goodenough-
Kanamori-Anderson (GKA) rules [22–24]. Figure 3.3 is a schematic description of
the magnetic exchange paths in our double perovskite system. For a dxy orbital,
the AFM exchange occurs with the NN dxy orbital through px and py orbitals of
19
the two intermediate oxygen ions . The same dxy orbital interacts ferromagnetically
with the NN dyz and dxz (but not dxy) orbitals, via the intermediate p orbitals. The
(a) (b)
Figure 3.3: Possible exchange paths of Ba2NaOsO6 : (a) NN AFM exchange path, (b) NN FMexchange path. Reprints from [18]
exchange term in the Hamiltonian therefore consists of two parts and has a form
based on t-J model [25]:
Hexchange = HAFM +HFM
= J∑
α=xy,xz,yz
∑i,j
(Si,α · Sj,α −1
4ni,αnj,α)
− J ′∑
α=xy,xz,yz
∑i,j
Si,α ·∑α′ 6=α
Sj,α′ +3
4nj,α
∑α′ 6=α
nj,α′
, (3.7)
where ni,α represents the occupation number at site i ’s α orbital, J and J ’ correspond
to AFM and FM interaction constants, respectively.
Then finally the interaction between the 5d electrons’ electric quadrupole mo-
ments are taken into account. This quadrupolar interaction in XY plane has the
form [18]
HXYquad = V
∑i,j∈XY
[−4
3(ni,xz − ni,yz)(nj,xz − nj,yz) +
9
4ni,xynj,xy
], (3.8)
20
3.3.3 Overall Hamiltonian
Including all the interactions discussed above and Zeeman interaction, we arrive at
the total Hamiltonian
Htotal = HZ +Hsoc +Hexchange +Hquad , (3.9)
where HZ = −gµBH · j and simply shifts the total energy without lifting the existing
energy degeneracy. One problem still remains regarding the Hamiltonian: all the
spin and occupation numbers involved are in the |ml,ms > subspace and have to
be projected onto the j = 3/2 quadruplets. Following are examples of results of
projections:
Si,xy =1
4jxi −
1
3jzi j
xi j
zi
ni,xy =3
4− 1
3(jzi )
2 ,
(3.10)
and we get a new representation of the total Hamiltonian in the j -basis eigenspace
by substituting the new spin and occupation number,
Htotal = HZ + Hexchange + Hquad . (3.11)
3.4 Mechanism of Mott Insulating
First-principles Density Functional Theory (DFT) calculations were performed to
numerically explore the band structure of Ba2NaOsO6 [26]. The strong SOC effect
and Hubbard on-site repulsion U are taken into account. The generalized gradi-
ent approximation (GGA) which considers the exchange-correlation energy does not
21
produce enough splitting to produce an insulating state [26]. However the electron
repulsion U can enhance the exchange splitting, and strong SOC is capable of split
up the degenerate t2g bands. With the inclusion of both U and SOC, a band gap
can be induced which explains the Mott insulating state.
A similar conclusion can be acquired by Local Density Approximation (LDA)
with SOC, Hund’s interaction, repulsion U and hopping effects. An intuitive schema
of the band structure evolution is shown in Figure 3.4. Strong SOC first lifts the six-
fold degeneracy ((2l+1)(2s+1) = 6) of t2g states. The inclusion of Hund’s interaction
further breaks the degeneracy by selecting the +1/2 spin direction as the ground
state. The key to creating the final energy gap is due to the Hubbard repulsion
U, which largely increases the gap between lowest energy band and excited state.
Without repulsion, the dispersion caused by inter-atomic hopping effect t would have
smeared the energy gap and results in a metallic state.
Figure 3.4: Energy level evolution of t2g band under the influence of SOC, Hund’s interaction,repulsion U and hopping effect t. Reprints from [27]
Furthermore, density-of-state (DOS) result shows that 5d and 2p states con-
tribute almost identically to the total DOS. Further more the three 5d orbitals (dxy,
22
dxz and dyz) contribute equally to DOS. This indicates an substantially large hy-
bridization between 5d and 2p orbitals and a strong covalent bonding between Os
and O. The large hybridization strongly influences the formation of magnetic order
and effective magnetic moments, as discussed in the next section.
3.5 [110] Ferromagnetism: SOC, Frustration, or
Multipolar Spin Interactions?
Perhaps the most intriguing property of Ba2NaOsO6 is the exotic weak ferromag-
netism with easy-axis [110]. The negative Curie temperature indicates that the
dominant exchange interaction is antiferromagnetic, but magnetization shows a weak
ferromagnetic moment. Furthermore, according to the Laudau theory, ferromagnetic
order usually chooses [100] direction as the easy-axis to minimize free energy. How-
ever in Ba2NaOsO6, the presence of strong SOC as well as electron correlation, the
frustration due to double perovskite fcc crystal structure, and the non-trivial multi-
polar spin interactions induced in quantum model make this material deviate from
the standard models.
According to the Hamiltonian of quantum model, the magnetic moment of the d
orbital electron is
M = P3/2[2S + (−l)]P3/2 , (3.12)
where P3/2 is the projection operator into the four fold subspace described in Sec-
tion 3.3.1. After projection, P3/2SP3/2 = j/3 and P3/2lP3/2 = 2j/3, which results
in a total magnetic moment of zero. In reality the total moment is non-zero due to
the strong hybridization between Os’s 5d orbital and O’s 2p orbital. Indeed, first
23
principle calculation [27] shows that about half of the t2g bands’ density is on the
oxygen ions which results in an incomplete cancellation between spin and orbital
angular momentums.
DFT calculation argues that the small magnetic moment and anisotropy from
bulk measurement is due to the cooperative effect of electron correlation and SOC
[26]. By comparing the ground state energy level, the A-type AFM state is slightly
less stable than FM state. However the most stable spin quantization direction (easy-
axis) is determined to be [111] instead of [110], and it also fails to produce the small
moment of 0.2µB/FU at zero-field and zero temperature. This quantitative discrep-
ancy is attributed to finite temperature in experiment. But, we will demonstrate
in our NMR results that the magnetic moments remain constant below T c. DFT
calculations also argue that the magnetic anisotropy is from the different occupation
numbers of three e2g orbitals at different directions of magnetization, but can not
provide quantitative comparison with the magnetization.
In contrast, local density approximation (LDA) [27] calculation shows that the
FM moments tends to vanish with a choice of finer k-point sampling. The ordered
FM moment acquired numerically is only 0.04µB, and is not zero only due to orbital
quenching.
The magnetic order problem is further complicated by several additional features
that we describe below. The Os ions form fcc structure, which is considered to be
highly frustrated. It is long suspected that highly frustrated structure can induce
large quantum fluctuation and affect the magnetic order and anisotropy properties.
Even orbital ordering can be subject to frustration [28]. To account for the frustra-
tion, quantum models are developed.
24
The problem is also approached by building quantum models [18] and considering
the competition among various interactions, as discussed in Section 3.3. The spin
operator after projection into the j = 3/2 eigenstates contains both linear and third
order components, shown in Equation 3.10. This induces higher-order spin exchange
interactions which can be thought of as couplings between multipole spin moments.
Since spin interactions are dependent on orbital states and spin is highly entangled
with orbital in the presence of strong SOC, the multipolar interactions are generically
strong. These interactions can give rise to a verry exotic physics [29, 30].
By applying mean-field method to the Hamiltonian in Equation 3.11, a phase
diagram of the ground state (T = 0) is acquired as shown in Figure 3.5a is obtained.
With fixed value of J, small values of FM interaction constant J ’ and quadrupole
coupling V result in an AFM state. With larger J ’ and V, a FM state with uncom-
pensated net ferromagnetic moment along [110] direction is acquired. The occurrence
of this FM phase is due to the dominant orbital interactions which breaks the time
reversal symmetry. Further, between AFM and FM[110] states, a small region of
FM [100] state is stabilized.
The analysis of phase diagram can be extended to finite temperature by adding
thermal fluctuation at the mean-field level. At low temperature, all the previously
established phases are stable against the thermal fluctuations. At high temperature,
system naturally displays paramagnetism. What is interesting is the existence of a
quadrupolar ordered state at the intermediate temperature and relatively large V,
due to the multipolar spin interactions aforementioned. Within this phase, orbital
ordering is formed which partially lifts the quadruplet degeneracy and leaves the
ground state to be Kramers doublet, which is the result of time reversal symmetry
in the absence of magnetic field. Figure 3.5b shows the revised phase diagram with
fixed J and J ’ (J ’ = 0.2J ). It is evident that before entering the FM [110] phase ob-
25
(a) (b)
Figure 3.5: (a) Ground state phase diagram (b) Finite temperature phase diagram. Data fromRef [18]
served in experiments, the system has to go through the quadrupolar phase. Within
the quadrupolar phase, the time-reversal symmetry is reserved but point-group sym-
metry is broken, i.e. cubic symmetry should be reduced to tetragonal or even or-
thorhombic. The existence of spin nematic ordering is also predicted in this phase.
The calculation of magnetic susceptibility shows that with large V and small J ’, the
system is in FM[110] state but with a negative (AFM) Curie temperature.
The quantum model predicts a cubic to tetragonal structural transition upon
entering the quadrupolar state from paramagnetic state with high transition tem-
perature (above room temperature). An important feature corresponding to the
tetragonal state as the result of the orbital ordering or quadrupolar phase is the
formation of two-sublattice [31]. Two possible scenarios are discussed: first model
considers identical spin orientation for each local electron and simple structural elon-
gation or contraction along z axis [Figure 3.6 (a)]; second model describes a canted
ferromagnetic order, with spins in the same plane point to the same direction, but
different from spin directions in the adjacent planes [Figure 3.6 (b)]. In the first
case, the difference in exchange interaction strength of spins within plane and be-
tween layers, due to the tetragonal structure, causes spin ordering within XY plane
with arbitrary angle. Then both thermal and quantum fluctuations lift this angle
26
Figure 3.6: Possible electron spin configurations. (a) Ferromagnetic order with tetragonal struc-ture. (b)Exotic canted ferromagnetism . Graph from [31].
degeneracy and select [110] direction as the magnetization direction. In the second
scenario, the spin orientations in the two-sublattice are symmetrical about [110] di-
rection. Therefore total spins result in not just an uniform magnetization along [110]
axis but also a staggered magnetization perpendicular to [110] in XY plane. This
corresponds to an exotic canted-ferromagnetic state.
The biggest discrepancy between the first-principle DFT method and quantum
model calculations is the existence of structural distortion. DFT simulations suggest
the conservation of cubic symmetry within the FM ground state, while quantum
model predicts a structural distortion which reduce the spatial symmetry to tetrag-
onal. Therefore, local probes sensitive to both spin and structural degree of freedom
like NMR are highly sough for to investigate and verify different theoretical models.
27
Chapter Four
NMR in Condensed Matter and
Experimental Techniques
4.1 Basic Concepts and Principles of NMR
First we consider the simplest scenario where nucleus are isolated from the lattice,
and only interact with applied magnetic field and each other. Fundamental principles
and detection methods of NMR can be understood intuitively by considering such
system under the classical description.
4.1.1 Isolated Nuclear Spins and NMR Resonance
For a single nucleus, the interaction between an applied magnetic field H0 and the
spin angular momentum I is called Zeeman interaction, described by the following
Hamiltonian
HZ = −µ ·H0 = −γ~H0 , (4.1)
where µ = γ~ is the magnetic moment of the nuclear spin, γ is the unitless nuclear
gyromagnetic ratio.We can assume that the applied filed is parallel to the z-axis
of spin quantization, i.e. H0 = H0z, then Zeeman interaction effectively lifts the
nucleus’ spin ground state with (2I +1) fold degeneracy and creates (2I +1) evenly
spaced eigenstates, characterized by the magnetic quantum number m
Em = −γ~mH0 , (4.2)
where m = −I,−I + 1, ...I − 1, I. Fig. 4.1 illustrates the Zeeman energy splitting for
23Na nucleus with I = 3/2. A resonance in NMR involves absorption and emission
of electromagnetic radiations with energy corresponding to that between adjacent
Zeeman energy levels
∆E = γ~H0 = ~ω0 , (4.3)
29
I = 3/2
H0 = 0 H
0 > 0
m = -3/2
m = 3/2
m = 1/2
m = -1/2E0
Em
Figure 4.1: Energy splitting of 23Na nucleus under the applied field. Degeneracy is lifted by anonzero applied field into 2I+1 = 4 levels.
ω0 is called the natural frequency, or Larmor frequency, and is typically in the order
of a few to several hundreds of MHz, within the radio frequency range.
More specifically, the resonance, i.e. transition between adjacent energy levels
(Em → Em±1), is induced if an exact ∆E amount of energy is transmitted to the
nucleus, and can be intuitively understood by describing the motion of the nucleus
from the classical point of view. In NMR, resonance is invoked by applying an oscil-
lating field H1 perpendicular to H0 with frequency matching the Larmor frequency.
Experimentally, H1 is generated by a RF coil with central axis perpendicular to H0,
as shown in Fig. 4.2. In the rotating frame of reference which rotate around z axis
with Larmor frequency, H1 is effectively static. The torque exerted on the nucleus
is
τ = µ×H1 , (4.4)
and it rotates the moment in the plane (XZ plane in Fig. 4.2) perpendicular to H0.
If H1 is turned on for time τ , the rotated angle θ follows
θ = γH1τ . (4.5)
We call the pulse applied to rotate the moment by 90 degree a π/2 pulse (Fig. 4.2(b)),
30
and the pulse that inverts the moment to -z direction a π pulse (Fig. 4.2(c)).
H1
H0
x
z
yH
1
H0
μ
H1
H0
(0)
μ(π/2)
μ(π)
Figure 4.2: Schematic of the effect of the oscillating field H1. (a) Without H1, spin is alignedwith H0. (b) After the application of a π/2 pulse, spin is in x axis. (c) A π pulse flips the spin to-z axis.
H1 is turned off after the moment is rotated angle θ away from H0, and then
the moment will precess freely around the z axis with Larmor frequency. The free
precession induces oscillating magnetization that is picked up by the same NMR coil.
This provides the possibility of acquiring an NMR signal.
4.1.2 FID Spin Echo, T1 and T2 in Pulsed NMR
Next we consider a collection of identical spins, and deduce the two common detection
methods of NMR signal: Free Induction Decay (FID) and Spin Echo.
FID and T2
Fig. 4.3 illustrates the process of FID. A π/2 pulse is first applied to rotate all the
spins into the x axis. Typically the pulse length is a few µS. Then the spins precess
around the z axis, but each spin senses a slightly different magnetic field than others
and has a unique Larmor frequency. With longer elapsed time, the net magnetization
31
in XY plane dephases and as a consequence the induced oscillating magnetization
picked up by the coil decays. The magnitude of the measured transverse magnetiza-
tion decays exponentially as a function of time
M(t) = M(0)e− t
T2 , (4.6)
where the exponent T2 is the so-called spin-spin relaxation time. The main factor
y
x
t = 0 t’ > 0 t’’ > t’
π/2 t
M (t)
(a) (c)(b)
Figure 4.3: Illustration of the Free Induction Decay.
that contributes to the decoherence process is the spin-spin coupling between nu-
cleus. However, in reality there always exists certain degree of local internal field
inhomogeneity at the nuclear site, which causes different precession frequency and
therefore magnetization decay. The total effective relaxation time, T ∗2 , is resulted
from the combination of both factors
1
T ∗2=
1
T spin−spin2
+1
T inhomo2
. (4.7)
32
Hahn Spin Echo
Another commonly used detection method is the spin echo which essentially con-
tains the same resonance information as FID. A so-called Hahn echo involves the
application of both π/2 and π pulses and the sequence of events is shown in Fig.
4.4. It also starts with the π/2 pulse and spins precess in XY plane and dephase as
described in FID process. After a time τ (typically a few µS), a π pulse is applied
by the same coil and it effectively invert the direction of all the spins in XY plane.
This is equivalent to let all the spins precess in the opposite direction, and therefore
after another time τ after the π pulse, all the spins will come back to the original
position at 2τ time ago, and point to the same direction. After this, they continue
precessing and begin to dephase again.
Figure 4.4: Illustration of the Spin Echo.
If we measure the transverse magnetization via the coil starting right after the
π pulse, the magnetization will increase first and then decay again in a symmetric
fashion. The second half of the signal is exactly a FID signal.
33
4.1.3 Fourier Transform Spectrum
The magnetization measurements of FID or spin echo as a function of time are NMR
signals in the time domain. The time domain signal resulted from the precession of
all the nuclear spins which have different Larmor frequency, and is a combination
of sinusoidal functions with different frequency. Thus, the frequency distribution of
the spin emsemble can be acquired by Fourier transforming the time domain signal
into the frequency domain, and is called a NMR spectrum. Since ω0 = γH0, the
spectrum also describes the magnetic field distribution of the nucleus. Fig. 4.5 shows
a typical transformation between the time domain signal and NMR spectrum. Notice
that in reality it rarely happens than a single shot of FID or spin echo pulse sequence
can generate a NMR signal with good signal to noise ratio (SNR). Usually multiple
measurements are taken and averaged to reduce the random electronic and thermal
noises. S/N is proportional to√N with N representing the number of repeated
measurements.
（a） （b）
Figure 4.5: A typical Fourier transformation from the (a) time domain signal to (b) frequencydomain NMR spectrum. Data acquired from 209Bi nuclei in Bi2Se3. In plot (a), the green and redcurves represent real and imaginary detection channels, respectively. The white curve is the totalmagnitude of the signal.
34
4.1.4 Spin-Lattice Relaxation Time T1
So far we have only considered the spin-spin interaction which induces the T2 relax-
ation effect. But in a solid state sample, nuclear spins are also interacting with the
lattice and electrons. The magnetization vector M of nuclei lies in the direction of
H0 at equilibrium, i.e. lowest energy state. With the application of the transverse
oscillating field H1, the z component of magnetization Mz become smaller than the
equilibrium value M0 due to reorientation. The spin-lattice interaction results in
a mechanism called ”spin-lattice relaxation” where spins emit energy to the lattice
and relax back to the equilibrium position. For example, after applying a π/2 pulse
to a spin 1/2 nucleus at equilibrium, the spin relaxes back to z axis from XY plane
following the equation
Mz(t) = M0(1− e−t/T1) , (4.8)
where spin-lattice relaxation time T1 is defined as the time to change the Mz by a
factor of e.
There are different methods of measuring T1. The most commonly used method
is the full saturation measurement. The strategy is to saturate the nuclear spins
with a π/2 pulse first, wait for a time tR which corresponds to the variable t in
Eq. 4.8, and then acquire the FID/echo signal. Since the measured magnetization
is a function of tR, by measuring the amplitude of FID/echo signal at different tR
values, we can obtain a magnetization recovery profile, as shown in Fig. 4.6. By
fitting the curve to Eq. 4.8, we can obtain the value of T1.
Typically for a solid material, T1 ranges from a few milliseconds to dozes of
seconds, and is much longer than T2. Therefore the decay in time domain FID or
echo signal as in Fig. 4.5 (a) is mainly governed by T2 effect.
35
1.0
0.8
0.6
0.4
0.2
0.0
M (
t) / M
0
0.001 0.01 0.1 1 10
t (s)
Figure 4.6: A typical semi-log plot of T1 recovery profile, measured on proton nucleus in pyruvicacid at temperature of 220 K. Hollow squares are measured magnetization as a function of waitingtime tR, and the solid line is a fit with Eq. 4.8.
In the full saturation method, usually enough time (Twait > 5T1) has to be allowed
between each data acquisition. There are situations where T1 is long and the NMR
signal is weak, therefore the measurement time for the regular full saturation method
becomes impractically long. A faster and as accurate method is the progressive
saturation, which does not require the establishment of the thermal equilibrium prior
to each measurement [40–42]. The recovery equation in the progressive saturation
method is dependent on the spin tipping angle θ and is given by,
M(t) = M0
[1− (cos θ − 1)e−t/T1
1− cos θe−t/T1
]. (4.9)
4.1.5 Bloch Equations
Based on the previous discussions on T1 and T2, the transverse component of nucleus’
magnetic moments decays due to spin-spin coupling, while the z component tends to
36
recover back to equilibrium state. Combining the relaxation effects with the torque
from applied field, the so-called Bloch equations are derived to describe the motion
of spin vectors under any applied field H and have the form
dMz
dt=M0 −Mz
T1
+ γ(M ×H)z
dMx
dt= −Mx
T2
+ γ(M ×H)x
dMy
dt= −My
T2
+ γ(M ×H)y .
(4.10)
Bloch equations are classical interpretation of spin motion and have some limita-
tions, but nevertheless provide a general and intuitive way of understanding nuclear
resonance and relaxation phenomena.
4.2 NMR in Solids
As in real solid materials, situations are drastically different from the isolated nucleus
system, due to the fact that nuclear spins not just interaction with applied field
and each other, but are also coupled to the environment in many different ways.
This coupling is weak compared to the electron-electron coupling, but nevertheless
provides useful local information about the material’s electronic, orbital, magnetic,
and structural degrees of freedom. Because of the low energy involved in the coupling,
NMR serves as a powerful yet non-invasive probe of the study of solid state materials.
37
4.2.1 Full Hamiltonian
Solid state NMR is more complex compared to liquid NMR due to the non-vanishing
dipolar and quadrupolar interactions that broaden the spectrum and reduce the line
resolution. The extra effects can also affect the spin relaxation processes. The full
Hamiltonian of a nuclear spin takes into account its interactions with external field,
electronic environment, other nuclear spins and the nearby electrons, and is described
by
H = HZeeman +Hn−n +HQ +Hhf , (4.11)
HZeeman describes the Zeeman interaction introduced before. Hn−n is the dipolar
interaction between nuclear spins. Quadrupole Hamiltonian HQ represents the in-
teraction between nucleus’ electric charge and its electronic environment. The final
term Hhf describes the interaction with electrons via the hyperfine coupling.
4.2.2 Dipolar Field
Each nuclear spin carries magnetic moment and creates a dipolar magnetic field Hd.
The Hn−n arises from the interaction between the moment µ of the nuclear spin and
the dipolar field from its neighboring nuclear spins, and has the form
Hn−n =µ0
4π
∑i
[µ · µir3i
− 3(µ · ri)(µi · ri)r5i
], (4.12)
where µi is the magnetic moment of a nuclear spin at a position ri away from µ.
The dipolar field contributes both to the NMR line width and spin-spin relaxation
T2. The local dipolar field sensed by the nuclei is different at each nucleus site, and
38
results in a continuous spread in precession rates. This spread manifests itself ex-
perimentally in the form of the probability distribution of precession frequency/local
field, i.e. the NMR spectrum. A larger variance in the dipolar field spread would
give rise to a broader NMR spectrum.
We can use dipolar field to estimate T2, which is due to the dephasing caused
by different precession rates ω = γHdipolar. The dephasing is significant by the time
τ such that ωτ = γHdipolarτ = 1. Therefore τ is a value comparable to T2. Since
Hdipolar ∼ µ/r3, we get the estimation of T2 to be
T2 =1
γHdipolar
=r3
γ2~, (4.13)
This number is approximately in the order of 100 µS.
4.2.3 Quadrupole Splitting
Any nuclei with spin I > 1/2 have a non-spherical charge distribution. The in-
teraction between the nuclei and the nearby electric field therefore depends on the
spatial orientation of the charge. Fig. 4.7 shows the comparison of electrostatic en-
ergy when charge is oriented in different directions. I = 1/2 nuclei have a spherical
charge distribution and no energetically favorable charge distribution.
The quadrupole Hamiltonian can be expressed as the electrostatic energy of the
nuclear charge distribution within a electric potential field V(r) generated by nearby
electrons
HQ =
∫ρ(r)V (r)dτ , (4.14)
39
+
+
--
+
+
--+ +
Higher Energy Lower Energy
Figure 4.7: Energy difference caused by different orientation of the charge density distribution.Arrangement on the left is less energetically favorable because of the closer distance between thepositive nuclear charge and nearby positive electrons, although the average distance between nucleusand electrons are the same in both configurations.
with the integration over the space. A Taylor expansion on the potential term gives
HQ =
∫ρ(r)V (0)dτ+
∫ρ(r)
(∑i
∂Vi∂xi
)xidτ+
1
2!
∫ρ(r)
(∑i,j
∂2Vi∂xi∂xj
)xixjdτ+O(x3) ,
(4.15)
with i, j = x, y, z. We define the first and second partial derivative to be
Vi =∂Vi∂xi
, Vij =∂2Vi∂xi∂xj
, (4.16)
and name the second-rand tensor Vij electric field gradient (EFG).
The first term in Eq. 4.15 represents the electrostatic energy of a point charge. It
merely provides a constant shift to the full Hamiltonian and can be neglected from
our discussion. The second term represents the electric dipole interaction. This term
vanishes because of the definite parity of nuclear states and the overlap of nuclear
center of mass and center of charge.
The third therm is the quadrupole interaction. Since we can always find a set of
40
principle axes for potential V such that
Vij = σij , (4.17)
from Laplace’s equation ∆V = 0 we can get
∑i
Vii = 0 , (4.18)
If the nucleus is in an environment with cubic symmetry, all three axes are equivalent
and
Vxx = Vyy = V zz . (4.19)
Combined with Eq. 4.18, all three diagonal components of EFG are zero. The sensi-
tivity of EFG to the local symmetry makes NMR a useful tool in detecting structural
transition and local distortion.
For convenience we define quantities Qij to be
Qij =
∫(3xixj − δijr2)ρdτ , (4.20)
and rewrite the Hamiltonian by replacing the xixj terms with Qij
HQ =1
6
∑i,j
(VijQij + Vijδij
∫r2ρdτ)
=1
6
∑i,j
VijQij ,
(4.21)
in which the second term vanishes according to the Laplace’s equation.
This form of Hamiltonian is too cumbersome to handle due to the integral over
the entire space in the definition of Qij. Using the Wigner-Eckark theorem [33],
41
which transforms the spherical tensor operator Qij on the angular momentum basis
into the product of reduced matrix element and Clebsch-Gordan coefficient, we can
express this Hamiltonian using the nuclear spins oprators
HQ =eQ
6I(I − 1)[Vzz(3I
2x − I2) + Vyy(3I
2y − I2) + Vzz(3I
2z − I2)] ,
where Q is the so-called quadrupole moment of the nucleus and e is the charge of a
proton. Knowing that∑
i Vii = 0, HQ can be further simplified
HQ =eQVzz
4I(I − 1)[Vzz(3I
2z − I2) + (Vxx − Vyy)(I2
x − I2y )] . (4.22)
Without loss of generality we can also choose the EFG principle axes such that
|Vxx| ≤ |Vyy| ≤ |Vzz|, and customarily define the EFG asymmetry factor η as |Vxx −
Vyy|/Vzz, the field gradient q as Vzz/e and the characteristic frequency νQ as 3eQVzzh2I(2I−1)
.
Considering both Zeeman and quadrupole Hamiltonian, the energy levels of the
nucleus when η = 0 and Vzz is along the direction of H0 are
Em = −γ~H0m+hνQ
6[3m2 − I(I + 1)] .
The energy levels are shifted from the Zeeman values in a way that the space between
each pair of (Em, Em−1) energy levels is no longer equal. For a NMR spectrum, the
quadrupole interaction does not shift the central line of the resonance for half integer
spin number, but creates 2I quadrupole satellites with equal splitting νq between each
pair of adjacent lines. A typical NMR spectrum with quadrupole effect is shown in
Fig 4.8. A strong enough quadrupole interaction can lift the nuclear spin degeneracy
even in the absent of the external applied field H0. The detection technique is called
Nuclear Quadrupolar Resonance (NQR), or “zero field NMR”. It is a valuable tool
42
Ma
gn
itu
de
(A
.U.)
79.279.078.878.678.478.2
Frequency (MHz)
Copper signal
from probe
νqνq
νq νq
νqνq
51V NMR Spectrum
H = 7 T
Figure 4.8: 51V (I = 7/2) NMR spectrum in V3Si at 220 K and H0 = 7T . There are 2I = 7 peakswith equal distance νq in frequency between adjacent lines.
to investigate a material’s properties without the influence of a magnetic field.
4.2.4 Hyperfine Interaction
The last term in the full Hamiltonian depicts the hyperfine interaction between nu-
clear spin and both the spin and orbital degrees of freedom of surrounding electrons.
The hyperfine Hamiltonian can be expressed as
Hhf =µ0
4πγe~
∑i
[−µ ·Li
r3i
+µ · Sir3i
− 3(µ · ri)(Si · ri)r5i
]
where γe = 28024.95 MHz/T is the electronic gyromagnetic ratio, Li and Si are the
orbital and spin angular momentum operator of an electron at distance ri away from
the nucleus µ positioned at origin. In solid state material, the expectation value of
43
all the three components of Li is zero due to the crystal field, i.e. a complete orbital
quenching. Therefore the first term which describes the coupling of nucleus to the
orbital degree of freedom of the electrons vanishes. The second and third terms
represent the magnetic coupling between an nucleus and an electron. This is true
when the electron is in a nonzero angular momentum state, like p-state, d -state,
etc. If the electron is in s-state and in close distant to the nucleus, the electron
wave function is nonzero at the nucleus site and there exists a contact hyperfine
interaction
Hcontacthf = −µ0
4πγe~
∑i
[8π
3µ · Siδ(ri)
].
In systems with strong electron correlation and/or spin-orbital coupling, the hy-
perfine Hamiltonian is further complicated by the presence of complex interactions,
including double exchange, superexhcange, etc. A more empirical way is to treat the
hyperfine interaction as the coupling between a nucleus and a local internal hyperfine
field Hhf generated by the electron orbital and spin degrees of freedom, and express
the Hamiltonian as Hhf = −µ ·Hhf . The internal field can be expressed in the
form of a logical hyperfine coupling tensor K, and Hhf = K ·J , where J is the total
angular momentum of the electrons. The complication to the hyperfine Hamiltonian
due to intricate orbital and magnetic ordering is therefore reflected in the form of
tensor K. Therefore the determination of microscopic structure and hyperfine tensor
usually relies on experimental results.
More specifically, using the hyperfine tensor, the Hamiltonian describing the
electron-nucleus interactions is
He−n = −γ~I ·∑N
KN · (gµBSN) , (4.23)
where KN denotes the second-rank tensor specifying the interaction between one
44
(a) (c)(b)
Figure 4.9: Sketch illustrates NMR shift with positive hyperfine tensor constant in differentmagnetic order. (a) Hhf is in the same direction with H0 with the presence of ferromagneticorder, and therefore shift the peak to higher frequency. (b) In an antiferromagnet, the anti-parallelelectron spins create two sublattices of magnetically inequivalent nuleus sites, and therefore the linesplits into two peaks with same absolute shift from ω0. (c) A more complicated incommensuratespin structure could result in a broad local field probability distribution.
nuclear spin I and the N’th electron spin SN . g is the electronic g-factor with
each component gi along the principal crystal axis i ∈ x, y, z, and µB the Bohr
magneton.
The effect of the internal field on nuclear sites induced by hyperfine interactions
on NMR spectrum is reflected in the shift the resonance frequency, and is called the
Knight shift. Fig. 4.9 shows a few commonly observed NMR shifts due to different
electron magnetic order. Here the orbital contribution to the shift, defined as the
orbital shift, is neglected due to quenching in most situations. Even if the orbital
angular momentum is not completely quenched, its contribution is substantially
smaller compared to the Knight shift.
4.2.5 Summary
Fig. 4.10 summaries the effect of different interactions on the energy level of the I
= 3/2 nucleus and the resulting NMR spectrum. The application of the external
field H0 lifts the degeneracy through Zeeman interaction. The hyperfine interaction
45
Figure 4.10: Energy shift resulted from different parts of Hamiltonian for I = 3/2 nucleus.
creates an additional local magnetic field and shifts the resonance to lower or higher
frequency, depending on the orientation of the internal field. Nonzero quadrupole
splitting into 2I = 3 satellites happens for spatial symmetries lower than cubic.
Eventually the spin-spin interaction results in a continuous distribution of local fields
and therefore broadened linewidth of NMR spectrum.
4.3 NMR Experimental Realization
4.3.1 Building Blocks
Fig. 4.11 demonstrates a simple but complete set-up for NMR apparatus. Pulse
generator first sends a continuous sinusoidal wave into the radio frequency power
amplifier with a fixed and pre-determined frequency. The amplified signal is trans-
mitted into the NMR coil with the help of the duplexer, and absorbed by the sample.
46
Pulse Generator Software
Pre-AmpPower Amplifier
Spectrometer
Probe & SampleTransmission Reception
Duplexer
Figure 4.11: Building blocks of a NMR experiment set-up. Green and blue blocks represent thetransmission and reception parts of the instrumentation, respectively.
The NMR signal generated by the nuclei is picked up by the same coil and get ampli-
fied by the pre-amplifier. Eventually the amplified signal arrives at the spectrometer
and is stored on the PC using specific software.
4.3.2 NMR Probe and Resonance Circuit
The proper design of the NMR probe is of vital importance in acquiring good SNR
and stable low temperature environment. A typical NMR probe consists of three
parts (Fig. 4.12): the probe head provides connections between external electronics
and in-magnet components/sample; the main body is the main part for heat insula-
tion and accommodates capacitors for the NMR circuit; the probe foot mounts the
sample and coil, as well as goniometer and temperature controlling parts if needed.
The diameter and length of the probe are limited by the bore size of magnet and
47
cryostat. The RF coil containing the sample must resonate at the Larmor frequency
of the nucleus in order to rotate the spins. This is realized in NMR with a simple LC
circuit composed of an inductor and two variable capacitors. In the design shown in
Fig. 4.12, both capacitors are mounted at the bottom of the probe and are cooled
down with the coil during a low temperature experiment. This is called bottom
tuning and the capacitors have to be enclosed in vacuum space to avoid arcing at
low temperature. Compared with top tuning where the capacitors are exposed to
room temperature, the bottom tuning set-up can reduce thermal noise and effectively
improve SNR.
The resonance circuit is used to ensure the sufficient power transmission to the
nuclei at the desired frequency. One capacitor is called the tuning capacitor and
the other matching capacitor. The tuning capacitor is responsible to alter the cir-
cuit resonance frequency which is determined by ν = 1/2π√LC, and the matching
capacitor matches the circuit impedance to 50 Ω. Two types of commonly used res-
onance circuits are parallel-tuned series-matched(PTSM) and series-tuned parallel-
matched(STPM), as shown in Fig. 4.13. The PTSM is usually capable of reaching
higher tuning frequencies, and is necessary if experiments to be run at high field
and for certain nucleus with high gyromagnetic ration. For example, to detect the
NMR signal of 1H with γproton = 45.5774 MHz/T at H0 = 10T, tuning has to reach
455.774 MHz. Using a large range variable capacitor (1 ∼ 120 pF), the upper tuning
limit of STPM configuration is only ∼200 MHz. On the other hand, the advantage
of STPM circuit is that it has a larger frequency range, and is the configuration we
mostly used if the desired frequency can be reached.
48
Head
Body
Foot
Figure 4.12: Sketch of the probe used in our NMR lab at Brown. The probe is placed in thecenter of a 9 T superconducting magnet with the sample positioned at the center of the magneticfield. Total length of the probe is about 100 cm and is suitable for temperature ≥ 4 K. Designedand drew by Adam Straub.
49
CT
CM
CT
CM
L
L
(a) (b)
Figure 4.13: Diagram of NMR resonance circuit with (a) parallel-tuned series-matched and (b)series tuned parallel matched configurations.
4.3.3 NMR Coil
Commercially designed NMR apparatus for chemical structure determination or
medical imaging usually only detects high resolution nuclei, like 1H and 13C. There-
fore the same resonance circuit in the NMR probe is applicable to different exper-
iments and samples. In contrast, the choice of nucleus to be detected in the solid
state NMR is constrained by the material being studied, and for each sample the
probe has to be modified accordingly in order to reach the desired frequency.
The difference in the subjects under study and spatial geometry lead to different
choices for RF coils. In the chemical/medical applications, samples are directly insert
into the RF coil through the bore of magnet. Therefore the central axis of the coil
has to be parallel to the H0, meanwhile the generated oscillating field H1 needs to
be perpendicular to H0 in order to rotate spins. Such complications require specific
and sophisticated coil design, and some commonly used types are Saddle coil [34,35],
Alderman-Grant coil [36] and Bird Cage coil [37], etc. The saddle pair coil design
is shown in Fig. 4.14(a). Solid state NMR has the flexibility of using much simpler
solenoid coil in most cases, like the solenoid in Fig. 4.14(b), due to the geometric
50
freedom of coil orientation. In reality, coils have to be customized for different sample
sizes in an effort to increase the filling factor (sample volume divided by coil volume).
Higher filling factor can significantly improve the signal-to-noise ratio when signal is
weak.
H1H
1
H0
H0
H1
H0
(a)
(c)
(b)
H1’
H1
H0
(d)
H1
’
Figure 4.14: Several typical designs for NMR RF coil: (a) Saddle coil, (b) Solenoid coil, (c) Spiralsurface coil and (d) Meanderline surface coil.
Medical imaging and solid state NMR cross their paths for coil choice at the
surface coil. A surface coil [38] generates oscillating fields in directions both perpen-
dicular to H0 and parallel to H0, shown as H1 and H ′1 in Fig. 4.14(c) and (d). 3D
electromagnetic simulation shows thatH1 is much larger thanH ′1 in magnitude, and
therefore the coil orientation in Fig. 4.14(c) provides a better SNR than theH1 ‖H0
arrangement. In medical applications, the advantage of surface coil is its applicabil-
ity to in vivo magnetic imaging. The application of surface coil in solid state NMR
relies on its characteristics that the distribution of magnetic field is concentrated
near the coil surface. More specifically, the field magnitude decays as a function of
the distance from the surface with a rate proportional the spacing between adjacent
turns. Therefore a surface coil with small spacing has the advantage of high filling
factor for flat and thin shaped samples, e.g. thin film semiconductors and optical
51
coating.
In principle the magnetic field generated by surface coils is less homogeneous than
solenoid and can induce additional line broadening. However, in many situations of
the solid state NMR spectrum measurement, the linewidth broadened by dipole-
dipole interaction overwhelms the field inhomogeneity and the choice of coil will
have no discernible affect on the spectrum lineshape.
4.3.4 Other Apparatus
Spectrometer
The spectrometer we used for all of our experiments is the MagRes2000 designed by
A. Reyes to specifically meet the need of solid state NMR. Its advantage includes
fast repetition rate, large working range of frequency, real time data acquisition
and analysis, intelligent experimental control and user-friendly programmable pulse
sequences.
Generation of short pulse with frequency and duration set by user with software
is realized by precise gating and phasing the input RF signal with the spectrometer.
The spectrometer is also responsible for the data acquisition by converting the ana-
log NMR signal to the digital form. In additional, the MagRes2000 model allows
sophisticated programming manipulation of pulse sequence and subroutines.
52
Duplexer
As stated in the preceding discussion, the transmission of high voltage RF signal to
the sample and the reception of NMR magnetization is accomplished by the same
NMR coil. This is realized by the usage of a duplexer that permits the bi-directional
power transmission over the same electronic path. Fig. 4.15(b) shows the block
Signal to Pre-Amp
Resonance
Circuit
Crossed Diodes A
Crossed Diodes B
Ground
RF from
Ampli!er
λ/4 Cable
R
RF from
Ampli!er
Signal to
Pre-Amp
RF to
Resonance
Circuit
0 o
90 o 0 o
90 oIN
IN
ISO
ISO
Coupler Coupler
(a)
(b)
50 Ω
Figure 4.15: Block diagram of the two commonly used electronic “traffic control” circuits. (a)λ/4 wave set-up. (b) Commercial duplexer assembly.
diagram of the duplexer circuit we use for our NMR experiment. It consists of two
commercial 3 dB directional couplers and provides low loss signal transmission, high
power tolerance and large frequency range.
Another network is the so-called quarter-wave circuit as shown in Fig. 4.15(a) [39].
The crossed diodes serve as switch which opens for large signals (RF input) and
closes for small signals (NMR signal). When the high power RF is transmitted from
amplifier, both diodes A and B are shorted. If the resonance circuit is well tuned
to the frequency of the RF input, its impedance is much smaller compared to the
circuit resistance R. However, since diodes B is still shorted, the power would skip the
53
resonance circuit and directly go to the ground. The key here is the λ/4 cable, which
transforms the impedance of the load to be the same as the characteristic impedance
of the transmission line. Therefore it transforms the diode B’s impedance to a high
value and effectively isolate the connection to pre-amp from the transmission line.
This way all the RF power is injected into the resonance circuit. When the weak
NMR signal is picked up and transmitted, both diodes switches are open and the
signal can only travel along the path to pre-amp.
The quarter-wave circuit is rarely used nowadays due to its larger signal loss from
the probe to the pre-amp. However, the commercially available duplexer suffers from
severe signal loss at low frequency (. 15MHz). In situations of measuring nucleus
with low gyromagnetic ratio at low magnetic field, the quarter-wave circuit is a
compromised but still feasible solution.
4.4 Experimental Set-Up for Ba2NaOsO6 NMR
The NMR measurements on Ba2NaOsO6 were done at Brown University with mag-
netic field up to 9 T and at the NHMFL in Tallahassee, FL at higher fields. In both
laboratories high homogeneity superconducting magnets were used. The tempera-
ture control was provided by 4He variable temperature insert. The NMR data were
recorded using the state-of-the-art laboratory-made NMR spectrometer (MagRes).
The spectra were obtained, at each given value of the applied field, from the sum
of spin-echo Fourier transforms recorded at constant frequency intervals. We used a
standard spin echo sequence (π/2− τ − π) with typical values of 4µS − 8µS − 4µS.
Shape of the spectra presented are independent of the duration of time interval τ .
54
The shift was obtained from the frequency of the first moment of spectral distribution
of set of triplet lines using a gyromagnetic ratio of 23γ = 11.2625 MHz/T. The same
gyromagnetic ratio was used for all frequency to field scale conversions.
The sample was mounted to one of the crystal faces and rotated with respect to
the applied field about an axis using a single axis goniometer. The rotation angle,
for applied fields below 9 T, was inferred from the signal of two perpendicularly
positioned Hall sensors. In addition, to ensure that data was taken with no external
pressure applied, the mounted sample was placed in a solenoid coil with cross sec-
tional area significantly larger than that of the sample. In this way, no pressure is
exerted on the sample as coil contracts on cooling.
55
Chapter Five
Identification of Phase Transition:
Temperature Dependence of NMR
Spectra
In this chapter we demonstrate the NMR spectra and spin-lattice relaxation of
Ba2NaOsO6 at various temperatures and magnetic fields. As we have discussed in
last chapter, the magnetic ordering and structural symmetry can manifest themselves
in the form of NMR shift and quadrupole splitting, respectively. Also the temper-
ature and field evolution of the spectra can provide further information about the
microscopic nature of phase transitions.
5.1 Temperature Evolution of NMR Spectrum
We first inspect the temperature (T) dependence of 23Na NMR spectra in Ba2NaOsO6
to establish the sensitivity of our measurements to putative lattice distortions, or-
bital order, and magnetism. These 23Na (with nuclear spin I = 3/2) spectra reveal
the distribution of the hyperfine fields and the electronic charge and are thus a sen-
sitive probe of both the electronic spin polarization (local magnetism) and charge
distribution (orbital order and lattice symmetry), as described in detail in section
4.2. Figure 5.1 shows Na spectra with varying temperatures observed at 9 T . The
temperature was lowered while crystal [001] axis (c) is aligned along the external
field direction. X-axis of the plot denotes the shift of the spectrum in relative to the
natural frequency ω0 = γH0 = 9 T × 11.2625 MHz/T = 101.3635 MHz.
From the spectra, we can clearly identify three different magnetic phases. For
T > 12 K, the spectrum consists of a single narrow NMR line, evidence of the
paramagnetic (PM) state. Below 8 K, an ordered state is developed and eventually
stabilized at lowest temperatures. Spectra within this region split into 6 peaks as
two sets of triplet lines, labeled as I and II, that are well separated in frequency.
The emergence of the two sets of triplet lines indicates the appearance of long-range
57
Ma
gn
itu
de
(a
rb. u
nits)
-2.00 -1.50 -1.00 -0.50 0.00
ω − ω0 (MHz)
H = 9T
4.2K
6 K
8 K
5 K
7 K
9 K
10 K
10.5 K
11 K
12 K
13 K
15 K
20 K
II I
δq
Paramagnetic
Phase
Transition:
Nematic ?
Ferromagnetic
Phase
Figure 5.1: Temperature evolution of 23Na spectra at 9 T with magnetic field applied parallel to[001] crystalline axis. Splitting into 2 sets of triplet lines (labeled as I and II) is evident at lowertemperatures. Zero of frequency is defined as ω0 = 23γ H0, the zero NMR shift frequency.
magnetic order (LRO) and the existence of two inequivalent magnetic sites in the
lattice. The fact that they are well separated in frequency indicates that the magnetic
order is commensurate, as discussed in section 4.2.4. Furthermore, both sets of lines
are shifted to frequencies below that of spectra in the PM state. This demonstrates
that net local magnetic fields on both Na sites are of the same sign, indicating that
the LRO order is likely ferromagnetic (FM).
Besides the splitting into the two triplets that reflects the appearance of two dis-
tinct magnetic sites in the low T phase, we observe additional splitting of each set of
the spectral lines into three peaks. This splitting, labeled as δq in Fig. 5.1, originates
from quadrupole interaction, implying changes in electronic orbitals and/or local lat-
tice symmetry. As described in section 4.2.3, for nuclear sites with spin I > 1/2, such
as 23Na with I = 3/2, and non-zero EFG, quadrupole interaction between nuclear
spin and EFG splits otherwise single NMR line to 2I lines. Thus, 23Na spectral line
splits in three in the presence of non-zero EFG. At sites with point cubic symmetry
58
EFG is zero, as is the case for Na nuclei in high temperature PM phase. Therefore,
observed splitting of the Na spectra into triplets indicates breaking of the point cu-
bic symmetry, caused by local distortions of electronic charge distribution, in the
magnetically ordered phase. As a matter of fact, significant distortions onset above
the transition into the magnetic state. Confirmation and quantitative measures of
the structural symmetry breaking will be discussed in Chapter 6.
Now we discuss the temperature range between the established PM and FM
phases, for 8 K < T < 12 K. During this transition region, we observe significant
line broadening and development of the quadrupole peaks at the same time. Line
broadening, indicated by increase in second moments or full width at half maximum
(FWHM), suggests the growth of magnetic inhomogeneity toward FM order at the
ordered state. Quadrupole splitting is evidence of structural distortion and occurs
before the appearance of magnetic splitting (two sets of peaks). It is possible that
some smaller effect of structural distortion already exists at PM state but is hidden
within the linewidth of the peak. One surprising feature in the transition region is
the inequivalent lineshape of I and II sets. Set II starts from much higher frequency
compared to its position at ordered state, and initially only appears as a single broad
peak. In contrast, quadrupole peaks from set I show up at much higher temperature.
It reflects some exotic magnetic and/or orbital ordering which might correspond to
the quadrupolar/nematic phase suggested in quantum models [18,31].
5.2 Field dependence of spectrum
With the different magnetic phases established in several temperature ranges at H0
= 9 T, next we discuss the field dependence of our spectra. The same temperature
59
dependence experiments were repeated at H0 = 7 and 15T, and the corresponding
spectra are shown in Figure 5.2. Temperature dependence of spectra at both fields
show similar features as 15 T. All three magnetic phases can be clearly identified.
5.2.1 Uniform and staggered local fields.
To further compare the NMR results across different fields in order to deduce mi-
croscopic nature of the LRO magnetism, we plot the temperature dependence of the
shifts of triplet sets I and II [Figure 5.3a] as well as the first moments of the entire
spectrum [Figure 5.3b] of all the applied fields together.
This data can then be used to further infer two quantities: the local uniform
(Hu) and staggered (Hstag) fields, defined as the following:
Hu =1
2[〈HI〉+ 〈HII〉] ,
Hstag =1
2[〈HI〉 − 〈HII〉] ,
(5.1)
where the bracket denotes the first moment, or weighted average, of the spectrum,
i.e. 〈HI〉 and 〈HII〉 represent the first moment of triplet I and II, respectively, the
uniform field Hu is the first moment of the full spectrum, and the staggered field
Hstag is the difference between the first moments of set I and II. From the definition
of Knight shift, 〈HI〉 and 〈HII〉 denote the local field sensed by two magnetically
inequivalent 23 Na sites projected along the direction of applied field, therefore Hu
is the common local field at both sites, and Hstag denotes the difference of local field
at two sites. As discussed previously, NMR spectrum is the probability distribution
of local field, and the shift of a NMR signal ∆ω in unit of frequency can by related
60
Ma
gn
itu
de
(a
rb. u
nits)
ω − ω0 (MHz)
(a)
Ma
gn
itu
de
(a
rb. u
nits)
ω − ω0 (MHz)
(b)
Figure 5.2: (a) Temperature dependence of 23Na spectra at 7 T, with field parallel to [001]. (b)Temperature dependence of 23Na spectra at 15 T, with field parallel to [001].
61
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
0.0
0.2
ω −
ω0 (
MH
z)
201816141210864
Temperature (K)
7 T
9 T
15 T
(a)
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
ω −
ω0 (
MH
z)
201816141210864
Temperature (K)
7 T
9 T
15 T
(b)
Figure 5.3: Temperature dependence of the frequency shift of (a) PM state and triplets I andII (solid symbols) and (b) the first moment of the entire spectrum, at various applied fields. Thefrequency scale is defined by subtracting ω0, the zero NMR shift frequency. Solid and dashed linesare guide to the eyes.
62
to the magnetic field unit Hlocal by the simple conversion
Hlocal =(ω − ω0)
γ=
∆ω
γ, (5.2)
therefore the uniform and staggered fields can also be expressed in terms of uniform
and staggered magnetization with the unit of magnetic field. To better describe the
physical meanings, throughout this thesis we will express the uniform and staggered
components of shifts in the unit of magnetic field, and use the units of frequency and
field interchangeably for single peak shift and quadrupole splittings.
With the above definitions, we can plot the temperature dependence of the ab-
solute values of the uniform and staggered fields at all three applied fields, leaving
aside the quadrupole splitting for now, as displayed in Fig. 5.4. It is evident that
both uniform and staggered fields remain constant as a function of temperature in
the ordered state below 8 K. Interestingly, both Hu and Hstag are of the same order
of magnitude. This clearly indicates that there exists large extend of non-uniform
structural and/or magnetic order, and suggests that local probes are required to
determine the nature of the LRO phase.
5.2.2 Transition temperature
The transition temperature (TC) from PM to FM state measured from specific heat
experiment [11] is 6.3 K, significantly lower than our NMR measurement. For exam-
ple at 9 T, the temperature dependence of spectra suggests the onset of LRO occurs
in the vicinity of 10 K. However, our data measured at different applied magnetic
fields indicate that the transition temperature is dependent on the applied field and
increases with the increasing field.
63
0.11
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
|H
u | (
T)
2018161412108642
Temperature (K)
H || [001]
7 T
9 T
15 T
(a)
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
Hstag (
T)
24222018161412108642
Temperature (K)
H || [001]
7 T
9 T
15 T
(b)
Figure 5.4: Temperature evolution of (a) uniform field and (b) staggered field at 7, 9 and 15 T.Dashed lines are guide to the eyes.
64
0.11
0.10
0.09
0.08
0.07
0.06
0.05
0.04
0.03
0.02
0.01
0.00
|H
u | (
T)
2018161412108642
Temperature (K)
12
10
8
6
Tc (
K)
151050
H0 (T)
H || [001]
7 T
9 T
15 T
Figure 5.5: Linear fits to uniform field in order to extract transition temperatures. Solid lines arelinear fits to the uniform local field in PM and transition regions. Dashed lines are guide to theeyes. Inset plots the field dependence of Tc and the red solid line is a linear fit to Tc.
A consistent way of determining the transition temperature for each applied
field is to find the crossing point of the plotted solid lines in Fig. 5.5. Different
magnetic phases can be characterized by the temperature dependence of (inverse)
susceptibility, whose counterpart in NMR is the uniform local field we just defined.
Therefore for each applied field, by applying linear fits to the uniform field of PM
and ordered low temperature phases separately, the temperature coordinate of the
crossing point of the two fits can be defined as the transition temperature Tc. All
the acquired values are listed in Table 5.1.
H ‖[001] TC0 T 6.3 K [11]7 T 10.2 K9 T 11.1 K15 T 13.3 K
Table 5.1: Transition temperature from PM to low temperature magnetic state at various appliedfields.
Evidently, transition temperature increases with increasing applied field. Inset to
65
Fig. 5.5 plots Tc values as a function of applied field, including transition temperature
from heat capacity measurement at zero field [11]. A linear fit to this curve shows
that Tc increases with applied field with a rate of 0.44 K/T. The increase of Tc
is significant and approximately scales as the magnetic energy associated with the
applied field. This indeed confirms the magnetic nature of the transition.
With the transition temperatures defined, the temperature evolution of local
fields can be plotted in the reduced temperature coordinate, T/Tc, displayed in
Figure 5.6 (b) and (d), after subtracting the orbital shifts which are constant and
negligible, as discussed in section 4.2.4. Well below Tc, Hu increases with increasing
H, while Hstag remains constant. From magnetization measurement, the FM mo-
ments do not saturate up to applied field of 5 T [11]. Our data demonstrate that the
magnetization is not fully saturated even with field up to 15 T. In addition, we plot
in Figure 5.6 (a) and (c) the T variation of the NMR shift, quantity proportional to
the Pauli spin susceptibility (χs):
K =H‖hf
H0
∝ 〈S〉H0
∝ χs (5.3)
Because the hyperfine coupling constant for Na is negative, we plot the absolute
value of the shift to reflect the spin susceptibility. Below Tc, both uniform (χsu) and
staggered (χsstag) susceptibilities decrease with the increasing applied field.
5.3 Spin-Lattice Relaxation Rate
The spin-lattice relaxation time (T1) is measured at the frequency of central peak
for both sets of triplets, at various temperature values ranging from 40 K (PM state)
66
100
90
80
70
60
50
40
30
20
10
| Hu
| (m
T)
1.61.41.21.00.80.60.40.2
b a
T/Tc
1.00.80.60.40.2
T/Tc
1.0
0.8
0.6
0.4
0.2
0.0
Hsta
g/
H(%
)
c
7 T
9 T
15 T
1.00.80.60.40.2
50
40
30
20
10
0
(m
T)
Hsta
g
d
7 T
9 T
15 T
T/Tc
1.2
1.0
0.8
0.6
0.4
0.2
|Hu | / H
(%
)
1.61.41.21.00.80.60.40.2
T/Tc
H || [001]
7 T
9 T
15 T
H || [001]
7 T
9 T
15 T
Figure 5.6: The absolute value of the relative uniform internal field, (a) Hu/H (reflecting uniformspin susceptibility (χs
u)) and (c) relative staggered internal field, Hstag/H (reflecting staggered spinsusceptibility (χs
stag)), as function of reduced temperature for variousH‖[001]. The shift is correctedfor a small field dependent orbital contribution as determined in the paramagnetic state( Section7.1). Absolute value of the uniform internal field, Hu, (b) and the staggered internal field, Hstag,(d) as a function of reduced temperature for various H‖[001]. Lines are guide to the eyes.
to 4 K (ordered state). The relaxation rate divided by temperature, 1/T1T , as a
function of temperature is plotted in Fig. 5.7 in semi-log style, with values crossing
more than two decades.
At high temperature PM state, 1/T1T increases with decreasing temperature.
Above 13 K the temperature dependence of T1 can be well described either by an
exponential formula
1
T1
= 4.43 · e∆kT , (5.4)
where ∆/k = 7.88±0.62 K, or a power function
1
T1
= 35.4 · T n , (5.5)
67
0.01
2
3
4
5
6
7
8
9
0.1
2
3
4
5
6
7
8
9
1
2
1/T
1T
(s
-1K
-1)
4 5 6 7 8 9
10
2 3 4 5
T (K)
15
10
5
0
T1 (
s)
40302010
T (K)
Triplet I Triplet || PM State
Triplet I Triplet II PM State
Pow = 6.02
Pow = 6.51
Figure 5.7: Temperature evolution of 1/T1T at H0 = 9T and H0 ‖ [001]. Red filled circlesrepresent 1/T1T values at PM state. Green and blue filled circles are 1/T1T values measured at thecentral transition frequency of triplet I and II, respectively. Solid lines are fit to power functions,with different color representing different exponents used as described in text. The dashed linesare guide to the eyes. Inset Spin-lattice relaxation time as a function of T.
with n = -0.6±0.14. The fit is shown as the yellow line on Fig. 5.7.
Below transition temperature, T1 measurements are done separately at the cen-
tral transition frequencies of triplet I and II. Both T1s increase significantly with
decreasing temperatures, and the T1 values of triplet II are much longer than that of
the triplet I at the same temperature in the ferromagnetic region. In the magnetic
ordered phase of a magnetic insulator with T → 0, the main relaxation channel is
the hyperfine interaction between nuclei and the unpaired electrons [46, 47] (5d Os
electrons in this case), i.e. AIi · Sj. Meanwhile at intermediate temperatures when
magnetic orders, i.e. magnons, are not fully developed, and the average phonon num-
bers increase with increasing temperature, the dominating relaxation mechanism is
the spin-phonon interaction. Nuclear spin would relax back to equilibrium and trans-
68
fer the energy to electron spins (the “lattice”) in the form of the excitation of electron
spin waves (magnon), or excitation of lattice vibration (phonon). The fact that we
observe two different T1 values for two triplets implies different strength of hyperfine
field or spin-phonon coupling, and thus the existence of two inequivalent structural
or magnetic environments for 23Na ions.
The first-order spin-magnon relaxation processes including hyperfine interaction
of nuclear spin to one, two or three-magnons [44], and the spin-phonon relaxation pro-
cess including interaction of nuclear spin to one, two or three-phonons are illustrated
in Fig. 5.8. In the one phonon/magnon process (direct process), the energy of spin re-
laxation is transferred to a single phonon/magnon energy. For two phonon/magnon
process (Raman process), the relaxation energy is the difference between an absorbed
phonon/magnon and an emitted phonon/magnon. Three phonon/magnon process
involves one absorbed phonon/magnon and two emitted phonons/magnons, with the
total energy difference between absorption and emission being the relaxation energy.
Furthermore, for spin-phonon relaxation processes, the energy can be absorbed by ei-
ther the acoustic or the optical branches of phonon, and the temperature dependence
of relaxation time is also different for Kramers and non-Kramers systems.
In either situations, the temperature dependence of relaxation rate 1/T1 follows
power law
1
T1
∝ Tα . (5.6)
The temperature dependence of relaxation rate corresponding to different pro-
cesses are listed in Table. 5.2. The 1/T1T curves below Tc (T ≤ 8 k) are fit
with Eq. 5.6, and the best fits result in (1/T1)I ∝ T 6.02±0.06 for triplet I and
(1/T1)II ∝ T 6.51±0.013 for triplet II. The α values are close to 6, and indicate the
69
K K1 K2 K1K2
K3(a) (c)(b)
Figure 5.8: The process of nuclear spin-lattice relaxation involving (a) one-, (b) two- and (c)three-magnon/phonon. Red arrows represents the nuclear spin flip.
Process 1/T1DependenceOne Magnon Forbidden [48]Two Magnon T 2ln(T ) [47]
Three Magnon or more T 7/2 (FM) or T 5 (AFM) [44]One Phonon T [49]
Two Phonon (acoustic branch) T 7 [50]Three Phonon (acoustic branch) T 7 [51]
Two Phonon (optical branch) + Kramers T 6 [52]Two Phonon (optical branch) + non-Kramers T 4 [52]
Table 5.2: Temperature dependence for different spin-relaxation processes for magnetic insulatorat low temperature.
two optical phonon relaxation process in Kramers system. It implies that the or-
dered state of Ba2NaOsO6 is a two-fold Kramers doublet, consistent with specific
heat measurement. In addition, as mentioned in section 3.5, Quantum models [18]
predicted that the Kramers doublet ordered state is a result of degeneracy lifting by
orbital ordering formed in the quadrupolar phase. In order to excludes the possibil-
ity of insensitivity of fitting to the exponent values, we also show in Fig. 5.7 the fits
with scaling exponent α = 7 (light blue lines) and 5 (orange lines), which obviously
deviates from our data.
70
5.4 Summary
In this chapter, we first demonstrated the temperature evolution of 23Na spectra,
which clearly identified three distinct phases. The two groups of six peaks in total
at lowest temperature state indicate two Na sites in inequivalent environments and
possible local symmetry breaking. NMR results from various field strengths confirms
the magnetic nature of the phase transition, and provide a way of defining transition
temperature. Furthermore, the temperature evolution of the spin-relaxation time
points to a ordered state with degenerate Kramers doublet. The experimental obser-
vations provide valuable information towards interpreting the microscopic nature of
the material of interests. The rest of the thesis will then focus on building a quanti-
tative understanding of the structural symmetry transition and the low temperature
magnetic/orbital ordering associated with the abnormal magnetic anisotropy.
71
Chapter Six
Observation of Local Symmetry
Breaking
From crystal field theory, octahedral complexes of transition metals often encounter
Jahn-Teller effect which tends to lift the ground state degeneracy (four-fold in this
case) via the geometrical distortion. This distortion, which will reduce the spatial
crystalline symmetry, is able to lower the ground state energy by forming certain
orbital orders, as discussed in section 3.1. Although quantum models [18, 31] argue
that in Ba2NaOsO6 system, the multipolar spin interactions due to strong SOC ef-
fect can lead to a spontaneous splitting of the degenerate quadruplet and avoid the
Jahn-Teller effect, as we discussed in section 3.2, it would still reduce the structural
symmetry from cubic at high temperature to tetragonal in the quadrupolar state.
The doublet ordered state observed from specific heat measurement [11] indeed in-
dicates that the degeneracy is lifted and the ordered state is two-fold degenerate.
Lack of evidence of structural phase transition from X-ray diffraction might be due
to the weak scattering power of oxygen ions and suggest a subtle nature of distor-
tion. Therefor the observation of local structural distortion of Ba2NaOsO6 from
NMR is of vital importance in correctly interpreting the mechanism driving the ex-
otic magnetism observed. In this chapter we will discuss the structural distortion
inferred from NMR quadrupole splitting and discuss all the possible scenarios that
can quantitatively account for the experimental observations.
6.1 Angle Dependence of Quadrupole Splitting
In the low temperature regime, we observed the splitting of the spectra into sets I and
II, that reflects the appearance of two distinct magnetic sites in the low T phase, and
in addition splitting of each set of the spectral lines into three peaks. This splitting,
labeled as δq in Fig. 5.1, originates from quadrupole interaction, implying lowered
local lattice symmetry with non-zero EFG, as discussed in section 4.2.3. Therefore,
73
observed splitting of the Na spectra into triplets indicates breaking of the point
cubic symmetry, caused by local distortions of electronic charge distribution, in the
magnetically ordered phase. As a matter of fact, significant distortions onset above
the transition into the magnetic state. To confirm this finding, we measured low T
spectra as a function of strength and orientation of the applied magnetic field.
We compare δq, the average separation between two adjacent quadrupolar satel-
lite lines, in fields ranging from 7 T to 15 T at 4 K, deep into the LRO phase (see
Fig. 6.1). δq values are shown in Table 6.1. Comparison reveals that δq varies by
less than 2 %, which is of the order of the error bars.
H0 (T) δq (MHz)7 0.19039 0.193515 0.1937
Table 6.1: Quadrupole splitting values in different fields.
Furthermore, we measured the spectra at 15 T and 8 K as a function of the of
the angle (θ) between H and [001] crystalline axis as plotted in Fig. 5.1A. In this
case, H0 was rotated in the (110) plane of the crystal which contains three high
symmetry directions: [001], [111], and [110]. The angle dependence of the splitting
δq is displayed in Fig. 6.1B. Notice that in our experiment, measurements show
that 90 ([110] direction) is a local maximum in δq value and the curve is strictly
symmetrical about this angle.
From Eq.4.22, the quadrupole Hamiltonian is represented as
Hq =e2qQ
4I(2I − 1)[3I2
Z − I(I + 1) +1
2η(I2
+ + I2−)] . (6.1)
In our experiment, crystal is rotated to various angles relative to the applied field,
74
H = 15 T
Ma
gn
itu
de
(a
. u
.)
-1.6 -1.2 -0.8 -0.4 0.0
ω − ω0 (MHz)
-2.0
90o
80o
70o
60o
50o
40o
30o
20o
10o
0o
200
160
120
80
40
0
(K
Hz)
10060200
Angle from [001] (deg.)
[001]
[110]
[111]
40 80
B A
T = 8 K
δq
Figure 6.1: Local cubic symmetry breaking in the ordered phase. (A) 23Na spectra in lowtemperature ordered state as a function of the angle between the applied magnetic field and [001]crystalline axis. (B) The mean peak-to-peak splitting (δq) between any two adjacent peaks of thetriplets I and II. Solid line is the fit to |(3 cos2 θ − 1)/2|
and it is convenient to use the frame of reference Oxyz, where Oz is parallel to field
H. Without loss of generality we can transform the spin components into the Oxyz
IZ = Iz cos θ + Ix sin θ . (6.2)
First we assume the local symmetry is tetragonal so that VXX = VY Y and η = 0.
The quadrupole Hamiltonian under the applied field coordinate then has the new
form
HQ =e2qQ
4I(2I − 1)1
2(3 cos2 θ − 1)(3I2
z − I(I + 1))
+3
2sin θ cos θ[Iz(I+ + I−) + (I+ + I−)Iz]
+3
4sin2 θ(I2
+ + I2−) ,
(6.3)
where now I± = Ix ± Iy are also in the Oxyz coordinate. By treating HQ as a
perturbation and using the second-order perturbation theory, we can get the energy
75
shift on top of the Zeeman energy level
Em =1
4hνQ(3 cos2 θ − 1)(m2 − 1
3I(I + 1)) , (6.4)
and then infer the distance in frequency between adjacent quadrupole satellites:
δq =Em − Em−1
h=
1
2νQ(3 cos2 θ − 1) , (6.5)
where for brevity we define
νQ =3e2qQ
2hI(2I − 1), (6.6)
If δq originates from quadrupole interactions the dependence should follow the
|(3 cos2 θ − 1)/2| function. This is indeed observed in Fig. 6.1B. The resulting
quadrupole splitting νq is about 190 KHz. Both the observed insensitivity of δq to
the strength of the magnetic field and its dependence on θ indicate that δq splitting
originates from structural distortions.
The fit to the angle dependence of δq indeed confirms the quadrupole nature of
the observed NMR line splitting. However it assumes that the asymmetry factor η
vanishes and limits the structural symmetry to tetragonal. A more general treatment
to the perturbation problem including non-zero η will result in a more complex form
of energy change, after a tedious derivation [32]:
Em =1
4hνQ
[(3 cos2 θ − 1) + η sin2 θ cos 2φ
](m2 − 1
3I(I + 1)
), (6.7)
and the frequency shift would be
δq =Em − Em−1
h=
1
2νQ(3 cos2 θ − 1 + η sin2 θ cos 2φ
). (6.8)
76
Since applied field is rotated in (110) plane, φ = 45o and cos 2φ = 1. Fitting
this new equation directly to Fig. 6.1B will again produce η = 0. However, we
have assumed that the largest component of EFG, VZZ , is parallel to the c-axis of
crystalline principle of axis. θ is in definition the angle between H0 and VZZ . If
instead VZZ is in the direction of a or b-axis, then Eq. 6.8 will need to be revised
accordingly. Fig. 6.2(a) demonstrates the tetragonal case where VZZ is along c-axis
and η = 0. If VZZ is along a(b)-axis as shown in Fig. 6.2(b), we denote the new
set of angles in the coordinate system of quadrupole principle axis as θ′ and φ′, and
use θ and φ represent the orientation of H0 vector in the principal crystallographic
axes. The relation between these two sets of spherical coordinates follows
cos θ′ = sin θ cosφ
sin θ′ cosφ′ = cos θ
sin θ′ sinφ′ = sin θ sinφ ,
(6.9)
where φ = 45o. Then Eq. 6.8 can be transformed to
δq = νQ
(3
2cos2 θ − 1 + η(
3
2cos2 θ − 1
2)
). (6.10)
Fitting with the above equations gives similar νq about 190 KHz and η = 0.87.
The fitting curve is precisely the same as the tetragonal case. Therefore a local
structure with symmetry less than tetragonal can also explain the angle dependence
data. The distinction between the two scenarios can be revealed by rotating the
applied field in (010) plane, as we will show in section 6.5.
Regardless of the angle θ no more than 3 lines are observed per set (I or II). This
indicates that the principal axes of the EFG coincide with those of the crystal. In a
material with cubic symmetry, it is thus possible to stabilize three different domains,
77
Figure 6.2: Illustration of the rotation of the applied field making angle θ with respect to theprincipal axis of the EFG, for VZZ along (a) c-axis and (b) a-axis.
with the principle axis of the EFG (Vzz) pointing along any of the 3 equivalent
crystal axes. Further, local magnetic field has to be parallel to Vzz in each domain.
The facts that the splitting is the largest for H‖[001] (Fig. 6.1B), and that only 3
peaks per set are observed for H‖[110] imply that two domains are plausible. One
domain is characterized by pure uniaxial 3z2 − r2 distortions where Vzz is in [001]
direction, while the other is distinguished by x2− y2 distortions where Vzz is then in
the (110) plane. Thus, these distortions can be a signature of staggered quadrupolar
order with distinct orbital polarization on two sub-lattices [18]. Our finding, that
structural distortion is present in the LRO phase, is in contrast to the predictions
made by the first-principles DFT calculation [26, 27]. This is important in so far
that it clearly shows that quantum models based on complex multipolar interaction
generating high-order spin exchange provide correct description of the nature of
emergent phases in Mott insulators with the strong SOC [18,31].
78
6.2 Distortion in Tetragonal Structure
So far, we find two classes of point symmetries that satisfy the angle dependence
constraint imposed by our data. First class is comprised of tetragonal distortions
of oxygen octahedra with η = 0, suggesting the elongation or compression of the
four Oxygen ions in the XY plane is symmetrical about [001] direction in order to
guarantee VXX = VY Y . In this section we calculate the amount of distortion that can
account for the experimentally acquired EFG values based on this axially symmetric
model.
With η = 0 and H ‖ [001], we have θ = 0 and the quadrupole Hamiltonian in
Eq. 6.3 can be simplified to
HQ =(eQ)(eq)
4I(2I − 1)[3I2
z − I(I + 1)] . (6.11)
For nuclear spin I = 3/2, as is the case of 23Na, the energy eigenstates of HQ are
given by,
Em =(eQ)(eq)
4I(2I − 1)[3m2 − I(I + 1)] , (6.12)
and the frequency shift from the central line is
νmq = ωm→m−1 =(eQ)(eq)
h 4I(2I − 1)[3(2m− 1)] =
1
2
(eQ)(eq)
h, for |+ 3/2〉 → |+ 1/2〉
0, for |+ 1/2〉 → | − 1/2〉
− 1
2
(eQ)(eq)
h, for | − 1/2〉 → | − 3/2〉 .
(6.13)
79
Therefore the quadrupole splitting δq between different quadrupole satellites is
δq =1
2h(eQ)(eq) =
1
2h(Quadrupole moment)× (EFG) . (6.14)
Evidently, in this case equal splitting is observed between quadrupole satellites lines,
as observed in our experiment. Further, we can estimate the value of the EFG using
experimentally determined value of the splitting, that is δq ≈ 190 KHz for H‖[001],
(EFG) =2hδqeQ
=2× 4.136× 10−15 eV · s× 190× 103 s−1
0.12× e× 10−28 m2= 1.31× 1020 V/m2 .
(6.15)
Next, this value can be used to estimate particular lattice distortions in our material.
In oxygen octahedra surrounding Na nuclei the EFG takes on the following form [54],
EFG =2q
4πε0
2a3 − 1
b3− 1
c30 0
0 − 1a3 + 2
b3− 1
c30
0 0 − 1a3 − 1
b3+ 2
c3
, (6.16)
where a, b and c are the O-Na bond distance along x, y and z axis. In materials with
cubic symmetry such as Ba2NaOsO6 in the paramagnetic state, a = b = c, therefore
EFG equals zero, leading to vanishing splitting, δq.
The observed δq is the largest for field applied in the [001] direction, as shown
in Fig. 6.1. In this case the simplest model, accounting for the splitting of the Na
line into three equally spaced quadrupole satellite lines, involves distortions of the
O octahedra surrounding Na nuclei solely along the [001] direction. In this case,
80
q = 2e, a = b 6= c, and we obtain
EFG =2q
4πε0
1a3 − 1
c30 0
0 1a3 − 1
c30
0 0 −2( 1a3 − 1
c3)
. (6.17)
Therefore, the principal axis of the EFG (Vzz ≡ eq) is given by,
Vzz = ± 8e
4πε0
(1
a3− 1
c3
)(6.18)
(1
a3− 1
c3
)= ±4πε0
8e× 1.31× 1020 V/m2 = ± 0.01137× 1030 m−3 . (6.19)
In Ba2NaOsO6, a = 2.274 A, distortions along the c crystalline axis of the order
of 4 % can account for the observed δq, that is
1
c3=
1
a3± 0.01137
c = 2.181A (−4.1%), for compression
c = 2.385A (4.9%), for elongation
. (6.20)
Notice that the percentage values here are relative to the O-Na bond distance. They
would be -1.1% and 1.3% in relative to the lattice constant 8.2870 A.
81
6.3 Point Charge Calculation
6.3.1 Motivation
We concluded in last section that a 4.1% compression or 4.9% elongation of the two
O sites along c axis can quantitatively account for the observed quadrupole splitting
under the assumption of axially symmetric tetragonal distortion. The problem is
that these values are only relative to the a and b crystal axes, and in principle more
combinations can exist to arrive at the same δq. Therefore a more first-principle
method is sought for.
Furthermore, in the case of orthorhombic distortion with the same quadrupole
splitting and non-zero η, by choosing again H ‖ [001], the quadrupole Hamiltonian
in the coordinate system defined by the principal axes of the EFG is given by
HQ(x, y) =eQVzz
4I(2I − 1)
[(3I2
z − I2) + η(I2
+ − I2−
2)
]. (6.21)
In this case, solving for spin 3/2 nuclei produces two doubly degenerate energy
levels [33],
E±3/2 = hνQ
(1 +
η2
3
)1/2
E±1/2 = −hνQ(
1 +η2
3
),
(6.22)
and the splitting is given by,
δq =(eQ)(Vzz)
2h
(1 +
η2
3
)1/2
. (6.23)
82
Thus, the value of δq is specified by both Vzz and the asymmetry parameter. We
cannot determine both parameters to extract distortion information simply with a
δq value. One common way for data interpretation in this situation is to assume η
= 0 which would induce an error of about 16% in δq estimation. However the η here
is relatively large and the fact that it is not zero defines our model, therefore we
turn to the point charge model which allows us to calculate Vzz and η concurrently.
This model is applicable to all the structural symmetry due to its first-principle
nature and can also provide a more complete set of solutions to both tetragonal and
orthorhombic models.
6.3.2 Method Description
The point charge model calculates the electron density at nuclear sites by taking
into account all the surrounding charges, which are treated as point charges of zero
radius that carry the appropriate ionic charge. The EFG is then evaluated by the
following equation:
Vij =∑µ
∑k
qk3(krµi )(krµj )− δij(krµ)2
(krµ)5, (6.24)
where∑
µ and∑
k are sums over multiple unit cells and all atoms within a single
unit cell, respectively, q the charge carried by the ion being considered and r denotes
the position of an ion relative to the central point. Vij with i, j ⊂ x, y, z corresponds
to the (i, j) component of the 3×3 EFG tensor, and is derived through a Taylor
expansion from the electrostatic Hamiltonian [33].
Thus, this model allows us to numerically calculate all the EFG elements based
on Eq. 6.24 as long as the ion positions within the lattice are defined. Considering
83
the periodic nature of crystal structure, one could denote the position of certain unit
cell by translation of three primary vectors. More precisely, within the 3-dimensional
crystalline coordinate, if we define origin as the position of the target Na ion whose
EFG values are to be calculated, all the other cells/lattice points in the lattice would
then be represented by:
r′ = µ1a1 + µ2a2 + µ3a3 , (6.25)
where ai are primary vectors of the lattice and µi are integer primary indices (µi =
0, 1, 2 ...).
After a unit cell is located using primary vectors, each atom in the unit cell can
be accessed with the basis representation:
ri = xia1 + yia2 + zia3 , (6.26)
where xi, yi and zi are fractions between [0,1) and represent the position of the ith
atom relative to basis origin. Fig. 6.3 illustrates an example of accessing a Na ion at
position (µ1, µ2, µ3) = (1, 1, 0) and (xi, yi, zi) = (0.5, 0, 0.5).
B
O
C
A
y
z
x
Origin
(xi, y
i, z
i)
(u1, u
2, u
3)
Figure 6.3: Schematic of the primary and basis vectors to locate a Na ion from origin. Os, O andBa ions are not shown.
84
Combining Eq. 6.25 and 6.26, the position of any atom within the lattice can be
described by:
ri = (µ1 + xi)a1 + (µ2 + yi)a2 + (µ3 + zi)a3 . (6.27)
As distortions in Ba2NaOsO6 only occur on Oxygen sites and are small, the
primary vectors can still be defined in a cubic fashion with a1 = (1, 0, 0), a2 =
(0, 1, 0), and a3 = (0, 0, 1). The basis is much more complex and is attached in
Appendix A.
With the primary vectors and basis, EFG could be numerically calculated by
Eq. 6.24. For example, the 3 components of the first row in the EFG tensor would
be
Vxx =∑µ
∑k
qk3(krµx)(krµx)− (krµ)2
(krµ)5
Vxy =∑µ
∑k
qk3(krµx)(krµy )
(krµ)5
Vxz =∑µ
∑k
qk3(krµx)(krµz )
(krµ)5
. (6.28)
6.3.3 Programming Procedure
In this section the algorithm of point charge calculation is briefly described. It is
evident from Eq. 6.28 that EFT is proportional to 1/r3 and therefore the contribution
from ions decays quickly as distance increases. Indeed, the numerical results show
that ions in unit cells beyond two lattice constants have little contributions to the
EFG values. Thus all the calculation in our method contains 4×4×4 = 64 unit cells,
85
symmetrical about the Na atom at origin. The numerical procedure is comprised of
the following steps:
1. Set up a 3-dimensional array containing 64 unit cells. Each unit cell includes
a total of 89 ions, consisting of 13 Os, 14 Na, 8 Ba and 64 O ions. Each ion
within our model can be accessed by Eq. 6.27.
2. Set up and initialize the 3×3 EFG tensor. Assign charge values to each ion.
3. Iterate over all the combinations of primary indices µ1, µ2 and µ3. For each set
of µ1, µ2 and µ3, go through each ion using basis indices and sum over EFG
tensor components by Eq.6.24.
4. Calculate asymmetry factor η and quadrupole splitting δq based on the acquired
EFG.
The local distortion of Oxygen ions are reflected numerically by altering the basis
indices. For example, in order to test the effect of 2%’s elongation along c-axis of
the Oxygen ion above the Na site at origin, its original position (0, 0, 1/4) should
be modified to (0, 0, (1+2%)/4).
6.4 Structural Equivalence of Na Ions
With cubic symmetry, calculation shows that all nine components of the EFG are zero
as expected, and therefore δq = 0. When local distortion arises, some components
become non-zero and quadrupole splitting rises. In this section we consider two
possible scenarios involving distortions of the O2− octahedra surrounding Na+ ions
as depicted in Fig. 6.4. In these models, we allow the oxygen to move by different
86
A B
Na
Os
O
y
z
x
a
b
Figure 6.4: Schematic of the proposed lattice distortions. (A) Two structurally distinct Nasites are generated by elongation, or compression, of one O2− octahedron along [001] direction andits concurrent compression, or elongation, in XY plane. (B) One structurally distinct Na site innon-cubic environment is produced by elongation/compression of O2− octahedra.
amounts along each of the three axes and so constrain it to the cubic axes of the
perovskite reference unit cell, as suggested in [18]. These two models differ by the
number of structurally distinct Na sites induced by the distortions. Moreover, we
point out that we cannot distinguish between displacement of the actual O atoms
and distortions of the O charge density.
6.4.1 One Structurally Distinct Na
In this first model we assume each NaO6 octahedra went through the same form
of distortion, and the three pairs of O ions along x, y and z axis are independently
distorted. That is, each pair of O ions can either elongate or compress symmetrically
about the central Na site by an arbitrary amount along the Na-O bond direction.
Therefore we are not making any prior assumptions about the structural symmetry,
and the only constraints are from the experimentally observed δq = 190 KHz and
η = 0 or 0.87. The schematic of this model is shown in Fig. 6.4(B). In the actual
simulation, we describe distortion in percentage relative to the Na-O distance (2.274
87
A), and define elongation displacement to be positive and compression displacement
to be negative.
The simulation is run to produce combinations of distortions in all three axes
which can produce the quadrupole splitting constrains. Parameter space consists of
three numbers, δa, δb and δc, corresponding to distortion along crystalline a, b and c
axis, respectively. Results show that many combinations can give rise to the desired
values of δq and η, and some of them are listed in the following table:
δa δb δc η vaa vbb vcc δq (KHz) vzz Direction
0.4% 2.2% 1% 0.8768 -2.395 38.89 -36.49 189.81 b1.4% 1.4% 1.8% 0 25.9 25.9 -51.8 190.09 c2.2% 0.4% 1% 0.8768 38.89 -2.395 -36.49 189.81 a3% 3% 4% 0 26.01 26.01 -52.02 190.91 c
-0.2% -0.2% -0.32% 0 -25.82 -25.82 51.63 189.5 c
-0.4% -0.4% -3% 0 -26.01 -26.01 52.02 190.91 b-1% -2.6% -0.4% 0.8769 2.394 -38.89 36.49 189.8 c
-1.8% -1.8% -1.4% 0 -25.9 -25.9 51.8 190.09 c-2.6% -1% -0.4% 0.8769 -38.89 2.394 36.49 189.8 a0% 0% -0.34% 0 -25.63 -25.63 51.25 188.1 c0% 0% 3% 0 25.52 25.52 -51.03 187.28 c
Table 6.2: Sample results of point charge calculation with two structurally distinct Na sites.Program loops through δa, δb and δc values within the range of (-5%, 5%) and returns combinationsof parameters that can produce δq in the vicinity of 190 KHz.
The vaa, vbb and vcc are EFG components along the a, b and c axis of the lattice
coordinate, and the vzz is the one with the largest absolute value by definition.
It is evident from Table. 6.2 that both tetragonal (η = 0) and orthorhombic (η =
0.87) distortions are possible in order to reproduce the δq value. EFG component vzz
can point to any of the three cubic axes in the unit cell, depending on the relative
strengths of δa, δb and δc.
88
6.4.2 Two Structurally Distinct Na
In this second model, two structurally different Na sites are generated by elongation,
or compression, of one O2− octahedron along [001] direction and its concurrent com-
pression, or elongation, in XY plane, as illustrated in Fig. 6.4(A). This model also
naturally accounts for the appearance of two magnetically different Na sites, that is
the appearance of distinct frequency shifts for triplet I and II, even if the LRO state
is a simple ferromagnet where all the electron spins on Os7+ ions are assumed to
point in the same direction. Transfer hyperfine field from Os electronic spins to Na
nuclei is mediated by O2− ions via its p-d hybridization with well localized 5d orbital
of Os7+. Evidently, shorter the distance between O2− and Os7+ ions, stronger the
hybridization and thus transfer hyperfine field at the Na site. Thus, the internal field
at the Na site in the lower plane (in Fig. 6.4A) consists of a sum of two stronger
and four weaker fields, while the field at Na in the upper plane of a sum of four
stronger and two weaker fields. Consequently, NMR signal from the lower plane Na
will appear at smaller absolute frequency shift (as is the case for triplet I), while
that from the upper plane at the larger absolute frequency shift (triplet II).
Point charge approximation results with this modes shows that, in order to pro-
duce equal δq for both Na sites as observed from NMR, the relative magnitude of the
elongation has to be equal to that of the compression, i.e. distortions must satisfy the
relation |δa| = |δb| = |δc|. For example, we will consider the values in Table. 6.3. The
(0.185% , 0.185% , -0.185%) distortion corresponds to octahedra a in Fig. 6.4(A)
can produce the same δq as the (-0.185% , -0.185% , 0.185%) distortion, depicted
by octahedra b. It is very unlikely that such distortions will occur, as electrostatic
energies associated with elongation and compression of the octahedra by the same
relative amount are very different.
89
δa δb δc η vaa vbb vcc δq (KHz) vzz Direction-0.185% -0.185% 0.185% 0 27.9 27.9 -55.9 190.03 c0.185% 0.185% -0.185% 0 -27.9 -27.9 55.9 190.03 c
Table 6.3: Sample results of point charge calculation with one structurally distinct Na sites.
6.5 Orthorhombic Symmetry Revealed by In-Plane
Rotation
Results of the point charge approximation concludes the existence of a single struc-
tural environment for all the 23Na nuclei. However it is not able to distinct be-
tween tetragonal and orthorhombic point symmetry, since both scenarios can pro-
duce the experimentally observed δq in Table 6.2 and the angular dependence plotted
in Fig. 6.1. By definition, in orthorhombic structure, C4 rotation symmetry is bro-
ken and the Na-O bond distances along three principle axes are unique (a 6= b 6= c).
Consequently the asymmetry factor η 6= 0. In order to discern the point symmetry,
another angular dependence experiment is done by rotating H0 from the [001] to
[100] (or [010]) direction, within the (010) or (101) plane. The spectrum and δq as a
function of angle is plotted in Fig. 6.5 together with the previous diagonal rotation
case.
The equation for the angular dependence of δq with in-plane rotation depends
on the selection of Vzz orientation. Different from the diagonal scenario, here we
have possible inequivalent situations, as shown in Fig. 6.6. Similar to the diagonal
rotation, we need to convert Eq. 6.8 from the quadrupole principle axis (θ’, φ’) into
the laboratory frame of reference (θ, φ).
[Vzz ‖ c] : In this case, θ = θ′ and φ = 0, therefore cos 2φ = 1 and Eq. 6.8
90
20
19
18
17
16
15
14
13
12
11
10
9
8
7
6
5
4
3
2
1
0
Ma
gn
itu
de
(a
. u
.)
0-0.5-1.0-1.5-2.0
ω − ω0 ( MHz )
c-axis [001]
a(b)-axis
[100]
[101]
[101]
c-axis [001]-
-
0.260.240.220.200.180.160.140.120.100.080.060.040.020.00-0.02-0.04-0.06
δq (
MH
z)
20016012080400-40
Angle From [001] (deg.)
[001]
[110]
[100]
[111]
In-plane rotation Diagonal rotation
[101]
Figure 6.5: (a) Angular evolution of spectrum with in-plane rotation. (b) Angular dependence ofδq for both in-plane and diagonal rotation. Dashes lines are fits to the modified version of Eq. 6.8depending on geometry, as discussed in text.
becomes
δq =νQ2
(3 cos2 θ − 1 + η sin2 θ) . (6.29)
[Vzz ‖ a] : Here θ = π2− θ′ and φ′ = 90. Eq. 6.8 is therefore modified to be
δq =νQ2
(3 sin2 θ − 1− η cos2 θ) . (6.30)
[Vzz ‖ b] : In this situation, θ′ = 90 and φ′ = θ. Eq. 6.8 is then
δq =νQ2
(η cos 2θ − 1) . (6.31)
Both Eq. 6.29 and 6.30 can fit the experimental result, shown as the green dashed
line in Fig. 6.5, and the resulting η = 0.87, which is consistent with the diagonal
rotation result and proof of the orthorhombic structure. As a matter of fact, it is
obvious that η can not be 0 just by examining the form of all the three equations.
91
a
b
c
H0
|| Vzz
θ φ’ = 0 deg.
( a )
a
b
c
H0 θ φ’ = 90 deg.
( b )
|| Vzz
θ’
a
b
c
H0 θ
φ’ = θ
( c )
θ’ = 90 deg.
|| Vzz
Figure 6.6: Three possible orientations of Vzz: (a) Vzz ‖ c, (b) Vzz ‖ a and (c) Vzz ‖ b
δq = 0 corresponds to θ ∼ 45 in experiment, which can only be possible with a
non-zero η term.
6.6 Discussion
We calculated distortions using the point charge approximation that can give rise to
the observed quadrupole splitting and asymmetry factor. Both models with one and
two structurally distinct Na sites can satisfy the constraints with various combina-
tions of small Oxygen local displacement, but we reasoned that the second model
with two different Na sites is unlikely due to its stringent symmetry requirements.
Understanding the distinction between different distortion models not just clarifies
the structural symmetry, but also fundamentally affects our interpretation of the
magnetic LRO. As aforementioned, two structurally distinct Na sites naturally ex-
plains the two triplets in spectrum and implies simple magnetism like ferromagnetism
with small moments. On the other hand, if there exists only one structurally equiva-
lent Na, then two magnetically distinct Na sites would arise from more exotic LRO,
like ferrimagnetism or canted (anti-)ferromagnetism.
Also it is worth pointing out that the purely uniaxial tetragonal lattice distortions
92
can also be present above the magnetic transition, but with much smaller distortions
which do not show up in the form of quadrupole splitting. In the PM state, the
width of the NMR spectra allows us to place an upper limit on such distortions. For
example, T = 20 K at applied field H0 = 9 T [Fig.5.1] is the lowest temperature that
shows no visible quadrupole splitting. FWHM at this temperature is about 20 KHz
and any potential quadrupole features will be covered up by the broad linewidth if
δq < 5 KHz . From the same point charge procedure, we infer that the limit equals
to 0.02% of the respective lattice constant, as any distortions that exceed this value
would cause visible broadening/splitting of the NMR spectra. This is in agreement
with the observed tetragonal distortion at temperatures above Tc by the latest x-ray
scattering measurements [55].
93
Chapter Seven
Numerical Simulation of Low
Temperature Magnetic Order
In this Chapter, we focus on understanding the temperature, field and angle depen-
dence of the local magnetic field. Incorporated with the preceding discussions of
local spatial symmetry and various forms of interactions, several models of magnetic
order will be tested.
For Na nuclei in Ba2NaOsO6 the main source of the local field is the interaction
with unpaired 5d1 Os electron spin via transferred hyperfine effect. In order to deter-
mine the microscopic nature, i.e. the microscopic electronic spin configuration from
NMR, it is crucial to have a good understanding of the electron-nucleus interaction.
From Eq. 4.23, the hyperfine interaction Hamiltonian is
He−n = −γ~I ·∑N
KN · (gµBSN) . (7.1)
In solids the major contributions to the interaction tensor are from the dipolar in-
teraction tensor D and transferred hyperfine tensor A. In Ba2NaOsO6, the hyperfine
interaction is transferred between Os electron and Na nuclei through the mediation
of the intermediate O ions. Thus, K contains two parts
K = D + A , (7.2)
which will be examined separately in the following sections. From Eq. 4.23 and 7.2
we can naturally define the local field as
Hloc = (D + A) · (gµBSN) , (7.3)
and therefore express the Knight shift as
K =Hloc · hH0
, (7.4)
95
where h denotes the unit vector pointing to the direction of applied field.
7.1 PM State: Orbital Shifts and Hyperfine Ten-
sor
At paramagnetic state, hyperfine shift at a given temperature is given by,
Khfα (T ) = Korbit
α + gαAααχ(T ) , (7.5)
where Korbitα denotes field dependent orbital contribution to the shift, α the principal
axis of the shift tensor, gi the electronic g-factor, Aαi the transfer hyperfine tensor,
and χ the electron spin susceptibility, i.e. 〈Si〉/H the net average electronic spin
projected along the field direction divided by H. Due to the cubic symmetry at PM
state, dipolar tensor is zero, as discussed in details in the next section.
Meanwhile, at PM region, the temperature dependence of both spin susceptibility
and Knight shift follows the Curie-Weiss law as
χα(T ) =Cχα
T − θ+ χ0
α
Khfα (T ) =
CKα
T − θ+K0
α ,
(7.6)
where Cχα and CK
α are Curie constants, and χ0α and K0
α are the temperature indepen-
dent orbital part of susceptibility and Knight shift, respectively. Combining these
two equations by eliminating the temperature parameter T from the relation, we get
96
at a given temperature,
Khfα =
CKα
Cχα
(χ(T )α − χ0α) +K0
α , (7.7)
which indicates a linear relation between Knight shift and the spin part of magnetic
susceptibility. Comparing Eq. 7.7 and 7.5, it is clear that the hyperfine tensor is just
the ratio between susceptibility and shift Curie constants.
gαAαα =CKα
Cχα. (7.8)
Plotting Khfα vs (χ(T )α − χ0
α) with T being the implied parameter gives us the
so-called Clogston-Jaccarino plot [56], as illustrated in Fig. 7.1. From Clogston-
Jaccarino plot we can determine hyperfine coupling constants and orbital shifts.
More precisely, the slope of the graph is related to the strength of the hyperfine
coupling, while the zero intercept gives the orbital shift for a particular orientation
of the applied magnetic field, evident from Eq. 7.7. Temperature dependence of
the bulk susceptibility (χ) can be fit to the Curie-Weiss behavior with a constant
offset, ascribed to Van Vleck paramagnetism, as described in Ref. [11]. In NMR
data linear behavior, of the form of Eq. 7.7, is found for T > 25 K, from which we
infer gαAαα in the units of [T/µB] as shown in Table 7.1. As sample is cubic in the
paramagnetic state, transfer hyperfine tensor defined in crystalline axes coordinate
system is diagonal with all diagonal elements being equal, i.e. gαAαα = −0.460 T/µB.
H ‖ Korbitα gαAαα(T/µB)
[001] -0.0216 % -0.460[111] -0.0253 % -0.458[110] -0.0211 % -0.437
Table 7.1: Hyperfine coupling constants and orbital shifts for different orientations of the appliedmagnetic field (H = 7 T) for 23Na in Ba2NaOsO6.
97
-0.08
-0.07
-0.06
-0.05
-0.04
-0.03
-0.02
-0.01
K (%
)
6x10-4543210
H = 7 T
χ (EMU/moleOe)
H || [110]
|K | for
H || [111]
H || [100]
-0.458 T/μB
-0.460 T/μB
-0.437 T/μB
Figure 7.1: 23Na Clogston-Jaccarino plots at 7 T applied field applied along three differentdirections as denoted. Solid lines are linear fits to the data for temperature ranging from 80 K to200 K.
7.2 Dipolar Interaction Tensor
The dipolar magnetic field induced by an Os unpaired electron spin SN at the posi-
tion of a Na nuclear spin I is:
Hdipolar = −µ0
4πgµB
1
r3[3(SN · r) · r − SN ] , (7.9)
where g is the unit-less electronic g-factor (g = -2.0023) and r is the unit vector
pointing from the electron to the nucleus.
Therefore, the direct dipolar coupling Hamiltonian between a Na nucleus spin Ii
98
and an Os unpaired electron spin Sj can be expressed as:
HD,ij = −γN~I ·H ijdipolar
=µ0
4πγN~gµB
1
r3ij
[3(Sj · r) · (r · Ii)− Sj · Ii]
= −µ0
4πγSγN
~2
r3ij
[3(Sj · r) · (r · Ii)− Sj · Ii] ,
(7.10)
where γS and γN are the gyromagnetic ratio of the Na nucleus and the Os unpaired
electron, respectively. Notice that the I and S are defined as dimensionless operators.
It can be written in terms of the second-rank dipolar coupling tensor Dij,
HD,ij = (γN~Ii) · Dij · (γS~Sj) . (7.11)
With components of Dij following the relation:
Dm,nij =
µ0
4πr3ij
[δmn − 3(rij,m · rij,n)] , (7.12)
where m, n represent x, y, z axes and rij,m, rij,m are the corresponding components
of vector rij. More specifically
Dij =
µ0
4πr3ij
·
3rxij r
xij − 1 3rxij r
yij 3rxij r
zij
3ryij rxij 3ryij r
yij − 1 3ryij r
zij
3rzij rxij 3rzij r
yij 3rzij r
zij − 1
. (7.13)
To get a sense of how much the dipolar interaction can contribute to the overall
interaction tensor, or the local field at Na sites, we calculated the dipolar tensor
between a Na nucleus at position (0,0,0) and one of its nearest neighbor Os moment
at position (d/2, 0, 0), where d is the lattice constant. Notice that the moment of the
99
Os electron is expressed in the unit of µB, and we need to convert the unit of dipolar
tensor elements from SI unit T 2
Jto T
µB, which is also consistent with the convention.
The dipolar tensor resulted from a pair of neighboring Na and Os turned out to be:
DNNij =
−0.026 0 0
0 0.013 0
0 0 0.013
T/µB . (7.14)
Next we consider the dipolar tensor in the complete Ba2NaOsO6 lattice under dif-
ferent crystal symmetries.
7.2.1 Cubic Symmetry
In our sample, Os atoms form face-centered cubic symmetry at high temperature.
To calculate the coupling tensor, we consider the interaction between one Na nu-
cleus and all Os atoms in N unit cells distributed symmetrically to the Na atom,
as shown in Figure 7.2. Large lattice size is chosen to guarantee the convergence
of the calculation. By summing up all the coupling tensors of each pair of Na and
Os, we can acquired the total effective dipolar interaction tensor at the position of
the Na, Di =∑
j Dij. In our case, without considering any local or bulk structural
distortion, the resulting tensor Di =
0 0 0
0 0 0
0 0 0
, which is expected due to the
cubic symmetry. So the dipolar interaction will not affect the NMR shift as long as
the cubic structural symmetry is conserved.
100
7.2.2 Reduced Symmetry
When we take into account the possible distortions at low temperature, if the dis-
tortion happens on the oxygen sites, the dipolar interaction between Na and Os will
not be affected and remains zero as in the previous case. But if we consider the
distortion happens on the Os sites, it will alter the tensor values quantitatively. For
example we assume a tetragonal distortion which only affects the c-axis (elongation-
compression model). By adding an exaggerated 10%’s distortion along c-axis, the
new dipolar tensor is
Di =
0.0014 0 0
0 0.0014 0
0 0 −0.0028
T/µB . (7.15)
Compared to the total shift tensor calculated later, the dipolar tensor contributes
very little to the overall interaction even when we assume a relatively large distor-
tion on the Os sites. Therefore the dipolar effect can be neglected in the future
calculations.
7.3 Ordered State LRO
In this section we center our discussion on the magnetic order observed at the ordered
state. The goal is to find out the specific spin configuration that can quantitatively
account for 1) the small ferromagnetic order from magnetization measurement, 2)
magnetically inequivalent nuclear spins sites and 3) magnetic anisotropy with an
unusual [110] easy-axis. We rely on the lattice sum method and simulate NMR shift
based on various feasible magnetic models inspired by both experimental observation
101
(a) 8 unit cells (b) 544 unit cells
Figure 7.2: Spatial distribution of Os atoms around one Na atom.
and theoretical predictions. We conclude that the most likely scenario is the canted
FM order which can explain all of the above experimental observations.
7.3.1 Magnetic Anisotropy
Fig. 5.1 illustrated the angel dependence of spectra at ordered state, from which we
have examined different models regarding the structural distortion. In this section
we focus on inspecting the uniform and staggered local fields as a function of angle,
as demonstrated in Fig. 7.3. This is essential for ensuring that we understand what
components of anisotropic magnetic susceptibility are being measured. For H‖[100],
Hstag reaches its maximum value, while Hu is at its minimum. In principle, Hu
should scale as bulk magnetization (M). As evident in Fig. 7.3B, this is not the
case here. This finding reveals that despite the fact that the net magnetization is
aligned with the [110] axis, the local fields, i.e. spin expectation values, are not. This
very fact was predicted to arise as a direct consequence of the local cubic symmetry
breaking driven by complex interaction in this class of materials [18].
102
9080706050403020100
Angle (θ)
40
30
20
10
5
Hsta
g (mT
)
0.21
0.20
0.19
0.18
0.17
M (
µb
/FU
)
100500
120
115
110
105
100
Triplet I
Triplet II
1st Moment
Hstag
[001] [111]
[110]
[001]
[111]
[110]
|Hu
(mT
)
|
M
Hu
140
120
100
80
60
|Hu| (
mT
)50
15
25
35
45
H = 15 T70
90
110
130
B A
Angle (θ (T = 8 K
Figure 7.3: Angular dependence of the uniform and staggered internal fields. (A) The magnitudeof the internal field associated with triplet I and II (solid symbols), and the first moment of theentire spectrum (uniform field, Hu) as a function of the angle between the applied magnetic fieldand [001] crystalline axis at 8 K and 15 T. Open symbols depict the angular dependence of thestaggered field, Hstag. Dotted lines are guide to the eyes. Solid lines are calculated fields fromthe canted Ferromagnetic spin model described in the later section. Comparison of the angulardependence of bulk magnetization from Ref. [11] (open symbols) and Hu determined from ourNMR measurements.
The existence of nonzero staggered field reveals two distinct Na sites, corre-
sponding to the two triplets in spectra. The uneven angel dependent shifts of the
two triplets cause the observed magnetic anisotropy of Hµ and Hstag. Since the in-
ternal field at the Na sites are generated by the hyperfine interaction between Na
and Os ions, we can calculate the local field distribution using the so-called lattice
sum method to determine the microscopic nature of the magnetic phase, by taking
into account different spin configurations, i.e. magnetic ordering, and the orientation
of the applied magnetic field relative to the lattice. By comparing the simulation
results with the experimental findings plotted in Fig. 7.3 and spectra in Fig. 5.1,
we can discern the possible spin configurations. One assumption we made is that
the moment is exclusively localized on Os site, although it is argued that the spin
density is also distributed to O [27]. However, our model is still valid because we can
account for the complexity of the spin density in A. For simplicity, we treat moment
as S = 1/2 localized on Os as was done in [31].
103
7.3.2 Lattice Sum Method
This numerical method is similar to the model employed in the point charge calcu-
lation, except that now we are interested in calculating the local field distribution
at 23Na sites due to hyperfine interaction. The calculation is constrained by the
experimental observations including the angle dependence of the bulk sample mag-
netization measurement and local field inferred from NMR.
More specifically, our method reconstructs the exact Ba2NaOsO6 lattice, includ-
ing both Na and Os sites while omitting Ba and O atoms. The internal field at each
Na site i is then given by,
H iint = h ·
∑〈j〉
Aj · µj (7.16)
where h is a unit vector in the applied field direction, Aj the symmetric 3×3 hyperfine
coupling tensor with the jth nearest-neighbor Os atom and µj its magnetic moment.
This equation sums over all the nearest-neighbor Os atoms.
Aj: The quadrupole splitting observed from spectrum below transition temper-
ature proves the existence of local lattice distortion, and will give rise to nonzero
off-diagonal terms in hyperfine tensor. Therefore in the simulation elements of Aj
can be set as fitting parameters.
µj: Each Os atom is assigned with a magnetic moment, whose magnitude and
orientation is model dependent. For example, if a simple ferromagnetic order is
assumed, each Os atom would carry the same moment. But in a model which
assumes anti-ferromagnetic, then a Os ion would carry a moment that is in opposite
direction to its neighbors. Thus our simulation includes the functionality to assign
different magnetic orders to the Os sublattice.
104
After iterating through the entire lattice and acquiring all the H iint values, the
NMR spectrum is generated by treating each peak as a Gaussian distribution with
variance σ. This way both simulated spectra and shift angle dependence can be di-
rectly compared with experimental data. A pseudo-code is described in Appendix. B.
To sum up, depending on the specific model under consideration, the results are
affected by several factors, including the hyperfine tensor symmetry, spin configura-
tion and its dependence on applied field, and local structural symmetry.
7.3.3 Microscopic Models of Magnetic Phase
Many magnetic orders are proposed based on different experimental findings [6,
11, 15] and theoretical models [18, 26, 27, 31], including weak ferromagnetism, anti-
ferromagnetism, canted (anti-)ferromagnetism, etc. As evident from our experimen-
tal data, NMR technique is able to simultaneously detect both structural/orbital
and spin degree of freedoms, and therefore putting stricter constrains and resulting
in a more definite solution to the problem. That is, the identification of magnetic
order has to be compatible with the observations of structural distortions.
From the earlier discussion, we recognize two possible scenarios of structural
symmetry breaking. One describes two forms of local distortion on octahedra which
creates two structurally inequivalent Na sites. Although this is very unlikely due
to the electrostatic energy involved, it can naturally explain the appearance of two
sets of triplets in spectrum, with the assumption of identical moment carrier by
each Os ions, i.e. ferromagnetic order. The other scenario is the indistinguishable
distortion on each octahedron. Then a more complex magnetic order is required
to reproduce the magnetic splitting. Such restrictions would provide guides to the
105
choices of models to be studied.
Another complication is the anisotropy of susceptibility. Bulk measurements
such as magnetization are not certain which components of the magnetic moments
are being measured. It is feasible that as an effort to minimize the free energy, spin
orientations may be adjusted as applied field changes directions, instead of being
pinned at fixed positions. Typically orientations which maximize susceptibility will
be favors, resulting in the alignment of spins with applied field [18].
We discuss in details the following models by taking into accounts all the stated
factors.
Model I: Single Structural Distortion + Canted Ferromagnetism
This second model considers the possibility that all the Na nuclei are in the same
structural environment, but there exists two magnetically inequivalent sublattices,
suggested by the quantum model [31]. Moments in each layer of XY plane are
parallel to each other, forming ferromagnetic order, but moments in the neighboring
layers point to a different direction, and in addition two adjacent layers’ moments
are symmetric about [110] axis, as shown in Figure 7.4. Lattice effectively forms
two sublattices with a C4 screw axis along the z axis. Moments arranged in this
fashion will sustain an uniform magnetic moment in [110] direction which provides
an overall shift to the NMR spectrum, meanwhile form a staggered pattern in the
direction perpendicular to the uniform moment. This model naturally accounts for
the appearance of two magnetically inequivalent Na sites, that is the appearance
of distinct frequency shifts for triplet I and II. The internal field at the Na site
in one plane consists of a sum of the four Os on the same layer, and thus equals
106
magnetic moments (Type A), and two Os above/below in neighboring layers with
magnetic moment pointing in a different direction (Type B). Na in the next plane
will then sense four Type B and two Type A Os moments. This generates two sets of
inequivalent Na sites and causes the magnetic splitting in spectrum between triplet
I and II, as two types of moments induce different local fields at the Na site. This
scenario of staggered magnetization was indeed predicted to be possible if we have
a negative in-plane coupling constant, J ′ < 0 [18]. Φ(A) and Φ(B) are the angle
between the two types of moments and [110] (or applied field direction if moments
follow the field), and Φ(A) = Φ(B) = Φ (Figure 7.4(b). By setting this angle as
a free parameter we will have control over the ratio of magnitude between uniform
and staggered local fields seen by Na sites.
a
Na
Os
O
y
z
x
b
[110]
x
y
φ
φ
Type A Spin
Type B Spin
Figure 7.4: Spin orientation in staggered layer model. (a) Schematic of the spin model consistentwith our data and proposed in [31] and (b) in the XY plane.
First we will try to fix the orientation of the magnetic moments in XY plane and
vary angle Φ, magnitude of moments and the hyperfine tensor to fit our data. In
this set-up the Kac, Kbc and Kcc components are not relevant anymore due to the
zero z component of the magnetic moment. Best fit is shown in Figure 7.5. The
angle dependence of the simulated results does mimic the experimental data, but
fails to make a quantitative fit. Next we allow the moments to follow the applied
field, which is probable due to free energy minimization. The two sets of moments
107
150
140
130
120
110
100
90
80
70
60
50
9080706050403020100
45
40
35
30
25
20
15
10
5
0
Triplet I Triplet II
|Hu| (
mT
)
Angle (θ)
Hsta
g (mT
)
Figure 7.5: Data fitting for two staggered layer model with fixed spin orientation.
are always kept in the same plane with the applied field, which rotates from [001]
to [110] direction. The orientation of the two sets of spin is calculated using the
rotation matrix with axis of rotation u = (−1/√
2, 1/√
2, 0):
R =
(cos θ + 1)/2 (cos θ − 1)/2
√2 sin θ/2
(cos θ − 1)/2 (cos θ + 1)/2√
2 sin θ/2
−√
2 sin θ/2 −√
2 sin θ/2 cos θ
.
When applied field is rotated θ angle away from [001] direction, the two sets of
moments have the following form:
SA =[(− sinφ+ sin θ cosφ)√
2,(sinφ+ sin θ cosφ)√
2, cos θ cosφ]
SB =[(sinφ+ sin θ cosφ)√
2,(− sinφ+ sin θ cosφ)√
2, cos θ cosφ]
. (7.17)
Using this set-up, the best fit is achieved and plotted as the solid lines in Fig. 7.3(A).
The simulated spectra at various angles are also plotted together with the experi-
mental spectra as comparison in Fig. 7.6. We find that data (Hu and Hstag as the
applied field is rotated in the (110) plane), is best accounted for by the following pa-
108
ω (MHz)
Ma
gn
itu
de
(a
rb. u
nit)
0ο
80ο
70ο
60ο
50ο
40ο
30ο
20ο
10ο
90ο
[001]
[110]
[111]
168.8168.4168.0167.6167.2
Figure 7.6: Comparison of angular dependence of simulated and experimental spectra.Yellowcurves are the actual NMR spectra, and the blue curves are simulated results. Rotation is in (110)plane. µ = 0.6 µB , φ = 67 and tensor with the form in Eq. 7.18 are used.
rameters: the angle φ ≈ 67, moment µ ≈ 0.6µB, which is the value of the effective
moment deduced from the fit to a Curie-Weiss behavior in the PM state in [11], and
transfer hyperfine coupling tensor in the units of (T/µB),
K ≈
0.45 −0.13 −0.14
0.13 0.45 0.16
−0.14 0.16 0.48
(T/µB) . (7.18)
The moment value of 0.6µB is specified for H0‖[001], and as direction of H0 is
rotated in the (110) plane, we adjust the value of the moment to scale as bulk
magnetization at a given direction. In this case, angle φ remains independent of the
direction of H0 and spins remain in-plane that follows the rotation of H0.
109
At first sight it is surprising that both the quadrupole and magnetic splittings
appear to vanish toward the same orientation. We show in the later chapter that at
θ = 55, only a single peak is observed in the ordered state. From the simulation
we can clarify that the apparently concurrent disappearance of both magnetic and
quadrupole splitting is coincidental, as shown in Figure 7.7. The absolutely value of
quadrupole splitting, as discussed in section 6.1, becomes 0 at angle 54.7, satisfying
the relation 3cos2θ − 1 = 0. On the other hand, with more granularity of the angle
θ in the lattice sum simulation, the minimum magnetic splitting is ∼ 22.9 kHz at
angle 56.2. The intrinsic linewidth in the ordered state is ∼ 50 kHz, and a splitting
smaller than the linewidth will not be visible. Therefore from Figure 7.7 we can
predict that between 52.6 and 57.6, only one peak can be observed experimentally,
which explains the spectrum at θ = 55.
100
80
60
40
20
0
Ma
gn
etic S
plit
tin
g (
kH
z)
60.057.555.052.550.0
Angle from [001] (deg.)
20
15
10
5
0
Qu
ad
rup
ole
Sp
litting
(kH
z)
56.2o
54.7o
Linewidth = 50 kHz
57.6o
52.6o
Figure 7.7: Comparison of angle dependence of magnetic (red circles, plotted to the left axis)and quadrupole (blue crosses, plotted to the right axis) splitting from simulation. Dashed linecorresponds to the 50 kHz linewidth.
Our initial guess for diagonal values of A was those determined in the PM state,
because regardless of the exact nature of the low temperature distortions, their domi-
110
nant effect on A is to induce finite off-diagonal terms, while changes of diagonal terms
are minor (Appendix C). Thus, we constrained the values of diagonal elements of
A to be close to those found in the PM state. As lattice distortions emerge at low
temperature, there is no a priori reason to believe that diagonal terms are equal in
FM and PM phase. However, we have no independent way of knowing the exact
strength of the elements of A, as our data only senses the product of AµB. Never-
theless, we can consider two different scenarios. In the first scenario, we assume that
the diagonal elements of A are close to those found in the PM state, as described
above, and find that the effective FM moment of 0.6µB, for H‖[001]. This assump-
tion seems reasonable, as the main effect of lattice distortions, which do not exceed
0.8% of the lattice constant, is to induce non-zero off-diagonal terms and not to
drastically change values of the diagonal terms of A. The moment that we deduced
corresponds to that found from the bulk magnetization data in the PM state, and is
three times larger than that found in the FM state in [11]. Small moment found in
the FM state from bulk measurements can be explained by large canting angle φ that
we determined. That is, small moment is due to partial cancellation of nonparallel
magnetic moments.
Alternatively, if moment is assumed to be 0.2µB, which is the value of the moment
in the FM state as determined from bulk magnetization [11], we find that all the
elements of A displayed in Eq. 7.18 are multiplied by a factor of three. Evidently,
in either case, off diagonal terms, induced by the O octahedra charge distribution
distortions around Na atoms, arise. We emphasize that symmetry of the inferred A
tensor does not reflect neither local tetragonal nor orthorhombic symmetry of the
distorted O octahedra (See Appendix. C for details). This implies that spin-
spin interactions are mediated by complex multipolar interactions, as theoretically
predicted in [31].
111
As magnetic field is rotated and spin-plane follows H, it is also possible that
the moment value remains constant at 0.6µB and it is variation of the angle φ that
accounts for the observed angular dependence of Hu and Hstag (and bulk magneti-
zation as well). In this case, we find that the angle φ varies from 65 to 71 as H
is rotated from [001] to [110] direction. Moreover, the inferred value of φ, allows as
to estimate the ratio of intra-plane (J′) to inter-plane (J) coupling constant [31].
In Ref. [31] the authors deduced that in the canted-FM state spins lie in the (XY )
plane with angles given by,
tanφA = − J ′
4|J |−
√1 +
J ′
4|J |, φB =
π
2− φA, for J ′ < 0 . (7.19)
Even though this result was derived for zero magnetic field, we showed that angle φ
does not vary significantly on the strength of the applied field, justifying the use of
the above expression. Thus, we find that J′/J ≈ 4.
Furthermore, we see no evidence of domain formation and significant lag of the
magnetic moment behind the field. Indeed, in Fig. 7.8, we plot integrated spectral
intensity, reflecting total number of nuclei, as a function of the angle θ. The facts
that variation in the intensity does not exceed 25 % and that the intensity at [001]
and [110] is the same, demonstrate that there are no significant domains in which
moments are oriented away from the applied field, i.e. local magnetization follows
the field direction.
Model II: Two Inequivalent Distortions + Ferromagnetism
This model introduces two inequivalent Na sites naturally from non-uniform distor-
tion. For example one set of Na interacts with two elongated O atoms in c-axis and
112
60200
[001] [110]
[111]
40 80
2
3
4
Inte
nsity (
a.u
.)
Angle (θ (
Figure 7.8: Integrated intensity of the entire spectrum, plotted in Fig.5.1, as a function of theangle between the applied magnetic field and [001] crystalline axis at 8 K and 15 T.
four compressed O atoms in XY plane, while the other set of Na is surrounded by 6
O atoms distorted in the opposite direction, as shown in Figure 6.4(A). And in this
model, each Os carries the same spin which follows the direction of applied field and
their magnitude is determined by magnetization measurement, but two sets of Na
interact with Os through different hyperfine tensors.
Unlike point charge method, the hyperfine tensor used in the internal field cal-
culation does not directly involve ion positions. Therefore we reflect the distortion
of O ions in terms of percentage change of value of the hyperfine tensor. That is,
the hyperfine tensors for both Na sites are kept to be diagonal, but each diagonal
term will be changed depending on the direction and extend of local distortion. For
example, for Na type A we might increase KAzz by 30% and decrease KA
xx and KByy by
20% from their original values, and for Na type B decrease KAzz by 40% and increase
KAxx and KB
yy by 10%. By tuning these percentage values as fitting parameters, we
eventually get a good fit for both magnetic splitting and shift as shown in Figure 7.9.
The hyperfine tensors used to get this fitting are of the form:
113
Na
Os
Oy
z
x
S || [110]
Electron Spin
|Hu| (
mT
)
Angle (θ)
Hsta
g (mT
)
(a) (b)150
140
130
120
110
100
90
80
70
60
50
9080706050403020100
45
40
35
30
25
20
15
10
5
0
Triplet I Triplet II
Figure 7.9: Fit to the angular dependence of internal fields with FM model. (a) Sketch of thetwo inequivalent distortion with ferromagnetism model. (b) Fits to the angular dependence. Linesand markers used the same way as in Fig. 7.3(A).
KA =
0.138 0 0
0 0.138 0
0 0 0
, KB =
0.552 0 0
0 0.552 0
0 0 0.84
However this model is unlikely to be true. In order to get a good fit, Kzz compo-
nents have to be changed as much as 100%. Considering the local distortion is merely
a few percent calculated from quadrupole model, this drastic alteration in value is
improbable. Combining the results of point charge and lattice sum calculations, we
can rule out the possibility of uneven distortions. Therefore, the magnetic order has
to be more complicated than the uniform ferromagnetism.
7.4 Conclusion
We found that LRO state is the exotic canted two-sublattice FM state. Such state is
predicted to occur due to multipolar spin-spin interactions for spins on the frustrated
fcc lattice. Moreover, formation of the two-sublattice magnetic state is driven by the
staggered quadrupolar order [18,31]. Besides, we reveal that despite the fact that the
114
net magnetization is aligned with the [110] axis, the local fields, i.e. spin expectation
values, are not. The component of the magnetic moment perpendicular to the net
magnetization is substantial in magnitude due to the large canting angle. This is
the first direct observation of such an exotic quantum state. Our observation of
both the local cubic symmetry breaking and appearance of two-sublattice exotic FM
phase is in line with theoretical predictions based on quantum models with multipolar
magnetic interactions. Thus, our findings clearly demonstrates that such microscopic
models are correct theoretical framework for predicting emergent quantum phases in
Mott insulators with the strong SOC.
115
Chapter Eight
Transition Region: Possible
Nematic Order
From all the preceding discussions we have clarified the magnetic transition from
the high temperature PM phase to the canted ferromagnetic ordered state, whereas
little attention was paid to the intermediate transition region. In this chapter the
transition region is attentively examined in order to understand the development of
FM order.
8.1 Temperature Dependence of Transition Re-
gion Spectrum
To gain further insight into the transition region, experiments are done with H0 =
15 T and much finer (∼ 0.2 K) step size within the transition region, as shown in
Fig. 8.1.
From the temperature dependence of spectra in Fig. 5.1 and 5.2 we observe that
for all the values of H0, the transition region spans over ∼ 2 K below Tc before the
establishment of two identical triplets I and II. The further scrutinization for the
transition region in Fig. 8.1 shows some interesting features of lineshapes within the
transition region. The developments of the two triplets are substantially different.
That is, as temperature is lowered from PM state into the transition region, triplet
I experiences certain degree of line broadening but the quadrupole satellites are still
visible during the entire transition region. Meanwhile triplet II starts as a broad
singe peak and shifts to lower frequency rapidly, and quadrupole splitting does not
show up until the magnetically ordered state. Furthermore, the integrated intensity
of triplet II is substantially lower than that of triplet I, and the difference vanishes
upon entering the FM phase. During the experiments, we carefully accounted for
the effects of RF power and T1 at all temperatures, and assured that the difference
117
25
20
15
10
5
0
Ma
gn
itu
de
(a
rb. u
nits)
-2.0 -1.6 -1.2 -0.8 -0.4 0.0 0.4 0.8
ω − ω0 (MHz )
25 K
14.1 K
14 K13.6 K
14.3 K
14.5 K14.7 K14.9 K
15.3 K15.6 K
16 K16.3 K
16.5 K
17 K
18 K
20 K
15.1 K
10 K
6 K
11 K
11.5 K11.8 K
12.3 K
13.3 K12.8 K
Figure 8.1: Temperature evolution of 23Na spectra at 15 T with H0 ‖ c. Zero of frequency isdefined as ω0 = 23γ H, the zero NMR shift frequency. Red, green and blue colors represent PM,transition and FM phases, respectively.
of intensity is not due to experimental conditions.
Also we have measured the spectra as a function of temperature with θ (the angle
between H0 and c-axis) ≈ 50 and 55 where the peaks merge into one according
to the angular dependence measurement in Fig. 6.1. Especially for θ in the vicinity
of 55, both the quadrupole and magnetic splittings varnish at the ordered state.
Within the transition region, a second broad peak with lower frequency compared
to the peak inherited from PM state develops. It shifts to lower frequency with
decreasing temperature. The linewidth grows broader initially and becomes narrower
as it approaches the ordered state. Meanwhile the higher frequency peak is gradually
covered up by the lower frequency line as temperature decreases. Spectra at both
angles are shown in Fig. 8.2.
118
22
20
18
16
14
12
10
8
6
4
2
0
Ma
gn
itu
de
(a
rb. u
nits)
-1.6 -1.2 -0.8 -0.4 0.0 0.4
ω − ω0 (MHz )
H 0 = 15 T
50 Degree25 K
8 K
10 K
12 K
13 K
13.5 K
14 K
14.5 K
15 K
15.2 K
15.4 K
15.6 K
15.8 K
16 K
16.2 K
16.4 K
16.6 K
17 K
20 K
30
25
20
15
10
5
0
-1.2 -0.8 -0.4 0.0 0.4
Ma
gn
itu
de
(a
rb. u
nits)
ω − ω0 (MHz )
H 0 = 15 T
55 Degree
20 K
14 K
6 K8 K9 K10 K
11 K12 K12.5 K
13 K13.1 K13.2 K13.3 K13.4 K
13.5 K13.6 K13.7 K
13.9 K13.8 K
14.1 K14.2 K14.3 K14.4 K14.5 K14.6 K14.7 K
14.8 K15 K15.5 K
16.5 K
18 K
Figure 8.2: Spectrum as a function of temperature at θ = 50 and 55
These observations suggest significant difference of the local fields and therefore
magnetic order sensed by the two 23Na sites during transition. Triplet II discerns
a broad lineshape with smaller intensity, suggesting the existence of a distribution
of inhomogeneous internal fields. More specifically, upon cooling from PM state,
certain domains in the material develops (canted-)FM order but with much smaller
magnitude of magnetization compared to the well developed ordered state. Therefore
the magnetic splitting during transition region is smaller initially and grows as the
FM order enhances with decreasing temperature. The broad linewidth observed for
the lower peak indicates that the development of FM order is accompanied with
large static spin inhomogeneity. Furthermore, the portion of domains, or volume,
that develop the FM order is also small at the onset of transition, reflected in the
small intensity of what develops into the triplet II. We define the ratio of integrated
intensity between the triplet I and II as
rI =IIIII
, (8.1)
and the ratio as a function of temperature with H0 ‖ [001] is plotted in the inset of
Fig. 8.3.
119
Notice that, spectra shown in this chapter so far are acquired in a separate run
compared to those plotted in Fig. 5.1 and 5.2. These new measurements have a
higher Tc compared to those determined in 5.2b. With careful calibration we found
out that the temperature readings of the new sets of data are shifted to higher values
by a constant ∼ 3.5 K. The reason is that, during the new measurements, a large
temperature gradient is induced between the thermometer and sample, due to a
much larger Helium flow. Taken into account this shift, we plot the NMR shift of
all the field orientation as a function of temperature in Fig. 8.3.
-1.8
-1.6
-1.4
-1.2
-1.0
-0.8
-0.6
-0.4
-0.2
ω −
ω0 (
MH
z)
242220181614121086420
Temperature (K)
1.2
1.0
0.8
0.6
0.4
0.2
0.0
In
teg
rate
d In
ten
sity R
atio
222018161412108642
Temperature (K)
H0 = 15 TH0 || [001]
0 Degree
50 Degree
55 Degree
H0 = 15T
Figure 8.3: First moment shift as a function of temperature at various field orientations. Solidlines are guide to the eyes. Zero of frequency is defined as ω0 = 23γ H0, the zero NMR shiftfrequency. [Inset] Integrated intensity ratio as a function of temperature with H0 ‖ [001].
8.2 Existence of Nematic Order during Transition
Region
Quantum models [18,31] have successfully predicted the geometrical distortions and
canted FM order observed experimentally by our NMR work. It also proposes that,
120
in the quadrupolar state, spin-lattice coupling can induce a so-called spin “uniaxial
nematic phase” [57, 58] with broken lattice rotation symmetry but not the time
reversal. In the spin nematic phase, two principle axes are equivalent but the third
one is distinguishable. Similar to Ba2NaOsO6, in the parent compounds of the iron-
based superconductors, structural transition from tetragonal to orthorhombic is also
observed with a magnetic transition into the stripe SDW order [59–61], and the
nematic order is proposed to be a result of the orbital ordering of Fe 3d electrons
and further induces the stripe SDW order [62,63].
Mean field theory [31] suggests that if the spin-phonon coupling exceeds certain
threshold strength, a nematic phase would appear, accompanied by a tetragonal
to orthorhombic structural transition. The calculated phase diagram is shown in
Fig. 8.4. In Section 5.3 we have concluded that the dominating spin-lattice relaxation
channel is the coupling of nuclear spins to the optical branch of phonon. This
clearly indicates strong spin-phonon coupling which would result in a phase crossing
through the nematic phase, as indicated by the red arrow in the phase diagram. The
nematic phase would correspond to the transition region observed in the temperature
evolution of NMR spectra. The transition region spans over ∼ 2 K in temperature
which can be utilized in estimating the specific strength of the spin-phonon coupling.
We then examine the temperature evolution of the NMR spectral lineshapes in
different field orientations, shown in Fig. 8.5. The red, blue and green filled markers
represent the second moment of the entire spectrum at angle θ = 0, 50 and 55,
respectively. Furthermore, when θ = 0, since the two triplets are well separated in
frequency even during part of the transition region, the second moment of the two
triplets are calculated and plotted independently, shown as the filled triangles in the
figure. The transition temperatures, Tc, are determined in the same way as Fig. 5.5,
using the crossing point of two linear fits to the PM and ordered phase. The linear
121
Phase Crossing
Figure 8.4: Theoretically proposed phase diagram based on quantum model with mean fieldtheory. Red arrow schematically shows the possible phase transition from PM to FM state via anintermediate nematic phase with decreasing temperature. Reprint from Ref. [31]
fits are shown as solid lines in the inset of Fig. 8.5. This way we are able to plot the
second moment in a reduced temperature scale.
The second moment σ2, or the linewidth, represents the spread of the local mag-
netic field distribution, just like the variance in a Gaussian distribution. For all
three orientations, the linewidth grows rapidly upon entering the transition region,
indicating a broad distribution of electron spin polarizations. For θ = 0 and 50
the second moment keeps growing until it reaches a constant value at the FM state.
At these two orientations, both quadrupole and magnetic splittings are temperature
dependent and contribute to σ. But for θ = 55, both quadrupole (3cos2 θ − 1 = 0;
see Section 6.1) and FM magnetic splittings are small in ordered state due to geom-
etry , therefore only the putative nematic order contributes to the second moment.
Indeed, at this “magic angle”, σ drops significantly after it reaches the peak at about
half way into the transition region. Thus, whatever the nature of spin order is during
transition, it increases the inhomogeneity of static spin distribution from PM state,
and eventually gives way to the more homogeneous FM order.
122
0.52
0.48
0.44
0.40
0.36
0.32
0.28
0.24
0.20
0.16
0.12
0.08
0.04
(σ
2)1
/2 (
MH
z )
1.61.51.41.31.21.11.00.90.80.70.60.50.40.3
T/Tc
0.60
0.50
0.40
0.30
0.20
0.10
0.00
(σ
2)1
/2 ( M
Hz )
20161284
Temperature (K)
55 Degree Entire Line 50 Degree Entire Line 0 Degree Entire Line 0 Degree Triplet I 0 Degree Triplet II
Possible Nematic Order
Figure 8.5: Square root of the second moment, measuring electronic spin polarization, of the 23NaNMR spectra (filled symbols) as a function of reduced temperature at θ = 0, 50 and 55. Dashedlines are guide to the eyes. [Inset] Square root of the second moment as a function of temperature.Solid lines are separate linear fits to transition region and PM state in order to extract the transitiontemperature at different field orientations.
8.3 Relaxation Rate
The spin-lattice relaxation rate 1/T1 is measured at θ = 55 and 15 T as a function
of temperature, and is plotted in Fig. 8.6 together with the θ = 0 rate at 9 T, as a
function of reduced temperature. The 9 T and 15 T relaxation rates are the same at
PM state, while the rate at 15 T is significantly lower than that of both triplets of
9 T. It has been proven that the spin-quadrupole interaction can contribute to the
relaxation process [64–66], via the lattice vibration, i.e. phonon. We have confirmed
in Section 5.3 that the main relaxation channel in 9 T is the spin-phonon interaction,
therefore at 15 T where the quadrupole splitting vanishes, relaxation rate naturally
increases due to the lack of spin-lattice coupling.
A power law fit to the transition region gives the exponent in range of 9.5 ∼
123
3
4
5
6
0.1
2
3
4
5
6
1
2
3
4
5
6
10
2
1/T
1 (
S-1)
3.53.02.52.01.51.00.50.0
T/T C
15 T
55 Degree
9 T
0 Degree Triplet I
0 Degree Triplet II
0 Degree PM
Figure 8.6: Temperature evolution of the relaxation rate at 15 T and θ = 55, together with 9 Tand θ = 0, in reduced temperature scale. Dashed lines are guide to the eyes. Solid line is a powerlaw fit.
11, in contrast to the T 6 relation before. This temperature dependence with expo-
nent larger than 9 was previously observed in PM state [67] and attributed to the
combination of Raman and Orbach [68] processes. More specifically, the relaxation
is a two-phonon process with energy transferred to the lattice. This energy is the
difference between two excited states, as discussed in section 5.3. It is a Raman
process if the energy difference is lower than Debye temperature, and otherwise an
Orbach process.
124
Chapter Nine
Resistively Detected NMR of
Topological Insulator
So far in this thesis we have been focusing on Ba2NaOsO6, material with the pres-
ence of both strong SOC and electron correlations. Consider the Hubbard model
Hamiltonian including strong SOC
H = −t∑i,j;σ
(c+i,σcj,σ + h.c) + U
∑i,σ
ni,σ(ni − 1) + λ∑i
Li · Si , (9.1)
where t is the electron hopping energy, U is the Hubbard repulsion and λ is the
SOC strength. Ba2NaOsO6 corresponds to a group of materials with U t and
λ t. The cooperative effect of strong electron correlation and SOC give rise to the
insulating nature and novel magnetic properties. While in this chapter, we consider
the materials with U t but still possess strong SOC. Such group of materials
are insulators with bulk excitation gap, but different from regular band insulator
by preserving surface metallic states, and are called topological insulator (TI). In
this chapter we present some interesting, although preliminary, results of quantum
oscillation (QO) and resistively detected NMR (RDNMR). The momentum and spin
of electrons in TI are always perpendicular to each other due to strong SOC, known
as “spin-momentum locking”, and we aim to create electron spin flipping which can
induce detectable resistance change. The goal is to manipulate the nuclear spins
and in turn induce a flip of the electron spin through the local hyperfine interaction
between electronic and nuclear spins.
9.1 Sample Preparation and Method Description
To show proof of principle, we tested three TI samples of Sb doped Bi2Se3,with
doping percentage 0%, 18.5% and 25%. Bi2Se3 is a classic example of the TI, but
large bulk conductivity hinders the detection of surface metallic state. J. Analytis
126
etc. showed that, by reducing the bulk carrier density with doped Sb, the lowest
Laudau level can be reached with applied magnetic field as low as 4 T [72]. The
same samples doped with Sb to limit the bulk conductivity, provided by I. Fisher’s
group, were used for our RDNMR measurements. Each square shaped sample was
cleaved in-situ into two pieces with fresh surfaces just before the experiment. One
piece (Sample A) is used for QO and RDNMR experiments with four conducting
leads places on each corner of the surface, in order to measure resistivity, and the
other piece (Sample B) is used for regular bulk NMR measurement. Samples were
never exposure to air for more than 30 minutes.
Experimental set-up for sample A is shown in Fig. 9.1. Sample is placed on top
of a meanderline surface RF coil (coil A), which is then mounted on a sample holder.
This surface coil is used as antenna, and can also be used as RF coil to penetrate
the entire sample. The sample holder is part of a single-axis goniometer and allows
rotation relative to the applied field. On top of the sample another surface coil
(coil B) is placed in contact with the sample surface, used to send RF power via
the regular NMR set-up described in Section.4.3.1. Coil B is made of thin 56 gauge
(12.7 µm in diameter) copper wire with small inter-turn spacing < 5 µm. As we have
explained in Section.4.3.3, the distribution of magnetic field generated by a surface
coil is concentrated near the coil surface, within distance approximately proportional
to the spacing between the turns. This way the RF coil only excites the nuclear spins
within ∼10 µm from the surface and minimizes the bulk spin polarization. Four leads
on the sample are connected to a Lakeshore AC 370 resistance bridge for resistivity
measurement.
During the QO and RDNMR measurements, sample A is cooled down to ∼30mK
and temperature is closely monitored to prevent electron heating induced by exci-
tation current and/or RF power injected onto the sample. There are two configura-
127
Figure 9.1: QO and RDNMR experimental set-up..
Sample
I+ I-
V+
V-
Sample
I+ V-
V+
I-
(a) (b)
Figure 9.2: (a) Parallel and (b) diagonal configurations for resistance measurement.
tions available for the sample surface resistance measurement, as shown in Fig. 9.2.
The parallel configuration measures the horizontal resistance of the sample surface,
and the diagonal set-up measures the diagonal resistance. Notice that this Van der
Pauw type of configuration with four-point probe method was used instead of Hall
bar set-up, due to the fact that we have no a priori knowledge of the surface current
direction.
Next we briefly describes the procedures of QO and RDNMR measurements.
128
9.1.1 Quantum Oscillation
The detection of two-dimensional surface state of TI was first reported by J. Analytis
etc. [72]. As mentioned in last section, the success of measuring the metallic surface
state while avoiding the normally overwhelming effect of bulk resistivity [73–75] relies
on the doping of replacing Bi with Sb [76] to reduce the carrier density.
For majority of the QO and RDNMR experiments, the applied field H0 is per-
pendicular to the sample surface, i.e. H0 ⊥ c. The measured resistance value is
comprised of two components. First is the field dependent longitudinal magne-
toresistance [77], Rxx, which is an even function of the applied field. The second
component is the transverse quantum Hall resistance, Rxy, whose sign depends on
the orientation of the perpendicular part of the field, H⊥. That is, if two resistance
measurements are done with the same magnitude of applied field but opposite field
orientations, H[001] and H[001], the measured resistance values would be
Rpos = Rxx +Rxy
Rneg = Rxx −Rxy ,
(9.2)
and therefore we can extract the magnetoresistance and Hall resistance through a
linear combination
Rxx =1
2(Rpos +Rneg)
Rxy =1
2(Rpos −Rneg) . (9.3)
The quantum oscillation, or Shubnikov-de Hass oscillation [79], is a periodic
oscillation of resistance in inverse field with the period τ = 2πe/~Ak, where Ak is
129
the cross-sectional area of Fermi surface. A common technique used to extract the
oscillation is the background subtraction. More specifically, a 2nd or 3rd polynomial
function is fitted to the field dependent Rxx and Rxy curves as the background, and
then subtracted from the original curve.
9.1.2 RDNMR
It is extremely hard to acquire NMR signal from 2D systems due to the small num-
ber of active nuclear spins. A technique used to circumvent this obstacle in GaAs
quantum wells is the resistively detected NMR [80,81]. It relies on the fact that the
longitudinal resistance Rxx is a function of the Zeeman energy gap, which depends
on both the external field and internal hyperfine field, i.e.
∆EZ = gµB(H0 +HHF) . (9.4)
In our experiment, coil B sends the RF wave, either in continuous or pulsed form,
to the sample surface. If the RF wave is at the Larmor frequency, it will depolarize
the nuclear spin and reduce HHF, and induces a resistance change. This percentage
of the resistance change is therefore proportional to the Knight shift.
As aforementioned, two forms of RF wave are employed in our RDNMR mea-
surements: continuous-wave (CW) [82] and pulsed [83]. The CW method sends RF
signal continuously into the sample with power low enough to prevent heating. The
pulse method sends a single RF pulse with short period of only a few µS’s, to cause
coherent spin manipulation.
130
9.2 Results and Discussion
In this section we demonstrate some results from the bulk NMR, quantum oscillation
and RDNMR measurements, and discuss the implication of the data in conjunction
to each other in a qualitative fashion. More quantitative interpretation would require
further experimental evidence.
9.2.1 Bulk NMR Spectrum
The spectrum measured with the regular NMR for all three samples is shown in
Fig. 9.3 at 4 K. For pure sample, 7 quadrupole peaks of 201 Bi nuclei with I = 9/2
are visible on top of a broad background. There should be 2 more peaks on the sides
and covered up by the broad line. For doped sample, Sb induces larger inhomogeneity
and the quadrupole features do not show up in the spectra. The Knight shift of the
peak frequency is ∼ 0.5%, for all the three samples. It provides an estimation of
resistance change expected for RDNMR measurements, based on Eq. 9.4.
5.04.03.02.01.00.0-1.0-2.0-3.0-4.0
ω − ω0 (MHz)
Magnit
ude (
arb
. unit
s)
H = 14 T 2
0
8
6
4
2
0
2.52.01.51.00.50.0-0.5-1.0-1.5-2.0
Magnit
ude (
arb
. unit
s)
Magnit
ude (
arb
. unit
s)
3.02.01.00.0-1.0-2.0
ω − ω0 (MHz)ω − ω0 (MHz)
H = 11 T
0 % Sb18.5 % Sb
25 % Sb
(a) (c)(b)
H = 12.77 T
Figure 9.3: Bulk NMR spectra for three samples with different Sb doping percentage.
131
9.2.2 Quantum Oscillation
The resistance of the pure sample is measured as a function of field, and shown
in Fig. 9.4(a). Oscillation feature does not show up until the field exceeds ∼5 T,
indicating the 3D quantum limit. That is, beyond this field, a gap is opened in
the gapless surface bands with the surface states of top and bottom being coupled,
and the transport properties are governed by the quantized surface states. The
quantized surface states can be observed experimentally, as shown in Fig. 9.4, in the
form of quantum oscillation as discussed in section 9.1.1. The Rxx and Rxy values
are extracted via the linear combination of Eq.9.3, and plotted as a function of field
in Fig. 9.4(b). By subtracting the 2nd order polynomial background, the oscillating
parts, ∆Rxx and ∆Rxy are acquired and plotted as a function of inverse field, 1/B,
shown in Fig. 9.4(c) and (d).
-1.2
-1.1
-1.0
-0.9
-0.8
-0.7
-0.6
-0.5
-0.4
-0.3
-0.2
-0.1
0.0
0.1
R (
Ohm
)
-20 -16 -12 -8 -4 0 4 8 12 16 20
B (T)
-0.70
-0.65
-0.60
-0.55
-0.50
-0.45
-0.40
-0.35
-0.30
-0.25
-0.20
-0.15
-0.10
-0.05
Rxx (
Ohm
)
181716151413121110987654321
B(T)
0.55
0.50
0.45
0.40
0.35
0.30
0.25
0.20
0.15
0.10
0.05
0.00
Rxy (O
hm
)
0.08
0.06
0.04
0.02
0.00
-0.02
-0.04
-0.06
ΔR
xx
(O
hm
)
0.160.140.120.100.080.06
1/B (T-1
)
N = 2 N = 3 N = 4
-0.06
-0.04
-0.02
0.00
0.02
0.04
ΔR
xy
(O
hm
)
0.160.140.120.100.080.06
1/B (T-1
)
N = 2 N = 3
(a) (b)
(c) (d)
Figure 9.4: Quantum oscillation of the pure sample. (a) The resistance as a function of field,ranging from -18 to 18T. (b)Rxx and Rxy as a function of field. Blue lines are 3rd order polynomialfit. (c) and (d) are the background-subtracted resistance value plotted as a function of inverse field.
132
The periodic oscillating of ∆Rxx and ∆Rxy are evident. We mark the Landau
indices N = 2, 3, 4 on the plot. N = 1 would correspond to a higher field beyond our
experimental capability. For ∆Rxx, we observe the oscillation period ∼ 0.044 T−1,
and the positions of dips are |1/B|= 0.064 T−1 for N = 2, and |1/B|= 0.108 T−1
for N = 3, and |1/B|= 0.152 T−1 for N = 4. ∆Rxy curve shows the same period
of oscillation, although the dip positions (|1/B|= 0.084 T−1 for N = 2, and |1/B|=
0.130 T−1 for N = 3) are different. Similar oscillation features are visible among all
the samples.
9.2.3 RDNMR
From the RDNMR measurement, we have observed resistance change in the vicinity
of Larmor frequency in both the CW and pulse methods. For both methods, we
sweep a few MHz’s frequency range in a few hundreds to thousands steps, and record
resistance value at each step. And in some of the measurements, especially with the
CW method, multiple resistance values are recorded at the same frequency. The
data plotted here are averaged values and the resistance fluctuation is represented
by the error bar (detail see Appendix D). Some of the significant results are shown
in Fig. 9.5.
9.3 Discussion
Although so far only preliminary results are obtained, we have successfully observed
the periodic quantum oscillation and resistively detected NMR. The temperature and
circuit-tuning induced effects have been carefully monitored and accounted for. The
133
0.6382
0.6380
0.6378
0.6376
0.6374
R (
Oh
m)
128126124122120
Frequency (MHz)
0.5194
0.5192
0.5190
0.5188
0.5186
0.5184
R (
Oh
m)
82807876
Frequency (MHz)
0.4320
0.4318
0.4316
0.4314
0.4312
R (
Oh
m)
120118116114112
Frequency (MHz)
0.0945
0.0940
0.0935
0.0930
0.0925
0.0920
R (
Oh
m)
10410310210110099
Frequency (MHz)
Pure Sample
H = 14.9 T
Pulsed Method
25% Sb Sample
H = 17.9 T
CW Method
25% Sb Sample
H = 11.6 T
CW Method
18.5% Sb Sample
H = 17 T
CW Method
Figure 9.5: Change of the resistance as a function of frequency at various fields for differentsamples. Blue circles denote the averaged value at a frequency point, and the red lines are errorbars. Black lines are natural frequency positions of Bi.
percentage change of resistance in RDNMR appears to be very small. In all the CW
results shown in Fig. 9.5, the resistance changes about 0.1% and ∼ 1 mΩ in absolute
sense, several orders of magnitude smaller compared to earlier RDNMR detection
in the quantum Hall state [82]. However in quantum Hall materials where the edge
state charges are responsible for the Hall conductivity, the quantized resistance is
Rxx = h/(Ne2) = 25.8/N kΩ [84], where “filling factor” N can take on integer (1, 2,
3, . . . ) of fractional values (1/3, 2/5, 3/7, . . . ). The resistance change in RDNMR
in this case is induced by transition between different plateaus (N values) in the
Hall resistance, and would correspond to a value in the order of kΩ. In our case,
the change of resistance is proportional to the number of electron spins flipped in
surface state, and is expected to be much smaller compared to quantum Hall effect.
However there are still unanswered questions regarding the our results of the
RDNMR measurement. First of all, the peak of resistance change does not always
134
align with the peak of bulk NMR spectrum. This could due to the different nature
of electron spin order in bulk and surface states. Lastly, it is still unclear what is
the relaxation rate of nuclear spins within the surface state and how would it affect
our measurements. Further systematic measurements are expected to clarify these
questions and provide quantitative interpretation to the 2D surface system.
135
Chapter Ten
Summary and Outlook
Many intriguing characteristics, especially the unusual [100] easy-axis of magneti-
zation and magnetic anisotropy, arise in BNOO, a Mott insulator with both strong
SOC and electron correlation. The theoretical framework is approached by both
DFT calculation and quantum models with deviating conclusions, focusing on the
existence of structural distortion. NMR is a valuable tool to experimentally investi-
gate the microscopic nature of BNOO, thanks to its sensitivity to spin, orbital and
structural degrees of freedom.
Temperature evolution of the NMR spectra reveals the existence of the magnetic
phase transitions. At high temperature the system is in the paramagnetic phase,
characterized by the appearance of a single peak and also the Curie-Weiss law. At
low temperature, two well separated triplet of peaks clearly identifies a commen-
surate magnetic order, and the negative Knight shifts from both triplets indicates
ferromagnetism, in agreement with the magnetization measurement. Furthermore,
the quadrupole splitting confirmed from angular dependence experiment revealed a
structural phase transition for the first time. Quantum models suggest tetragonal
symmetry at temperatures above Tc and extends well beyond it, and orthorhombic
symmetry in the ordered state. We place upper bound on such tetragonal distor-
tion by considering the possibility that some small geometrical distortion is masked
within the NMR linewidth, induced by the dipole-dipole coupling. The T1 values
as a function of temperature below transition recognize a Kramers doublet as the
ordered state, in agreement with the specific heat results and theoretical findings.
In 3d systems, this lifting of energy degeneracy is usually due to the Jahn-Teller
effect. However, quantum calculation also suggests that this lifting is from a type
of orbital ordering and the atomic displacements are a secondary parameter. The
relaxation data also confirms the existence of strong spin-phonon coupling from the
temperature dependence, which is necessary in inducing a nematic phase prior to
137
the FM order formation.
The angular dependence of uniform and staggered internal fields imposes strin-
gent constraints on the possible nature of both the structural and magnetic ordered
states. It is a surprise at the first glance that, at certain angle (θ ≈ 60 ), the
staggered field varnishes. Indeed, our numerical calculation reveals that, with the
existence of one structural environment for 23Na nuclei, the only model that can
account for all the experimental results is an exotic canted FM order. Such state is
predicted to occur due to multipolar spin-spin interactions for spins on the frustrated
fcc lattice. Meanwhile, this order is accompanied by a hyperfine tensor with symme-
try that cannot be accounted for by neither tetragonal nor orthorhombic structure.
This might be a result of some complex orbital ordering as suggested by theory,
which induces additional anisotropy in the transferred hyperfine interactions. How-
ever it is hard to include this orbital ordering into the tensor-based interpretation
of symmetry, since the hyperfine tensor is merely a phenomenal parameter. In addi-
tion, our model requires the orientation plane containing the electron spins to follow
the applied field direction. This can be explained as a way for magnetic moments to
minimize the free energy.
The large distribution of second moments during the transition region could indi-
cate the presence of the spin nematic order induced by strong spin-phonon coupling,
although our current data does not provide a direct identification of such phase
and therefore could not draw a definitive conclusion. Whatever the distribution of
electron spin polarization is, it should induce substantially larger magnetic inhomo-
geneity reflected in the broad distribution of local fields at 23Na sites.
In conclusion, our observation of the local cubic symmetry breaking into the
orthorhombic structure and appearance of two-sublattice exotic FM phase for the
138
first time in Ba2NaOsO6 is in line with theoretical predictions based on quantum
models with multipolar magnetic interactions. Future research could focus on ac-
quiring a more complete picture of the angular dependence of relaxation rate. It
should provide enlightening information on interpreting the relaxation mechanism
during transition, and further infer the lattice/magnetic orders. Furthermore, NMR
under pressure would be very useful as a systematically controlled artificial lattice
strain could alter the anisotropy induced by the putative nematic order, similar to
pressurized resistivity measurement done on FeSc [69–71].
139
Appendix A
Ion Positions in a Ba2NaOsO6 Unit
Cell
It is important in both point charge and lattice sum simulations to assign the posi-
tions of all the ions within an unit cell. We list the position, occupation ratio and
charge value in the following tables.
Na x y z ratio Charge Na x y z ratio Charge0 0 0 0 1/8 +1 1 1 0 0 1/8 +12 0 1 0 1/8 +1 3 1 1 0 1/8 +14 1/2 1/2 0 1/2 +1 5 1/2 0 1/2 1/2 +16 0 1/2 1/2 1/2 +1 7 1/2 1 1/2 1/2 +18 1 1/2 1/2 1/2 +1 9 0 0 1 1/8 +110 1 0 1 1/8 +1 11 0 1 1 1/8 +112 1 1 1 1/8 +1 13 1/2 1/2 1 1/2 +1
Table A.1: Position, occupation ratio and charge values of 14 Na ions.
Ba x y z ratio Charge Ba x y z ratio Charge14 1/4 1/4 1/4 1 +2 15 3/4 1/4 1/4 1 +216 1/4 3/4 1/4 1 +2 17 3/4 3/4 1/4 1 +218 1/4 1/4 3/4 1 +2 19 3/4 1/4 3/4 1 +220 1/4 3/4 3/4 1 +2 21 3/4 3/4 3/4 1 +2
Table A.2: Position, occupation ratio and charge values of 8 Ba ions.
Os x y z ratio Charge Os x y z ratio Charge22 1/2 0 0 1/4 +7 23 0 1/2 0 1/4 +724 1/2 1 0 1/4 +7 25 1 1/2 0 1/4 +726 0 0 1/2 1/4 +7 27 1 0 1/2 1/4 +728 0 1 1/2 1/4 +7 29 1 1 1/2 1/4 +730 1/2 1/2 1/2 1 +7 31 1/2 0 1 1/4 +732 0 1/2 1 1/4 +7 33 1/2 1 1 1/4 +734 1 1/2 1 1/4 +7
Table A.3: Position, occupation ratio and charge values of 13 Os ions.
141
O x y z ratio Charge O x y z ratio Charge35 1/4 0 0 1/4 -2 36 0 1/4 0 14 -237 3/4 0 0 1/4 -2 38 0 3/4 0 1/4 -239 1/2 1/4 0 1/4 -2 40 1/4 1/2 0 1/2 -241 1/2 3/4 0 1/2 -2 42 3/4 1/2 0 1/2 -243 1 1/4 0 1/4 -2 44 1/4 1 0 1/4 -245 1 3/4 0 1/4 -2 46 3/4 1 0 1/4 -247 0 0 1/4 1/4 -2 48 1/2 0 1/4 1/2 -249 0 1/2 1/4 1/2 -2 50 1 0 1/4 1/4 -251 0 1 1/4 1/4 -2 52 1/2 1/2 1/4 1 -253 1 1/2 1/4 1/2 -2 54 1/2 1 1/4 1/2 -255 1 1 1/4 1/4 -2 56 1/4 0 1/2 1/2 -257 0 1/4 1/2 1/2 -2 58 3/4 0 1/2 1/2 -259 0 3/4 1/2 1/2 -2 60 1/2 1/4 1/2 1 -261 1/4 1/2 1/2 1 -2 62 1/2 3/4 1/2 1 -263 3/4 1/2 1/2 1 -2 64 1 1/4 1/2 1/2 -265 1/4 1 1/2 1/2 -2 66 1 3/4 1/2 1/2 -267 3/4 1 1/2 1/2 -2 68 0 0 3/4 1/4 -269 1/2 0 3/4 1/2 -2 70 0 1/2 3/4 1/2 -271 1 0 3/4 1/4 -2 72 0 1 3/4 1/4 -273 1/2 1/2 3/4 1 -2 74 1 1/2 3/4 1/2 -275 1/2 1 3/4 1/2 -2 76 1 1 3/4 1/4 -277 1/4 0 1 1/4 -2 78 0 1/4 1 1/4 -279 3/4 0 1 1/4 -2 80 0 3/4 1 1/4 -281 1/2 1/4 1 1/2 -2 82 1/4 1/2 1 1/2 -283 1/2 3/4 1 1/2 -2 84 3/4 1/2 1 1/2 -285 1 1/4 1 1/4 -2 86 1/4 1 1 1/4 -287 1 3/4 1 1/4 -2 88 1/4 1 1 1/4 -2
Table A.4: Position, occupation ratio and charge values of 54 O ions.
142
Appendix B
Lattice Sum Algorithm
Here we list an example of lattice sum procedure
(1) By choosing the number of Na sites in each lattice direction, we initialize the
crystal structure as shown in Figure B.1 (N = 2 here). The yellow spheres are
Na atoms and blue atoms are Os atoms. In order to have the correct boundary
conditions, the 8 Na atoms in the center (unrendered yellow spheres) are the ones
actually used for calculation. The program iterates through all the 8 atoms, and for
each of them calculates the local field generated by hyperfine interaction with the
neighboring six moments from Os sites.
Figure B.1: Schematic for lattice construction in lattice sum method. Blue spheres are Os ionsand yellow spheres are Na ions. The internal field from hyperfine coupling is calculated on thepositions of the eight unrendered Na ions.
(2) Next we will need to set up our hyperfine tensor K, which is a 3x3 matrix.
The program provides the freedom of setting all six tensors separately or all together
for convenience.
(3) Then we should assign the magnetic moments to each Os atoms. Because each
Os atoms has the same main component parallel to the each other in the direction
of applied field, we only need to assign the additional small moments to the lattice.
144
The main component will be directly included in the calculation.
Here is the pseudo-code for a typical calculation: - Iterate through each Os site
to be calculated in the pre-determined order based on magnetic order, and assign
the amplitude of the moment as the value to each site:
for Os atom at (i,j,k): Lattice[i][j][k] = m*phaseN . //phase = -1 if AFM.
- Assign values to all the HF tensor elements and distortion settings.
- For each angle θ of applied field, iterate through each Na site (e.g. 8 sites in
Figure B.1), and for each of them, iterate through each of the 6 nearest Os atoms
and calculate local field/frequency shift generated from HF interaction:
for angle θ:
for ith Na:
for jth Os (6 in total):
~H iloc = Kij · (M(θ)h|| +mjh⊥) (B.1)
save the result in an array with the size of number of Na atoms
calculate the line splitting and first moment shift (∆ωi = γh · ~H iloc)
plot simulated angle dependence.
145
Appendix C
Hyperfine Tensor Symmetry
In the following calculation of hyperfine tensor, we often need to transform the hy-
perfine tensor from the bond coordinate to the crystalline coordinate, which requires
a rotation operation. Mathematically speaking, to transform a tensor K from coor-
dinate system S into coordinate system S’, the relation between the new tensor form
K′ and the original tensor is:
K′ = RT · K ·R . (C.1)
R here is the rotation matrix which reflects the relation between the two sets of
base vectors in the two coordinate systems. Assuming the coordinate system S’ is
acquired by rotating an angle θ around axis of rotation ~u in S, then the rotation
matrix R has the form:
R =
cos θ + u2
x (1− cos θ) uxuy (1− cos θ)− uz sin θ uxuz (1− cos θ) + uy sin θ
uyux (1− cos θ) + uz sin θ cos θ + u2y (1− cos θ) uyuz (1− cos θ)− ux sin θ
uzux (1− cos θ)− uy sin θ uzuy (1− cos θ) + ux sin θ cos θ + u2z (1− cos θ)
.(C.2)
where R[i]T is the transposed form of ith row.
C.1 Tetragonal symmetry
In tetragonal symmetry, assuming c-axis is the inequivalent axis, the hyperfine inter-
action between the Na site and the four Os sites on x-y plane has the same hyperfine
tensor in the crystalline coordinate. The two Os along c-axis is either further away
or closer to the Na, which results in a quantitative difference in the hyperfine tensor,
but reserves the same symmetry.
147
z
x y
K1
K6
K5
K4K3
K2
Figure C.1: Schematic of coordination of the nearest neighbor Os sites around an Na nucleus.
Now we start with the hyperfine tensor between Na and the +x Os moment. The
tensor has the form:
K1 =
Kaa Kab Kac
Kba Kbb Kbc
Kca Kcb Kcc
, (C.3)
and K2, K3, K4, K5 and K6 corresponds to the tensors between Na and Os along +y,
-x, -y, +z and -z axes, respectively, as illustrated in Fig. C.1
Tetragonal structure has C4 symmetry, with axis of rotation ~u = (0, 0, 1) and
θ = 90. The corresponding rotation matrix can be calculated from Equation C.2
and has the form
R =
0 −1 0
1 0 0
0 0 1
. (C.4)
148
Then K2, K3 and K4 can be calculated based on Equation C.1:
K2 =RT ·K1 ·R =
Kbb −Kba Kbc
−Kab Kaa −Kac
Kcb −Kca Kcc
K3 =RT ·K2 ·R =
Kaa Kab −Kac
Kba Kbb −Kbc
−Kca −Kcb Kcc
K4 =RT ·K3 ·R =
Kbb −Kba −Kbc
−Kab Kaa Kac
−Kcb Kca Kcc
. (C.5)
Now define the +z tensor in the bond coordinate to be:
K5 =
K ′aa K ′ab K ′ac
K ′ba K ′bb K ′bc
K ′ca K ′cb K ′cc
, (C.6)
whose values are different from K1 but probably not by too much, since the possible
distortions along c-axis are very small. The bond coordinate for the +z Os can
be transformed into the crystalline coordinate by using a rotation matrix with ~u =
(0, 1, 0) and θ = 90.
R =
0 0 1
0 1 0
−1 0 0
. (C.7)
149
By using Equation C.1 again, we get K5 in the crystalline coordinate:
K5 =RT ·K5 ·R =
K ′cc −K ′cb −K ′ca
−K ′bc K ′bb K ′ba
−K ′ac K ′ab K ′aa
, (C.8)
and from mirror symmetry we can easily get:
K6 =
K ′cc K ′cb K ′ca
K ′bc K ′bb K ′ba
K ′ac K ′ab K ′aa
. (C.9)
If we assume each Os carries the same moment ~m (pure ferromagnetic ordering),
then we get the internal local field
Hlocal =∑i
Ki × ~mi =
K11 K12 0
−K12 K22 K23
0 K32 K33
× ~m , (C.10)
where K11 ≡ 2(Kaa + Kbb + K ′cc), K12 = 2(Kab −Kba), K22 = 2(Kaa + Kbb + K ′bb),
K23 = 2K ′ba, K32 = 2K ′ab and K33 = 4Kcc + 2K ′aa.
In our simulation regarding the angle dependence, the magnetic order is deter-
mined to be canted ferromagnetic. Each Os has the same component ~m‖ parallel to
the applied field direction, and also has an alternating component ~m⊥. Let’s assume
that site 1, 5 and 6 carry the moment ~mA, and site 2, 3, and 4 carry moment ~mB,
150
where ~mA = ~m‖ + ~m⊥, ~mB = ~m‖ − ~m⊥. We get
Hlocal =
K11 K12 0
−K12 K22 K23
0 K32 K33
× ~m‖ +
2Kbb − 2K ′cc −2Kba −2Kac
−2Kab 2Kaa − 2K ′bb −2Kbc − 2K ′ba
−2Kca −2Kcb − 2K ′ab 2Kcc − 2K ′aa
× ~m⊥ .
(C.11)
this obviously does not have the symmetry we acquired from the result of lattice
sum simulation.
C.2 Orthorhombic Symmetry
In the case of orthorhombic crystalline symmetry, we have three sets of inequiva-
lent hyperfine tensors: ±x, ±y and ±z. Each pair of moments satisfies the mirror
symmetry. Using the same K1 and following the same calculation procedures in
tetragonal’s case, we get:
K1 =
Kxaa Kx
ab Kxac
Kxba Kx
bb Kxbc
Kxca Kx
cb Kxcc
, K2 =
Kybb −Ky
ba Kybc
−Kyab Ky
aa −Kyac
Kycb −Ky
ca Kycc
K3 =
Kxaa −Kx
ab −Kxac
−Kxba Kx
bb Kxbc
−Kxca Kx
cb Kxcc
, K4 =
Kybb Ky
ba −Kybc
Kyab Ky
aa −Kyac
−Kycb −Ky
ca Kycc
K5 =
Kzcc −Kz
cb −Kzca
−Kzbc Kz
bb Kzba
−Kzac Kz
ab Kzaa
, K6 =
Kzcc Kz
cb Kzca
Kzbc Kz
bb Kzba
Kzac Kz
ab Kzaa
, (C.12)
151
here the superscripts x, y and z correspond to the three crystal axes.
In the pure ferromagnetic ordering case, we get the internal local field to be:
Hlocal =∑i
Ki × ~mi =
K11 0 0
0 K22 K23
0 K32 K33
× ~m , (C.13)
where K11 = 2(Kxaa+Ky
bb+Kzcc), K22 = 2(Kx
bb+Kyaa+Kz
bb), K23 = 2(Kxbc−Ky
ac+Kzba),
K32 = 2(Kxcb −Ky
ca +Kzab) and K33 = 2(Kx
cc +Kycc +Kz
aa).
If the magnetic order is canted, as discussed in last section, with ~mA = ~m‖+ ~m⊥
for Os sites 1, 5, and 6, ~mB = ~m‖ − ~m⊥ for sites 2, 3, and 4, then the overall tensor
is:
Hlocal =
K11 0 0
0 K22 K23
0 K32 K33
× ~m‖ +
2Kz
cc − 2Kybb 2Kx
ab 2Kxac
2Kxba 2Kz
bb − 2Kyaa 2Ky
ac + 2Kzba
2Kxca 2Ky
ca + 2Kzab 2Kz
aa − 2Kycc
× ~m⊥ .
(C.14)
The only way this will satisfy the symmetry of the simulated tensor is to have
Kxab = −Kx
ba, which is unlikely.
152
Appendix D
Method of Data Averaging for
RDNMR Data
In continuous wave (CW) RDNMR measurements, multiple measurements were done
at the same frequency. We want to average these measurements and represent the
data value as the average value and the distribution as error bar (one standard
deviation). Assuming for each frequency there are m measurements, the averaged
resistance would be
R =
∑mi=1 Ri
m, (D.1)
and standard deviation is
SD =
√∑mi=1(Ri − R)2
m, (D.2)
and by doing this, for each frequency we have a distribution of resistance with µ =
R, σ = SD.
Furthermore, sometimes the step size of frequency sweep is very small and makes
plotted data points too dense. For this reason it’s reasonable to average the resistance
data over the frequency domain. For every consecutive n frequencies, the averaged
resistance would be
R =
∑ni=1 Ri
n, (D.3)
Assuming the n points being averaged are independent random variables, the new
averaged value has variance
V ar(R) = V ar(
∑ni=1 Ri
n)
=1
n2
n∑i=1
V ar(Ri), (D.4)
where Var(Ri) is variance = SD2 in which SD is already known. Therefore after this
154
“frequency averaging” the standard deviation (error bar) would be
SD =√V ar(R) =
1
n
√√√√ n∑i=1
V ar(Ri) . (D.5)
The function implemented to realized this averaging works for any value of n. From
the final equation we can see that, assuming all the n random variables are identically
independent, the standard deviation is proportional to 1√n. So the larger n is, the
smaller standard deviation will be, and this is qualitatively correct according to the
central limit theorem (CLT). However in our situation, n can not be chosen to be
too large, otherwise the n Ri random variables will not be independent and we will
have to take into account the covariance between them, which deviates from our
motivation. The appropriate choice of n for best visualization purpose in under this
constraint and is case sensitive.
155
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156
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