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Tailoring asymmetry in light-matter interaction with atomic-scale lattice modulations

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Content Tailoring Asymmetry in Light-Matter Interaction with Atomic-Scale Lattice Modulations
by
Boyang Zhao
A Dissertation Presented to the
FACULTY OF THE USC GRADUATE SCHOOL
UNIVERSITY OF SOUTHERN CALIFORNIA
In Partial Fulfillment of the
Requirements for the Degree
DOCTOR OF PHILOSOPHY
MATERIALS SCIENCE
May 2024
Copyright 2024 Boyang Zhao

ii
Acknowledgments
The author acknowledges Dr. Jayakanth Ravichandran, chair of the dissertation defense
committee, for his support and instructions on the research. The author also acknowledges Dr.
Alan E. Willner, Dr. Yu-Tsun Shao, and Dr. Zhenglu Li for being the dissertation defense
committee members, Dr. Jahan M. Dawlaty, Dr. Andrea Martin Armani, Dr. Paulo Branicio, and
Dr. Shaama Sharada for being the qualification exam committee.
The author is grateful for the support of his parents for bringing me to the world, raising me
with imperishable love, and guiding me to the gallery of science. The author is thankful to his
teachers and professors for educating me with their knowledge and passion, and to his friends for
their accompany through even the toughest days and nights.
The author gratefully acknowledges the use of facilities at’s Lab, Core Center for Excellence
in Nano Imaging at the University of Southern California, SLAC National Accelerator Laboratory
(Dr. Sang-Jun Lee, Dr. Jun-Sik Lee, Dr. Chang-Tai Kuo, Dr. Donghui Lu, Dr. Makoto Hashimoto,
Dr. Ming-Fu Lin, Dr. Sharon Bone), Oak Ridge National Laboratory (Dr. Raphael P. Hermann,
Dr. Michael E. Manley, Dr. Bryan C. Chakoumakos, Dr. Rama K. Vasudevan), Advance Light
Source of Lawrence Berkeley National Laboratory (Dr. Simon J. Teat, Dr. Nick S. Settineri), and
Advanced Photon Source of Argonne National Laboratory (Dr. Haidan Wen) for the results
reported in this report. The author also acknowledges the collaborative work and insightful
discussion with Dr. Rohan Mishra, Dr. Mikhail Kats, Dr. Han Wang, Dr. Nuh Gedik, Dr. Patrick
E. Hopkins, Dr. Rafael Jaramillo, Dr. Stephen Cronin, Dr. Brent C. Melot, Dr. Raphaële J. Clément,
and Dr. Di Xiao.

iii
The author appreciates the mentoring from Shanyuan Niu, Thomas Orvis, and Yang Liu during
the startup of the doctorate project. The author also thanks the technical assistance from Jieyang
Zhou, Qinai Zhao, Zhengyu Du, Huandong Chen, Shantanu Singh, Mythili Surendran, Harish
Kumarasubramanian, and Claire Wu in collaboration in the related projects.
This work was supported by the Army Research Office (ARO) under award number W911NF19-1-0137 and via an ARO MURI program with award number W911NF-21-1-0327, the National
Science Foundation of the United States under grant number DMR-2122070 and 2122071, and the
USC Provost New Strategic Directions for Research Award.

iv
Table of Contents
Acknowledgments...................................................................................................................... ii
List of Tables............................................................................................................................ vii
List of Figures......................................................................................................................... viii
Abstract .................................................................................................................................... xii
Chapter 1. : Introduction........................................................................................................... 1
1.1 Background and motivation........................................................................................ 1
1.2 Crystal Structure and Light-Matter Interaction Asymmetry....................................... 4
1.3 Quasi-1D Perovskite Chalcogenides........................................................................... 6
1.4 Unconventional Optical, Electronic, and Thermal Properties.................................... 9
1.4.1 Underestimated optical anisotropy. ........................................................................ 9
1.4.3 Glass-like thermal conductivity and lattice disorder ............................................ 11
Chapter 2. Material Synthesis and Characterization .............................................................. 14
2.1 Chemical Vapor Transport Crystal Growth.............................................................. 14
2.2 Crystal Orientation Characterization ........................................................................ 15
2.2.1 X-ray diffraction (XRD) ....................................................................................... 15
2.2.2 In-plane and out-of-plane orientations.................................................................. 17
2.2.3 Large area reciprocal space mapping.................................................................... 19
2.3 Atomic Scale Structure Analysis.............................................................................. 21
2.3.1 Single crystal X-ray diffraction (SC-XRD).......................................................... 21
2.3.2 Electron distribution and structure determination................................................. 22
2.3.3 X-ray Studies with Synchrotron Radiation........................................................... 24
2.4 Optical Spectroscopy ................................................................................................ 25
2.4.1 Wavelength dispersion of refractive index ........................................................... 25
2.4.2 Fourier-transform infrared spectroscopy (FTIR).................................................. 26
2.4.3 Reflectance/transmittance spectrum anisotropy ................................................... 28
2.4.4 Polarization Resolved Raman Spectroscopy ........................................................ 29
Chapter 3. Giant Modulation of Refractive Index from Correlated Disorder in BaTiS3........ 31
3.1 Introduction............................................................................................................... 31
3.2 Orientation Controlled Crystal Growth of BaTiS3.................................................... 33
3.2.1 Sealed ampoule growth of A1+xTiX3 (A = Sr, Ba; X = S, Se) .............................. 33
3.2.2 Optimized crystal growth...................................................................................... 34
3.2.3 Crystal morphology .............................................................................................. 35
3.3 Structure Reconfiguration via Single Crystal Diffraction ........................................ 36
3.3.1 Precession map and superstructure ....................................................................... 36
3.3.2 Refined BaTiS3 structure ...................................................................................... 38

v
3.3.3 Symmetry breaking and non-thermal disorder. .................................................... 41
3.3.4 Electron density distribution and a-b plane displacements................................... 42
3.3.5 Solid-state Nuclear Magnetic Resonance (ssNMR) ............................................. 44
3.4 Correlated Disorder by Electron Spectroscopy......................................................... 46
3.4.1 Diffuse scattering in electron diffraction .............................................................. 46
3.4.2 Scanning Transmission Electron Microscope....................................................... 47
3.4.3 Correlated Ti a-b plane displacements.................................................................. 49
3.5 First Principles Calculations..................................................................................... 51
3.5.1 Ti a-b plane displacements and optical anisotropy ............................................... 51
3.5.2 Origin of the optical anisotropy ............................................................................ 53
3.5.3 Disordered a-b plane displacements ..................................................................... 54
3.6 Conclusions............................................................................................................... 55
Chapter 4. Phase Transitions and Emergent Polar Vortices in BaTiS3 .................................. 57
4.1 Introduction............................................................................................................... 57
4.1.1 Topological defects............................................................................................... 57
4.1.2 BaTiS3 phase transitions – TiS6 dipole ordering................................................... 58
4.2 BaTiS3 Phase Transitions.......................................................................................... 59
4.2.1 Optical phase transitions....................................................................................... 59
4.2.2 Precession map analysis........................................................................................ 60
4.2.3 Charge-density-wave-like phase transition........................................................... 61
4.2.4 Chiral-achiral structure transition ......................................................................... 66
4.3 Dipole Ordering and Polar Vortices ......................................................................... 68
4.3.1 TiS6 dipoles along the c-axis................................................................................. 68
4.3.2 TiS6 polar vortices along the a-b plane ................................................................. 70
4.3.3 Vortex network and symmetry.............................................................................. 74
4.4 Vortex Phase Stability and Evolution....................................................................... 76
4.4.1 First-principles calculations.................................................................................. 76
4.4.2 Vortex network phase evolution ........................................................................... 78
4.5 Conclusions............................................................................................................... 79
Chapter 5. Chemical Tunability of the Optical Anisotropy in BaTiX3 (X = S, Se)............... 81
5.1 Introduction............................................................................................................... 81
5.2 Crystal Synthesis and Characterizations................................................................... 82
5.2.1 Crystal growth and chemical analysis................................................................... 82
5.2.2 Orientation determination by X-ray diffraction.................................................... 84
5.2.3 Precession map analysis by SC-XRD................................................................... 85
5.2.4 Crystal structure of BaTiSe3 ................................................................................. 87

vi
5.3 BaTiSe3 Optical Anisotropy ..................................................................................... 91
5.3.1 Polarization-dependent Raman anisotropy ........................................................... 91
5.3.2 Fourier-transform infrared spectroscopy .............................................................. 92
5.4 Conclusions............................................................................................................... 95
Chapter 6. Optical Anisotropy of Incommensurate Hexagonal TMPCs Sr1+xTiS3 ................ 96
6.1 Introduction............................................................................................................... 96
6.2 Crystal Structure Determination ............................................................................... 98
6.2.1 Optimized crystal growth...................................................................................... 98
6.2.2 Out-of-plane X-ray diffraction.............................................................................. 99
6.3 Optical Anisotropy.................................................................................................. 107
6.3.1 Fourier transform infrared spectroscopy............................................................. 107
6.3.2 Refractive indices and colossal birefringence..................................................... 109
6.4 Lattice Modulation Induced Optical Anisotropy .................................................... 112
6.4.1 Electronic structure variation.............................................................................. 112
6.4.2 Colossal optical anisotropy................................................................................. 115
6.5 Conclusions............................................................................................................. 117
Chapter 7. Conclusions and Perspectives............................................................................. 119
7.1 Summary................................................................................................................. 119
7.2 Perspectives............................................................................................................. 121
7.2.1 Novel optical properties...................................................................................... 121
7.2.2 Dynamics study with asymmetric light............................................................... 122
References.............................................................................................................................. 123

vii
List of Tables
Table 3.1: A1+xTiX3 (A = Sr, Ba; X = S, Se) of different shapes and orientations.................. 33
Table 3.2: Comparison between candidate space groups for 300K BaTiS3. ........................... 37
Table 3.3: Comparison between candidate space groups for 300K BaTiS3. ........................... 38
Table 3.4: Atomic coordinates and equivalent isotropic atomic displacement parameters for
300K BaTiS3. ................................................................................................................................ 39
Table 3.5: Anisotropic atomic displacement parameters for 300K BaTiS3............................. 41
Table 4.1: Data collection and refinement statistics of BaTiS3 platelet at different
temperatures.................................................................................................................................. 61
Table 4.2: Atomic coordinates and isotropic atomic displacement parameters for 220K
BaTiS3........................................................................................................................................... 62
Table 4.3: Anisotropic atomic displacement parameters for 220K BaTiS3............................. 63
Table 4.4: Atomic coordinates and isotropic atomic displacement parameters for 130K
BaTiS3........................................................................................................................................... 65
Table 4.5: Anisotropic atomic displacement parameters for 130K BaTiS3............................. 66
Table 4.6: Refinement results of 220K-BTS: dipolar displacement analysis of BaS12 and
TiS6. Displacement expressed as polar vectors following the crystal structure of P3c1.............. 71
Table 5.1: Single crystal synchrotron diffraction studies on BaTiSe3 ..................................... 83
Table 5.2: Crystal structure of needle-shaped single crystal BaTiSe3.
[64]
............................... 84
Table 5.3: Anisotropic atomic displacement parameters (ADPs) of needle-shaped single
crystal BaTiSe3.
[64]
....................................................................................................................... 85
Table 6.1: Data collection, intensity statistics, and refinement statistics of SC-XRD on
Sr9/8TiS3. ....................................................................................................................................... 98
Table 6.2: Atomic coordinates and equivalent isotropic atomic displacement parameters for
R3c Sr1.125TiS3............................................................................................................................... 99
Table 6.3: Anisotropic atomic displacement parameters for R3c Sr1.125TiS3........................... 99
Table 6.4: Comparison between Sr and Ba............................................................................ 107

viii
List of Figures
Figure 1.1: Left, human eye imaging......................................................................................... 1
Figure 1.2: Light as an electromagnetic wave. .......................................................................... 2
Figure 1.3: Double refraction and optical anisotropy. ............................................................... 4
Figure 1.4: Comparison of the absolute birefringence of a variety of anisotropic crystals, 2-
D materials, and quasi-1D A1+xTiS3 (A = Sr, Ba) crystals. ............................................................ 5
Figure 1.5: Quasi-1D structure of BaTiS3.................................................................................. 7
Figure 1.6: The linear birefringence, linear dichroism, and normalized dichroism of BaTiS3.
......................................................................................................................................................... 8
Figure 1.7: Commensurate (incommensurate if x is an irrational number) lattice modulation
of Sr1+xTiS3. .................................................................................................................................... 8
Figure 1.8: Calculated optical anisotropy of P63mc-BaTiS3.................................................... 10
Figure 1.9: Electronic phase transitions of BaTiS3. ................................................................. 11
Figure 1.10: Lattice disordered of BaTiS3 in the neutron scattering study of BaTiS3............. 12
Figure 2.1: Illustrations of chemical vapor transport............................................................... 15
Figure 2.2: Bragg diffraction. .................................................................................................. 16
Figure 2.3: Out-of-plane XRD of (a) BaTiS3 needles, (b) BaTiS3 100 platelet, and (c)
BaTiS3 001 platelet. ...................................................................................................................... 17
Figure 2.4: Out-of-plane and in-plane diffraction for thin film. .............................................. 18
Figure 2.5: In-plane orientation determination of (100) and (001) BaTiS3 plates................... 19
Figure 2.6: Reciprocal space mapping of different geometry.................................................. 20
Figure 2.7: Rotational XRD Mapping ( -2 scan while varying ), and a sign of
superstructure................................................................................................................................ 21
Figure 2.8: Single crystal diffraction. ...................................................................................... 22
Figure 2.9: Real space structure determination and refinement of single crystal X-ray
diffraction...................................................................................................................................... 23
Figure 2.10: Schematic of a synchrotron radiation source....................................................... 25
Figure 2.11: Refractive index as a function of frequency........................................................ 26
Figure 2.12: Illustration of the polarization-dependent infrared reflection and transmission
geometry. ...................................................................................................................................... 27
Figure 2.13: Schematic representation of an FTIR spectrometer. ........................................... 28
Figure 2.14: Representative optical spectroscopy of BaTiS3 crystals by polarizationdependent FTIR. ........................................................................................................................... 29

ix
Figure 2.15: Raman spectroscopy............................................................................................ 30
Figure 3.1: Comparison of the birefringence of BaTiS3 against other birefringent crystals.... 33
Figure 3.2: BaTiS3 crystals with different morphologies and orientations.............................. 36
Figure 3.3: Reciprocal-space precession maps for h0l- and hk1-type reflections from
synchrotron single-crystal diffraction. .......................................................................................... 37
Figure 3.4: Schematic of the BaTiS3 crystal structure projected onto the a-b plane from the
refined diffraction results.............................................................................................................. 40
Figure 3.5: Electron density maps analyses of Ti01 (in chain A) and Ti02 (in chain B) in
BaTiS3........................................................................................................................................... 43
Figure 3.6: Static 47/49Ti solid-state NMR (ssNMR) spectra as direct evidence of the a-b
plane displacements...................................................................................................................... 45
Figure 3.7: Electron diffraction pattern observed in TEM with streaky diffuse scattering. .... 47
Figure 3.8: STEM and atomic displacement analysis of BaTiS3............................................. 48
Figure 3.9: Ti displacement vector map calculated from the HAADF-STEM image of
BaTiS3 along [001]-zone axis overlaid onto the original HAADF image .................................... 49
Figure 3.10: Linear correlation map for the Ti a-b plane displacement vectors...................... 50
Figure 3.11: Optical anisotropy enhanced by Ti a-b plane displacements.............................. 52
Figure 3.12: Optical anisotropy of BaTiS3 structures with and without ordered Ti a-b plane
displacements................................................................................................................................ 52
Figure 3.13: Electron distribution and optical anisotropy........................................................ 53
Figure 3.14: Optical anisotropy of disordered ab initio molecular dynamics (AIMD)
simulations.................................................................................................................................... 55
Figure 4.1: Optical anisotropy as a function of temperature.................................................... 60
Figure 4.2: CDW order evolution revealed by single crystal X-ray diffraction of BaTiS3 ..... 61
Figure 4.3: “Ferroic” ordering of c-axis dipolar TiS6 octahedra along the a-b plane in BaTiS3.
....................................................................................................................................................... 69
Figure 4.4: Electron density map analysis on off-centric atomic displacements..................... 72
Figure 4.5: Polarization Textures in P3c1-BaTiS3................................................................... 73
Figure 4.6: Experimental symmetry-controlled vortex -anti-vortex polarization network in
CDW phase of BaTiS3 .................................................................................................................. 75
Figure 4.7: Vortex network of P3c1BaTiS3 ............................................................................. 76
Figure 4.8: First-principles calculations of the symmetry-controlled vortex-anti-vortex
polarization network in the CDW phase of BaTiS3 ...................................................................... 77
Figure 4.9: Temperature-driven melting and crystallization of dipolar textures in BaTiS3..... 79

x
Figure 5.1: BaTiSe3 crystal growth.......................................................................................... 83
Figure 5.2: Characterization of BaTiSe3 single crystals .......................................................... 83
Figure 5.3: XRD study of BaTiSe3. ......................................................................................... 84
Figure 5.4: Characterization of BaTiSe3 single crystals .......................................................... 85
Figure 5.5: Refined crystal structure of BaTiSe3. .................................................................... 90
Figure 5.6: Electron density map of BaTiSe3 near Ti01 and Ti02 projected along the a-b
plane.............................................................................................................................................. 91
Figure 5.7: Room-temperature polarization-resolved Raman study of BaTiSe3 ..................... 92
Figure 5.8: Polarization-resolved FTIR (a) transmittance and (b) reflectance spectra show
large anisotropy between the ordinary (⟂ c) and the extraordinary (|| c) polarization ................. 93
Figure 5.9: Optical anisotropy of BaTiSe3 ............................................................................... 94
Figure 6.1: Sr1+xTiS3 crystal structure and morphology .......................................................... 97
Figure 6.2: Sr1+xTiS3 crystal 00l XRD. Indices are labeled as hkl1l2 of incommensurate
lattice a, b, c1, c2 according to 2 ............................................................................................... 100
Figure 6.3: Precession map of STS crystal along the 0kl, h0l, and hk0 reciprocal planes..... 101
Figure 6.4: Structural modulation in Sr9/8TiS3 ....................................................................... 105
Figure 6.5: Structural modulations in Sr9/8TiS3. .................................................................... 107
Figure 6.6: Polarization-dependent infrared a) transmittance and b) reflectance of Sr9/8TiS3.
..................................................................................................................................................... 108
Figure 6.7: Total signal (R + T) from reflectance and transmittance spectra of Sr9/8TiS3
crystal for polarization perpendicular and parallel to the c-axis................................................. 109
Figure 6.8: Uniaxial anisotropic optical properties of Sr9/8TiS3 ............................................ 110
Figure 6.9: Comparison of the birefringence of two representative hexagonal quasi-1D
chalcogenides (Sr9/8TiS3 and BaTiS3) with highly anisotropic 2D materials (h-BN and MoS2).110
Figure 6.10: Comparison of the refractive index of A1+xTiX3 (A = Sr, Ba; X = S, Se)......... 111
Figure 6.11: Electronic structure and optical properties of modulated Sr9/8TiS3................... 113
Figure 6.12: Ti-L2,3 EELS ...................................................................................................... 114
Figure 6.13: Crystal-field splitting by Ti–Ti distance modulation. ....................................... 115
Figure 6.14: Calculated complex dielectric function for polarization perpendicular ( ⊥)
and parallel (||) to the c-axis of the hypothetical stoichiometric SrTiS3 and modulated
Sr9/8TiS3, compared to the experimental results (black dashed line).......................................... 116
Figure 6.15: Optical anisotropy comparison.......................................................................... 117

xi

xii
Abstract
Vision, a graceful gift from nature, utilizes the ubiquitous yet mysterious electromagnetic
radiation – light to perceive wonders. The pupil, lens, and retina spatially resolve the intensity and
frequency of incoming light to perceive matter that emits or reflects. Such intuitive optical
processes lead to generations of imaging, sensing, and even free space telecommunications.
However, the waveform of electromagnetism also introduces polarization and phase to fulfill the
quantum states of light. Polarimetry, which decodes the polarization state of light unveils richer
photon characteristics and is an important ingredient for quantum computing or communication
with photons. Towards this goal, crystalline optics with considerable optical anisotropy that
spatially splits opposite linearly, or circularly polarized light are the fundamental building blocks
of polarimetry.
Quasi-one-dimensional (quasi-1D) perovskite chalcogenide, BaTiS3, adopts chains of faceshared TiS6 octahedra along the c-axis and is recently discovered to adopt giant linear optical
anisotropy. We further explored BaTiSe3 and Sr1+xTiS3, which fall into the class of isostructural
A1+xTiX3 (A = Sr, Ba; X= S, Se) quasi-1D chalcogenides to expand the magnitude and spectrum
range of optical anisotropy. While investigating the origin of the giant infrared (IR) optical
anisotropy of these materials, we discovered rich structural anomalies that break the symmetry of
A1+xTiX3 and give rise to the optical anisotropy. Such as the antiparallel chains displacements,
antipolar TiX6 octahedral displacements, disordered a-b plane Ti displacements in BaTiS3 and
BaTiSe3, and the incommensurate lattice modulations in Sr9/8TiS3. First-principles calculations,

xiii
when accounted for the observed structural anomalies, show optical anisotropy consistent with
experiments, both real and imaginary parts, bringing A1+xTiX3 to the record-high IR birefringence.
At the same time, electrical, thermal, and optical investigation as a function of temperature
discovered charge-density-wave (CDW) phase transitions in BaTiS3 and BaTiSe3. Structural
analysis revealed the displacements of TiX6 chains, dipole ordering of TiX6-octahedra, and more
importantly, the a-b plane polarization vortex lattice to be the defining characteristic of the CDW
phase. First-principles calculations show the phase stability of observed a-b plane polarization
vortices of BaTiS3. During heating and cooling, the polarization vortices melt and nucleate at very
similar temperatures, revealing the low energy barrier of such transition, while demonstrating
reliable control of the polarization vortex lattice.
BaTiS3 demonstrates another chiral phase at lower temperatures. The chiral BaTiS3 breaks
down the polarization vortices while introducing optical activity with the broken symmetry.
Similarly, circularly polarized light sensitivity can be selectively enhanced by controlling the
underlying helical polarizations, such as in ZrOS. We carefully examine the helical polarizations
and resolve the handedness of the chiral phonons in ZrOS by the pseudo-angular-momentum
transfer in Raman spectroscopy.

1
Chapter 1. : Introduction
1.1 Background and motivation
“Let there be light” and there was light. Light, separating us from darkness, gifts us vision.
Beyond human vision, our efforts to generate, control, and detect light never cease. For
ophthalmologists[1], light is a colorful ray through pupils and lenses, whose projection on the retina
[Figure 1.1] lets us perceive the shape and the color of the surroundings that emit or reflect light.
One thus invented geometric optics and color-sensitive photodetectors. For physicists[2], light is
electromagnetic waves [Figure 1.2] traveling at the speed of light, carrying quantized energy, or
photons, from point A to point B. One thus designed diffractometers and spectrometers to lay out
the spectrum dispersion beyond vision. Our deepened understanding of how light interacts with
matter provides us with richer approaches to the mystery of nature.
Figure 1.1: Left, human eye imaging. Source: fortworthastro.com. Right: Geometric optics imaging
imitating human eyes. Source: hyperphysics.phy-astr.gsu.edu.

2
Despite being almost perfect for visible light sensing, human eyes lack sensitivity to
polarization, a fundamental characteristic of electromagnetic waves. Many animals, such as fish,
insects, birds, crabs, and even shrimps, have “secret” polarization filters for well-tuned
polarization vision[3], helping them see beyond “color” for navigation[4], searching food [5]
,
adaptation to aquatic lifestyle[6], underwater vision[7] and even camouflage[8]. The study of such a
“secret” revealed how to leverage the asymmetry of light: measuring and interpreting the
polarization of light: linearly, circularly, or elliptically polarized light using polarization-resolved
optics. A common ingredient for polarization-resolved optics are anisotropic crystals, or
birefringent crystals in general.
Figure 1.2: Light as an electromagnetic wave. Source: science.nasa.gov.

3
Double refraction, observed as early as 1669[9], is a phenomenon of splitting light into two rays
while passing through a birefringent crystal, in this case, calcite [Figure 1.3(a)]. Such anisotropic
“bending” of light was due to the anisotropic refraction of the transverse[10,11] electromagnetic
wave polarized parallel (extraordinary) and perpendicular (ordinary) to the optical axis of CaCO3,
facilitating polarization control with birefringent optics [Figure 1.3(b)]. Bragg[12] originated such
anisotropic refraction of light in calcite-type carbonates (RCO3; R = Mg, Zn, Fe, and others) as the
interaction of dipole excitations around the oxygen atoms within the planar carbonate ions (CO3
2-
),
which are all oriented perpendicular to the optic axis of the crystal[12–15], and thus unveiled the
connections between the asymmetry of light-matter interaction and the asymmetry in the atomistic
crystal structure.
The change of the magnitude and phase of the light can thus be expressed in terms of the electric
field of the electromagnetic field[2]:
(,) = [0
(−)
] = [0
(
2
0
−)
] = [0
(
2(+)
0
−)
] =
0
−2/0[
(
2
0
−+
2(−1)
0
)
],
where , , , , and 0 are the distance, time, wavevector, angle frequency, and wavelength
in the vacuum of a propagating electromagnetic wave, respectively. Then real and imaginary part
of the refractive index, = + , determines the phase retardation 2( − 1)/0 and
amplitude attenuation
−2/0 of the electromagnetic wave propagating in a transparent medium.
They are commonly known as refractive index () and extinction coefficient (). The magnitude
of their asymmetry is thus quantified as the difference between the refractive indices, both real

4
(, or birefringence) and imaginary (, or dichroism), exhibited by materials. Crystals with
such asymmetries, such as calcite (CaCO3), are generally recognized as birefringent crystals.
Figure 1.3: Double refraction and optical anisotropy. (a) Birefringent CaCO3 crystal showing double
refraction. (b) Polarization control by optical anisotropy.
In modern optics, besides polarization control[16–18] (i.e. wave plates, retarders, polarizing beam
splitters, etc.), birefringent crystals with large birefringence are also sought after for nonlinear
optics and quantum optics (e.g. for phase matching[19–21] and production of entangled photons[22]),
micromanipulation[23], and as a platform for unconventional light-matter couplings, such as
Dyakonov-like surface polaritons[24] and hyperbolic phonon polaritons[25–27]. Beyond the
transmittance window, the dichroism window of birefringent crystals demonstrates anisotropic
photo-response, i.e. photocurrent, which can be engineered to realize polarization-sensitive
photodetectors[28]. Therefore, the search for birefringent crystals across various transparency
regions and the exploration of birefringence tuning mechanisms have been an active area of
photonics research.
1.2 Crystal Structure and Light-Matter Interaction Asymmetry
Carbon is lustrous and opaque in the form of graphite but turns transparent when presented as
diamonds. Similarly, the crystalline structures of matter play a pivotal role in light-matter

5
interactions. Although long been discovered as anisotropic to light, the mystery of controlling such
asymmetry in the light-matter interaction was first revealed by considering the crystalline
structures of calcite (CaCO3) crystals.
The anisotropic nature of bonding in CaCO3, specifically the anisotropic neighborhood of CO3
2-
,
led to the birefringence consistent with a simple geometric analysis of polarizability based on
atomic positions[29]. A wide range [Figure 1.4] of birefringent crystals[30–38] have been achieved
based on this principle and facilitates a wide range of birefringent optics. Some, e.g., calcite
(CaCO3), quartz (SiO2), rutile (TiO2), YVO4, ⍺-BBO, and LiNbO3 are commercially available as
optical substrates for asymmetric optics but fall shorts of infrared (IR) optical transparency,
especially for mid-wave infrared (MWIR) and long-wave infrared (LWIR) wavelength spectrum.
Figure 1.4: Comparison of the absolute birefringence[39] of a variety of anisotropic crystals, 2-D materials,
and quasi-1D A1+xTiS3 (A = Sr, Ba) crystals.

6
Although fabrication approaches such as nano optics[40,41] and liquid crystals[42] can be used to
engineer the enhanced birefringence, a fundamental breakthrough towards a class of tunable
optical asymmetry in the MWIR to LWIR spectrum range puts high demands on chalcogenide (i.e.,
compounds with S, Se, and Te) materials with novel crystalline structures. Towards this path,
layered “van der Waals” materials, or two-dimensional (2-D) materials, with strong intra-layer
bonding and weak inter-layer bonding, can feature some of the largest optical anisotropy[43–47]
,
however, their use in most optical systems is limited because their optic axis is out of the plane of
the layers and the layers are weakly attached, making the anisotropy hard to access in a
technologically or phenomenologically meaningful way.
Niu et al. (2018) reported[39,48] quasi-1D perovskite chalcogenides, BaTiS3, to be an ideal
candidate for highly anisotropic IR crystals, with abundant IR transparency and record hitting, at
that time, optical birefringence and dichroism. Such discovery opens up new opportunities to
engineer quasi-1D perovskite chalcogenides, A1+xTiX3 (A = Sr, Ba; X = S, Se), as birefringent
optics upon revealing the role of crystalline structure in the modulation of asymmetric light-matter
interaction.
1.3 Quasi-1D Perovskite Chalcogenides
Quasi-one-dimensional (Quasi-1D) materials show 1D chains of atoms with anisotropic
bonding, where differences in the bonding strength along and across the chain direction are
observed. Here, we studied a class of quasi-1D materials with weaker covalent bonds between
adjacent chains[49], as Figure 1.5 shows. The bonding between the chains, although weak, can be
visualized as strongly coupled two-dimensional (2D) sheets[50] and therefore are more compatible
in the morphology[51] to accessible anisotropy.

7
Figure 1.5: Quasi-1D structure of BaTiS3. Face-shared TiS6 octahedra forms quasi-1D chains along the caxis. Ba2+ ions bonds [TiS3]
2-
chains along the a-b plane.[39]
BaTiS3 was reported[39,48] as a BaNiO3-type[52] quasi-1D hexagonal perovskite chalcogenide
with face-shared (TiS6) octahedral chains, has Δ = 0.76 at long-infrared frequencies [Figure 1.6]
where it is fully transparent (both along and across the optic axis)—a record at the time. These
quasi-1D hexagonal chalcogenide single crystals[51] and thin films[53] have the potential to be
grown with different orientations to enable easy access to their anisotropic properties. Therefore,
they are an attractive and largely unexplored class of materials to achieve higher birefringence and
dichroism.

8
Figure 1.6: The linear birefringence, linear dichroism, and normalized dichroism of BaTiS3.
[39]
Moreover, BaTiS3 is chemically tunable towards a class of isostructural quasi-1D perovskite
chalcogenides, A1+xTiX3 (A = Sr, Ba; X = S, Se), but adopts various structure modulations.
Sr1+xTiS3 (X = S, Se), whose off-stoichiometry x can be 0.05 ~ 0.22[54], has incommensurate[55] or
commensurate[56] lattice modulations between the Sr and TiX3 sublattices [Figure 1.7], not only
accepting more valence electrons from the extra Sr but also periodically distort the TiS6-chains
along the c-axis, bringing chemical stoichiometry tunability. Similar incommensurate structure
modulation was also observed in Ba1+xTiS3
[57,58] and Sr21/19TiSe3
[59]. On the other hand,
stoichiometric BaTiSe3 is reported to have a similar crystal structure as BaTiS3, potentially with
comparable[60] birefringence at longer wavelengths[61]
.
Figure 1.7: Commensurate (incommensurate if x is an irrational number) lattice modulation of Sr1+xTiS3.
TiS6 octahedron of “octahedral” and “trigonal” structures are colored blue and orange, respectively.

9
1.4 Unconventional Optical, Electronic, and Thermal Properties
However, as we further investigated the unconventional optical[51,62–64], thermal[51,65], and
electronic[51,31,66] properties of A1+xTiX3, we not only realized the inconsistency between observed
properties and the calculated properties from the reported crystal structures but also discovered the
structural complexity as a function of temperature during phase transitions. This highlights the
importance of understanding the role of atomic-scale lattice modulations towards asymmetric
light-matter interactions.
1.4.1 Underestimated optical anisotropy.
The measured giant optical anisotropy was qualitatively explained in terms of the anisotropic
distribution of elements with large differences in electronic polarizability[39]; however, the
theoretically predicted and experimentally observed values of optical anisotropy were not in
agreement[48] First-principles, density-functional theory (DFT) calculations of Δ for BaTiS3 with
P63mc space group[60] show that it is moderately birefringent [Figure 1.8(a)], much lower than the
experimentally measured values for BaTiS3
[39]. Furthermore, the theoretically predicted dichroism
shows an incorrect spectral dependence compared to the measured spectrum [Figure 1.8(b)]. Using
different exchange-correlation functionals within DFT does not ameliorate the situation, leaving
an open question on the existence of further lattice modulations in the BaTiS3 single crystals.

10
Figure 1.8: Calculated optical anisotropy of P63mc-BaTiS3. (a) Birefringence, and (b) dichroism using
different methods within DFT compared to the experimental spectra.[62]
1.4.2 Electronic phase transitions
Electrical transport measurements on bulk single crystals of BaTiS3 along the c-axis [Figure
1.9(a)] identified two different phase transitions from their non-monotonic and hysteretic transport
behavior: Upon cooling, the electrical resistivity increases, and the system undergoes a phase
transition at 240 K (Transition II) featuring a resistivity jump; On further cooling, it continuous to
increase till 150 K, after which the material undergoes another transition that we call Transition I
with a sharp drop in resistivity. Transition II (240-260 K) hints at the emergence of a CDW state
from a high-temperature semiconducting phase; while at Transition I (150-190 K), the CDW order
is suppressed and the system switches to a more conductive state.[66]

11
Figure 1.9: Electronic phase transitions of BaTiS3. (a) Representative temperature-dependent electrical
resistivity of BaTiS3 crystal along the c-axis. Abrupt and hysteric jumps in resistance are shown near 150-
190 K (Transition I), and 240-260 K (Transition II). Inset shows an optical microscopic image of the BaTiS3
device used for Hall measurements; (b) Temperature dependence of the mobility, µ, and carrier
concentration, n, of the dominant carrier, extracted from Hall measurements. The solid and dashed lines
illustrate the data taken from a warming and cooling cycle, respectively.[66]
The Hall mobility evolution in BaTiS3 [Figure 1.9(b)] followed a non-monotonic, but an overall
increasing trend as temperature decreases suggesting the transport is likely largely phonon-limited.
The non-monotonic trend is manifested by a notable drop after Transition II and then a substantial
increase in mobility values after Transition I, as the temperature is decreased. Moreover, the
modulation of electrical resistances during both transitions [Figure 1.9(a)] was predominantly due
to the variations in Hall mobility, rather than changes in carrier concentration. All these transport
observations in BaTiS3 are consistent with two-phase transitions that lead to a sequence of
electronic phases, starting from a high-temperature semiconducting phase that transitions to a
reduced-mobility CDW phase at intermediate temperatures, and finally to a high-mobility phase
at low temperatures.[66]
1.4.3 Glass-like thermal conductivity and lattice disorder
Sun et al. (2019) detected an anomalous trend of increasing thermal conductivity with
increasing temperature over the 70-400 K temperature range that is unexpected for a single crystal

12
but is more like the thermal conductivity of amorphous materials such as glass. The investigation
of such a trend via neutron scattering of BaTiS3 revealed a lattice disorder never resolved in the
reported P63mc structure of BaTiS3.
Figure 1.10: Lattice disordered of BaTiS3 in the neutron scattering study of BaTiS3. Anisotropic distribution
ellipsoids for 90 percent probability from neutron scattering at (a) 100 K and (b) 300 K from different
perspectives. The Ti distributions along the c-axis (stretched blue ellipsoids) decrease in size with
increasing temperature, the opposite of what is expected for thermal vibrations; (c) Derived atomic
displacement parameters versus temperature. Note that the Ti-U33 parameter decreases with increasing
temperature. The inset shows a schematic for the bimodal distribution of Ti atoms residing in shallow
potential wells.[65]
Neutron scattering of BaTiS3 was analyzed by the pair distribution function to quantify the local
structures. The resulting anisotropic displacement parameters at 90% probability are illustrated in
Figure 1.10(a-b). At 100 K [Figure 1.10(a)] the anisotropic displacements of the Ti atoms are
elongated along the c-axis with a large atomic displacement parameter Ti-U33, compared to Ti-U22,
which is more consistent with typical thermal amplitudes. The S atoms are similarly elongated
along the c-axis with S-U33 compared to S-U22. On heating, however, these large U33 values
decrease for Ti and S atoms, as shown in Figure 1.10(c). This behavior is opposite to that expected
for thermal vibrations. The U33 and U22 parameters for the Ba atoms, on the other hand, are smaller
and show the expected increase in heating from 100 to 300 K (Figure 1.10(c)). Such an
anomalously large temperature dependence of the anisotropic ADPs much beyond thermal

13
broadening was attributed to the disorder of equivalent Ti displacements along the c-axis at room
temperature which adopted atomic tunneling at low temperatures, e.g., 2.4 K.
In this work, we will employ high-resolution diffraction studies to learn the atomic scale crystal
structure of the room temperature A1+xTiS3 and study their variation as a function of temperature.
The lattice modulations of such remodeled A1+xTiS3 crystal structures show an anisotropic impact
on the electronic structure near the Fermi level and thus enhance the asymmetry in the light-matter
interaction near the band gap energy. Moreover, higher-order electric dipole ordering, e.g.,
topological and ferroelectric symmetries are found to dominate the structure characteristics of the
unconventional phase transitions in BaTiS3, revealing the coupling between crystal symmetry and
novel dipolar structures towards a versatile light-matter interaction investigation.

14
Chapter 2. Material Synthesis and
Characterization
2.1 Chemical Vapor Transport Crystal Growth
Chemical vapor transport (CVT) is a process where solid precursors are “vaporized” in the
presence of a gaseous transport agent and deposited elsewhere in the form of crystals, as shown in
Figure 2.1. Common transport agents include halogens and halogen compounds[67], and we used
I2 for A1+xTiX3. The precursors and transport agent are sealed in ampoules to stay away from the
oxygen and/or water from the atmosphere. The reaction takes place at the reaction temperature in
a tube furnace with a single or double heating zone. The precursors are on the source side and the
grown crystals are on the sink side.
Transport in such a mini chamber is governed by two processes, convection, and diffusion.
Various parameters such as temperature, transport direction, mass transport rate, and transport
agent must be optimized for a successful CVT process. Though larger crystals can be obtained by
increasing the transport rates favoring convection, the crystals are more likely to be
inhomogeneous and prone to be defective. Thus, optimization for each chemical system is vital.
Depending on the free energy of the reaction between the species, the source and sink temperature
can be altered. A reaction that is exothermic indicates transport from cold to hot end and the reverse
is expected for an endothermic reaction. Also, if the reaction between the species is highly
exothermic or endothermic, minimal transport is expected to take place.

15
Figure 2.1: Illustrations of chemical vapor transport. (a) A two-zone furnace with controlled temperature
gradient from the hot zone, T2, to cold zone, T1. Adopted from literature[68]. (b) A1+xTiX3 crystal growth
within sealed ampule.
2.2 Crystal Orientation Characterization
2.2.1 X-ray diffraction (XRD)
Preliminary structure analysis is carried out by X-ray diffraction (XRD). Crystals are regular
arrays of atoms, and X-rays are waves of electromagnetic radiation whose wavelength is
comparable to the interatomic distances (Figure 2.2). Atoms thus scatter X-ray waves, primarily
through the electrons of atoms. A regular array of scatterers produces a regular array of spherical
waves. Although these waves cancel one another out in most directions through destructive
interference, they add constructively in a few specific directions, determined by Bragg's law[69]:
2 sin() =
where is the spacing between diffracting planes, is the incident angle, is any integer, and
is the wavelength of the beam. These specific directions appear as spots on the diffraction pattern
called reflections. Thus, X-ray diffraction results from the X-ray impinging on a regular array of
scatterers (the repeating arrangement of atoms within the crystal) carries size information of the
atomic lattices.

16
Figure 2.2: Bragg diffraction. Picture courtesy: Wikipedia.
In crystallography, the basis and lattice are treated separately. For a perfect crystal, the lattice
gives the reciprocal lattice, which determines the positions (angles) of diffracted beams, and the
basis gives the structure factor ℎ which determines the amplitude and phase of the diffracted
beams:
ℎ = ∑
[2(ℎ++ )]
=1
,
where the sum is over all atoms in the unit cell,
is the scattering factor of the -th atom sitting
at position (
,
,
)·(a, b, c), diffraction intensity ℎ = |ℎ|
2
. When there the unit cell follows
specific symmetry, e.g., a 21-screw axis along the c-axis, have two atoms (
,
,
) and (1 -
, 1
-
, 0.5 +
),
ℎ =
[2(ℎ++
)] +
{2[ℎ(1−
)+(1−
)+(0.5+
)]}
.
When ℎ = = 0,
ℎ =
2
+
∙
2
= [1 + (−1)

]
2

,
forcing ℎ = 0 when l = 2n + 1. Extinct reflections of 00l, where l = 2n + 1, is thus a distinct
sign of the c-axis.
Because A1+xTiX3 crystals of larger size take the form of platelets, the shape and orientation of
these crystals are analogous to oriented films growing on substrates. Thin-film XRD with a high-

17
resolution point detector is then adopted to verify and characterize the orientation of A1+xTiX3
crystals. Both the angles and the extinction (intensity) of reflections are used to determine the
orientation of BaTiS3.
One could use the thin film out-of-plane XRD and the rocking curve to determine the crystal
orientation and the crystal crystallinity, For example, the BaTiS3 needle, BaTiS3 (001), and BaTiS3
(100) are shown in Figure 2.3(a-c) respectively. BaTiS3 needle and BaTiS3 (100) have the same
0k0 orientation and a full width at half maximum (FWHM) of 0.013° and 0.011° in the rocking
curve. The rocking curve of BaTiS3 (100) is asymmetric and has a stronger background. This is a
sign of small angle twinning domains across the crystal. BaTiS3 (001) has a 00l orientation, where
001 and 003 are extinct. The rocking curve has a FWHM of 0.013°, comparable crystal quality as
other morphologies.
Figure 2.3: Out-of-plane XRD of (a) BaTiS3 needles, (b) BaTiS3 100 platelet, and (c) BaTiS3 001 platelet.
The inset shows the corresponding rocking curves.
2.2.2 In-plane and out-of-plane orientations
Upon resolving the out-of-plane orientation of A1+xTiX3, the in-plane orientation could be
further determined regarding to the spatial orientations between the in-plane and out-of-plane
orientations. Figure 2.4 shows the diffraction modes for thin-film XRD[70]. A conventional -2
out-of-plane diffraction characterizes only the crystal planes perpendicular to the out-of-plane

18
orientation. Although the in-plane orientation, lying perpendicular to the out-of-plane direction, is
best detected by in-plane diffraction, the limited thickness of the crystal makes it difficult in
practice. Instead, “asymmetric” planes, which can be decomposed into an out-of-plane plus an inplane orientation, are measured by tilted geometry. Pole figure analysis is achieved by mapping
such tilting axes. A tilting about the rotational axis of the sample would show the corresponding
repetition rate of these asymmetric planes.
Figure 2.4: Out-of-plane and in-plane diffraction for thin film. Pole figure analysis utilizes the rotational
symmetry of asymmetric out-of-plane diffraction.[70]
Figure 2.5 shows the pole figure analysis of BaTiS3 crystals that determines the in-plane
orientation of BaTiS3 plates. Figure 2.5(a-b) illustrates the diffraction geometries of (100)- and
(001)-oriented BaTiS3. When 002 is the out-of-plane orientation. The Pole figure of 102 reflections
is mapped around the 6-fold symmetry parallel to 002 to determine the in-plane orientations. The
angle between 002 and 012, =26.8°, is calculated from the crystal structure and is applied to
achieve Figure 2.5(c). Although a certain amount of asymmetry in the reflections comes from the
misalignment, the 60° in-plane rotation between 012 reflections verified the 6-fold rotation about
002. On the contrary, when 020 is the out-of-plane orientation, the 6-fold axis is now lying inplane. The 60° -rotation between 200 and 020 orientations now best describes such symmetry. A

19
= 60° tilting between 020 and 200 best describes the 6-fold symmetry of such BaTiS3 (100)
crystal. The pole figure then shows only 2-fold symmetry as in Figure 2.5(d). The c-axis is now
perpendicular to the -tilting plane.
Figure 2.5: In-plane orientation determination of (100) and (001) BaTiS3 plates. (a) XRD geometry for
BaTiS3 (001); (b) XRD geometry for BaTiS3needle and BaTiS3 (100); (c) Pole figure of BaTiS3 (001)
crystal; (d) Pole figure of BaTiS3 (100) crystal. [51]
2.2.3 Large area reciprocal space mapping
Reciprocal space map (RSM) is a high-resolution X-ray diffraction method to resolve the
reciprocal space intensity distribution around a specific diffracted peak. These maps can reveal
additional information beyond that provided by a line scan such as high-resolution rocking curves.
Figure 2.6 shows the geometries of RSM accessing different regions of the reciprocal space via
consecutive 2θ scans while varying the tilting angles of the goniometer: skew geometry (tilting
with the χ-axis), co-planar geometry (tilting with the ω-axis), and in-plane geometry (in-plane φrotation), representative of qx-qz, qy-qz, and qx-qy RSMs[70]
.

20
We used RSMs to exclude[51,64] or characterize[63] incommensurate superlattices along the
primary axes[57,58] by XRD. For A1+xTiX3, rotational XRD mapping is done in the a-b and a-c
planes. This is a series of -2 scans with a stepped angle. Once the c-axis is aligned perfectly
parallel to the out-of-plane orientation, a ζ mapping will demonstrate all reflections between 100
and 010 in the a-b plane [Figure 2.7(a)], or the a-c plane [Figure 2.7(b)] if the a-axis is parallel to
the out-of-plane. In Figure 2.7(a), 010 is first aligned to the out-of-plane orientation, then rotated
the b-axis parallel to rotational plane. θ-2θ scans from 10° to 60° are carried out for every 1°
between = - 6° to 66°. Peaks are indexed based on the 2 and relative to out-of-plane
orientation (100 series in this case) and plotted in the 2 space. This verified the six-fold axis to
be perpendicular to the rotational plane. Similar mapping is done for the a-c plane for χ = -5° to
75°, in Figure 2.7(b), with no sign of incommensurate or superlattice.
Figure 2.6: Reciprocal space mapping[70] of different geometry.

21
Figure 2.7: Rotational XRD Mapping (-2 scan while varying ), and a sign of superstructure. (a) a-b
plane. A 60° rotation from 100 to 010 is characterized. Peaks showing up match with the diffraction pattern
of BaTiS3; (b) a-c plane. No incommensurate superstructure shows up along the a- or c-axis. The structure
matches with a stoichiometry BaTiS3.
[51]
However, limited by the accessible area of the RSM, an exhaustive search for weak reflections
is not available at relatively high resolution. We thus seek out single crystal X-ray diffraction to
further determine the structure of A1+xTiX3.
2.3 Atomic Scale Structure Analysis
2.3.1 Single crystal X-ray diffraction (SC-XRD)
Single-crystal X-ray Diffraction is a non-destructive analytical technique that provides detailed
information about the internal lattice of crystalline substances, including unit cell dimensions,
bond lengths, bond angles, and details of site-ordering. When being refined to match the reciprocal
space structure factor of a real space structure, atomic-scale crystal structure could be determined
at high credence. Figure 2.8 shows the geometry of a single crystal diffractometer[71]. Singlecrystal diffractometers use either 3- or 4-circle goniometers. These circles refer to the four angles
(2, , , and ) that define the relationship between the crystal lattice, the incident ray, and the
detector. Samples are mounted on thin glass fibers or MiTeGen crystal loops which are attached

22
to brass pins and mounted onto goniometer heads. Adjustment of the X, Y, and Z orthogonal
directions allows accurate centering of the crystal within the X-ray beam to avoid crystal
translation during rotations.
SC-XRD uses a high-power collimated X-ray source, the diffracted X-ray beams impinging on
the detector are detected regardless of the crystal-detector distance. A two-dimensional (2D)
detector covering a specific 2θ range, e.g., 2θ1-2θ2, would detect all reflections falling within the
reciprocal space whose source-detector (ω-2θ) angles fall into 2θ1-2θ2 upon efficiently crystal
rotation. By recording the statistically average angles and intensities of the diffracted beams, one
not only establishes a one-to-one relationship between XRD intensity and 2θ but also records the
reciprocal vectors of these reflections, Ihkl. We can then use the relationship between the real space
structure and the reciprocal space structure to map out the crystal structure.
Figure 2.8: Single crystal diffraction. Almost full coverage is achieved. Ideally, all reflections within
achievable 2θ are detected. Source: serc.carleton.edu
2.3.2 Electron distribution and structure determination
Electron density map (2Fobs - Fcal) analysis is a built-in function of the structure refinement of
single crystal diffraction[72], revealing both the observed structure factor (Fobs) and the refinement
error distribution (Fobs - Fcal). One can therefore use Fobs to solve the real space electron density

23
distribution which best matches the symmetry and basis of the observed unit cells [Figure 2.9],
and the refinement residual, e.g., R1 = | Fobs - Fcal | / Fobs for subtle structure adjustment (or
refinements). Based on the discrete model of the diffraction structure factor,
ℎ = ∑
[2(ℎ++ )]
=1
,
where
is the scattering factor of the -th atom sitting at position (
,
,
) within the unit
cell. Because electrons dominate the scattering of X-ray, real space electron charge density ()
is achieved by the forward Fourier transformation of the discrete diffraction space:
=
1

∑ |ℎ|
∞ −2(ℎ++)
ℎ,,
−∞
,
where (ℎ) is the phase referred to as an arbitrary origin of phases. Because X-ray diffraction
measures only the reflection intensity |ℎ|
2
, promising combination
(ℎ)
is refined based on
crystallographic symmetry to reconstruct that best match ℎ.
Figure 2.9: Real space structure determination and refinement of single crystal X-ray diffraction. Source:
www.fzu.cz
However, in practice, resolution truncation must be introduced to maintain adequate reflection
intensity ℎ/ℎ, completeness (at least 99% in this work) and redundancy (the number of
independent observations per unique reflection). The truncation limit ℎ and

24
ℎ corresponds to the smallest distances (ℎℎ, typically measured in Å) between
crystal lattice planes that is accurately resolved in the diffraction pattern:
≅ =
1

∑ ∑ ∑ |ℎ|
−2(ℎ++)
=

=
ℎ
ℎ=ℎ
.
Therefore, the higher order reflections |ℎ|
2
that are accounted for, the smaller ℎℎ one can
achieve, the more accurate becomes.
2.3.3 X-ray Studies with Synchrotron Radiation
In order to achieve higher X-ray fluence to resolve the weaker reflections, shorter wavelengths
for better spatial resolution, lower divergence for a more accurate peak profile, and wavelengthdependent spectroscopy study, we carried out diffraction and spectroscopy studies with
synchrotron radiation in the Stanford Synchrotron Radiation Lightsource (SSRL) of the SLAC
National Accelerator Laboratory, the Advanced Light Source (ALS) of the Lawrence Berkeley
National Laboratory, and the Advanced Photon Source (APS) of the Argonne National Laboratory.
These synchrotron radiation sources consist of four main components: a linear accelerator
(linac), a booster ring, a storage ring, and the beamlines [Figure 2.10]. The linac is responsible for
the first stage of stored beam generation, producing a stream of electrons that are accelerated by
electric fields to the full storage-ring energy. Once in the storage ring, the electrons are bent around
a circular path using magnetic fields generated by electric coils or permanent magnets. In fact,
rather than being a storage ‘ring’, the synchrotron resembles a polygon: the corners of the polygon
are the bending magnets that determine the path of the beam, whilst on the edges or ‘straight
sections’ specialized devices can be inserted. These ‘insertion devices’ take the form of wigglers
or undulators which, together with bending magnets, provide the radiation sources for the
beamlines. The radiation is produced when the particle beam changes direction in the magnetic

25
devices (i.e., the velocity vector changes, giving an effective acceleration), monochromated by
channel-cut Si(111), and focused by toroidal mirrors to achieve desired flux, spot size, and
divergence.
Figure 2.10: Schematic[73] of a synchrotron radiation source showing the linac, booster, and storage rings,
and the magnetic devices that produce X-ray radiation.
2.4 Optical Spectroscopy
2.4.1 Wavelength dispersion of refractive index
In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its
frequency. The absolute refractive index which describes the ratio between the phase velocity of
the magnetic wave traveling in vacuum and medium, varies with the frequency (or wavelength)
of light. Classical optics expresses the dispersion of the refractive index as the spontaneous
emission of electromagnetic-driven dipole vibrations. Then the complex form of dielectric
function = ′ + ′′ follow Kramers–Kronig relations:
′ = 1 +
2

∫
Ω′′(Ω)
Ω2−2
+∞
0
Ω

′′ = −
2

∫
′(Ω)
Ω2−2
+∞
0
Ω

26
, where Ω is the angular frequency variable running through the whole integration range, and
denotes the Cauchy principal value. This tells us the dispersive and absorptive properties of the
medium are not independent. Considering a simple one-resonance model, then one derives the
refractive index dispersion[2]:

2−1
2+2
=

2
30
∑

0
2−2−

, where is the number of contributing electrons per unit volume,
is the elementary charge,
0 is the vacuum permittivity,
is the electron rest mass, 0 and are the resonant frequency
and damping force parameter summed over resonant mode by oscillator strength
. One can
derive the dispersion of refractive index based on the resonant frequency, thus absorptive
properties of a material. Figure 2.11 is a qualitative plot of the dispersion of the refractive index
for dipole vibrations.
Figure 2.11: Refractive index as a function of frequency[2]
. 01, 02, 03 are resonant frequencies of
exciton excitation, outer shell electron excitation, and inner shell electron excitation. KE is the static
dielectric constant.
2.4.2 Fourier-transform infrared spectroscopy (FTIR)
The reflectance and transmittance spectra of A1+xTiX3 crystals are measured by Fourier
transform infrared spectroscopy (FTIR). Figure 2.12 shows the geometry of the polarizationdependent reflectance and transmittance measurements. Infrared spectroscopy was performed
using an infrared interferometer (Bruker Vertex 70) connected to an infrared microscope

27
(Hyperion 2000). A 15× Cassegrain microscope objective (numerical aperture = 0.4) was used for
both transmission and reflection measurements at normal incidence. Time domain interferogram
which leads to the frequency spectrum upon Fourier transform was achieved by the KBr beam
splitter and motorized mirrors in the FTIR spectrometer.
Figure 2.12: Illustration of the polarization-dependent infrared reflection and transmission geometry.
Within the IR interferometer, an incandescent source of light emits a bright ray in the mid-wave
infrared (MWIR) wavelength range. Once collimated, the beam is split in half by the beam splitter.
After traveling a fixed and controlled distance, the coherent beams are reflected from fixed and a
motorized mirror and converge back into an interfered beam out of the FTIR interferometer, as
Figure 2.13 shows. In reflection mode, the interfered beam is directed to the upper optical path of
the microscope, passes through the linear polarizer, and focal lens, reflects from the surface of the
sample, goes back to the lens, and arrives at the liquid-nitrogen-cooled MCT detector. Polarizer.
In transmission mode, the interfered beam is directed to the lower optical path of the microscope
and passes through the linear polarizer and a condenser to become a converging beam toward the
sample. Transmitted light is then collected by the same focal lens configured in the reflection
mode, analyzed by another linear polarizer when necessary, and arrives at the detector.

28
Figure 2.13: Schematic representation of an FTIR spectrometer.
The resulting interferogram is a function of time (optical path difference over the velocity of
the motorized mirror) and the values outputted by this function of time are said to make up the
time domain. The time domain is Fourier transformed to get a frequency domain, which is
deconvolved to produce an FTIR spectrum. The theoretical frequency resolution is thus determined
by the length of the optical path and the lower rate of the motorized stage.
2.4.3 Reflectance/transmittance spectrum anisotropy
The asymmetric light-matter interaction of linearly anisotropic crystals can be detected by
polarization-resolved transmittance and reflectance spectrum. Figure 2.14(a-b)[39] show the
normal-incidence transmittance and reflectance spectra of BaTiS3 (100)-oriented crystal platelets
with linear polarization parallel and perpendicular to the c-axis of the crystal. (100)-oriented
BaTiS3 platelet shows obvious polarization-dependent IR spectra, showing difference absorption
shoulders, 0.27 eV and 0.76 eV when linearly polarized parallel and perpendicular to the optical
axis (along the c-axis). (100)-oriented BaTiS3 is therefore optically anisotropic in-plane. Fabry–
Pérot fringes indicate sufficient smoothness of the surfaces to achieve interference between the

29
two surfaces. Consistent with the transmittance shoulders, Fabry–Pérot fringes of the reflectance
spectrum also disappear at the respective wavelength.
Figure 2.14: Representative optical spectroscopy[62] of BaTiS3 crystals by polarization-dependent FTIR. (a)
Transmittance, and (b) reflectance spectra of BaTiS3 crystal with polarization parallel and perpendicular to
the optical axis along the c-axis.
2.4.4 Polarization Resolved Raman Spectroscopy
Aside from the refractive index of light towards the reflection, transmission, and absorption of
light from materials, light is also effectively scattered by the electrical polarizability of particles,
e.g., ions. The oscillating electric field of a light wave acts on the charges within a particle, causing
them to move at the same frequency. The particle, therefore, becomes a small radiating dipole
whose radiation we see as scattered light. Such is a process of Rayleigh scattering.

30
In crystalline materials, the elastic arrangement of atoms or ions adopts quantized collective
vibrations, known as phonons. Rayleigh scattering can evolve the excitation or the relaxation of
specific vibrational energy states, as shown in Figure 2.15(a). The resulting radiation is now of
higher (anti-Stokes Raman scattering) or lower frequency (Stokes Raman scattering) than the
incident light, whose energy difference, Elaser - Rscatter known as Raman shift, reflects the energy
differences between the vibrational modes of materials, as shown in Figure 2.15(b). Such
vibrational mode spectrum, or Raman spectroscopy is therefore a fingerprint of the crystalline
structure of materials.
Figure 2.15: Raman spectroscopy. (a) Schematic of the energy states involved in Raman scattering. Raman
shift reflects the energy loss or gain equal to the vibrational mode energy. This is a different process than
infrared absorption. Source: Wikipedia. (b) Illustration of Raman scattering. As monochromatic light is
scattered from a material, elastic Rayleigh scattering, anti-Stoke Raman scattering, and Stoke Raman
scattering have different wavelengths and are analyzed by a spectrometer. Source: www.edinst.com.
Since Raman scattering involves a virtual energy state of the oscillating charges, it is sensitive
to the electrical polarizability of particles. Raman spectroscopy resolves the polarization sensitivity
between light and phonon, and can therefore provide detailed information on the symmetry of
Raman active modes.

31
Chapter 3. Giant Modulation of Refractive
Index from Correlated Disorder in BaTiS3
3.1 Introduction
The asymmetry of the light-matter interactions originates from the anisotropy of the crystalline
structure. The latter is commonly described by the presence of a periodic order, expressed as a
finite periodic unit cell with symmetric atomic structures[75]. Although real crystalline materials
possess disorders in various forms across multiple length scales[76]. While disorder can be
detrimental to physical properties, it can also lead to emergent physical properties and influence
phase transitions[77]. The latter is especially true for correlated disorder, which signifies the
presence of short-range correlations between some structural features in an otherwise disordered
structure[78]. For example, correlated disorder leads to large electro-mechanical coupling in
ferroelectric alloys[79,80], spin frustration and glassy behavior in magnets[81], colossal magnetoresistive effects in manganites[82], unconventional metal-insulator transitions in disordered metals
and semiconductors[83], enhanced superconducting phase fluctuations[84]
, and enhanced Li-ion
conductivity in oxide alloys[85]
.
Correlated disorder dramatically affects the static or zero-frequency dielectric response of
materials. For instance, relaxor ferroelectric alloys exhibit static dielectric constants that are orders
of magnitude higher than unary ferroelectric compounds, with the correlated disorder being the
origin of the enhanced response[80]. Similar enhancements in the high-frequency optical properties

32
of materials, such as their refractive index (), have not been observed with the disorder.
Furthermore, given the strong correlation between the refractive index of a material and its
nonlinear optical response[86], new approaches to enhance the refractive index could also result in
additional functionalities such as large optical anisotropy and nonlinearities. Here, we show that
correlated disorder results in a large change in the refractive index along specific crystallographic
directions in a quasi-one-dimensional (1D) hexagonal chalcogenide, BaTiS3, and leads to giant
optical anisotropy that has been reported previously[39] but remained poorly understood[48,60]
.
These results suggest that correlated disorder can be used as an additional degree of freedom[87] to
design high refractive index optical materials for communication and sensing applications.
Optical anisotropy is characterized by birefringence () and dichroism (), which are,
respectively, the differences in the real () and imaginary () parts of the complex refractive index
between two crystallographic directions. Organic materials demonstrate stereoisomerism[88], and
one can leverage large dipolar rearrangements in these materials to achieve dramatic changes in
anisotropic optical properties such as optical activity, birefringence and dichroism, and nonlinear
optical properties[89]. In crystalline materials, an anisotropic crystal structure and favorable
polarizability of the component elements are desirable for optical anisotropy[2]. However, the
theoretically predicted and experimentally observed values of optical anisotropy were not in
agreement[48]. First-principles, density-functional theory (DFT) calculations of Δ for BaTiS3 with
P63mc space group show that it is moderately birefringent [Figures 3.1][60], on the same level as
other birefringent crystals such as rutile (~0.25)[30], but much lower than the experimentally
measured values for BaTiS3 (~0.76)[39]. Such a dismal mismatch led us to reconsider the accuracy
of the crystal structure of BaTiS3.

33
Figure 3.1: Comparison of the birefringence of BaTiS3, which was experimentally measured[39] (blue solid
line) and theoretically calculated from first principles using the previously reported P63mc space group
(dashed blue line) against other birefringent crystals.[62]
3.2 Orientation Controlled Crystal Growth of BaTiS3
3.2.1 Sealed ampoule growth of A1+xTiX3 (A = Sr, Ba; X = S, Se)
Single crystals of A1+xTiX3 were grown by chemical vapor transport with iodine as transporting
agent. O2-free sample preparation is achieved by storing and handling the starting materials,
barium sulfide powder (Sigma-Aldrich, 99.9%), barium selenide powder (Sigma-Aldrich,
99.99%), strontium sulfide powder (Alfa Aesar, 99.9%), titanium powder (Alfa Aesar, 99.9%),
sulfur pieces (Alfa Aesar, 99.999%), selenium powders (Sigma-Aldrich, 99.99%) and iodine
pieces (Alfa Aesar 99.99%) in a nitrogen-filled glove box. Stoichiometric quantities of precursor
powders with a total weight of 1.0 g were mixed and loaded into a quartz tube of diameter 19 mm
and thickness 2 mm along with around 0.75 mg∙cm-3
iodine inside the glove box. To further
minimize point defect formation such as sulfur vacancies and oxygen substitution in sulfur during
the growth, around 1 mg∙cm-3
extra sulfur is added to the precursor. The tube was then capped
with ultra-torr fittings and a bonnet needle valve to avoid exposure to the air, evacuated, and sealed

34
using a blowtorch, with oxygen and natural gas as the combustion mixture. The sealed tube was
about 12 cm in length and was loaded and heated in a Lindberg/Blue M Mini-Mite Tube Furnace
or MTI OTF-1200X-S-II Dual Heating Zone 1200C compact split tube furnace.
3.2.2 Optimized crystal growth
We performed a systematic investigation of the effect of growth parameters on the resultant
crystals’ orientation and size to determine the optimized conditions for BaTiS3 (Table 3.1). We
observed a sensitive dependence for the orientation of the crystals on the growth temperature of
BaTiS3, suggesting a subtle temperature dependence of the surface energy for the different facets.
Further, the surfaces that possess large surface energies (enthalpic component) tend to be more
stable at higher temperatures as the entropy plays a more dominant role. Based on this argument,
a temperature below 1020℃ is too low to strike a balance between nucleation and growth to form
large-area crystalline surfaces, thus making polycrystalline powders the dominant product. As the
growth temperature is raised above 1020℃, the (100) surface becomes most favored amongst all
the terminations, and the (100) orientation of BaTiS3 crystal nuclei becomes more stable and
sustains appreciable growth rate, forming BaTiS3 needles. When the temperature goes up above
1055℃, the (001) orientation is also favored. Between 1020℃ and 1055℃, a certain amount of
growth along <001> is allowed to form a larger (100) surface, enabling BaTiS3 platelets with (100)
orientation. We usually find larger crystals at higher temperatures, where crystals of both
morphologies coexist, but sintering is found to be dominant at temperatures higher than 1060 ℃
where free-standing crystals were not observed.

35
Table 3.1: A1+xTiX3 (A = Sr, Ba; X = S, Se) of different shapes and orientations
Crystal Form Reaction Temperature Cooling Temperature Cooling Method
BaTiS3 Needle 1020 -1060 ℃ 960 ℃ Furnace Cool
BaTiS3 (100) Platelet 1040 -1060 ℃ 960 ℃ Furnace Cool
BaTiS3 (001) Platelet 1055 -1060 ℃ 960 ℃ Furnace Cool
BaTiSe3 Needle & Platelet 945 ℃ 945 ℃ Air Quench
Sr9/8TiS3 Needle & Platelet 1050 - 1060 ℃ 1010 ℃ Furnace Cool
3.2.3 Crystal morphology
The morphology of a crystal describes the set of crystallographic planes that show on the crystal
surface and is, therefore, an external representation of the internally ordered atomic
arrangement[90], or crystal structure. Figure 3.2(a-b) show the schematic projected views of the
crystal structure of BaTiS3 for (010) (later mentioned as the equivalent orientation (100)), and
(001) crystal orientations respectively, defined by the orientation of the projected surfaces.
Following the terminating edges, (100) and (001) BaTiS3 crystals correspond to the rectangular
(or needle-like) and the hexagonal-shaped BaTiS3 crystals.
In addition to the reported needle-like and platelet-like crystals with (100) orientation,[39] we
also synthesized (001)-oriented BaTiS3 crystals. Figure 3.2(a-c) show the SEM image of the
BaTiS3 needle, and (100)- and (001)-oriented BaTiS3 platelets, respectively. Compared to the
needle and rectangular platelet-shaped BaTiS3 (100) crystals, BaTiS3 (001) crystals show distinct
hexagonal faceting, suggesting hexagonal symmetry for this face. The typical thickness of the
platelets was ~ 5-20 m, while needles were ~ 20-50 m thick.

36
Figure 3.2: BaTiS3 crystals with different morphologies and orientations: (a) Projected view of the crystal
structure of (100)-oriented BaTiS3 visualizing quasi-1D TiS6 chains along the c-axis. Thickness one unit
cell; (b) Projected view of the crystal structure of (001)-oriented BaTiS3. Thickness one unit cell; Scanning
electron microscopy (SEM) images and crystal orientations of (c) BaTiS3 needle, (d) (100)-oriented BaTiS3
platelet, and (e) (001)-oriented BaTiS3 platelet. Crystal shape and surface terraces can be used to distinguish
them.
3.3 Structure Reconfiguration via Single Crystal Diffraction
3.3.1 Precession map and superstructure
Synchrotron single crystal X-ray diffraction (SC-XRD) studies were performed on crystalline
BaTiS3 samples — that were synthesized using a previously reported method[39,51]. We observe
weak Bragg reflections suggesting a lower symmetry than P63mc. The integrated reciprocal space
structure can be best represented by h0l and hk1 precession maps shown in Figure 3.3. Even though
the previously assigned P63mc space group[39,65] captures the main reflections, we observe weaker
symmetric superlattice reflections (yellow dotted circles in Figure 1b) corresponding to a √3 ×
√3 × 1 trimerized supercell of the P63mc lattice.

37
By including these superlattice reflections, we could resolve the unassigned superlattice
diffractions as with a P63cm space group of a ~11.7 Å × 11.7 Å × 5.8 Å unit cell. Table 3.2 compares
a series of other candidate space groups including the reported P63mc and potential symmetry
breakings of P63cm, which are P3c1, P31m, and Cmc21. P63cm matches the observed structure
factors much better than P63mc by adopting a lower refinement residual of R1 = 0.0184 than R1 =
0.0514. While further lowering the symmetry does not improve the refinement, whose refinement residual
is only marginally different than P63cm. We therefore concluded that P63cm best captures the periodic
order of the room temperature BaTiS3.
Figure 3.3: Reciprocal-space precession maps for h0l- and hk1-type reflections from synchrotron singlecrystal diffraction. Satellite reflections caused by a Ba-TiS3 superstructure are highlighted with yellow
dotted circles while weak reflections denoting disordered displacements are shown with red dotted
circles.[62]

38
Table 3.2: Comparison between candidate space groups for 300K BaTiS3.
Refinement BaTiS3 300K
Space group P63mc* P63cm P3c1 P31m Cmc21
#
R1, wR2 [all data] 0.0514,
0.1669
0.0184,
0.0419
0.0202,
0.0490
0.0190,
0.0452
0.0202,
0.0478
R1, wR2 [I > 4σI]
0.0513,
0.1665
0.0178,
0.0415
0.0189,
0.0481
0.0178,
0.0443
0.0185,
0.0465
GoF 1.310 1.151 1.142 1.106 1.104
Merohedral
Twining
-1 0 -1 0 -1 0
0 0 -1 2
-1 0 -1 0 -1 0
0 0 -1 2
0 1 0 1 0 0
0 0 -1 -4
0 1 0 1 0 0
0 0 -1 -4
0 1 0 1 0 0
0 0 -1 -6
BASF 0.83947 0.50069
0.25504
0.26264
0.24637
0.28082
0.23559
0.21605
0.13101
0.18149
0.12913
0.21265
0.13661
*
a, b, c (Å); ⍺, β, ɣ (°): 6.7287(14), 6.7287(14), 5.8471(13); 90.00(3), 90.00(3), 120.00(3).
#
a, b, c (Å); ⍺, β, ɣ (°): 20.226(4), 11.677(2), 5.849(2); 90.00(3), 90.00(3), 90.00(3).
3.3.2 Refined BaTiS3 structure
The refined room temperature BaTiS3 structure of the P63cm space group [Table 3.4] shows
subtle but periodic, atomic displacements. P63cm-BaTiS3 is projected onto the a-b plane in Figure
3.4. The previously degenerate TiS6 chains are now split into two types, each occupying different
positions [right insets of Figure 3.4] along the c-axis and with antiparallel Ti off-centering along
the c-axis. They are labeled as TiS6-chain A at the edges of the unit cell and TiS6-chain B inside
the unit cell; each unit cell thus has one chain A and two chains B. The Ti atoms are also displaced
along the c-axis from the S6 centroid, downwards by 0.167 Å in chain A and upwards by 0.147 Å
in chain B. These antiparallel off-center displacements result in a ferrielectric ordering, as opposed
to the ferroelectric ordering proposed for the P63mc space group[65]
.

39
Table 3.3: Comparison between candidate space groups for 300K BaTiS3.
BaTiS3
Introduce Ti a-b plane displacement
to BaTiS3
Wavelength 0.7288 Å
Temperature 300K
Space group P63cm P63cm
Cell dimensions
a, b, c (Å) 11.677(2), 11.677(2), 5.849(2) 11.677(2), 11.677(2), 5.849(2)
⍺, β, ɣ (°) 90.00(3), 90.00(3), 120.00(3) 90.00(3), 90.00(3), 120.00(3)
Volume (Å3
) 690.7(3) 690.7(3)
Density (g/cm3
) 4.059 4.059
Intensity Statistics
Resolution (Å) Inf. - 0.61 0.71 - 0.61 Inf. - 0.61 0.71 - 0.61
Completeness (%) 97.8 100.0 97.8 100.0
Redundancy 44.04 31.43 44.04 31.43
Mean I / σI 77.68 56.89 77.68 56.89
Reflections
Rσ, Rint 0.0165, 0.0498 0.0165, 0.0498
θmin, θfull, θmax (°) 2.014, 25.242, 35.791 2.014, 25.242, 35.791
Friedel fraction max, full 0.953, 0.915 0.953, 0.915
Refinement
Resolution (Å) Inf. - 0.61 Inf. - 0.61
No. reflections 1117 1117
No. I > 2σI 1079 1079
No. parameters 32 40
No. constraints 1 1
R1, wR2 [all data] 0.0184, 0.0419 0.0172, 0.0386
R1, wR2 [I > 4σI] 0.0178, 0.0415 0.0167, 0.0382
GooF 1.151 1.065
Merohedral Twin Matrix (
1 0 0
0 1 0
0 0 1
), (
-1 0 -1
0 -1 0
0 0 -1
) (
1 0 0
0 1 0
0 0 1
), (
-1 0 -1
0 -1 0
0 0 -1
)
BASF 0.49931, 0.50069 0.50731, 0.49269

40
Figure 3.4: Schematic of the BaTiS3 crystal structure projected onto the a-b plane from the refined
diffraction results. The ellipsoids used to show Ba, Ti, and S atoms reflect their refined atomic displacement
parameters (ADPs). The trimerized √3 × √3 × 1 superstructure of TiS6 chains leads to a lower symmetry
P63cm space group, which is the result of antiparallel displacements along the c-axis. The insets on the right
visualize the positions of TiS6 chains A (brown) and B (blue) with respect to adjacent Ba atoms. Chain A
displaces down while chain B displaces up from a “fixed” Ba lattice. Moreover, Ti atoms also move away
from the centroid of the S6 octahedron antiparallelly between chain A and chain B. [62]
Table 3.4: Atomic coordinates and equivalent isotropic atomic displacement parameters for 300K BaTiS3.
Ato
m
x/a y/b z/c Uiso (Å2
)
Occupanc
y
Symmetr
y Order
BaTiS3
Ti01 0
* 1
0.92656(15
)
0.02028(18
)
1 6
Ba01 0.33433(2) 1
0.25818(16
)
0.01980(7) 1 2
Ti02 1/3* 2/3 0.06632(17
)
0.0266(2) 1 3
S001 0.16588(4) 1 0.7052(3) 0.01482(16
)
1 2
S002 0.16782(4) 0.49998(4
)
0.7901(2) 0.0175(2) 1 1
Introduce Ti
a-b plane
displacemen
t to BaTiS3
Ti01
-
0.0091(19)
*
1
0.92695(18
)
0.0162(18) 1/3 2
Ba01 0.33433(2) 1
0.25829(16
)
0.01982(7) 1 2
Ti02 0.3455(6)* 2/3 0.0655(2) 0.0190(9) 1/3 1
S001 0.16589(4) 1 0.7054(3) 0.01481(15
)
1 2
S002 0.16782(3) 0.49998(4
)
0.7903(2) 0.01748(19
)
1 1
* Refined average Ti01, Ti02 a-b plane displacement magnitudes, (x/a - x0
/a) * a, are 0.106 Å, 0.142 Å.

41
3.3.3 Symmetry breaking and non-thermal disorder.
Despite the significant improvement in the refinement residual with the P63cm space group, we
find systematic deviations in the intensity of the reflection spots between the measured and refined
diffraction patterns. For example, we observe weak signals of certain symmetry-extinct reflections
(h0l or 0kl peaks whose l = 2n + 1) that are highlighted by red dotted circles in Figure 3.3. The
existence of such reflections suggests the presence of subtle displacements that may locally distort
the P63cm symmetry. As atomic displacement parameters (ADPs) capture the average atomic
displacements about the statistical centroid of atomic positions[91], large ADPs, well beyond the
thermal vibrations with the opposing trend as a function of temperature than thermal vibrations,
are another sign of the presence of subtle displacements that are not thermally driven. The
anomalously large ADPs along the c-axis (U33) reported earlier[39,65] have already been resolved
by refining the diffraction data with the P63cm structure [Table 3.5]. Nevertheless, the ADPs of Ti
along the a-b plane (U11 and U22) are still larger than that of Ba and the lighter S atoms[92]
,
suggesting potential displacements of Ti atoms within the TiS6 octahedra in the a-b plane
(ellipsoidal atoms in Figure 3.4 covers the 90% possibility region finding corresponding atoms, a
result of ADPs refinement).

42
Table 3.5: Anisotropic atomic displacement parameters for 300K BaTiS3.
Ato
m
U11 (Å2
) U22 (Å2
) U33 (Å2
) U23 (Å2
) U13 (Å2
) U12 (Å2
)
BaTiS3
Ti01 0.0259(2) 0.0259(2) 0.0090(4) 0 0
0.01296(11
)
Ba01 0.01649(8) 0.01560(9) 0.02701(11
)
0
0.00567(3
)
0.00780(5)
Ti02 0.0297(2) 0.0297(2) 0.0204(5) 0 0
0.01487(11
)
S001 0.01247(18
)
0.0194(2) 0.0149(3) 0 -0.0003(3) 0.00972(12
)
S002 0.01238(19
)
0.01231(18
)
0.0240(5)
-
0.0011(3
)
0.0004(2) 0.00334(15
)
Introduce Ti
a-b plane
displacemen
t to BaTiS3
Ti01 0.019(3) 0.021(2) 0.0087(4) 0 -0.0014(9) 0.0107(10)
Ba01 0.01651(8) 0.01562(9) 0.02702(10
)
0
0.00569(3
)
0.00781(4)
Ti02 0.0159(15) 0.0220(13) 0.0210(5) 0.002(3)
-
0.0020(14
)
0.0109(7)
S001 0.01244(17
)
0.0194(2) 0.0149(3) 0 -0.0003(3) 0.00969(12
)
S002 0.01239(18
)
0.01231(17
)
0.0240(4)
-
0.0010(2
)
0.0004(2) 0.00337(14
)
3.3.4 Electron density distribution and a-b plane displacements
To understand the nature of this weak symmetry breaking and large ADPs, we examined the
difference between the observed and refined electron densities, also known as the Fourier
difference map [Figure 3.5(b)]. The Fourier difference map shows small but definite, deviations
in the partial occupancy of Ti away from the centroid of the two types of TiS6 chains. This
deviation is highlighted in Figure 3.5(a) using electron density maps projected onto the a-b plane
about the refined Ti A and Ti B sites. Here, we observe statistically significant, three-fold
degenerate Ti atomic displacements (as compared to random displacements indistinguishable from
the thermal vibrations) having an occupancy ~21% away from the centroid. The magnitude of such

43
a-b plane Ti displacements is obtained from the refined structure [Table 3.4]. This structure further
lowers the refinement residual (R1 = 0.0171) and the ADPs of corresponding Ti atoms [Table 3.5],
suggesting that these displacements are relevant. We do not discern whether the a-b plane Ti
displacements are static or dynamic as they are likely to have a similar effect on the high-frequency
optical response. The temperature-dependent phonon dispersion[93] of the a-b plane off-centering
mode may have implications on inelastic scattering[65] and thermal properties[51]
.
Figure 3.5: Electron density maps analyses of Ti01 (in chain A) and Ti02 (in chain B) in BaTiS3. (a) a-b
plane projection of the observed electron density map near Ti01 (top) and Ti02 (bottom). Electron density
below 1 e/Å3
is comparable with the noise level and thus whitened; (b) a-b plane cross-section of the Fourier
difference contour map across the centroid of Ti01 (top) and Ti02 (bottom), S atoms not within this crosssection are overlaid for orientation referencing; (c) An anharmonic core electron distribution for Ti reveals
disordered non-thermal Ti a-b plane displacements towards S atoms. This is modeled by Ti01 (Ti02 is
analogous) a-b plane disorder in SC-XRD refinement.[62]

44
3.3.5 Solid-state Nuclear Magnetic Resonance (ssNMR)
X-ray diffraction relies on (core and valence) electron density to probe the positions of the
nuclei, so we resorted to techniques directly sensitive to the nuclei to confirm these observations.
Solid-state nuclear magnetic resonance (ssNMR) spectroscopy is sensitive to the short-range
structure around the Ti nuclei of interest. As such, it provides a window into deviations from
symmetry in the Ti coordination environment. Titanium has two NMR-active isotopes, 47Ti and
49Ti, with remarkably similar gyromagnetic ratios (γ) of −1.5105 and −1.51095 rad·s−1·T−1
,
resulting in very similar NMR resonant frequencies and overlapping signals. Two types of
interactions dictate 47/49Ti ssNMR lineshapes: the chemical shift anisotropy (CSA) resulting from
a non-spherical electron density around the 47/49Ti nucleus, and first- and second-order quadrupolar
interactions between the quadrupolar moment of 47Ti (I=5/2) and 49Ti (I=7/2) nuclei and the local
electric field gradient (EFG). While both CSA and quadrupolar effects are accounted for in the
following analysis, we focus our discussion on the latter as they dominate and provide insights
into the centrosymmetry (through the quadrupolar coupling constant, ) and asymmetry (through
the quadrupolar asymmetry, ) of Ti local environments.

45
Figure 3.6: Static 47/49Ti solid-state NMR (ssNMR) spectra as direct evidence of the a-b plane displacements
(Collaborated with Raphaële J. Clément of UC Santa Barbara). Data collected on BaTiS3 at 270 K and
18.8 T, overlaid with first-principles guided fits of 47/49Ti NMR spectra derived from (a) the P63cm BaTiS3
structure obtained from SC-XRD and with no Ti a-b plane displacements, and (b) the BaTiS3 structure
obtained from DFT calculations, and exhibiting 0.213 Å Ti a-b plane displacements. [62]
To address issues of low sensitivity and wide excitation ranges associated with 47/49Ti ssNMR[94]
,
and to minimize quadrupolar broadening effects, static 47/49Ti ssNMR spectra were acquired at low
temperature (270 K) and at a high magnetic field (18.8 T, 800 MHz for 1H) using a WURSTQCPMG experiment (full acquisition details are described in the Experimental section). The 47/49Ti
ssNMR spectrum obtained on BaTiS3 is shown in black in Figure 3.6, where the normally
continuous spectrum is instead represented by an envelope of spikelets resulting from the QCPMG
experiment. To facilitate the assignment of experimental 47/49Ti ssNMR data, first principlesguided fits of the ssNMR lineshape (red line) were performed using two structural models: the
undistorted BaTiS3 structure (P63cm space group) obtained from SC-XRD [Figure 3.6(a)], and a
BaTiS3 structure with Ti a-b plane displacements obtained by freezing the Γ5 distortion mode and
optimized from first principles using the Vienna Ab initio Simulation Package (VASP)[95,96]
[Figure 3.6(b)].

46
The experimental 47/49Ti ssNMR spectrum contains three main peaks, labeled “left”, “center”,
and “right”, which is inconsistent with the axially-symmetric NMR properties of the two Ti local
environments (Ti01 and Ti02) in the undistorted BaTiS3 structure, resulting in a poor fit [Figure
3.6(a)]. In contrast, a much better fit of the spectrum is obtained using the distorted BaTiS3
structural model [Figure 3.6(b)]: in this structure, the close to axially symmetric ( = 0.2) Ti01
environment has a large quadrupolar coupling constant, , resulting in a double-horn signal
spanning the “left” and “right” peaks of the experimental spectrum, and the Ti02 environment has
a very high quadrupolar asymmetry, = 0.7, leading to the “center” peak observed
experimentally. These results confirm the presence of large Ti a-b plane displacements in the
BaTiS3 structure, resulting in less symmetric Ti environments.
3.4 Correlated Disorder by Electron Spectroscopy
3.4.1 Diffuse scattering in electron diffraction
All the methods discussed so far provide evidence for anomalous Ti a-b plane displacements,
but the nature of its order is unclear. To check the local ordering, if any, of these displacements,
we carried out electron diffraction and imaging studies. Figure 3.7(a) shows a diffraction pattern
viewed along the [11̅0]-zone axis obtained using a transmission electron microscope (TEM).
Between the Bragg spots, we observe diffuse scattering rods (highlighted by blue arrows) that are
specifically oriented along the 110-direction in reciprocal space. Line profile along the green
dotted boxes are fitted as a convoluted pseudo-Voigt function[97] of Gaussian
1
√2

−
(−)
2
22 and
Lorentzian

1
(2+(−)2)
distribution, as shown in Figure 3.7(b-c). These streaks suggest the

47
presence of strain or disorder normal to the (110)-planes. We do not observe any streaks between
the Bragg reflections along 001. Diffuse rods were also observed along 100 in Figure 3.7(d) but
are weaker in intensity, fitted in Figure 3.7(e-f). These diffraction patterns suggest that the
observed Ti a-b plane displacements could possess correlated short-range ordering.
Figure 3.7: Electron diffraction pattern observed in TEM (collaborated with Rohan Mishra of WashU) with
streaky diffuse scattering. The a-b plane diffuse scatterings are labeled with blue arrows. Line cuts along
the green dashed boxes are extracted and fitted in detail. (a) [11̅0]-zone axis, the line-cut and quantitative
fitting along (b) 110 and (c) 001, corresponding to the correlation along [110] and [001], respectively; (d)
[010]-zone axis, the line-cut and quantitative fitting along (e) 100 and (f) 001, corresponding to the
correlation along [100] and [001], respectively. Fitted values and standard deviations of and are listed
above the corresponding figures.[62]
3.4.2 Scanning Transmission Electron Microscope
To directly visualize the local ordering of these a-b plane displacements, we performed atomic
scale imaging using an aberration-corrected scanning transmission electron microscope (STEM).
Figure 3.8(a) shows a high-angle annular dark field (HAADF) STEM image of BaTiS3 viewed

48
along the [001]-zone axis. In this imaging mode, the intensity is proportional to the square of the
average atomic number (Z
2
) of the columns[98]. Thus, the heavier Ba atomic columns appear
brighter than the lighter Ti columns, while the lightest S columns are almost invisible due to the
dynamic-range constraints of the detector. We extracted the position of the atomic columns by
fitting 2D Gaussians as shown in the small field-of-view HAADF images of three representative
regions in Figure 3.8(b). We observe that the Ti atomic columns are displaced away from the
centroid position defined by the six adjacent Ba columns.
Figure 3.8: STEM and atomic displacement analysis of BaTiS3 (collaborated with Rohan Mishra of WashU).
(a) HAADF-STEM image along the a-b plane projection. The local structural motifs from the HAADF
image show (b) Ti a-b plane displacement towards adjacent S atoms. Ti displacements are determined from
the centroid of the surrounding hexagonal Ba sublattice.[62]
The Ti off-center displacements along the a-b plane have been extracted for the entire region
shown in Figure 3.8(a) and are overlaid as displacement vectors on a vector-color map in Figure
3.9(a). While most of the Ti atomic columns are displaced away from the centroid, several Ti
atomic columns have nearly zero displacements and are largely present at the boundaries of the
domains having columns with displacements along different orientations. Figure 3.9(b) shows a
histogram of the magnitude of Ti off-center displacements obtained from Figure 3.9(c). By fitting

49
a Gaussian curve to the histogram, the average displacement of Ti atomic columns along the a-b
plane is derived to be 0.11 Å. The average Ti displacement measured from STEM is in good
agreement with SC-XRD refinement, where a-b plane displacements of TiS6-chain A and chain B
are 0.104 Å and 0.142 Å [Table 3.5], with an average displacement of 0.129 Å.
Figure 3.9: (collaborated with Rohan Mishra of WashU) (a) Ti displacement vector map calculated from
the HAADF-STEM image of BaTiS3 along [001]-zone axis overlaid onto the original HAADF image in
Figure 3.8(a). The direction and magnitude of Ti displacements are represented by arrows. The direction is
also indicated by the color wheel for easier visualization; b) Histogram of the magnitude of Ti a-b plane
displacements obtained from the HAADF image in Figure 3.78(a). Aside from a small number of Ti atoms
that do not undergo off-centering in the a-b plane, resembling boundaries of domains with displacements,
the average displacement is ~0.11 Å. The distribution is fitted with a Gaussian; (c) Linear correlation map
of Ti a-b plane displacement vectors. The color map shows the correlation coefficient of nearest-neighbor
displacement vector pairs overlaid onto the HAADF image in Figure 3.8(a). [62]
3.4.3 Correlated Ti a-b plane displacements
It is apparent from the displacement vector color map shown in Figure 3.9a that the orientation
of Ti displacements in neighboring unit cells are somewhat aligned giving rise to a domain-like
structure, while displacements farther away from any unit cell are randomly oriented. To quantify
the correlation between the orientation of the Ti displacements, we calculated the Pearson
correlation coefficient (Pearson’s r)
[99], which evaluates the linear correlation between any pair of
vectors with a value varying from -1 to 1, where 0 indicates no correlation (disorder), 1 shows a

50
perfect correlation, i.e., the vectors are parallel, and −1 signifies a negative correlation, i.e., the
vectors are antiparallel.
The distribution of the short-range correlation between every Ti atomic column with its six
nearest neighboring columns is shown as a color map in Figure 3.9(c), with red, blue, and white
representing positive, negative, and zero correlation, respectively, to the neighboring cells. A
histogram of the linear short-range correlations for the entire region in Figure 3.8(a) shows that
most Ti a-b plane displacements have a strong positive correlation with their nearest neighbors
with a mean value of ~0.6 [Figure 3.10(a-b)]. In contrast, the long-range correlation obtained by
averaging the correlation between any Ti column displacement with all the other displacements in
the entire region shown in Figure 2d, shows no meaningful correlation [Figure 3.10(c-d)], in
support of the P63cm space group observed from the macroscale XRD measurements.
Figure 3.10: Linear correlation map for the Ti a-b plane displacement vectors (collaborated with Rohan
Mishra of WashU). (a) Correlation coefficient distribution histogram for nearest neighboring pairs; (b)

51
Color map for correlation coefficient of long-range displacement vector pairs overlaid on HAADF-STEM
image; (c) Correlation coefficient distribution histogram for long-range displacement pairs. Scale bars in
(a) and (c) are 5 nm. [62]
3.5 First Principles Calculations
3.5.1 Ti a-b plane displacements and optical anisotropy
To gain insights into the effect of the Ti a-b plane displacements on the optical properties, we
performed first-principles density functional theory (DFT) calculations[100]. As mentioned before,
the P63cm structure does not have any a-b plane displacement of Ti atoms. A group symmetry
analysis shows that freezing the Γ5 distortion mode in the P63cm structure leads to the off-centering
of Ti atoms along the a-b plane. These displacements are ordered with neighboring Ti atoms
aligned antiparallel. Freezing the Γ5 distortion mode lowers the energy of the system, as shown in
Figure 3.11(a), which explains the presence of a-b plane displacements in the experiments.
We then calculated the complex dielectric function, ( + ), of BaTiS3 with different
amplitudes of the Γ5 distortion mode frozen to the P63cm structure. The dielectric function was
calculated along the c-axis, which we refer to as the extraordinary axis, and perpendicular to it,
which are the ordinary axes. The real parts of the ordinary ( ) and the extraordinary (
)
refractive indices calculated in the transparent range of BaTiS3 for photon energies smaller than
0.3 eV (light wavelength > 4 μm); and their difference, which is the birefringence (Δ = − ),
as a function of the average Ti displacement in the a-b plane is presented in Figure 3.11(b-c),
respectively. We find that Δ increases with increasing Ti a-b plane displacements until it plateaus
for displacements > 0.3 Å. We also find that
remains almost unchanged with increasing Ti a-b
plane displacements. So, the dominant contribution to the increase in Δ comes from a decrease

52
in the magnitude of with increasing Ti a-b plane displacements. These trends are also observed
for the wavelength-dependent Δ [Figure 3.12(b)] and dichroism (Δ = − ), [Figure 3.12(a)]
with varying displacements.
Figure 3.11: Optical anisotropy enhanced by Ti a-b plane displacements (collaborated with Rohan Mishra
of WashU). (a) Energy of BaTiS3 as a function of the Γ5 distortion mode associated with ordered a-b plane
displacements of Ti, as shown in atomic models in the inset; (b) Real part of refractive index along the
ordinary () and extraordinary (
) axes, and (c) birefringence (Δ = − ) of BaTiS3 (averaged
between 4-16 μm wavelengths) as a function of average Ti a-b plane displacements. [62]
Figure 3.12: Optical anisotropy of BaTiS3 structures with and without ordered Ti a-b plane displacements
(collaborated with Rohan Mishra of WashU). (a) Dichroism and (b) birefringence of BaTiS3 as a function

53
of Ti displacement in the a-b plane obtained by freezing the Γ5 distortion mode with different magnitudes.
[62]
3.5.2 Origin of the optical anisotropy
To identify the electronic origin of the enhancement Δ with Ti a-b plane displacements, we
investigated the evolution of the occupied electronic states. Δ in a crystal arises from the
distribution and orientation of valence electrons near the Fermi energy.[63,101] Thus, electron
redistribution near the Fermi energy introduced by the a-b plane Ti displacements can be expected
to change the anisotropic optical response of BaTiS3, as Figure 3.13 shows. The valence band is
primarily constituted by S-3p states, as confirmed by the atom- and orbital-projected density of
states (PDOS)[62]. We find that within 0.5 eV of the Fermi energy, the S-3p states have a nonbonding character, and are most likely to be polarized.[101]
Figure 3.13: Electron distribution and optical anisotropy (collaborated with Rohan Mishra of WashU). (a)
Spatial distribution of valence electrons within 0.5 eV below the Fermi energy as a function of Ti
displacements in a-b plane. The isosurface is set to an electron density of 0.0025 e/Å3
; (b) The evolution of

54
the integrated orbital projected density of states (S-px, S-py) and ordinary refractive index (no) as a function
of Ti a-b plane displacements; (c) The evolution of the integrated orbital projected density of states (Ti3
2) and extraordinary refractive index (ne) as a function of Ti a-b plane displacements; (d) The correlation
between the extraordinary (ne) and ordinary (no) refractive indices and S-px, S-py, and Ti-3
2 states
integrated within 0.5 eV of the Fermi energy.[62]
Ti displacements in the a-b plane result in a decrease in the density of electrons that are oriented
along the a-b plane and located within 0.5 eV below the Fermi energy, as shown in the integrated
electron density plots in Figure 3.13(a). These electrons primarily belong to S-3px and 3py orbitals.
The decreasing occupation of S-3px and 3py states with increasing Ti a-b plane displacements is
further confirmed from the PDOS integrated within -0.5 eV to the Fermi energy (set to 0 eV), as
shown in Figure 3.13(b). The decreasing electron distribution within the a-b plane has a direct
correlation with the decreasing with a-b plane Ti displacements, as shown in Figure 3.13(b, d).
3.5.3 Disordered a-b plane displacements
Finally, we have investigated the effect of the experimentally observed disordered Ti a-b plane
displacements on Δ, as opposed to the ordered Ti displacements simulated by freezing the Γ5
distortion mode. We performed ab initio molecular dynamics (AIMD) simulations starting with
BaTiS3 with ordered Γ5 distortions and equilibrated the structure at 800 K for 5 ps to randomize
the Ti displacements [Figure S12]. We then used 8 randomly selected snapshots and calculated
their dielectric function, Δ and Δ. The average values and standard deviations (Std.) of the
calculated Δ and Δ as a function of wavelength are shown in Figure 3.14(a-b), respectively.
These results are in excellent agreement with the experimental results, which show that the
hybridization of Ti and S states is, by large, determined by the local octahedral distortions, and is
less sensitive to the Ti and S atoms in adjacent TiS6-chains. Furthermore, we observe that the

55
magnitude of Ti a-b plane displacements correlates extremely well with charge redistribution and
the optical anisotropy in BaTiS3.
Figure 3.14: Optical anisotropy of disordered ab initio molecular dynamics (AIMD) simulations
(collaborated with Rohan Mishra of WashU). (a) Birefringence (Δ) and (b) dichroism spectra (Δ) of the
structure were obtained by averaging 8 snapshots from an AIMD run of BaTiS3 equilibrated at 800 K. The
averaged structure has randomized Ti a-b plane displacements. The standard deviation (Std.) of the 8
structures is shaded in orange and centered around the average shown by the red solid line. The black solid
line shows the experimentally measured anisotropy. [62]
3.6 Conclusions
BaTiS3, at room temperature, shows correlated disorder. Off-center Ti displacements within the
a-b plane have long-range disorder but are ordered along the c-axis. The ordinary () refractive
index in the partial (1.5 – 5 μm) and fully transparent region (> 5 μm) decreases while the
extraordinary refractive index (
) remains almost unchanged with increasing Ti a-b plane
displacements [Figure S11b, d]. The overall effect is a large increase of ~0.8 in the birefringence,
Δ. Our results suggest that the refractive index tensor of BaTiS3 is highly sensitive to the
magnitude of the Ti displacements and can potentially be tuned via external stimuli, such as

56
temperature, strain, or electric field, in a manner of “ferroic” switching between different
correlated Ti displacements modes. Therefore, the potential for reversible symmetry breaking by
a-b plane Ti displacements in the TiS6 octahedra in BaTiS3 and related perovskite chalcogenides
makes them a good platform to achieve tunable anisotropy, large non-linearity, and coupled
phenomena such as opto-elastic and electro-optic effects. The adjustable refractive indices in
BaTiS3 can facilitate miniaturized controlled retarders and variable waveplates for polarization
manipulation, and electro-optic modulators towards a broad range of mid-infrared photonic
applications.

57
Chapter 4. Phase Transitions and Emergent
Polar Vortices in BaTiS3
4.1 Introduction
4.1.1 Topological defects
Topological defects have been an active area of research in physics spanning cosmology [102,103]
and condensed matter [104,105]. The scaling limits of such defects remained in the mesoscale (e.g.
liquid crystals [106]), but recent advances in the atomic/nano-scale synthetic control and
characterization probes [107,108] for spins [109] and electric dipoles [110] have revealed nanoscale
topological defects such as vortices and skyrmions in low-dimensional ferroelectrics and
ferromagnets. As nanoscale vortex-anti-vortex textures in both magnetic and dipolar systems are
of interest for information storage and processing [111], the scaling limit of these structures is of
fundamental importance. The feature size of these topological structures is dictated by the
geometrical constraints, where the physical dimensions of magnetic nanodots [112] or nanoscale
ferroelectric heterostructures or superlattices [113] decide the length scale of the vortices. Moreover,
scaling down these boundary-limited topological structures to the atomic scale may warrant a new
mechanism not explored to date. Further, recent reports on sub-terahertz (THz) dynamics in
nanoscale polar vortices suggest the potential for strong light-matter interaction in the THz and far
infrared for atomic-scale topological structures [114]. Thus, scaling of polar topological structures
to atomic scale shall have fundamental scientific and broader technological relevance.

58
The clue to realizing atomic scale topological structures lies in understanding the Landau theory
of phase transitions for dipolar [31] and magnetic [115] topological structures. Here, the hypertoroidal moment is shown as the order parameter [116] in low dimensional ferroelectrics [117], and
so, materials with toroidal moments or more generally multi-pole moments are critical to scaling
down these topological structures [117]. Nevertheless, it is likely that the Curie temperature of
ferroelectric and ferromagnet largely sets the ordering temperature in low dimensional
ferromagnets and ferroelectrics. Recently, atomic scale collective polar toroidal order [118] and
Fermi surface nesting [119,120] were observed in candidate materials for excitonic insulators, 1TTiSe2
[121] and Ta2NiSe5
[122]. The strong electron-lattice coupling was also demonstrated by
photoinduced melting and recovery of the toroidal order in the Ta2NiSe5
[123,124] and TiSe2
[125]
.
One could therefore speculate the potential for hyper-toroidal moments in zero-filling
semiconductors with charge density wave-like phases, and hence, the possibility of non-trivial
dipolar textures.
4.1.2 BaTiS3 phase transitions – TiS6 dipole ordering
BaTiS3 (BTS) [51] adopts a BaNiO3-type perovskite-derived hexagonal structure at room
temperature [39], with quasi-one-dimensional (quasi-1D) face-shared TiS6 octahedra along the caxis. Analogous to the Ti off-centered displacement within the TiO6 octahedra in ferroelectric
perovskite oxides such as BaTiO3
[126] and PbTiO3
[127], BaTiS3 adopts a non-centrosymmetric
structure with Ti ions displaced from the centroid of S6 octahedron at room temperature [65,66]
. The
c-axis component of such displacement forms a periodic TiS6 dipole arranged as a superstructure.
The a-b plane component of such displacement is disordered but correlated in nature, which might
be a leading factor towards the two distinct phase transitions we recently observed [Figure 1.9] in

59
BaTiS3: (a) an unconventional zero-filling CDW transition near 240-250K and (b) a structural
transition around 150-180K [66]
.
In this chapter, we experimentally demonstrate the stabilization of polarization textures such as
vortices, due to the underlying symmetry in the charge-density-wave (CDW) phase of an electron
dilute d
0
semiconductor, single crystal BaTiS3 (BTS). Here, vortices emerge and dissolve with a
coincident, distinctive phase transition as a function of temperature. This observation follows our
recent discovery of a charge density wave-like phase in BTS[66]. We build a Landau model with
inputs from first-principles calculations to show that the atomistic dipolar topological structures
are stabilized by a series of coupled soft modes.
4.2 BaTiS3 Phase Transitions
4.2.1 Optical phase transitions
The phase transitions of BaTiS3 are further verified by a change in birefringence as a function
of the temperature shown in Figure 4.1. The large change in the birefringence across a similar
temperature range of the electronic transitions must have experienced an anisotropic structural
transition that reorganized the optical responses to the infrared spectrum range in the process.

60
Figure 4.1: Optical anisotropy as a function of temperature, (a) Unpolarized infrared reflectance spectrum
of (100) BaTiS3 while cooling (collaborated with Nuh Gedik of MIT).. The Fabry–Pérot fringes of both
ordinary and extraordinary polarizations are overlaid. Nevertheless, the frequency shift of Fabry–Pérot
fringes is obvious. (b) Fast Fourier transform is carried out to deconvolute the Fabry–Pérot fringes
frequencies of ordinary and extraordinary polarizations. The refractive index is extracted by measuring the
thickness of BaTiS3. Transitions of refractive index are observed across both phase transitions.
4.2.2 Precession map analysis
We carried out synchrotron XRD at three representative temperatures to track the structural
changes across the phase transitions and to accurately determine the structure for each phase.
Figure 4.2(a) shows the precession maps of BaTiS3 crystal projected onto the hk2 reciprocal plane
at 298 K, 220 K, and 130 K respectively, and Figure 4.2(b) illustrates a zoomed-in intensity cut
along the direction as indicated in the precession images. The corresponding crystal structures
were solved by single crystal refinement.

61
Figure 4.2: CDW order evolution revealed by single crystal X-ray diffraction of BaTiS3. (a) Reciprocal
precession images of BaTiS3 crystal along hk2 projection at 298 K, 220 K, and 130 K, respectively. (d) Xray intensity cut along the direction as indicated by the white dotted boxes in the precession maps. (c) Unit
cell comparison between the crystal structures solved from (a). (d) Resultant crystal structures match the
electronic phase transitions of BaTiS3.
4.2.3 Charge-density-wave-like phase transition.
At room temperature, a hexagonal array of reflection spots is observed in the precession image
[Figure 4.2(a)], which is consistent with the P63cm (formerly discussed in Section 3.2) space group
(a = b = 11.7 Å, c = 5.83 Å). Upon cooling to 220 K, additional superlattice reflections appear at
h+1/2 k+1/2 2 that are two orders of magnitude weaker in intensity than the primary reflections.
This indicates a 2 × 2 change in the periodicity of the lattice (a = b = 23.3 Å, c = 5.84 Å) associated
with Transition II, which is then refined (refinement residual R1 = 0.0420) as a phase transition
from P63cm to P3c1 (Table 4.1, Table 4.2, and Table 4.3).

62
Complementary to the resistivity anomalies observed from transport measurements, the weak
superlattice reflections in diffraction patterns provide one of the most convincing experimental
pieces of evidence of CDW formation in BaTiS3 below Transition II. These satellite peaks stem
from periodic lattice distortions that are directly associated with charge modulation[128–130]
.
Interestingly, the unit cell doubling in BaTiS3 takes place in the a-b plane, rather than along the
chain axis (c-axis), which is usually the case in many other classic quasi-1D CDW systems such
as NbSe3
[31] and BaVS3
[31]. Moreover, the space group symmetries of the two phases (P63cm and
P3c1) show a group/subgroup relationship. Hence, Transition II can be classified as a displacive
transition, according to Buerger’s classification of phase transitions[131], which tends to show
second-order or weak first-order thermodynamic characteristics, consistent with the small thermal
hysteresis observed from transport measurements.

63
Table 4.1: Data collection and refinement statistics of BaTiS3 platelet at different temperatures
BaTiS3 300K BaTiS3 220K BaTiS3 130K
Space group P63cm P3c1 P21
Cell
dimensions
a, b, c (Å) 11.671(2), 11.671(2),
5.833(2)
23.285(5), 23.285(5),
5.836(2)
13.431(3), 5.817(2),
13.431(3)
   ()
90.00(3), 90.00(3),
120.00(3)
90.00(3), 90.00(3),
120.00(3)
90.00(3), 120.00(3),
90.00(3)
Volume (Å3
) 688.1(3) 2740.0(15) 908.8(5)
Density (g/cm3
) 4.075 4.093 4.114
Intensity
Statistics
Resolution (Å) Inf. - 0.62 0.72 - 0.62 Inf. - 0.62 0.72 - 0.62 Inf - 0.60 0.70 - 0.60
Completeness
(%) 99.9 100.0 100.0 100.0 99.9 100.0
Redundancy 31.30 18.39 34.89 19.01 28.28 17.41
Mean I / I 34.18 18.13 22.42 10.27 32.42 17.86
Reflections
Rσ, Rint 0.0170, 0.0498 0.0315, 0.0691 0.0479, 0.0499
θmin, θfull, θmax () 2.015, 25.242, 34.703 1.035, 25.930, 35.40 1.751, 25.242, 36.160
Friedel fraction
full 0.990 0.999 0.999
Refinement
Resolution (Å) ~ (Inf. - 0.62) ~ (Inf. - 0.62) 50.0 - 0.60
No. reflections 1046 7983 8676
No. I > 2σI 916 6621 7161
No. parameters 30 185 187
No. constraints 1 1 1
R1, wR2 [all data] 0.0308, 0.0769 0.0420, 0.0692 0.0448, 0.0814
R1, wR2 [I > 4σI] 0.0272, 0.0734 0.0305, 0.0657 0.0321, 0.0764
GoF 1.126 1.058 0.969
Twinning -1 0 0 0 -1 0 0 0 -1 2 -1 0 0 0 -1 0 0 0 1 -4 1 0 1 0 1 0 -1 0 0 -6
BASF 0.53047 0.39520 0.09971
0.10131
0.21467 0.10736
0.19557 0.19508
0.12829
XNDP ~ ~ 0.001

64
Table 4.2: Atomic coordinates and isotropic atomic displacement parameters for 220K BaTiS3.
Atom x/a y/b z/c Uiso (Å2) Occupancy Symmetry Order
Ba01 0.50086(5) 0.16598(6) 0.1488(4) 0.0160(4) 1 1
Ba02 0.33230(8) 0.33184(3) 0.12095(9) 0.01285(14) 1 1
Ba03 0.16619(5) 0.16711(6) 0.6390(2) 0.0136(3) 1 1
Ba04 0.33424(9) 0.50007(5) 0.67561(6) 0.01412(17) 1 1
Ti01 0 0 0.4711(7) 0.0151(7) 1 3
Ti02 0.17477(12) 0.33363(13) 1.3939(8) 0.0172(6) 1 1
Ti03 2/3 1/3 -0.0228(9) 0.0200(10) 1 3
Ti04 0.49169(13) 0.49143(12) 0.3953(6) 0.0141(5) 1 1
Ti05 1/3 2/3 0.5072(5) 0.0164(4) 1 3
Ti06 0.33335(16) 0.17228(9) 0.8064(3) 0.0138(2) 1 1
S001 0.41913(17) 0.41791(19) 0.6766(8) 0.0120(7) 1 1
S002 0.0823(2) 0.3343(3) 1.1338(10) 0.0162(10) 1 1
S003 0.1654(3) 0.2502(2) 1.1191(10) 0.0131(8) 1 1
S004 0.2481(2) 0.4155(2) 1.1245(12) 0.0184(12) 1 1
S005 0.2502(2) 0.0849(2) 0.5836(8) 0.0091(10) 1 1
S006 0.4158(2) 0.1670(3) 0.5860(9) 0.0094(9) 1 1
S007 0.3337(3) 0.25135(17) 0.5774(4) 0.0099(3) 1 1
S008 0.4980(3) 0.4150(2) 0.1661(7) 0.0126(7) 1 1
S009 0.3321(3) 0.58292(15) 0.2306(5) 0.0120(5) 1 1
S010 0.5826(2) 0.2504(2) -0.2985(10) 0.0106(7) 1 1
S011 0.08327(19) 0.0824(2) 0.1900(7) 0.0105(6) 1 1
S012 0.4159(2) 0.4996(3) 0.1585(11) 0.0131(9) 1 1

65
Table 4.3: Anisotropic atomic displacement parameters for 220K BaTiS3.
Atom U11 (Å2
) U22 (Å2
) U33 (Å2
) U23 (Å2
) U13 (Å2
) U12 (Å2
)
Ba01 0.0121(5) 0.0106(4) 0.0253(11) 0.0010(4) 0.0058(5) 0.0057(4)
Ba02 0.0118(2) 0.0108(4) 0.0157(3) 0.0038(3) 0.0007(6) 0.0054(3)
Ba03 0.0122(4) 0.0120(4) 0.0169(8) -0.0035(4) -0.0060(4) 0.0064(3)
Ba04 0.0114(2) 0.0128(4) 0.0185(4) 0.0048(4) 0.0013(6) 0.0063(4)
Ti01 0.0218(9) 0.0218(9) 0.0018(16) 0 0 0.0109(5)
Ti02 0.0139(14) 0.0241(12) 0.0137(15) -0.0008(12) -0.0015(10) 0.0096(12)
Ti03 0.0217(12) 0.0217(12) 0.017(2) 0 0 0.0109(6)
Ti04 0.0157(15) 0.0103(14) 0.0109(13) 0.0001(9) -0.0012(9) 0.0026(7)
Ti05 0.0207(6) 0.0207(6) 0.0077(8) 0 0 0.0103(3)
Ti06 0.0174(5) 0.0154(10) 0.0093(5) 0.0002(7) -0.0002(12) 0.0087(10)
S001 0.0081(11) 0.0068(11) 0.0198(19) 0.0035(11) 0.0019(11) 0.0027(8)
S002 0.0144(13) 0.0173(15) 0.023(2) -0.0005(14) 0.0018(14) 0.0122(12)
S003 0.0104(14) 0.0085(12) 0.0203(19) -0.0019(12) -0.0009(11) 0.0046(10)
S004 0.0069(13) 0.0110(15) 0.035(3) 0.0029(16) 0.0046(14) 0.0031(12)
S005 0.0075(16) 0.0050(15) 0.012(2) -0.0018(13) -0.0011(11) 0.0011(14)
S006 0.0098(17) 0.0177(19) 0.0014(16) -0.0003(14) -0.0017(10) 0.0074(13)
S007 0.0144(7) 0.0123(14) 0.0075(7) -0.0011(10) -0.0036(17) 0.0101(13)
S008 0.0143(13) 0.0068(11) 0.0158(18) -0.0003(11) 0.0004(13) 0.0046(9)
S009 0.0130(8) 0.0052(13) 0.0161(11) 0.0005(12) 0.001(2) 0.0032(13)
S010 0.0110(16) 0.0099(16) 0.0081(15) -0.0001(12) 0.0010(14) 0.0031(11)
S011 0.0081(16) 0.0085(16) 0.0123(14) -0.0009(14) 0.0024(15) 0.0023(10)
S012 0.0103(14) 0.0119(15) 0.020(2) -0.0021(12) -0.0030(11) 0.0076(11)

66
4.2.4 Chiral-achiral structure transition
On further lowering the temperature to 130 K, the superlattice peaks disappeared and a new set
of reflections [Figure 4.2(a)] associated with a smaller 2
√3
×
2
√3
unit cell emerged (a = b = 13.4 Å,
c = 5.82 Å), which indicates a direct suppression of the CDW via the structural transition, refined
as a P3c1 to P21 (refinement residual R1 = 0.0448) phase transition. The low-temperature space
group P21 (Table 4.1, Table 4.4, and Table 4.5) is not a subgroup of P3c1, and we have observed
the large thermal hysteresis (~ 40 K) from transport measurements, both of which lead us to
conclude that the Transition I is a reconstructive transition[131], where first-order thermodynamic
characteristics, such as thermal hysteresis, coexistence of phases at equilibrium, and metastability,
are often expected.
Moreover, since P21 is a chiral space group, while P3c1 and P63cm are achiral, the onset
temperature of the asymmetry in the left and right circular Raman spectroscopy [Figure 2.21(b)]
of ~170K matches with the chiral-achiral phase transitions from P3c1 to P21 in BaTiS3. An
evolution of unit cell sizes in BaTiS3 is summarized in Figure 4.2(c).

67
Table 4.4: Atomic coordinates and isotropic atomic displacement parameters for 130K BaTiS3.
Atom x/a y/b z/c Uiso (Å2) Occupancy Symmetry Order
Ba01 0.16964(18) 0.0966(5) 0.83377(19) 0.0092(4) 1 1
Ba02 0.33529(19) 0.6180(3) 0.66598(17) 0.0083(3) 1 1
Ba03 0.16408(19) 0.1159(5) 0.33356(17) 0.0104(4) 1 1
Ba04 0.33066(18) 0.5974(5) 0.16634(19) 0.0097(4) 1 1
Ti01 0.4834(5) -0.0783(11) 0.4967(5) 0.0071(8) 1 1
Ti02 0.0157(5) 0.4179(12) 0.5016(5) 0.0141(10) 1 1
Ti03 0.5138(5) 0.2949(12) 0.9973(6) 0.0096(9) 1 1
Ti04 -0.0137(5) 0.7926(11) 0.0019(6) 0.0105(10) 1 1
S001 0.4196(6) 0.073(2) 0.8349(6) 0.0128(16) 1 1
S002 0.0835(7) 0.6528(18) 0.4145(6) 0.0079(17) 1 1
S003 -0.0847(6) 0.1486(19) 0.3346(6) 0.0094(17) 1 1
S004 0.1644(6) 0.137(2) 0.5826(7) 0.0091(14) 1 1
S005 0.4166(6) 0.1589(18) 0.5858(6) 0.0078(13) 1 1
S006 0.6651(6) 0.141(2) 0.5824(7) 0.0084(15) 1 1
S007 0.4151(6) 0.1534(17) 0.3342(5) 0.0068(16) 1 1
S008 0.4175(6) 0.086(2) 1.0830(6) 0.0143(17) 1 1
S009 0.3310(6) 0.5666(19) 0.9144(7) 0.0132(18) 1 1
S010 -0.1689(6) 0.5543(12) -0.0855(6) 0.0059(11) 1 1
S011 0.0808(6) 0.5578(13) 0.1663(5) 0.0060(11) 1 1
S012 -0.0817(6) 1.0685(14) 0.0824(6) 0.0071(11) 1 1

68
Table 4.5: Anisotropic atomic displacement parameters for 130K BaTiS3.
Atom U11 (Å2
) U22 (Å2
) U33 (Å2
) U23 (Å2
) U13 (Å2
) U12 (Å2
)
Ba01 0.0076(8) 0.0124(11) 0.0049(5) -0.0028(7) 0.0011(5) -0.0013(6)
Ba02 0.0088(7) 0.0082(8) 0.0086(7) -0.0033(7) 0.0048(5) -0.0013(7)
Ba03 0.0083(7) 0.0158(10) 0.0054(7) 0.0021(7) 0.0023(5) 0.0004(7)
Ba04 0.0094(7) 0.0121(11) 0.0089(6) 0.0024(7) 0.0055(5) 0.0018(6)
Ti01 0.011(2) 0.0016(14) 0.0109(17) 0.0018(16) 0.0071(17) 0.0024(17)
Ti02 0.008(2) 0.021(2) 0.0116(18) -0.001(2) 0.0040(17) -0.001(2)
Ti03 0.009(2) 0.010(2) 0.0118(19) 0.0005(18) 0.0061(16) -0.0017(19)
Ti04 0.013(2) 0.009(2) 0.0103(18) 0.0030(17) 0.0063(16) -0.0028(18)
S001 0.008(3) 0.023(4) 0.006(3) -0.001(3) 0.002(2) -0.002(3)
S002 0.011(3) 0.005(4) 0.008(3) 0.001(2) 0.005(2) 0.001(2)
S003 0.008(3) 0.009(4) 0.010(3) 0.000(2) 0.004(2) -0.001(2)
S004 0.007(3) 0.013(4) 0.004(2) 0.000(2) -0.0001(19) 0.001(2)
S005 0.011(3) 0.007(3) 0.008(3) 0.000(2) 0.007(2) 0.001(2)
S006 0.007(3) 0.010(4) 0.010(2) 0.000(3) 0.005(2) -0.001(2)
S007 0.009(3) 0.008(4) 0.002(2) -0.0008(19) 0.002(2) 0.000(2)
S008 0.009(3) 0.031(4) 0.006(3) 0.002(3) 0.006(2) 0.002(2)
S009 0.008(3) 0.023(5) 0.008(3) 0.003(3) 0.004(2) 0.003(3)
S010 0.006(2) 0.002(2) 0.007(2) -0.0017(17) 0.0018(19) -0.0025(17)
S011 0.011(3) 0.002(2) 0.005(2) -0.0009(17) 0.0039(19) -0.0019(18)
S012 0.010(3) 0.001(2) 0.010(2) -0.0014(17) 0.0056(19) -0.0002(16)
4.3 Dipole Ordering and Polar Vortices
4.3.1 TiS6 dipoles along the c-axis
Synchrotron single crystal X-ray diffraction [69,132] (SC-XRD) accompanied by accurate charge
density mapping [133] (formerly discussed in Section 3.2) was used to resolve the arrangement of

69
TiS6 dipoles of the 300K, 220K, and 130K BaTiS3. Figure 4.3 visualizes the TS6 dipoles along the
c-axis, whose direction and magnitude are calculated from the relative positions of the atoms in
TiS6 octahedra:
⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗ −
1
6
∑ (
⃗⃗⃗⃗⃗ )
6
=1
;
⃗⃗⃗⃗⃗⃗⃗ = ⃗⃗⃗⃗⃗ −
1
12
∑ (
⃗⃗⃗⃗⃗ )
12
=1
.
The resulting Ba displacements are much smaller than Ti displacements. Considering Ba2+ also
has less positive charge than Ti4+, we decided to neglect the electrical polarization of BaS12
cuboctahedra and focus only on dipolar TiS6 octahedra.
Figure 4.3: “Ferroic” ordering of c-axis dipolar TiS6 octahedra along the a-b plane in BaTiS3. Ti
displacements from the center of S6 octahedron (visualized in the inset) in each TiS3 chain are uniformly
pointing up (navy) or down (red) along the c-axis. The resulting TiS6 dipoles show different “ferroic”
ordering, which are (a) 300K hexagonal ferrielectric order of 1 × 1 × 1 unit cell, P63cm, (b) 220K CDWlike hexagonal antiferroelectric order of 2 × 2 × 1 unit cell, P3c1, and (c) 130 K lamella monoclinic
antiferroelectric order of √3 × √3 × 1 unit cell, P21.
We observed “Ferroic” ordering of the TiS6 dipoles along the c-axis. At room temperature
[Figure 4.3(a)], the TiS6 octahedron at the corners of the unit cell is polarized towards the opposite
direction than the TiS6 octahedron within the unit cell, the ratio between them is 1:2. Considering

70
the Ti-S6 displacement along the c-axis is relatively close, there is a net polarization along the caxis, thus making the P63cm BaTiS3 a ferrielectric crystal, which is antiparallel dipolar
displacements but has net non-zero net polarization. The CDW phase BaTiS3, however, enlarges
the repetition periodicity of antiparallel polarizations and balances the number of dipoles towards
either orientation along the c-axis, bringing the net polarization of P3c1 close to zero. The lowtemperature BaTiS3 also switches the dipole ordering but breaks the mirror symmetries along the
a-b plane, reducing the dipole ordering to a lamella-type antiferroelectric.
4.3.2 TiS6 polar vortices along the a-b plane
As we look closer at the a-b plane TiS6 dipoles, 300K-BTS do not show ordered a-b plane Ti
displacements and thus the electrical polarizations are only along the c-axis. 220K-BTS not only
rearranged the dipolar displacements along the c-axis but also adopted comparable magnitude a-b
plane displacements, which are later determined to follow vortex-anti-vortex ab-plane topology.
130K-BTS has the largest a-b plane displacements but does not adopt an obvious vortex structure,
recovering translational symmetry but losing mirror symmetry. Here, we focus on the structure of
the CDW phase between these two-phase transitions. The c-glide plane will replicate the upper
half of the unit cell in the lower half along the c-axis by mirroring about {110} planes [75]. We thus
focus on the upper half of the unit cell and use the glide operation along (110) for a thorough
structural description in later discussions.
We visualize the [001]-projected view of the electron density map for the CDW phase from the
measurements at 220 K in Fig. 1A by showing the upper unit cell of refined P3c1-BTS. The low
refinement residue (R1 = 0.0202) gives us accurate spatial electron density distribution resolution.
Intuitively, atoms are located at and refined as the centroid of the corresponding core electron

71
cloud. TiS6 chains circled by solid lines have the Ti off-centered from TiS6 octahedra in the a-b
plane. While other TiS6 chains circled by dash-dot lines have Ti centered and thus, similar to the
Ti positions from the structure measured at 300 K [66]. An orthogonal projection of representative
TiS6 chains shows both the Ti off-centering and without any off-centering in Fig. 1B. Ti
displacements are marked by the arrows colored blue or red based on the sign along the c-axis.
Considering the centroid of the S6 octahedron is relatively fixed, dipolar Ti displacements orient
the local electrical polarization antiparallel with them. A direct outcome is the observed valence
electrons near Ti are redistributed towards the direction of the Ti displacements as visualized in
Fig. 1B.
The observed electrical polarization is quantified by the dipolar Ti displacement vectors with
respect to the centroid of the S6 octahedron from the refined structure of the CDW phase of BaTiS3.
The orientation and magnitude of such dipolar displacements are reported in Table 4.6. We thus
decompose Ti displacements along the c-axis (colored blue or red according to the sign) and along
the a-b plane (arrows pointing towards the observed orientation), as illustrated in Fig. 4.5(b). We
discovered that the electrical polarization (antiparallel to Ti off-center displacements) forms
layered toroidal vortices along the a-b plane of BTS, as visualized in Fig. 4.5(a) overlaid with the
projected view of the upper and the lower layer of the CDW phase. We identify three distinct
patterns of toroidal dipolar structures arising here, where each toroid comprises six Ti off-centered
TiS6 octahedra around one without off-centering. The vortices have three distinct characteristics
for the orientation of the dipoles in the six TiS6 octahedra surrounding the one without offcentering. They are [Figure 4.5(c)] right-handed vortex, left-handed vortex, and head-to-head-andtail-to-tail anti-vortex, colored in red, blue, and grey respectively. Such atomic-scale electrical

72
polarization vortex-anti-vortex network turns out to have an intimate connection to the underlying
symmetry of the phase.
Figure 4.4: Electron density map analysis on off-centric atomic displacements. (a) The a-b plane electron
density contour map of the structure refined from SC-XRD measured at 220 K. Ba, Ti, and S are located at
the center of electron density peaks with the corresponding integrated number of electrons in structure
refinements. (b) Distorted, with respect to (c) undistorted TiS6 chains showing additional a-b plane
displacements.

73
Figure 4.5: Polarization Textures in P3c1-BaTiS3. (a) Ti a-b plane polarization pattern at 220 K derived
from SC-XRD, shown for the upper and lower layers of a unit cell. The TiS6 dipoles form atomic-scale
polarization vortices about the undistorted TiS6 chains. The frustration arising from the triangular lattice of
polarization vortices is relieved by the tiling of a periodic network of right-handed, left-handed, and
antivortices in the a-b plane. (b) Representation of polarization of TiS6 dipoles, visualized individually as
arrows (or dots when pointing only out-of-the-plane without any in-plane polarization), which form an inplane vortical polarization pattern in the a-b plane. Polarization vortices are colored blue, red, or grey based
on their handedness: right-handed, left-handed, and anti-vortices respectively. (c) Vorticity of polarization
vortex revealed their w = 1, vortex; and w = -2, anti-vortex topologies of TiS6 electrical dipoles.

74
Table 4.6: Refinement results of 220K-BTS: dipolar displacement analysis of BaS12 and TiS6. Displacement
expressed as polar vectors following the crystal structure of P3c1.
Atom Position (Follow Table S2) Displacement Vector Magnitude (Å)
x/a y/b z/c dx/a dy/b dz/c ⟂ c ∥ c
Ba01 0.33486(3) 0.33313(3) 0.45347(4) 0.00178 0.00017 0.00538 0.040 0.031
Ba02 0.33211(3) 0.49896(2) 0.89879(4) -0.00134 -0.00103 -0.00163 0.028 0.010
Ba03 0.66591(3) 0.49955(2) 0.43722(8) -0.00074 -0.00071 -0.00610 0.017 0.036
Ba04 0.16570(2) 0.16753(3) -0.07652(17) -0.00106 0.00104 0.00178 0.043 0.010
Ti01 0.33300(5) 0.17224(3) -0.23172(15) -0.00038 0.00555 0.02651 0.134 0.155
Ti02 0.49223(5) 0.49201(5) 0.1829(3) -0.00772 -0.00804 -0.02060 0.184 0.121
Ti03 1/3 2/3 1.5658(2) 0.00000 0.00000 -0.02700 0.000 0.158
Ti04 0 0 0.5966(4) 0.00000 0.00000 -0.02620 0.000 0.153
Ti05 0.15815(5) 0.33351(7) -0.32352(19) -0.00835 0.00032 0.02438 0.199 0.143
Ti06 0.666667 0.333333 0.6030(3) 0.00000 0.00000 -0.02990 0.000 0.175
4.3.3 Vortex network and symmetry
Figure 4.6 shows the symmetry of collective atomic displacements of the CDW-phase from
room temperature BaTiS3. The “anti-ferroic” displacements along the c-axis, of both TiS6 chain
motions and Ti off-center displacements characterize the 2 × 2 × 1 CDW unit cell of BaTiS3
(marked as solid rhombus). However, we also note the chain and off-center displacements along
the c-axis are spatially out-of-phase with each other (displacement orientations are spatially offset
by 1
3
(a - b)). Figure 4.6(a-b) illustrate a broken mirror symmetry at the dot-dash line that leads to
the observed trigonal P3c1 symmetry that adopts the a-b plane frustrated vortex- anti-vortex
polarization topology [Figure 4.6(c)].

75
Figure 4.6: Experimental symmetry-controlled vortex -anti-vortex polarization network in CDW phase of
BaTiS3. Experimentally observed (a) TiS6 chain motions along the c-axis, (b) Ti off-centric displacements
along the c-axis, and (c) Vortical polarizations along the a-b plane of CDW phase of BaTiS3, analogous to
the magnetic 2D Kagome spin ice [134] in the inset. (a-c) shares the same periodicity labeled by the black
solid unit cell but (a) adopts a different symmetry than (b) and (c). Displacement orientations along the caxis within the dotted boxes are out-of-phase and break one of the mirror symmetries bringing BaTiS3
towards P3c1.
Figure 4.6(c) visualizes the a-b plane displacements on top of the triangular lattice BaTiS3 layer.
Vortices adjacent to each other share a common off-centered Ti octahedra. As the lattice is
triangular in nature, the tiling of right and left-handed vortices experience frustration in
determining the handedness of the third site. This frustration is relieved by stabilizing
electrostatically high-energy anti-vortex connections (Fig. 2E). Their winding number is then
extracted as w = 1, w = 1, and w = -2, respectively. Considering the spatial distance between
adjacent vortices along the a-b plane is 1
3
(a - b) (~ 12.9 Å), their spatial distance along the c-axis,
1
2
c (~2.9 Å), as Figure 4.7 shows, which suggests a strong coupling between the layers of the polar
vortex network. Thus, polarization vortices and the constituent Ti displacements are strictly
antiparallel to adjacent layers along the c-axis, resulting in the P3c1 space group.

76
Figure 4.7: Vortex network of P3c1BaTiS3. (a) Spatial distribution of toroidal vortices along the ab-plane
and along the c-axis. Vortex-anti-vortex along the ab-plane is stacked on top of the counterpart of the glided
layers. (b) Ti a-b plane polarization pattern at 220 K derived from SC-XRD, shown for both the upper and
lower layers of a unit cell. The TiS6 dipoles form atomic-scale polarization vortices about the undistorted
TiS6 chains. The frustration arising from the triangular lattice of polarization vortices is relieved by the
tiling of a periodic network of right-handed, left-handed, and antivortices in the a-b plane.
4.4 Vortex Phase Stability and Evolution
4.4.1 First-principles calculations
To understand what stabilizes such a highly concentrated electrical polarization vortex network,
and the origin of this complex, layered polarization topology in single crystals of BaTiS3, we
performed first-principles calculations via density functional theory (DFT) and then used the
energy values to establish the Landau model to study the thermodynamics of the CDW phase of
BTS. Figure 4.8(a) shows the phonon band diagram m of the high-symmetry state of BTS, which
assumes the P63/mmc space group with centrosymmetry [52]. Phonon mode calculations identify

77
four unstable soft phonon modes near the Γ point, colored brown, green, blue, and pink in Figure
4.8(a). They can be classified into three modes: TiS6 chain motion along the c-axis (brown), Ti
off-center displacements along the c-axis (pink), and a-b plane Ti displacements (green and blue).
These modes can then be frozen into K3
(+)
, M2
-(+) and 3
(+) , representing the antiparallel
displacements of Figure 4.6(a-b) along the c-axis, and the vortex- anti-vortex polarization network
of Figure 4.6(c) along the a-b plane. The corresponding Landau energy landscape is then calculated
and illustrated in Figure 4.8(b). Although each contributes to lower binding energy than the
centrosymmetric P63/mmc, a P3c1 lattice adopting all K3
(+)
, M2
-(+) and 3
(+) leads to the more
stable BaTiS3 than the room temperature P63cm. An appreciable decrease in binding energy is led
by the coupling between TiS6 chain motion and vortex- anti-vortex dipole network, represented by
the negative second-order Landau terms between K3
(+) and 3
(+) in Supplementary V.
Displacement modes of Figure 4.6(a-c) are therefore coupled in a P3c1 lattice to stabilize the
vortex- anti-vortex polarization networks in BaTiS3.
Figure 4.8: First-principles calculations of the symmetry-controlled vortex-anti-vortex polarization network
in the CDW phase of BaTiS3 (in collaboration with Rohan Mishra of WashU). (a) The phonon band of
centrosymmetric BaTiS3 reflects rich symmetry-breaking dynamics. Accepting different soft phonon modes
K3
(+)
, M2
–(+)
, 3
(+) reproduces the displacement symmetries of Figure 4.6(a-c), respectively. (b) The strong

78
coupling between K3
(+)
, M2
–(+)
, 3
(+) brings down the P3c1-BaTiS3 to a ground state of trigonal lattice
BaTiS3.
4.4.2 Vortex network phase evolution
Next, we show how the transition between the intricate dipolar texture in P3c1 BaTiS3, the
vortex- anti-vortex polarization textures, and the high symmetry room temperature phase (P63cm)
occur as a function of temperature [Figure 4.9(a)]. This transition shows a pronounced hysteresis
in the electrical transport properties [66]. To understand how the electronic and structural
characteristics evolve across the phase transition, we performed a high-resolution synchrotron Xray diffraction study mapping specific Bragg and superlattice reflections during the heating and
cooling cycle across the phase transition.
The a-b plane dipolar displacements responsible for the vortex network are represented by the
1.5 2 0 superlattice reflection (hkl reflections with l = 0 exclude the symmetry breaking along the
c-axis) and are distinct from the underlying hexagonal or trigonal lattice, which are reflected by 2
2 0 Bragg reflection [Figure 4.9(b)]. Figure 4.9(c) shows the intensity evolution of 1.5 2 0 during
cooling and heating cycles, while 2 2 0 maintained relatively constant in Figure 4.9(d). As 1.5 2 0
breaks the a-b plane translational symmetry and is insensitive to the structural transitions along
the c-axis, the temperature dependence of the intensity of this reflection represents the melting and
crystallization of the vortex- anti-vortex dipolar network. This shows that the temperature in
between the transition uniquely determines the phase mixture of the room temperature and CDW
phases. This is unlike a classic nucleation-growth-driven first-order transition, where the
nucleation is stochastic in nature. To learn about the nature of the nucleation and growth process
in this phase transition, we fit the temperature dependency of the intensity of 1.5 2 0 superlattice
reflection with a modified JMAK relationship [135,136], whose details are discussed in the

79
Supplementary section. In line with the earlier deduction on the nature of the transition, transition
onset temperatures (T0) during heating and cooling cycles show negligible
overheating/undercooling around 250K, hinting low or even no energy barrier for nucleation,
unlike a classic nucleation and growth process.
Figure 4.9: Temperature-driven melting and crystallization of dipolar textures in BaTiS3. (a) Visualization
of the 3D dipole moment arrangement of BTS with P3c1 (220 K) and P63cm (300 K) symmetry. Markers
follow the convention used in Figure 4.5(b); (b) The hk0 precession map near 220 reflections, extracted
from synchrotron SC-XRD. Superlattice reflections at half-order positions for the CDW phase of BTS
(measured at 220 K) are pointed out by black arrows. Representative reflections, 2 2 0 and 1.5 2 0 are
chosen for tracking the temperature dependence of the Bragg reflection intensities; Temperature-dependent
Bragg reflection intensity for (c) 1.5 2 0 and (d) 2 2 0 are extracted between 230 K and 270 K. The cooling
and heating cycles are colored blue and red, respectively. Experimental data is marked as circular dots with
statistical error bars while KJMA fitting is shown as solid lines. The hysteresis temperature values noted in
(c) are T0 = 250.0 K, Tc = 242.5 K, Th =258.2 K. The intensity of 2 2 0 in (d) remains stable across the
transition.
4.5 Conclusions
In conclusion, we observe an intricate, topological polarization vortex network composed of an
atomic-scale toroidal arrangement of dipole moments in a dilute d-orbital semiconductor, BaTiS3.

80
Several groups in the past have also studied the physical properties of BaTiS3 compound at
cryogenic temperatures[28,39,137][9-10, 26], but no phase transition was reported before. This is
mainly because careful and advanced characterizations are needed to probe the subtle signatures
of intrinsic phase transitions while by that time no one was expecting their existence in such a
semiconducting system. Hence, good control over BaTiS3 single crystal synthesis, the employment
of high-flux XRD characterization, improvement on device fabrication, and strain-free transport
measurements have allowed us to observe those phase transitions reliably in this work.
We verified the stability of this topological polarization vortex network with first-principles
calculations. The similarities between the polar vortex structure and the 2D Kagome spin ice shall
inspire the discovery of novel polarization textures in polar chalcogenides with dipole. Lastly, the
coincidence of novel CDW-like electronic phase transitions around the phases with polarization
textures in zero-filling semiconductors such as 2D layered TiSe2
[121] and Ta2NiSe5
[122], besides
BaTiS3 prompts the possibility of tunable electronic and photonic functionalities.

81
Chapter 5. Chemical Tunability of the
Optical Anisotropy in BaTiX3 (X = S, Se)
5.1 Introduction
In the former discussions, hexagonal sulfides BaTiS3 with quasi-one-dimensional (quasi-1D)
chains of faced-shared TiS6 octahedra[39] have giant optical anisotropy between the inter-chain (ab plane) and intra-chain (c-axis) orientations[138,139], whose optical anisotropy can originate from
not only its anisotropic crystal structure but also structural modulations of the a-b plane correlated
disorder of the off-centric TiS6-octahedra[62]. During phase transitions, the linear optical anisotropy
is further enhanced by the ordering of a-b plane TiS6 dipoles. We, therefore, wonder whether
similar structure characteristics would also exist in its chalcogenide counterpart, BaTiSe3, that
would potentially lead to the chemical tunability of the optical structure by replacing S with Se.
In this work, we extend the class of quasi-1D birefringent chalcogenide crystals A1+xTiX3 (A =
Ba, Sr; X = S, Se) by demonstrating comparable optical anisotropy in single crystals of BaTiSe3,
grown for the first time. At the same time, we studied the structural similarities and differences
compared to BaTiS3 by carefully mapping the structure of BaTiSe3 using single-crystal X-ray
diffraction and Raman spectroscopy. The larger real part of the refractive index (no ~ 3.2, ne ~ 4.1),
compared to BaTiS3 (no ~ 2.61, ne ~ 3.37), with a greater birefringence of Δn ~ 0.9 in BaTiSe3
enables additional design flexibility for birefringent optics operating in the MWIR to LWIR.

82
5.2 Crystal Synthesis and Characterizations
5.2.1 Crystal growth and chemical analysis
BaTiSe3 crystals were synthesized using chemical vapor transport with iodine as the transport
agent in a sealed quartz ampoule. The synthesis process is similar to that used for BaTiS3
[138] and
Sr9/8TiS3
[139], with high-purity selenides in place of sulfide precursors. The synthesis temperature
is around 945 ℃ to complete the reaction while maintaining a steady crystal growth with the help
of an iodine transport agent. BaTiSe3 crystals preferably crystallize on top of the quartz ampule
wall and grow along it [Figure 5.1(a-b)], giving rise to a thin needle-like [Figure 5.1(c)] or thicker
rod-like [Figure 5.1(d)] geometrical shapes, the latter may achieve their size with much greater
concentration twin domain boundaries (or greater crystal mosaicity). Chemical composition
mapping via energy dispersion spectroscopy (EDS) attached to a scanning electron microscope
(SEM) reveals uniform chemical composition across the crystal [Figure 5.2(a)], with an overall
integrated atomic ratio Ba:Ti:Se around 2.1: 1.9: 6.0 [Figure 5.2(b)]. This is close to the nominal
stoichiometry of 1:1:3 well within the accuracy limits of EDS. Moreover, the 1:1:3 stoichiometry
is also validated by the X-ray diffraction studies, which will be discussed later.

83
Figure 5.1: BaTiSe3 crystal growth. (a) Illustration of sealed ampule growth of BaTiSe3. Presumed vapor
transport directions of BaTiSe3 are marked as green arrows; (b) Optical picture of resultant BaTiSe3 in the
sealed ampule. Crystals are prone to grow along the quartz wall from the hot zone (precursors near the end
of the ampule) to the cold zone (sealing neck of the ampule); (c) Needle-shaped BaTiSe3 crystals,
reproduced from Figure 1b. Flat surfaces reflect lower mosaicity, ideal for anisotropic optical properties
determination; (d) Rod-shaped BaTiSe3 crystals. Uneven surfaces and irregular terraces reveal larger
mosaicity in such crystals. SC-XRD of such crystals show similar structure but larger disorder, especially
along the c-axis.[64]
Figure 5.2: Characterization of BaTiSe3 single crystals. (a) Scanning electron microscope (SEM) image and
energy dispersion spectroscopy (EDS) mapping of BaTiSe3 crystal. Ba, Ti, and Se are uniformly distributed
across the crystal; (b) Quantitative chemical composition analysis of EDS spectrum. The atomic ratio of
Ba:Ti:Se is around 0.21:0.19:0.60, close to the theoretical 1:1:3. [64]

84
5.2.2 Orientation determination by X-ray diffraction
Crystal orientation is determined by out-of-plane X-ray diffraction (XRD) while varying the
tilting angle (χ) about the rotational axis parallel to the long edge [Figure 5.3(b)] of the needle,
shown in Figure 5.3(a). Observed XRD peaks match the reported P63/mmc structure[140] and are
indexed based on the 2θ position. The rocking curve of 020 has a full width at half maximum
(FWHM) of 0.07°, showing great crystallinity of needle-like BaTiSe3. Upon tilting about the longaxis of the needle-like BaTiSe3, the 100-series reflections are observed around 60° tilt from each
other, while 210 and 120 are observed ~ 30° tilt about 100 or 010. We thus confirmed the long
axis of the BaTiSe3 needle as the rotational axis (the c-axis). However, as we look closer at the
off-principle-axes (not along 100, 010, or 001) reflections, the room-temperature structure of
BaTiSe3 reveals periodic symmetry breaking similar to the supercell modulation in BaTiS3
[62,141]
.
Figure 5.3: XRD study of BaTiSe3. (a) The tilting angle resolved XRD of BaTiSe3 between 0° to 70 χtilting about the long axis of the (b) BaTiSe3 needle. Peaks are indexed based on the reported P63mmc unit
cell. Relative tilting angles between 100-series, 110-series, and 120-series reflections follow the
hexagonal/trigonal rotational symmetry. The c-axis is thus along the long axis of the BaTiSe3 needle; (c)
The rocking curve of the 020 peak. FWHM of 0.07° reveals a great crystallinity.

85
5.2.3 Precession map analysis by SC-XRD
Single-crystal X-ray diffraction (SC-XRD) resolves the crystal structure by mapping the
reciprocal space diffraction patterns to atomic-scale resolution[69]. SC-XRD was carried out on
BaTiSe3 crystals at room temperature (Table 5.1) with high-flux X-ray synchrotron radiation at
Lawrence Berkeley National Laboratory and Argonne National Laboratory. Figure 5.5 shows the
precession maps extracted from SC-XRD of BaTiSe3 along h0l and hk0 reciprocal planes.
Although major reflections are roughly consistent with the formerly reported[140] P63/mmc,
symmetric weak reflections are observed, whose periodicity matches exactly with a 2√3 × 2√3 × 1
superstructure.
Figure 5.4: Characterization of BaTiSe3 single crystals. (a) X-ray diffraction studies of BaTiSe3 crystals
show a superstructure of the formerly reported[140] P63/mmc along the a-b plane, represented by (left) hk0,
and (right) h0l or 0kl precession reciprocal maps from single-crystal X-ray diffraction (SC-XRD); As we
look closer at the intensity distribution of observed reflections, such as within the dashed box, reflections
in (b) show streak-like diffuse scattering within the a-b plane (pointed out by arrows) but are sharp along
the c-axis. Such is a sign of correlated a-b plane disorder. [64]

86
Table 5.1: Single crystal synchrotron diffraction studies on BaTiSe3
BaTiSe3 ALS Beamline 12.2.1 APS Beamline 15-ID-B
Crystal Form Rod (Mosaic Crystal) Needle
Crystal Cut Size 0.1 mm × 0.1 mm × 0.2 mm 0.02 mm × 0.02 mm × 0.03 mm
Wavelength 0.7288 Å 0.41328 Å
Temperature 300 K 300 K
Space group P31c P31c
Cell dimensions
a, b, c (Å) 21.0946(11), 21.0946(11), 6.0797(4) 21.107(4), 21.107(4), 6.076(2)
⍺, β, ɣ () 90, 90, 120 90, 90, 120
Volume (Å3
) 2342.9(3) 2344.3(12)
Density (g/cm3
) 5.385 5.382
Integration and Scaling
Resolution (Å) Inf- 0.45 0.55- 0.45 Inf - 0.55 0.65 - 0.55
Completeness (%) 99.5 99.4 88.1 77.9
Redundancy 18.35 13.24 20.29 2.48
Mean I / sI 14.21 7.12 92.40 51.66
Rmerge 0.0714 0.1436 0.0344 0.0242
Rσ 0.0282 0.0643 0.0070 0.0116
Statistics of Reflections
hmin, hmax -42, 42 -36, 36
kmin, kmax -42, 42 -36, 36
lmin, lmax -12, 12 -9, 5
Rσ, Rint 0.0493, 0.0550 0.0105, 0.0330
θmin, θfull, θmax () 1.143, 25.930, 46.785 0.648, 14.357, 22.067
Friedel fraction 0.986 0.896
Refinement
Cutoff Resolution 0.45 Å 0.55 Å
Observed reflections 17881 6610
I > 2σI reflections 11099 6537
No. parameters 153 153
No. constraints 1 1
R1, wR2 [all data] 0.0868, 0.1869 0.0268, 0.0722
R1, wR2 [I > 4σI] 0.0576, 0.1598 0.0266, 0.0719
GoF 1.058 1.064
Merohedral Twinning
(
1 0 0
0 1 0
0 0 1
), (
0 1 0
1 0 0
0 0 -1
),
(
-1 0 0
0 -1 0
0 0 -1
), (
0 -1 0
-1 0 0
0 0 1
)
(
1 0 0
0 1 0
0 0 1
), (
0 1 0
1 0 0
0 0 -1
),
(
-1 0 0
0 -1 0
0 0 -1
), (
0 -1 0
-1 0 0
0 0 1
)
Twin Fraction 0.25762, 0.24354, 0.24397, 0.25487 0.23419, 0.24435, 0.26565, 0.25581

87
5.2.4 Crystal structure of BaTiSe3
We then refine[132] BaTiSe3 in 21.1 Å × 21.1 Å × 6.1 Å unit cell as a P31c space group (Table
5.1 lists the refinement statistics including a refinement residual R1 as low as 2.66%). The P31c
BaTiSe3 breaks the translational, inversion, and mirror symmetries by adopting ordered c-axis offcentric TiSe6 octahedra in c-axis antiparallel displaced TiSe6-chains, which was also observed in
the room temperature crystal structure of BaTiS3
[62]. Figure 5.5 visualizes the resulting (Table 5.2
and Table 5.3) supercell-ordering of the antiparallel TiS6 chain displacements, whose P31c unit
cell is highlighted (solid line) in comparison with P63/mmc (dotted line). TiSe6 chains are colored
red (-c) and blue (+c) corresponding to the direction of chain displacement to the Ba atoms around
them along the c-axis, illustrated in Figure 1f. On top of the chain displacements, Ti atoms also
displace away from the centroid of TiSe6 octahedra, thus leaving behind antiparallel TiSe6 dipoles
along the c-axis.

88
Table 5.2: Crystal structure of needle-shaped single crystal BaTiSe3.
[64]
Atom x/a y/b z/c Uiso (Å2
) Occupancy Symmetry Order
Ba01 0.88750(2) 0.10891(3) 0.88212(9) 0.02293(13) 1 1
Ba02 0.44175(2) -0.11198(3) 0.40830(9) 0.02655(14) 1 1
Ba03 0.22431(2) -0.22106(3) 0.90769(9) 0.02709(15) 1 1
Se01 0.60849(3) 0.38781(4) 0.4491(2) 0.0191(2) 1 1
Se02 0.55401(2) 0.27911(4) 0.9492(2) 0.0190(2) 1 1
Se03 0.72124(4) -0.05796(4) 0.83719(19) 0.01702(16) 1 1
Se04 0.38788(3) 0.11302(3) 0.9532(2) 0.0177(2) 1 1
Se05 0.72291(4) 0.11228(3) 0.83603(19) 0.01715(15) 1 1
Se06 0.55389(2) -0.05567(4) 0.83676(19) 0.01801(16) 1 1
Se07 0.22060(3) -0.05681(3) 0.9538(3) 0.0177(2) 1 1
Se08 0.39054(3) -0.05512(3) 0.9538(3) 0.0174(2) 1 1
Ti01 0.666667 0.333333 0.7239(6) 0.0274(5) 1 3
Ti02 0.33260(6) -0.00017(8) 0.7289(5) 0.0250(5) 1 1
Ti03 0.66597(5) -0.00005(6) 0.5607(4) 0.0223(3) 1 1
Ti05 0.666667 0.333333 1.2241(6) 0.0272(5) 1 3
Se9A 0.94606(5) 0.05887(5) 0.3373(4) 0.0156(2) 0.749(4) 1
Ti5A 1 0 0.5602(8) 0.0217(4) 0.749(4) 3
Se9B 0.94145(12) 0.05419(12) 0.4339(9) 0.0144(7) 0.251(4) 1
Ti5B 1 0 0.7143(19) 0.0202(15) 0.251(4) 3

89
Table 5.3: Anisotropic atomic displacement parameters (ADPs) of needle-shaped single crystal BaTiSe3.
[64]
Atom U11 (Å2
) U22 (Å2
) U33 (Å2
) U23 (Å2
) U13 (Å2
) U12 (Å2
)
Ba01 0.02225(12) 0.02136(12) 0.0264(4) 0.00698(14) 0.00553(13) 0.01185(11)
Ba02 0.02596(13) 0.02190(17) 0.0344(4) -0.0067(2) -0.01504(17) 0.01395(14)
Ba03 0.02642(14) 0.02106(14) 0.0351(4) 0.0083(2) 0.01508(17) 0.01289(14)
Se01 0.01932(17) 0.01836(17) 0.0243(6) -0.0011(3) -0.0023(3) 0.01302(15)
Se02 0.01160(16) 0.01837(17) 0.0243(6) 0.0017(3) -0.00121(19) 0.00541(11)
Se03 0.0201(2) 0.0210(2) 0.0159(5) 0.0026(3) 0.0032(3) 0.01478(17)
Se04 0.0220(2) 0.01223(18) 0.0169(7) 0.0010(2) 0.0008(3) 0.00694(14)
Se05 0.0226(2) 0.01162(15) 0.0146(5) -0.00011(18) -0.0026(3) 0.00654(16)
Se06 0.01156(14) 0.0235(2) 0.0144(5) -0.0027(3) -0.0006(2) 0.00540(18)
Se07 0.01191(17) 0.0213(2) 0.0159(7) 0.0000(2) -0.0003(2) 0.00531(15)
Se08 0.0210(2) 0.0220(2) 0.0154(6) 0.0003(2) 0.0005(2) 0.01548(18)
Ti01 0.0307(4) 0.0307(4) 0.0208(14) 0 0 0.0154(2)
Ti02 0.0286(4) 0.0319(4) 0.0147(16) 0.0009(3) 0.0001(3) 0.0153(3)
Ti03 0.0287(4) 0.0291(3) 0.0088(9) -0.0009(3) -0.0006(3) 0.0141(3)
Ti05 0.0307(4) 0.0307(4) 0.0202(13) 0 0 0.0153(2)
Se9A 0.0189(2) 0.0178(2) 0.0154(6) 0.0001(3) -0.0004(3) 0.01319(19)
Ti5A 0.0276(5) 0.0276(5) 0.0100(10) 0 0 0.0138(3)
Se9B 0.0158(6) 0.0147(6) 0.018(2) -0.0040(9) -0.0056(9) 0.0113(5)
Ti5B 0.0239(11) 0.0239(11) 0.013(4) 0 0 0.0119(5)

90
Figure 5.5: Refined crystal structure of BaTiSe3. (a) BaTiSe3 adopts a P31c space group in a 2√3×2√3×1
supercell of the P63/mmc along the a-b plane. In P31c, TiS6 chains are displaced along the c-axis in opposite
directions, colored in red (downwards along the c-axis) and blue (upwards along the c-axis) as indicated in
(a) and (b). Besides, Ti atoms also move away from the centroid of Se6 octahedra along the c-axis. This is
analogous but of different periodicity compared to BaTiS3.
[64]
Moreover, as we examine the scattering profile between the Bragg reflections, diffuse scattering
arises. Figure 5.4(d) points out the streaky diffuse scattering between adjacent sharp Bragg peaks,
observed only along the a-b plane. This is analogous to the a-b plane diffuse scattering in
BaTiS3
[62]. Consequently, the corresponding electron density distribution of Ti atoms along the ab plane adopts the anharmonic trigonal displacements [Figure 5.6] in the same manner as the
correlated a-b plane displacements of room temperature BaTiS3, which has been shown as a
significant factor in achieving the giant optical anisotropy[62] of BaTiS3. We thus expect analogous
disordered Ti a-b plane displacements[62] in BaTiSe3 to play an important role in achieving optical
anisotropy, comparable to BaTiS3, across the visible to infrared wavelength range.

91
Figure 5.6: Electron density map of BaTiSe3 near Ti01 and Ti02 projected along the a-b plane. The offcentric electron distribution of Ti is an indicator of disordered a-b plane Ti displacements towards the
closest Se atoms. The average a-b plane off-centric displacements of Ti01 and Ti02 are refined as 0.155 Å
and 0.166 Å towards the black arrows. This is larger than the ~ 0.12 Å in BaTiS3
[62]
.
[64]
5.3 BaTiSe3 Optical Anisotropy
5.3.1 Polarization-dependent Raman anisotropy
In line with the lower symmetry room-temperature structure of BaTiSe3 (21.1 Å × 21.1 Å × 6.1
Å unit cell of P31c space group) than BaTiS3 (13.3 Å × 13.3 Å × 5.8 Å unit cell of P63cm space
group[62]), more active vibrational Raman modes[142] compared to BaTiS3 were observed in the
Raman spectra as shown in Figure 2a. Under 532 nm excitation, the Raman spectrum resolves at
least six Raman modes: 121 cm-1
, 158 cm-1
, 186 cm-1
, 227 cm-1
, 252 cm-1
, 315 cm-1
, marked by
dashed lines. This is a clear distinction from two major Raman modes observed in BaTiS3 under
the same conditions[143]
.
We studied the polarization-dependent Raman response by varying the polarization of the 532
nm excitation, as shown in Figure 5.7(b). We represent the polarization dependence of Raman
mode intensity (before background subtraction) as a polar plot in Figure 5.7(c) to show the
anisotropic Raman response. With the a-axis aligned parallel to the 0° polarization, 121 cm-1
and

92
315 cm-1
are strongest near 0°, while 186 cm-1
and 227 cm-1
(along with difficult-to-extract 158
cm-1
and 252 cm-1
) near 90°. Although a polarization analyzer is needed to map the exact Raman
tensors of observed vibrational modes, it is evident that qualitatively BaTiSe3 shows a highly
anisotropic Raman response for 532 nm excitation.
Figure 5.7: Room-temperature polarization-resolved Raman study of BaTiSe3. (a) Unpolarized Raman
spectrum of BaTiSe3. Major Raman modes are labeled by dashed lines. The inset shows the optical image
of the BaTiSe3 crystal whose crystallographic orientation is labeled; (b) Polarization-resolved Raman
spectrum with linearly polarized excitation light 0° to 360° with respect to the a-axis; (c) Area-under-thecurve Raman intensities of different polarization orientations. 121 cm-1 and 315 cm-1
show opposite
polarization dependence compared to 186 cm-1
and 227 cm-1
.
[64]
5.3.2 Fourier-transform infrared spectroscopy
Infrared (IR) reflectance and transmittance spectra were measured by polarization-resolved
Fourier transform infrared spectroscopy (FTIR). When incident light was linearly polarized
parallel or perpendicular to the uniaxial optical axis, which is the c-axis based on the trigonal P31c
crystal structure, anisotropic transmittance [Figure 5.8(a)] and reflectance [Figure 5.8(b)] spectra
were observed. The limited thickness uniformity, as shown in the inset of Figure 5.8(a), and the
crystal tilting perturbs the Fabry-Perot fringes in the transmittance spectra and leads to appreciable
scattering. Nevertheless, the anisotropic absorption edges of the extraordinary (|| c) and ordinary
(⟂ c) optical polarization were clear and extracted to be 0.20 eV and 0.33 eV. Further, the Fabry-

93
Perot fringes of the reflectance spectra [Figure 5.8(b)], measured on the smooth front surface, show
extinction at the energies consistent with the absorption edges seen in the transmission spectra
[Figure 5.8(a)].
As the refractive index is relatively constant within the transparent regime, the Fabry–Pérot
interference between the top and bottom surfaces of BaTiSe3 has a free spectral range[144] of:
∆ ≈

2 cos
=

2 cos
∙
1

,
where ν is the frequency, c is the vacuum velocity of light, n is the real part of the refractive index,
l is the crystal thickness ~21.33 µm, and θ is the incident angle. We thus carry out a fast Fourier
transform (FFT) for the Fabry–Pérot fringes on the reflectance spectrum on the significantly
smoother top surface [Figure 5.8(b)] to approximate the spectrum-averaged real part of the
refractive index (n)
[145] between the 0.075 eV and 0.2 eV. The inset of Figure 5.8(b) shows the
resultant FFT spectrum of refractive index, with a birefringence of ne – no ~ 0.9.
Figure 5.8: Polarization-resolved FTIR (a) transmittance and (b) reflectance spectra show large anisotropy
between the ordinary (⟂ c) and the extraordinary (|| c) polarization. The absorption energy of ordinary and
extraordinary are observed at 0.20 eV and 0.33 eV, consistent with the extinction of Fabry-Perot fringes.
The cross-section of BaTiSe3 is shown in the inset of (a). The crystal thickness was measured to be 21.33
μm. Inset of (b) shows FFT analysis on the reflectance spectra for the polarization-dependent refractive
index of BaTiSe3.
[64]

94
To fully quantify the degree of optical anisotropy, we then combined variable-angle
ellipsometry measurements over the spectral range of 210 nm to 2500 nm with the polarizationresolved reflection and transmission measurements over 1.5 μm to 17 μm and quantitatively
extracted the complex refractive index of BaTiSe3 for wavelengths from 210 nm (5.9 eV) to 17
μm (0.073 eV) in Figure 5.9(a). The resulting birefringence (Δn = ne – no) and dichroism (Δκ = κo
– κe) are then plotted in Figure 5.9(b). The dichroism becomes largest near 1 eV, within the shortwave infrared (SWIR) spectrum range, and the birefringence is as large as 0.9 across MWIR and
LWIR. As the low-loss regime of BaTiSe3 occurs below 0.20 eV, red spheres marked its spectrum
range in Figure 5.9(b).
Figure 5.9: Optical anisotropy of BaTiSe3 (in collaboration with Mikhail Kats of UW–Madison). (a) Optical
properties obtained from combining FTIR and ellipsometry, both real (n) and imaginary (κ) parts for
ordinary and extraordinary linear polarization. (b) The large dichroism (Δκ = κo – κe) peak across SWIR
leaves behind a giant birefringence (Δn = ne – no) up to 0.9 across the MWIR to LWIR wavelengths. [64]
BaTiS3 and BaTiSe3 share similar structural features, especially the existence of the a-b plane
Ti displacements and the disorder of such in-plane distortions. BaTiSe3, up to 0.9, slightly
surpasses the birefringence of BaTiS3, which is up to 0.76.

95
5.4 Conclusions
In this chapter, we demonstrated another member of the quasi-1D perovskite chalcogenide
A1+xTiX3 (A = Sr, Ba; X = S, Se), BaTiSe3, to be highly optically anisotropic at room temperature.
We synthesized and studied the single crystals of BaTiSe3, formerly reported[140] P63/mmc space
group in powder form. High-quality BaTiSe3 crystals adopt antiparallel displacements and dipolar
octahedron symmetry breaking along the c-axis whose periodicity follows a 2√3×2√3×1 supercell
of P31c symmetry. Despite the longer-range a-b plane structural complexity compared to BaTiS3,
BaTiSe3 crystals possess an analogous uniaxial optical axis parallel to the long axis of the needle,
with giant birefringence (Δn up to 0.9) in a broadband spectral region across the MWIR and LWIR
ranges. We anticipate the class of quasi-1D chalcogenides, ABX3 (A = Sr, Ba, ..., B = Ti, Zr, …,
and X = S, Se, …) to cover a wide spectral range of optical and optoelectronic anisotropy, leading
to miniaturized polarization resolved IR devices for sensing and telecommunications.

96
Chapter 6. Optical Anisotropy of
Incommensurate Hexagonal TMPCs Sr1+xTiS3
6.1 Introduction
In the search for materials of large optical anisotropy, TMPCs of BaNiO3-type hexagonal
structures such as BaTiS3 and BaTiSe3 have demonstrated giant birefringence in the MWIR and
LWIR region. We have observed lattice distortion of the quasi-1D TiX3 (X = S, Se) chain at
different periodicity as a function of temperature, which also induced the Ti off-centric
displacement within TiS6 octahedra both along the c-axis and along the a-b plane. The c-axis
displacements do not lead to optical anisotropy, while the a-b plane displacements preferentially
enhance the extraordinary refractive index while deducting the ordinary refractive index. Such
room temperature lattice modulation is disordered in nature and could not be efficiently resolved
by space group symmetry. However, another ubiquitous lattice modulation, incommensurate or
commensurate lattice modulation, of non-stoichiometric Sr1+xTiS3 has periodic, despite x can
sometimes be irrational, lattice modulations with well-defined (3+1)-D space group symmetry.
Niu et al. (2018) reported the growth of single crystal Sr1+xTiS3 via chemical vapor transport[139]
.
Sr1+xTiS3 adopts an incommensurate or commensurate ratio[55,56] [Figure 6.1] of excessive (x)
stoichiometry ratio of Sr atoms along the c-axis to accommodate the mismatched lattice size
between the Sr and TiS3 sublattices. A giant dichroic ratio up to ~20 is observed between the
absorptance of intra-chain and inter-chain polarizations[139], larger than BaTiX3 (X = S, Se). This

97
is a clear sign that Sr1+xTiS3 also has giant birefringence, whose optical tensors measurement is
limited by the size of the crystals. Moreover, the unexplored crystal structure of Sr1+xTiS3 crystals,
especially the incommensurate ratio x and the atomic scale lattice distortion of the incommensurate
modulations, prevents us from understanding what the stoichiometry ratio contributes to the
asymmetry in the light-matter interaction in Sr1+xTiS3.
Figure 6.1: Sr1+xTiS3 crystal structure and morphology. (a) Optical image of Sr9/8TiS3 platelet crystal.
Terraces near the edges make the thickness non-uniform. (b) Optical image of thin Sr9/8TiS3 needle and its
crystal structure; [63]
Here, we synthesized platelets [Figure 6.1(a)] and thin needles [Figure 6.1(b)] of Sr9/8TiS3 with
great quality and resolved their crystal structure in SC-XRD. The observed structural modulation
turns out as a new mechanism for dramatically enhancing the anisotropy of electronic
polarizability, far exceeding values that have been achievable by the anisotropic distribution of
individual ions with distinct polarizability. In quasi-1D chalcogenide Sr1+xTiS3, the structural
modulation controls the selective occupation of strongly oriented (anisotropic) electronic states
and hence leads to a birefringence of ~ 2.1, significantly larger than has been observed in
transparent regions of both quasi-1D and layered “van der Waals” materials to date [Figure 1.3].

98
6.2 Crystal Structure Determination
6.2.1 Optimized crystal growth
Crystal growth of Sr9/8TiS3 adopts the same sealed ampoule growth via chemical vapor
transport (CVT) method as BaTiX3 (X = S, Se), except for SrS (Sigma-Aldrich, 99.9%) instead of
BaS is used as the precursor[139]
. In order to get larger platelet crystals of Sr9/8TiS3, growth
temperature and cooling rate are optimized. STS crystals of larger size are synthesized at a reaction
temperature of 1060 ℃, held for 100 h, and slowly cooled down to 1010 ℃ at 5 ℃/h before
shutting down the furnace. The heating rate is set to be 100 ℃/h and the cooling rate is consistent
with the natural cool-down rate of the furnace.
Figure 6.1(a-b) show the optical image of the Sr9/8TiS3 crystal. Most Sr9/8TiS3 crystals feature
platelet morphologies but commonly with terraces on the surface. The width of Sr9/8TiS3 crystals
can go up to ~500 µm with thickness of ranging from ~10 µm to ~100 µm. Thinner crystals come
with optically flat surfaces and are thus ideal for optical anisotropy characterizations.
The orientation determination of Sr9/8TiS3 is consistent with that of BaTiX3, where out-of-plane
XRD determines the crystal orientation and pole figure analysis of off-axis reflections determines
the orientation parallel to the tilting / rotational axis, thus the in-plane orientation. The
characterized crystal orientation is consistent with the reported Sr9/8TiS3 orientation, where the caxis is parallel to the geometric long axis of the (110)-oriented Sr9/8TiS3. One interesting feature
of Sr9/8TiS3 is that the ends of the needles usually feature sharp angles instead of being flat or
uneven. This angle is consistent between different crystals and is also seen in the corners of platelet
crystals. Considering the edges and surfaces of a naturally grown crystal usually preserve special

99
crystalline edges and surfaces, the special corner angle could be a special surface in the crystal
structure of STS and may guide us in crystal structure determination.
6.2.2 Out-of-plane X-ray diffraction
The wider and thicker crystals make the XRD of any Sr1+xTiS3 orientation easy to carry out
with the help of manually aligning the large crystals at any orientations of the goniometer. Figure
6.2 shows the XRD along the 001 orientation of the crystal, measured along the geometrical long
axis. As this is parallel to the incommensurate vector q, we can then calculate the magnitude of q
to determine the incommensurate ratio x. One-dimensional incommensurate along the c-axis are
defined by two independent c-axes: c1 and c2, and the reciprocal vector
= ℎ
∗ +
∗ + 11
∗ + 22
∗
,
where
∗
,
∗
, 1
∗
and 2
∗
are reciprocal primary vectors, ℎ12 is the incommensurate index of
the corresponding reflections. As reported for Sr1+xTiS3
[146]
, 1
∗
is parallel to 2
∗
, and 1
∗ = 2
∗
for x = 1. Although 1
∗
and 2
∗
are mathematically free to define, we will define that 1
∗
and 2
∗
are reciprocal primary vectors of TiS3 and Sr sublattices, respectively.

100
Figure 6.2: Sr1+xTiS3 crystal 00l XRD. Indices are labeled as hkl1l2 of incommensurate lattice a, b, c1, c2
according to 2. The rocking curve of 0002 is shown in the inset. A FWHM of 0.01º shows good
crystallinity. [63]
All reflections in Figure 6.2 are indexed following such definition. We can thus calculate the
ratio between 1
∗
and 2
∗
. For example, comparing 0004 and 0032:
31
∗−22
∗
42
∗ =
5
6
.
Therefore, 1
∗
/ 2
∗= 16 / 9. As there are two layers of Sr for one unit cell of Sr sublattice, 1 + x =
22
∗
/ 1
∗ = 9 / 8. However, as this is not the only way to index all the reflections, further
characterizations are needed.
6.2.3 Single crystal X-ray diffraction
Single crystal X-ray diffraction (SC-XRD) of Sr9/8TiS3 shows rhombohedral symmetry. Figure
6.3 shows the h0l, 0kl, and hk0 reciprocal planes of the precession map. We saw hexagonal lattices
in hk0 plane, but the h0l and 0kl planes reduce the rotational symmetry to trigonal. In h0l and 0kl
planes, both satellite reflections and main show up. However, satellite reflections follow a different
symmetry from the main reflections. They seem to line up along a different axis, noted as dashed
lines. The angle of these axes with respect to
∗
are the same in h0l and 0kl planes. This could then

101
be a rhombohedral lattice, which can be defined either as a larger volume trigonal unit cell
(reciprocal axes
∗
,
∗
, orthogonal to
∗
) or rhombohedral unit cell (
∗
,
∗
,
∗ not
orthogonal with each other). The angle between the a-b plane and the c-axis of the rhombohedral
lattice defined the sharp angle of the needle tips mentioned in crystal morphology.
The crystal structure of Sr9/8TiS3 was resolved using single-crystal X-ray diffraction (SC-XRD).
Both 3D and (3+1)D modulation approaches were used to solve the modulated structure. The
resulting Sr54Ti48S144 structure of R3c (with refinement residual R1 = 0.0468) or Sr1.125TiS3
structure of R3̅m(00g)0s
[147] space groups reveal a similar modulated structure (visualized in
Figure 6.4(a-b)), which is consistent with the previously reported R3m(00g)0s Sr9/8TiS3
structure[148–152]. The details of the data collection, data reduction, and structure refinements are
listed in Table 6.1, and the best-resulting crystal structure is reported in Table 6.2 and Table 6.3.
Figure 6.3: Precession map of STS crystal along the 0kl, h0l, and hk0 reciprocal planes. Observed reflections
follow a trigonal symmetry but the main reflections do not line up along the primary axes, i.e., reciprocal
space unit principal axes not horizontal nor vertical in 0kl and h0l, but at an angle. Such principal axes are
then verified and labeled as rhombohedral axes
∗
,
∗
,
∗ beneath the precession maps, side by side
with the equivalent primary trigonal axes
∗
,
∗
,
∗
. Following a primary hexagonal unit cell, satellite
reflections in 0kl and h0l can be labeled according to the sublattice size of Sr and TiS3, whose extinction
revealed their sublattice symmetries of P3c1 and R3. [63]

102
In contrast to the hypothetical stoichiometric counterpart SrTiS3, the Sr9/8TiS3 lattice has
structural modulations consisting of blocks of face-shared octahedra (referred to as ‘O’) that are
separated by pseudo-trigonal-prismatic TiS6 units (referred to as ‘T’) along the c-axis as shown in
Figure 6.4. The structural modulation of Sr9/8TiS3 arises from an overall trigonal twist distortion
compared to the average unmodulated structure of SrTiS3. To accommodate excess Sr in the lattice
along the c-axis, Sr atoms undergo displacements within the a-b plane, resulting in the triangularshaped projection in Figure 6.4a. The Sr displacements are accompanied by twist distortion of TiS6
units from octahedral to trigonal-prismatic polyhedron. These different polyhedral units have
different Ti–Ti distances along the c-axis. By counting the stacking sequence of building blocks
(structurally classified as O and T), TiS6 chains with periodic [−(T−O−T) –(O)5−]2 successions
can be used to define the modulation periodicity of 16 units of TiS6 within every 18 Sr layers in
the Sr9/8TiS3 lattice [Figure 6.4b].

103
Table 6.1: Data collection, intensity statistics, and refinement statistics of SC-XRD on Sr9/8TiS3.
Sr1.125TiS3
Temperature 260 K
Radiation Type Mo K⍺
Wavelength 0.71073 Å
Space group R3c R-3c R32
Cell dimensions
a, b, c (Å) 11.4555(4), 11.4555(4), 47.6287(18)
α, β, γ () 90, 90, 120
Volume (Å3
) 5412.9(4)
Density (g/cm3
) 3.573
Reduction &
Scaling Point group symmetry: -3m1
Resolution (Å) inf1.47
1.47-
1.15
1.15-
1.00
1.00-
0.90
0.90-
0.84
0.84-
0.78
0.78-
0.74
0.74-
0.71
0.71-
0.68
Obs. Reflections 3999 4474 3526 3003 2575 2270 2122 1970 1328
Completeness (%) 99.6 100 100 100 100 100 100 100 97.7
Redundancy 15.8 17.7 13.9 11.9 10.2 9 8.4 7.8 5.2
Mean I / I 61.92 52.15 40.07 31.95 20.86 18.33 16.87 13.12 9.84
Rint 0.031 0.035 0.046 0.056 0.102 0.113 0.109 0.147 0.137
Rσ 0.012 0.013 0.017 0.022 0.035 0.042 0.045 0.064 0.088
Reflections I > 2σI
Rσ, Rint 0.0343, 0.0501 0.0233, 0.0512 0.0340, 0.0501
θmin, θfull, θmax () 2.097, 25.242, 30.504 2.097, 25.242, 30.504 2.097, 25.242, 30.504
Friedel fraction
full 0.938 ~ 1.000
Refinement
Weighting w = 1/[σ
2
(Fo
2
) + (0.0217P)
2 + 48.6113P], where P = (Fo
2 + 2Fc
2
) / 3
Resolution (Å) 50.0 - 0.70 50.0 - 0.70 50.0 - 0.70
No. of reflections 3465 1846 3690
No. of I > 2σI 2872 1622 2823
No. parameters 126 65 128
No. constraints 1 0 0
R1, wR2 [all data] 0.0468, 0.0659 0.0439, 0.0677 0.0532, 0.0679
R1, wR2 [I > 4σI] 0.0322, 0.0593 0.0358, 0.0637 0.0324, 0.0591
GoF 1.070 1.095 1.073
Twin Matrix (
1 0 0
0 1 0
0 0 1
), (
0 1 0
1 0 0
0 0 -1
) (
1 0 0
0 1 0
0 0 1
)
(
1 0 0
0 1 0
0 0 1
),
(
-1 0 0
0 -1 0
0 0 -1
)
BASF 0.50697, 0.49303 1 0.54625, 0.45375

104
Table 6.2: Atomic coordinates and equivalent isotropic atomic displacement parameters for R3c Sr1.125TiS3
Atom x/a y/b z/c Uiso (Å2
) Occupancy Symmetry Order
Sr01 0.6627(2) 0.69470(19) 0.47341(3) 0.0155(5) 1 1
Sr02 0.33830(19) 0.30412(18) 0.42312(3) 0.0110(4) 1 1
Sr03 -0.01735(15) 0.31715(15) 0.36505(8) 0.01654(14) 1 1
Ti04 0.333333 0.666667 0.67383(14) 0.0101(11) 1 3
Ti05 0.333333 0.666667 0.73435(14) 0.0119(12) 1 3
Ti06 0.333333 0.666667 0.42931(14) 0.0095(11) 1 3
Ti07 0.333333 0.666667 0.61528(19) 0.0073(3) 1 3
Ti08 0.333333 0.666667 0.3646(2) 0.0130(3) 1 3
Ti09 0.333333 0.666667 0.30027(14) 0.0095(11) 1 3
Ti0A 0.333333 0.666667 0.49553(13) 0.0078(11) 1 3
Ti0B 0.333333 0.666667 0.55666(13) 0.0058(10) 1 3
S00C 0.1646(5) 0.6675(5) 0.70429(10) 0.0092(9) 1 1
S00D 0.3253(5) 0.4973(5) 0.26680(10) 0.0090(9) 1 1
S00E 0.3094(5) 0.8278(5) 0.64432(10) 0.0078(8) 1 1
S00F 0.5036(5) 0.6683(5) 0.52486(10) 0.0107(9) 1 1
S00G 0.1478(5) 0.6437(5) 0.58459(10) 0.0102(9) 1 1
S00H 0.3403(5) 0.5053(5) 0.46237(10) 0.0118(10) 1 1
S00I 0.2345(6) 0.4788(5) 0.33233(12) 0.0255(12) 1 1
S00J 0.4295(6) 0.5745(5) 0.39707(12) 0.0179(10) 1 1
Table 6.3: Anisotropic atomic displacement parameters for R3c Sr1.125TiS3.
Atom U11 (Å2
) U22 (Å2
) U33 (Å2
) U23 (Å2
) U13 (Å2
) U12 (Å2
)
Sr01 0.0135(9) 0.0116(9) 0.0176(11) -0.0026(7) 0.0078(6) 0.0033(7)
Sr02 0.0133(8) 0.0113(8) 0.0119(9) 0.0036(6) 0.0042(6) 0.0087(7)
Sr03 0.0114(9) 0.0142(10) 0.0192(3) 0.0069(9) -0.0017(9) 0.0028(2)
Ti04 0.0099(15) 0.0099(15) 0.011(3) 0 0 0.0050(7)
Ti05 0.0111(17) 0.0111(17) 0.013(3) 0 0 0.0055(8)
Ti06 0.0088(16) 0.0088(16) 0.011(3) 0 0 0.0044(8)
Ti07 0.0071(4) 0.0071(4) 0.0077(8) 0 0 0.0035(2)
Ti08 0.0134(5) 0.0134(5) 0.0122(8) 0 0 0.0067(2)
Ti09 0.0087(16) 0.0087(16) 0.011(3) 0 0 0.0044(8)
Ti0A 0.0094(16) 0.0094(16) 0.005(3) 0 0 0.0047(8)
Ti0B 0.0078(15) 0.0078(15) 0.002(2) 0 0 0.0039(7)
S00C 0.008(2) 0.015(2) 0.010(2) -0.0017(16) 0.0022(16) 0.0096(17)
S00D 0.013(2) 0.005(2) 0.008(2) 0.0013(15) 0.0032(16) 0.0032(16)
S00E 0.0107(18) 0.007(2) 0.007(2) 0.0010(14) -0.0021(16) 0.0049(16)
S00F 0.010(2) 0.016(2) 0.006(2) 0.0006(16) -0.0018(16) 0.0059(17)
S00G 0.010(2) 0.018(2) 0.005(2) -0.0018(17) -0.0027(16) 0.0091(18)
S00H 0.017(2) 0.009(2) 0.010(2) -0.0042(16) -0.0035(17) 0.0069(18)
S00I 0.039(3) 0.006(2) 0.013(2) -0.0026(17) 0.013(2) -0.0026(19)
S00J 0.030(2) 0.018(2) 0.015(2) 0.0085(17) 0.0102(19) 0.0185(18)

105
Figure 6.4: Structural modulation in Sr9/8TiS3. Schematics representing the modulated Sr9/8TiS3 (R3c) lattice,
resolved from single-crystal XRD (SC-XRD) and viewed along the (a) [001] axis and (b) [100] axis. The
octahedral (O) and pseudo-trigonal prismatic (T) TiS6 units are highlighted in blue and orange, respectively;
Atomic-resolution HAADF-STEM images (in collaboration with Rohan Mishra of WashU) a Sr9/8TiS3
crystal along the (c) [001] axis and (d) [100] axis; (e) High-magnification HAADF-STEM image (left panel)
and simulated image (right panel) of Sr9/8TiS3 view along the [100] axis. A schematic of one column of
atoms is overlayed on the simulated image. A repeating pattern of three bright atomic columns, where the
Ti and Sr atoms overlap along the viewing direction, can be observed in these HAADF images. These triplet
atomic columns are highlighted with square brackets in the atomic model; (f) Line profiles across the
experimental and simulated STEM images, along the white boxes in (e), comparing the intensity variation
across a single atomic column. [63]
6.2.4 Electron Diffractions
To directly visualize the subtle structural modulations in Sr9/8TiS3, we performed atomically
resolved imaging using an aberration-corrected scanning transmission electron microscope
(STEM). Large field-of-view, high-angle annular dark-field (HAADF) images of the Sr9/8TiS3
crystal viewed along the [001] and [100] zone axis are shown in Figure 6.4(c-d). In this imaging
mode, the intensity of the atomic columns is approximately proportional to the square of the
effective atomic number of the column (Z
2
)
[153]. Along the [001] zone axis, the Sr atomic columns
appear as triangles due to their staggered arrangement along the c-axis [Figure 6.4(c)], which

106
matches well with the structural features in Figure 6.4(a). Along the [100] orientation, the Ti and
Sr columns overlap within the triple blocks of – (T–O–T) –, and therefore, they appear as bright
triplets in the HAADF images as they have a higher intensity than the Sr-only atomic columns
within the block of five octahedral units –(O)5− [Figure 6.4(d-e)]. We also observe periodic
distortions of the Sr atomic columns in the form of contraction and dilation of the Sr–Sr distance
between neighboring chains. A comparison of the intensity and spacing between atomic columns
in the experimental and the simulated HAADF images, as shown in Figure 6.4(f), shows excellent
agreement, and corroborates the modulation periodicity.
6.2.5 Lattice modulations in Sr9/8TiS3
To accommodate the excess Sr atoms in Sr9/8TiS3 (R3c), certain TiS6 units undergo a periodic
trigonal twist distortion resulting in the formation of TiS6 trigonal-prismatic polyhedra from the
octahedral polyhedra in SrTiS3. Figure 6.5(b-c) show two building blocks of the modulated
Sr9/8TiS3 lattice. Compared to face-shared octahedral TiS6 units (O) represented in blue, the
pseudo-trigonal prismatic units (T) in orange have elongated Ti–S bonds and have a ~30 ˚ rotation
of S3 triangles around the c-axis. As shown in Figure 6.5(a), Sr atoms show displacive modulation
about the average position along both the c-axis and the a-b plane, which leads to the twist
distortion of TiS6 units accompanied by altered Ti–Ti distances along the c-axis. When projected
along the [100] zone axis, Ti atoms periodically overlap with Sr as highlighted with dashed ellipses
in Figure 6.5(a).

107
Figure 6.5: Structural modulations in Sr9/8TiS3. (a) Schematic representing modulated Sr9/8TiS3. Three
identical TiS6 chains (1, 2, 3) are shown. They have a commensurate stacking sequence of [−(T−O−T) –
(O)5−]2, which defines the modulation periodicity. The dashed ellipses indicate the periodic (T−O−T)
segments, where Ti atoms overlap with Sr atoms, when projected along [100] zone axis; Schematic
representing building blocks for modulated Sr9/8TiS3 with top and side views. (b) Face-sharing octahedra
(O) are highlighted in blue, and (c) pseudo-trigonal prismatic (T) units are represented in orange. [63]
6.3 Optical Anisotropy
6.3.1 Fourier transform infrared spectroscopy
The optical anisotropy of Sr9/8TiS3 was measured in two steps: we first acquired polarizationdependent, normal-incidence reflectance and transmittance spectra of the platelet Sr9/8TiS3 in
Figure 6.1(b) using Fourier-transform infrared spectroscopy (FTIR), shown in Figure 6.6. The
thickness of the crystal was estimated to be 3.2 μm by fitting to the Fabry–Pérot fringes in the
spectra for each polarization and verified by cross-section scanning electron microscope (SEM)
imaging. The large difference in the reflectance and transmittance between the two polarizations
(parallel and perpendicular to the c-axis) is a clear indication of large optical anisotropy. Such
anisotropic absorption edges of intra-chain and inter-chain polarizations are ~ 5 µm and ~2.5 µm,

108
respectively. The wavelength where Fabry–Pérot fringes disappear in the reflectance spectra is
consistent with the transmittance.
Figure 6.6: Polarization-dependent infrared a) transmittance and b) reflectance of Sr9/8TiS3. Polarizations
are either perpendicular (blue) or parallel (orange) to the c-axis of the Sr9/8TiS3 crystal. [63]
The summation of reflectance (R) and transmittance (T) is also plotted in Figure 6.7. We can
see that for thin Sr9/8TiS3 crystals, R + T is close to 0.95 for polarization perpendicular to the caxis at a wavelength of 3 – 16 μm, while close to 0.85 for polarization parallel to the c-axis at
wavelengths longer than 6 μm. Such low loss (1 – R – T) Sr9/8TiS3 transmittance window extends
up to the cut-off wavelength of the detector at 17 μm proving that Sr9/8TiS3 is an ideal birefringent
crystal to manipulate polarizations with a transmittance window longer than 6 μm. We expect that
this low-loss region extends to longer wavelengths, limited by phonon (~27 μm)[154] or plasmon
resonance.

109
Figure 6.7: Total signal (R + T) from reflectance and transmittance spectra of Sr9/8TiS3 crystal for
polarization perpendicular and parallel to the c-axis. [63]
6.3.2 Refractive indices and colossal birefringence
To fully quantify the degree of optical anisotropy, we then combined variable-angle
ellipsometry measurements over the spectral range of 210 nm to 2500 nm with the polarizationresolved reflection and transmission measurements in Figure 6.6 and extracted the complex
refractive index of Sr9/8TiS3 for wavelengths from 210 nm to 17 μm [Figure 6.8(a)]. Three different
ellipsometry measurements were performed: with the c-axis parallel to the plane of incidence,
perpendicular to the plane of incidence, and at an off-axis angle; all this data was combined in a
simultaneous analysis. In the low-loss region > 6 μm, Sr9/8TiS3 has a birefringence (Δ) up to
2.1 [Figure 6.8(b)]. This is by far the largest birefringence among reported anisotropic crystals, to
the best of our knowledge [Figure 1.3]. Compared to layered 2D materials MoS2
[47] and h-BN[44]
,
Sr9/8TiS3 possesses large anisotropy across a broad low-loss region in mid-infrared [Figure 6.9],
and the in-plane optic axis is easy to exploit for practical optical components. As we discuss below,
this extreme birefringence is a result of the enhancement of the extraordinary index (
) from the
expected ≲3.4 in a hypothetical unmodulated stoichiometric crystal to ~4.5 in Sr9/8TiS3.

110
Figure 6.8: Uniaxial anisotropic optical properties of Sr9/8TiS3 (in collaboration with Mikhail Kats of UW–
Madison). a) The real (n) and the imaginary (κ) part of the refractive index of Sr9/8TiS3. The difference
between extraordinary (parallel to the c-axis) and ordinary (perpendicular to the c-axis) refractive index are
represented as birefringence and dichroism in b). [63]
Figure 6.9: Comparison of the birefringence of two representative hexagonal quasi-1D chalcogenides
(Sr9/8TiS3 and BaTiS3
[155]) with highly anisotropic 2D materials (h-BN[44] and MoS2
[47]). The transparent
regions of these materials are identified using symbols. [63]
6.3.3 Comparison with BaTiX3 (X = S, Se)
We, therefore, compare the optical anisotropy of A1+xTiX3 (A = Sr, Ba; X = S, Se). The
imaginary part of the refractive index (or extinction coefficient), κo and κe, of BaTiS3
[39], BaTiSe3,
and Sr9/8TiS3
[63] are compared with each other in Figure 6.10(a). Contrary to the κ peak featuring

111
a sharp drop of absorption that decays to ~ 0 near 4 μm in in Sr1+xTiS3, BaTiS3 and BaTiSe3 have
κ peaks at shorter wavelengths but decay gradually to zero until ~5 μm and ~ 6 μm. Such absorption
characteristic of BaTiS3 and BaTiSe3 leads to dichroism of a broader spectrum range but also cause
more loss (reflectance + transmittance < 100%) in the LWIR region compared to Sr9/8TiS3.
The room temperature birefringence (Δn) and the low loss regions (after the absorption edge)
of BaTiS3
[39], BaTiSe3, and Sr9/8TiS3
[63] are then listed for wavelengths from 1 µm to 17 µm in
Figure 6.10(b), along with a variety of IR birefringent crystals[30–38]. BaTiS3 and BaTiSe3 share
similar structural features, especially the existence of the a-b plane Ti displacements and the
disorder of such in-plane distortions. BaTiSe3, up to 0.9, slightly surpasses the birefringence of
BaTiS3, which is up to 0.76. Birefringence of Sr9/8TiS3, on the other hand, is at least an order of
magnitude higher than commercially available birefringent crystals such as rutile[30] and calcite[31]
[Figure 1.3], and three times as large as BaTiS3
[39], more than two times as large as BaTiSe3
[64]
.
Figure 6.10: Comparison of the refractive index of A1+xTiX3 (A = Sr, Ba; X = S, Se). (a) Extinction
coefficients, κo and κe, of Sr9/8TiS3, BaTiS3 and BaTiSe3. (b) The absolute birefringence value of a variety
of IR birefringent crystals (from the literature[30–39,63]), quasi-1D perovskite chalcogenides Sr9/8TiS3, BaTiS3,

112
and BaTiSe3 extends the largest birefringence to ~0.7 - 2.1. The symbols indicate low-loss regions of
BaTiSe3, BaTiS3, and Sr9/8TiS3.
[64]
However, the origin of such pronounced birefringence improvement from other quasi-1D
structures such as BaTiS3 and BaTiSe3 is unclear. Table 3.4 lists the comparison between Ba and
Sr. Neither the analogous ionic sizes nor electronegativity between Ba and Sr nor the lower atomic
polarizability of Sr explains the large optical difference between BTS and STS. So, such optical
anisotropy differences are only caused by the non-stoichiometry.
Table 6.4: Comparison between Sr and Ba
Properties Sr Ba
Electronegativity 0.95 0.89
Ionic Radii 1.44 Å 1.61 Å
Atomic Polarizability 197 C·m2·V−1 272 C·m2·V−1
6.4 Lattice Modulation Induced Optical Anisotropy
6.4.1 Electronic structure variation
To reveal the origin of the giant optical anisotropy in Sr9/8TiS3 and its relationship with
structural modulations, we performed first-principles density-functional theory (DFT) calculations.
To understand the formation of the modulated Sr9/8TiS3 phase instead of SrTiS3, we performed a
convex hull analysis using the DFT calculated energies of the two compounds and all possible
lower-order decomposition products. We find that modulated Sr9/8TiS3 is on the hull, and is thus
stable against decomposition, as opposed to stoichiometric SrTiS3, which is thermodynamically
metastable, being 45 meV/atom above the hull.

113
Next, we calculated the electronic structures of Sr9/8TiS3 [Figure 6.11(b)] and SrTiS3 [Figure
6.11(a)] to understand the effect of modulations on the optical properties. Both SrTiS3 and Sr9/8TiS3
are computed to possess an indirect bandgap. SrTiS3 has an indirect bandgap between the valence
band maximum (VBM) at Γ point and the conduction band minimum (CBM) at A point. The
topmost valence band and bottom of the conduction band of SrTiS3 show a relatively flat behavior
along all the paths in the Brillouin zone, except for the Γ−Z direction, which in the reciprocal space
corresponds to the direction parallel to the c-axis where the neighboring TiS6 octahedra have facesharing connectivity, while octahedral connectivity is broken along the ab-plane[156,157]. Compared
with SrTiS3, Sr9/8TiS3 shows similarly flat bands. The topmost valence bands and bottom
conduction bands arise from d-states and form the indirect bandgap between VBM at Γ and CBM
at T.
Figure 6.11: Electronic structure and optical properties of modulated Sr9/8TiS3 (in collaboration with Rohan
Mishra of WashU). Orbital-projected band structures for hypothetical stoichiometric (a) SrTiS3, and (c)
modulated Sr9/8TiS3. The thicker lines highlighted in red correspond to the contribution from Ti-3
2 states;
The Fermi energy is set to 0 eV. Spatial distribution of the valence electrons below the Fermi energy
showing (b) S-3p character in SrTiS3 and (d) Ti-3
2 character in Sr9/8TiS3, which are shaded in gray in (a),
(c). The isosurface is set to an electron density of 0.004 e/Å3
). [63]
In Sr9/8TiS3, the electrons introduced by excess Sr2+ cations occupy the nominally empty Ti d
states. Using DFT + Hubbard U calculations[158], with a U = 3.0 eV for the Ti atoms, we checked

114
for different magnetic orderings of the moments and found the paramagnetic configuration to be
the most stable. As shown in the calculated band structure of Sr9/8TiS3 in Figure 6.11(b), the
additional valence electrons preferentially occupy 3
2 states (highlighted in red) of Ti atoms in
Sr9/8TiS3. These selectively occupied energy states in modulated Sr9/8TiS3 can be corroborated by
atomic-resolution electron energy-loss spectroscopy (EELS), which exhibits subtle but distinct
differences between the pseudo-trigonal prismatic TiS6 units (T) and the octahedral TiS6 units (O)
[Figure 6.12].
Figure 6.12: Ti-L2,3 EELS (in collaboration with Mikhail Kats of UW-Madison). (a) HAADF-STEM image
showing the region used for EELS acquisition highlighted with the dashed white box. Scale bar is 1 nm.
The left-side panel shows the atomic structure with the modulations. The right-side panel shows the
HAADF intensity profile collected simultaneously with the EEL spectra. The octahedral sites (O) and
pseudo-trigonal prismatic (T) units are shaded in blue and orange, respectively; (b) Comparison between
Ti-L2,3 edges extracted from (T−O−T) and (O)5 segments. The L3 peak is labeled with a black arrow. The
L3/L2 ratio of the Ti from the center of the octahedra (O)5 unit is smaller compared to the Ti from the T-site
of the (T−O−T) segment indicating a reduced oxidation state of the former; (c) MLLS intensity profiles
showing the spatial variation of the fraction of the two reference Ti-L2,3 edges shown in (b). The spatial
variation in the two profiles shows a clear correlation with (T−O−T)−(O)5 periodicity schematically shown
in the atomic structure on the left side.[63]
Compared to the pseudo-trigonal prismatic TiS6 units (T), octahedral TiS6 units (O) have shorter
Ti-Ti distance, resulting in 3dz
2 states that are lower in energy. Thus, the Ti atoms in the O block
preferentially accept the additional electrons [Figure 6.13]. This also opens up a band gap between
the occupied 3dz
2 states of the octahedrally coordinated Ti atoms and the unoccupied Ti-3dz
2 states

115
of the Ti atoms with trigonal-prismatic coordination. The character of the edge states in modulated
Sr9/8TiS3 is in sharp contrast to that of SrTiS3 wherein the band gap is between the S-3p states in
the valence band and Ti-3dz
2 states in the conduction band [Figure 6.11(a)].
Figure 6.13: Crystal-field splitting by Ti–Ti distance modulation. (a) Ti-
2 orbital occupation as a function
of Ti position along the TiS6 chain. (b) Schematic of energy levels of d states according to PDOS analysis.
The Ti–Ti distance modulation arising from structural modulation lowers the
2 (1g
) states at the O sites.
[63]
6.4.2 Colossal optical anisotropy
With the electronic ground states computed, we then calculated the complex dielectric function
∥/⊥() = 1∥/⊥() + 2∥/⊥() for electric fields along (||) and perpendicular to (⟂) the c-axis.
The imaginary part ɛ2(ω) is obtained by calculating the direct transitions between occupied and
unoccupied states[159]. The real part 1(ω) is then extracted by a Kramers-Kronig transformation.
Figure 6.14(a-b) show the frequency-dependent dielectric functions of the stoichiometric SrTiS3
and the modulated Sr9/8TiS3, with very similar results perpendicular to the c-axis (1⊥ ), but
dramatic enhancement of the dielectric function parallel to the c-axis (1∥
).

116
Figure 6.14: Calculated complex dielectric function for polarization perpendicular (⊥) and parallel (||
)
to the c-axis of the hypothetical stoichiometric SrTiS3 and modulated Sr9/8TiS3, compared to the
experimental results (black dashed line). The index = 1 represents the real part of the dielectric function
and 2 represents the imaginary part.[63]
The enhancement of 1∥
is a consequence of the selective occupation of dz
2 states at the (O)5
segments in modulated Sr9/8TiS3, which we show in real space by using an isosurface plot of the
charge density arising from the occupied 3dz
2 band [Figure 6.10(d)]. The occupied 3dz
2 electrons
form a highly oriented blob and result in additional polarizability along the optic axis (1∥
). In
contrast, the electrons from the valence band states in stoichiometric SrTiS3 have an isotropic
character and are localized on the S atoms [Figure 6.11(c)]. The unoccupied conduction band is at
substantially higher energy compared to the 3dz
2 electrons, resulting in very few free carriers and
therefore low free-carrier absorption.
While an experimental comparison between Sr9/8TiS3 and SrTiS3 cannot be made due to the
metastable nature of SrTiS3, we did compare the optical properties of Sr9/8TiS3 to BaTiS3, and to
hypothetical SrTiS3 which is isostructural to BaTiS3 in Figure 6.15.

117
Figure 6.15: Optical anisotropy comparison. (a) Birefringence and (b) dichroism comparison between
experimental measurements for Sr9/8TiS3 (in black solid), BaTiS3 (in black dash), and DFT calculations for
Sr9/8TiS3 (in blue solid), and SrTiS3 (in red solid). All DFT calculations were performed at the GGA+U
level (U = 3 eV for Ti atoms). The hypothetical stoichiometric SrTiS3 has a similar optical anisotropy as
BaTiS3.
[63]
6.5 Conclusions
In this article, we demonstrate how subtle atomic-scale structural modulations of a bulk crystal
can dramatically change its optical properties. We synthesized and studied single-crystal plates of
modulated quasi-1D chalcogenide Sr9/8TiS3, a uniaxial material that we found to possess a record
birefringence (Δ > 2.1) in a broadband low-loss spectral region between = 6 μm and at least
17 μm, with the optic axis in-plane of the samples. Compared to stoichiometric unmodulated
SrTiS3, which is expected to have Δ ≲ 1, the extra Sr in Sr9/8TiS3 results in additional electrons
that selectively occupy localized anisotropic states (Ti-3
2), greatly enhancing the polarizability
of the material along the optic axis, thus resulting in a degree of optical anisotropy far larger than
has been demonstrated in any bulk material. The atomic-scale structural features of Sr9/8TiS3 were
resolved using single-crystal X-ray diffraction and directly observed with HAADF-STEM imaging,
and the resulting structural information was used to perform DFT calculations that clarified the
physical mechanism leading to the experimentally observed colossal optical anisotropy. We

118
anticipate that structural modulation in nonstoichiometric crystals will be a new tool in realizing
materials with large degrees of optical and optoelectronic anisotropy. Furthermore, the connection
between subtle structural modulations and large changes in the refractive index may enable a new
class of optical materials that can be tuned with an applied stimulus.

119
Chapter 7. Conclusions and Perspectives
7.1 Summary
Quasi-1D hexagonal perovskite chalcogenides A1+xTiS3 (A = Sr, Se; X= S, Se) were
investigated to adopt giant infrared birefringence that is record high to our knowledge across their
low-loss window in the mid-wave infrared and long-wave infrared. Although the large atomic
polarizability of a quasi-1D structure laid out the fundamentals to be optically anisotropic, atomic
scale lattice modulations determine the magnitude of their birefringence and dichroism.
BaTiX3 (X = S, Se) crystals adopt superstructures of their reported structures at room
temperature, which is a result of the periodic TiS6 chain displacements and the off-centric Ti-toS6-octahedra displacements along the c-axis. Such structure modulation model is verified to be
consistent in BaTiS3 and BaTiSe3 but of different periodicity. This is verified by the increased
number of active Raman modes in BaTiSe3. However, despite greatly modulating their crystal
structures, such superstructures do not contribute to a higher magnitude optical anisotropy, i.e.,
birefringence and dichroism, near their infrared band gap energy. Instead, the disordered subtle ab plane Ti-to-S6 displacements in BaTiX3 largely contribute to a distorted TiS6 octahedral, whose
rearranged electronic structure near the Fermi level largely contributes to the optical anisotropy
enhancement. We observed an increased birefringence as we cooled down the BaTiS3 crystal,
especially near the phase transition temperatures where the disordered a-b plane displacements are
frozen as ordered and larger magnitude a-b plane displacements. This mechanism has the potential

120
to be engineered as refractive index or birefringence tunable devices, such as electro-optic or
photo-elastic modulators.
At the same time, the off-centric Ti-to-S6 displacements separate the center of positively and
negatively charged ions and thus leave behind a dipole in each TiS6 octahedron. The stability of
such “densely packed” dipoles in BaTiX3 facilitates the investigation of how multi-pole
interactions interact and stabilize crystal structures in materials. The phase transition of BaTiS3
was investigated in-depth, revealing two phase transitions, one near 240-260 K, the other one near
150-180 K, both featuring rearrangements of the c-axis TiS6 dipoles across the a-b plane. More
importantly, aside from the obvious c-axis TiS6 dipoles, a-b plane polarizations arrange themselves
as polar vortices in the charge-density-wave-like phase of BaTiS3. Such atomic-scale polar vortices
are coupled and stabilized by the P3c1 symmetry, forming a polar vertex network to stabilize a
more comprehensive multi-pole polarization topology potentially facilitating the control of
topological structures in quasi-1D perovskite chalcogenides with twisted light.
Sr1+xTiS3 crystals adopt commensurate or incommensurate lattice modulations along the c-axis.
Such structure modulation is ubiquitously observed in non-stoichiometric quasi-1D hexagonal
perovskites, but its correlation with optical anisotropy was never investigated. We determined the
lattice modulations of Sr9/8TiS3 crystals, and the resulting structures revealed major electronic
structure variation compared to the theoretical stoichiometric SrTiS3: The TiS6 octahedra are
periodically distorted into the pseudo-trigonal prism, which is longer along the c-axis, separating
the Ti-Ti distances and splitting the Ti 3dz
2 orbitals. The additional electron from Sr occupies the
undistorted TiS6 octahedra Ti dz
2
orbitals which is preferably arranged along the c-axis. Such Ti

121
dz
2
orbitals thus give rise to additional optical anisotropy above the already largely anisotropic
stoichiometric SrTiS3 structure. Sr9/8TiS3.
7.2 Perspectives
7.2.1 Novel optical properties
The exploration of lattice modulations in novel chalcogenide crystals leads us to not only
interpret the properties of various materials systems but also predict novel optical properties in
anomalous crystal lattices. New possibilities arise including:
Similar stoichiometric quasi-1D materials, for example CsTaS3, CsMnI3, may also adopt offcentric TaS6 octahedra both along the c-axis and the a-b plane. They may also adopt giant optical
anisotropy that covers a different spectrum wavelength, enriching the electro-optics and photoelastic modulators. Above room temperature properties, phase transitions of such materials are
also of vital importance to the study of the physics of atomic-scale multi-pole interactions in
semiconductors, especially when there are topological structures stabilized as individual chargedensity-wave-like crystalline phases.
Chemical turnability control. We already observed the optical anisotropy and the phase
transitions featuring a-b plane dipole arrangements in BaTiSe3. More chemical tunability, for
example, BaTiS3-xSex, will give us more flexibility to the optical anisotropy and the richer
dynamics of the topological structure.
More topological structures with broken parity in symmetry, for example, helical structures in
ZrOS, will accommodate both helical polarizations with net pseudo-angular momentum electrical
polarizability and broken inversion symmetry. We expect ZrOS to adopt asymmetric light-matter

122
interactions to left- and right-handed circularly polarized light or even the orbital angular
momentum of light.
Non-stoichiometry control. Sr1+xTiS3 crystals with other stoichiometry, such as Sr10/9TiS3, and
Sr22/19TiS3 have been demonstrated experimentally. These crystals will show the control of optical
anisotropy in across a considerable birefringence range.
7.2.2 Dynamics study with asymmetric light
Moreover, the knowledge of asymmetric light-matter interaction also led us to study how
asymmetric light induces lattice modulations in materials. Although subtle and likely transient,
ultrafast pump-probe measurements could facilitate such investigation.
Optical pump optical probe measurements already verified the phase transitions of BaTiS3 from
another perspective, where transient reflectivity oscillation was observed to arise as the polar
vortices disappeared during heating and became stronger as the temperature went up. This is a
capable approach to looking into the phase transition mechanism of quasi-1D perovskite
chalcogenides. Asymmetric light such as linearly or circularly polarized light would give us the
freedom to asymmetric excitation.
Optical pump X-ray probe measurements would give us additional information on the lattice
reconfiguration while the relaxation of transient lattice modulations. This is a great approach to
the dynamics of the polarization vortices as a function of the temperature, leading us to understand
how the polarization vortices are formed and stabilized by lattice symmetries.

123
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Abstract (if available)

Abstract Vision, a graceful gift from nature, utilizes the ubiquitous yet mysterious electromagnetic radiation – light to perceive wonders. The pupil, lens, and retina spatially resolve the intensity and frequency of incoming light to perceive matter that emits or reflects. Such intuitive optical processes lead to generations of imaging, sensing, and even free space telecommunications. However, the waveform of electromagnetism also introduces polarization and phase to fulfill the quantum states of light. Polarimetry, which decodes the polarization state of light unveils richer photon characteristics and is an important ingredient for quantum computing or communication with photons. Towards this goal, crystalline optics with considerable optical anisotropy that spatially splits opposite linearly, or circularly polarized light are the fundamental building blocks of polarimetry.

Quasi-one-dimensional (quasi-1D) perovskite chalcogenide, BaTiS3, adopts chains of face-shared TiS6 octahedra along the c-axis and is recently discovered to adopt giant linear optical anisotropy. We further explored BaTiSe3 and Sr1+xTiS3, which fall into the class of isostructural A1+xTiX3 (A = Sr, Ba; X= S, Se) quasi-1D chalcogenides to expand the magnitude and spectrum range of optical anisotropy. While investigating the origin of the giant infrared (IR) optical anisotropy of these materials, we discovered rich structural anomalies that break the symmetry of A1+xTiX3 and give rise to the optical anisotropy. Such as the antiparallel chains displacements, antipolar TiX6 octahedral displacements, disordered a-b plane Ti displacements in BaTiS3 and BaTiSe3, and the incommensurate lattice modulations in Sr9/8TiS3. First-principles calculations, when accounted for the observed structural anomalies, show optical anisotropy consistent with experiments, both real and imaginary parts, bringing A1+xTiX3 to the record-high IR birefringence.

At the same time, electrical, thermal, and optical investigation as a function of temperature discovered charge-density-wave (CDW) phase transitions in BaTiS3 and BaTiSe3. Structural analysis revealed the displacements of TiX6 chains, dipole ordering of TiX6-octahedra, and more importantly, the a-b plane polarization vortex lattice to be the defining characteristic of the CDW phase. First-principles calculations show the phase stability of observed a-b plane polarization vortices of BaTiS3. During heating and cooling, the polarization vortices melt and nucleate at very similar temperatures, revealing the low energy barrier of such transition, while demonstrating reliable control of the polarization vortex lattice.

BaTiS3 demonstrates another chiral phase at lower temperatures. The chiral BaTiS3 breaks down the polarization vortices while introducing optical activity with the broken symmetry. Similarly, circularly polarized light sensitivity can be selectively enhanced by controlling the underlying helical polarizations, such as in ZrOS. We carefully examine the helical polarizations and resolve the handedness of the chiral phonons in ZrOS by the pseudo-angular-momentum transfer in Raman spectroscopy.

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Creator Zhao, Boyang (author)

Core Title Tailoring asymmetry in light-matter interaction with atomic-scale lattice modulations

School Andrew and Erna Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Materials Science

Degree Conferral Date 2024-05

Publication Date 05/17/2024

Defense Date 02/22/2024

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

Tag asymmetry,atomic-scale,crystal structure,light-matter interaction,OAI-PMH Harvest,X-ray single crystal diffraction

Format theses (aat)

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Ravichandran, Jayakanth (committee chair), Li, Zhenglu (committee member), Shao, Yu-Tsun (committee member), Willner, Alan E. (committee member)

Creator Email asyzzby@gmail.com,boyangz@usc.edu

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