infrared study of magnetic and electric excitations in...
TRANSCRIPT
INFRARED STUDY OF MAGNETIC AND ELECTRIC EXCITATIONS IN NOVELCOMPLEX OXIDES
By
KEVIN H. MILLER
A DISSERTATION PRESENTED TO THE GRADUATE SCHOOLOF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OFDOCTOR OF PHILOSOPHY
UNIVERSITY OF FLORIDA
2013
1
c© 2013 Kevin H. Miller
2
To my family
3
ACKNOWLEDGMENTS
My completion of the work in this dissertation would not have been possible without
the help of numerous people. I owe my deepest gratitude to my advisor, Prof. David B.
Tanner, whose knowledge has never ceased to impress me, and whose style of mentoring
promoted the independent and problem-solving style of thinking necessary to thrive in
this field of work. Dr. Tanner′s style of mentoring is undeniably collaborative at the same
time, with graduate students helping each other whenever needed. I am also thankful for
the guidance provided by the other four members of my committee: Professors Arthur
Hebard, Amlan Biswas, Christopher Stanton, and David Norton.
I will forever be in debt to Helmuth Berger of the EPFL in Lausanne, Switzerland
who grew all the crystals described in this dissertation. The timeliness of our collaboration
with Helmuth Berger combined with the high-quality of the crystals provided allowed me
to “hit the ground running” following my first two semesters of course work.
Additional acknowledgement is directed toward the many collaborators who
have provided me with the opportunity of experimental learning beyond the Tanner
Lab. Among such individuals must be mentioned Prof. Mark Meisel of UF′s Physics
Department for magnetic susceptibility measurements, Dr. G. Larry Carr and Prof.
Peter Stephens of the National Synchrotron Light Source (Brookhaven National Lab) for
infrared magneto-optics and x-ray diffraction, Dr. DJ Arenas of the University of North
Florida for Raman measurements, and Dr. Xiaoshan Xu of Oak Ridge National Lab for
lattice dynamical calculations. A special thanks goes to Prof. Roger A. Lewis and Evan
Constable of the University of Wollongong, Australia. Prof. Lewis graciously accepted
me into his laboratory for three months as an NSF EAPSI fellow to pursue an ambitious
project. During my stay in Australia, Prof. Lewis also provided me with some much
needed financial aid, thus going above and beyond what was expected of him.
I would also like to explicitly thank two peers of mine who have help sculpt my skills
as an experimental physicist. First is Dr. Catalin Martin, a postdoctoral researcher in the
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Tanner Lab whose technical savvy and problem-solving skills were an essential ingredient
aiding my transition into a line of work in which unexpected problems, almost on a daily
basis, are customary. Second is my former officemate, Dr. Xiaoxiang Xi, who was the
driving force behind the progression of my computing skills.
I would like to publicly acknowledge my numerous peers in the Tanner Lab with
whom I have been both learner and teacher. These include Dr. Dimitrios Koukis, Zahra
Nasrollahi, Naween Anand, Berik Uzakbaiuly, Evan Thatcher, Chang Long, and Luyi
Yan. Additional gratitude is expressed towards the Physics Departments Machine Shop,
specifically, Ed Storch and Marc Link. Finally, I would like to thank John Mocko of the
Physics Department for the equipment and advice he has provided for my numerous
outreach excursions to local grammar schools; these events, in which I shared with
elementary school children my fascination for physics, were a refreshing break from my
normal laboratory work.
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TABLE OF CONTENTS
page
ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
LIST OF TABLES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9
LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
CHAPTER
1 INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2 MULTIFERROICS AND MAGNETOELECTRICS . . . . . . . . . . . . . . . . 17
2.1 Fundamentals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172.1.1 Definitions and Applications . . . . . . . . . . . . . . . . . . . . . . 172.1.2 History and Renaissance . . . . . . . . . . . . . . . . . . . . . . . . 192.1.3 Contradicting Requirements . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Experimental Realization . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242.2.1 Infrared Activity in Multiferroics and Magnetoelectrics . . . . . . . 242.2.2 Electromagnons . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
3 EXPERIMENTAL METHODOLOGY . . . . . . . . . . . . . . . . . . . . . . . 28
3.1 Instrumentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.1.1 Fourier Transform Interferometry . . . . . . . . . . . . . . . . . . . 28
3.1.1.1 Detectors . . . . . . . . . . . . . . . . . . . . . . . . . . . 343.1.1.2 Sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
3.1.2 Terahertz Two-color Photomixing System . . . . . . . . . . . . . . . 373.2 Analytical techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39
3.2.1 Lorentz Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393.2.1.1 Fitting procedures . . . . . . . . . . . . . . . . . . . . . . 41
3.2.2 Kramers Kronig . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.2.3 Sum Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
4 MAGNETODIELECTRIC COUPLING IN Cu2OSeO3 . . . . . . . . . . . . . . 47
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474.2 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.3 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.3.1 Magnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 494.3.2 Reflectance and Transmittance Spectra . . . . . . . . . . . . . . . . 504.3.3 Kramers-Kronig Analysis and Optical Properties . . . . . . . . . . . 524.3.4 Oscillator-Model Fits . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
6
4.4.1 Magnetodielectric Effect . . . . . . . . . . . . . . . . . . . . . . . . 554.4.2 Anomalous Phonons . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.4.3 Assignment of Phonon Modes . . . . . . . . . . . . . . . . . . . . . 58
4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
5 OPTICAL PROPERTIES OF MULTIFERROIC FeTe2O5Br . . . . . . . . . . . 64
5.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 645.2 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 665.3 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
5.3.1 X-Ray diffraction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 685.3.2 Reflectance and Transmittance Spectrum . . . . . . . . . . . . . . . 685.3.3 Field-Dependent Transmittance . . . . . . . . . . . . . . . . . . . . 705.3.4 Determination of Optical Properties . . . . . . . . . . . . . . . . . . 725.3.5 Lorentz Oscillator Fits . . . . . . . . . . . . . . . . . . . . . . . . . 73
5.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 745.4.1 Group theory and Lattice Dynamics . . . . . . . . . . . . . . . . . . 745.4.2 52Bu Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4.3 53Au Modes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 765.4.4 Assignment of Modes . . . . . . . . . . . . . . . . . . . . . . . . . . 78
5.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
6 PHONON ANOMALY AND MAGNETIC EXCITATIONS IN Cu3Bi(SeO3)2O2Cl 85
6.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.1.1 Crystal Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 856.1.2 Beauty of Infrared Spectroscopy . . . . . . . . . . . . . . . . . . . . 866.1.3 Major Findings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
6.2 Experimental Procedures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.3 Results and Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
6.3.1 Zero Field Reflectance and Transmittance Spectra . . . . . . . . . . 896.3.2 Kramers-Kronig and Oscillator-model fits . . . . . . . . . . . . . . . 906.3.3 Optical Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 926.3.4 Powder X-Ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . 936.3.5 Magnetic Field-Dependent Transmission . . . . . . . . . . . . . . . 956.3.6 Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . 986.3.7 Magnetic Field-Dependent Capacitance . . . . . . . . . . . . . . . . 100
6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1026.4.1 Group Theory and Observed Modes . . . . . . . . . . . . . . . . . . 1026.4.2 Powder X-Ray Diffraction . . . . . . . . . . . . . . . . . . . . . . . 1046.4.3 Phonon Repulsion . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1056.4.4 Magnetic Excitations . . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.4.4.1 Nature and isotropy . . . . . . . . . . . . . . . . . . . . . 1086.4.4.2 H ‖ c field dependence . . . . . . . . . . . . . . . . . . . . 1096.4.4.3 H ⊥ c field dependence . . . . . . . . . . . . . . . . . . . . 112
6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
7
7 CONCLUSIONS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116
APPENDIX
A PRELIMINARY RESULTS ON SINGLE CRYSTAL Cu3(SeO3)2Cl2 . . . . . . . 119
A.1 Background and Crystal Structure . . . . . . . . . . . . . . . . . . . . . . 119A.2 Magnetic Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119A.3 Room-temperature Infrared and Raman Results . . . . . . . . . . . . . . . 119A.4 Temperature-dependent Infrared Spectra . . . . . . . . . . . . . . . . . . . 121
B CALCULATING SINGLE-BOUNCE REFLECTANCE . . . . . . . . . . . . . . 125
B.1 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125B.2 Formalism . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131
BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
8
LIST OF TABLES
Table page
2-1 Mechanisms of inducing ferroelectricity . . . . . . . . . . . . . . . . . . . . . . . 23
4-1 Oscillator parameters for Cu2OSeO3 at 20 K . . . . . . . . . . . . . . . . . . . . 57
4-2 Lattice dynamical calculations of Cu2OSeO3 . . . . . . . . . . . . . . . . . . . . 62
5-1 Oscillator parameters for FeTe2O5Br phonons along eA . . . . . . . . . . . . . . 75
5-2 Oscillator parameters for FeTe2O5Br phonons along eC . . . . . . . . . . . . . . 81
5-3 Oscillator parameters for FeTe2O5Br phonons along eB . . . . . . . . . . . . . . 83
6-1 Oscillator parameters of Cu3Bi(SeO3)2O2Cl at 7 K . . . . . . . . . . . . . . . . 115
A-1 Point group symmetry character table of C2h . . . . . . . . . . . . . . . . . . . . 121
9
LIST OF FIGURES
Figure page
2-1 Multiferroic coupling schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2-2 Spiral spin order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
2-3 First report of electromagnons . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
3-1 Schematic of a Michelson interferometer . . . . . . . . . . . . . . . . . . . . . . 29
3-2 The sampling of an interferogram using a Bruker 113v . . . . . . . . . . . . . . 33
3-3 Horizontal polarization of beamline U4IR at the NSLS below 100 cm−1 . . . . . 38
3-4 Two-color photomixer at the University of Wollongong . . . . . . . . . . . . . . 39
3-5 Kramers-Kronig integration procedure . . . . . . . . . . . . . . . . . . . . . . . 43
4-1 The dc magnetic susceptibility of Cu2OSeO3 . . . . . . . . . . . . . . . . . . . . 50
4-2 Temperature-dependent reflectance spectrum of Cu2OSeO3 . . . . . . . . . . . . 51
4-3 Broadband reflectance and optical conductivity of Cu2OSeO3 . . . . . . . . . . . 52
4-4 Temperature-dependent far-infrared transmission of Cu2OSeO3 . . . . . . . . . . 53
4-5 Broadband transmission of Cu2OSeO3 at 300 K . . . . . . . . . . . . . . . . . . 54
4-6 Temperature-dependent far-infrared optical conductivity of Cu2OSeO3 . . . . . 55
4-7 Lorentz oscillator fit of Cu2OSeO3 reflectance . . . . . . . . . . . . . . . . . . . 56
4-8 The dielectric constant of Cu2OSeO3 as extracted from the infrared . . . . . . . 58
4-9 Anomalous phonon parameters in Cu2OSeO3 . . . . . . . . . . . . . . . . . . . . 59
5-1 Single-crystal x-ray diffraction pattern of FeTe2O5Br at 300 K . . . . . . . . . . 69
5-2 Temperature-dependent reflectance spectrum of FeTe2O5Br . . . . . . . . . . . . 70
5-3 Broadband optical conductivity of FeTe2O5Br . . . . . . . . . . . . . . . . . . . 71
5-4 Measured mid-infrared transmission of FeTe2O5Br . . . . . . . . . . . . . . . . . 71
5-5 Calculated single-bounce reflectance in the mid-infrared of FeTe2O5Br . . . . . . 72
5-6 Lorentz oscillator fit of FeTe2O5Br reflectance . . . . . . . . . . . . . . . . . . . 74
5-7 Evidence for buried modes in FeTe2O5Br . . . . . . . . . . . . . . . . . . . . . . 77
6-1 The crystal structure of Cu3Bi(SeO3)2O2Cl . . . . . . . . . . . . . . . . . . . . . 86
10
6-2 Temperature-dependent reflectance spectra of Cu3Bi(SeO3)2O2Cl . . . . . . . . 91
6-3 Mid-infrared transmission of Cu3Bi(SeO3)2O2Cl at 300 K . . . . . . . . . . . . . 91
6-4 Oscillator fit of Cu3Bi(SeO3)2O2Cl reflectance . . . . . . . . . . . . . . . . . . . 93
6-5 Optical conductivity and loss function of Cu3Bi(SeO3)2O2Cl . . . . . . . . . . . 94
6-6 Temperature dependence of Cu3Bi(SeO3)2O2Cl lattice parameters . . . . . . . . 95
6-7 Reitveld refinement of Cu3Bi(SeO3)2O2Cl at 295 and 85 K . . . . . . . . . . . . 96
6-8 Isotropic magnetic excitation in Cu3Bi(SeO3)2O2Cl at 33.1 cm−1 . . . . . . . . . 97
6-9 Magnetic field dependence of excitation in Cu3Bi(SeO3)2O2Cl at 33.1 cm−1 . . . 98
6-10 Complementary techniques to probe a magnetic excitation in Cu3Bi(SeO3)2O2Cl 99
6-11 Isothermal magnetization of Cu3Bi(SeO3)2O2Cl at 5 K . . . . . . . . . . . . . . 100
6-12 Inverse of the magnetic susceptibility for Cu3Bi(SeO3)2O2Cl . . . . . . . . . . . 101
6-13 Field-dependent capacitance of Cu3Bi(SeO3)2O2Cl . . . . . . . . . . . . . . . . . 101
6-14 Normalized temperature-dependent capacitance of Cu3Bi(SeO3)2O2Cl . . . . . . 102
6-15 Evidence for phonon repulsion in Cu3Bi(SeO3)2O2Cl . . . . . . . . . . . . . . . 106
6-16 Isotropic oscillator strengths of the magnetic mode in Cu3Bi(SeO3)2O2Cl . . . . 109
A-1 Infrared reflectance of Cu3(SeO3)2Cl2 300 K . . . . . . . . . . . . . . . . . . . . 120
A-2 Raman ab spectra of Cu3(SeO3)2Cl2 300 K . . . . . . . . . . . . . . . . . . . . . 122
A-3 Temperature-dependent infrared reflectance of Cu3(SeO3)2Cl2 . . . . . . . . . . 123
A-4 New infrared modes in Cu3(SeO3)2Cl2 . . . . . . . . . . . . . . . . . . . . . . . 124
B-1 Refractive index from both Rs and Kramers-Kronig . . . . . . . . . . . . . . . . 129
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Abstract of Dissertation Presented to the Graduate Schoolof the University of Florida in Partial Fulfillment of theRequirements for the Degree of Doctor of Philosophy
INFRARED STUDY OF MAGNETIC AND ELECTRIC EXCITATIONS IN NOVELCOMPLEX OXIDES
By
Kevin H. Miller
May 2013
Chair: David B. TannerMajor: Physics
This dissertation describes the characterization of novel complex-oxide single crystals
that are candidates for the discovery of new multiferroic materials and the realization
of strong magnetoelectric coupling. The primary method of characterization is infrared
spectroscopy in either reflection or transmission geometry, as governed by the material’s
optical response over a given frequency range. Optical properties are estimated via
Kramers-Kronig relations and by fits to a Lorentz oscillator model. In certain materials,
infrared results have motivated further characterization techniques: namely, magnetic
susceptibility, x-ray diffraction, Raman spectroscopy, and terahertz spectroscopy. The
specific materials studied are identified along with a short statement describing the
major experimental findings resulting from each study. The Cu2OSeO3 system exhibited
anomalous behavior of its infrared active phonons across the ferrimagnetic ordering
temperature (Tc=60 K), which contributed to an abrupt change in the dielectric constant
at the onset of magnetic order. The FeTe2O5Br system displayed a highly anisotropic
phonon spectrum, which was corroborated by theoretical lattice dynamical calculations.
In the Cu3Bi(SeO3)2O2Cl system, 16 new infrared phonons were observed below 115 K
despite the lack of a structural transition, and 2 magnetic excitations were discovered
below the long range magnetic ordering temperature (Tc=24 K). The Cu3(SeO3)2Cl
system, which is still a work in progress, has shown drastic phonon anomalies near both
12
80 K and 40 K (suspected magnetic ordering temperature) suggesting the existence of a
rich phase diagram.
13
CHAPTER 1INTRODUCTION
This dissertation reports the characterization of novel complex oxide single crystals
grown by collaborators at the EPFL in Lausanne, Switzerland. The motivation for this
research falls into the broader category of multiferroic and magnetoelectric materials.1
Multiferroic materials exhibit the coexistence of at least two long range ferroic orders
(e.g., ferroelectricity and ferromagnetism) in a single homogeneous crystal. The orders
are often coupled, thus giving rise to potential device applications. Although multiferroic
materials were first studied in the 1960s in the former Soviet Union,[1, 2] experimental
and computational tools at the time were not advanced sufficiently to investigate fully
the wealth of knowledge and potential future applications provided by these materials.
However, a renaissance in multiferroic research that occurred near the turn of the 21st
century[3–5] unveiled a renewed appreciation for multiferroic materials aided by a
theoretical understanding of the microscopic mechanisms necessary for multiple orders
to coexist and couple in a single homogeneous crystal. It was subsequently discovered
that complex interactions (e.g., frustrated lattice geometries) at times led to “improper”
ordering of electric dipoles, thus sidestepping the often contradicting requirements for the
simultaneous existence of multiple ferroic orders (cf. Chapter 2). Therefore, additional
inspiration for this work is linked to the ongoing quest to discover and characterize new
multiferroic materials from the plethora of promising candidates, in this case, novel
complex oxide single crystals.
Modern interest in multiferroic materials is fueled from both the basic scientific and
the applied technological viewpoints. On the technological side, such materials present
an opportunity to combine multiple tasks, formally performed by separate materials, by
1 Henceforth, the term multiferroic will include magnetoelectric materials. The subtledifference between the two types of materials will be elucidated in Chapter 2.
14
one material. If one considers ferroelectric ferromagnetic multiferroics, for example, the
benefits would stem from making certain magnetic technologies electric-field controllable, a
clear advantage considering that electric fields are much easier to produce and manipulate.
On the scientific side, interest is driven by an advancement of knowledge about the
microscopic interactions that favor the existence of multiferrocity, as well as about the
mechanisms that give rise to the coupling of the ferroic orders. Fruitfulness emerging from
such theoretical understandings will continue to provide direction for the design[6] of new
materials possessing higher Tc′s (critical temperatures at which ferroic orders manifest
themselves) and stronger coupling.
Infrared spectroscopy, which is the primary method of characterization for this
work, represents a unique way to probe the phenomena associated with multiferroic and
magnetoelectric materials. First of all, infrared spectroscopy is a contactless probe of
materials′ properties; consequently, interface effects associated with electrical contacts
can be avoided. Second, infrared spectroscopy is extremely sensitive to a number of
signatures associated with the ferroic orders, namely, magnetic resonance modes of
ordered magnetic dipoles, soft mode behavior and structural transitions of ferroelectrics,
and lattice deformation of ferroelastics. Results from the infrared measurements of the
materials examined in this dissertation have at times suggested the need for alternative
experimental techniques providing both complementary and supplementary information.
Such techniques include single-crystal and powder x-ray diffraction, terahertz and
infrared magneto-optics, Raman spectroscopy, and dc magnetic susceptibility. The
independent variable common to all such techniques utilized in the work presented
here is the sample temperature, a necessary parameter considering that only one room
temperature multiferroic currently exists, BiFeO3 and analogues thereof.[7]
This dissertation is organized as follows: Chapter 2 presents a comprehensive
overview of multiferroic physics as well as a review of recent literature pertaining to
the characterization of multiferroics by infrared spectroscopy. Chapter 3 covers principles
15
of some experimental instrumentation utilized and also analytical techniques implemented
to interpret the data. Chapter 4, Chapter 5, and Chapter 6 contain the study of three
novel complex oxide single crystals. The report of each crystal system will begin with a
brief review of all pertinent information previously published. The specific experimental
instrumentation used in each study will be stated. The complex oxide single crystals
studied in Chapter 4, Chapter 5, and Chapter 6 are Cu2OSeO3, FeTe2O5Br, and
Cu3Bi(SeO3)2O2Cl respectively. Appendix A contains an uncompleted report of a fourth
and quite interesting single crystal system, Cu3(SeO3)2Cl, which was ongoing at the time
this dissertation was compiled. Appendix B develops the theory of a combined reflection
and transmission analysis technique, first reported in the seminal work of Zibold et al.[8]
that was implemented to scrutinize infrared results in three of the four crystal systems
studied.
In infrared and Raman spectroscopy, it is conventional wisdom to report the
frequency of light in the units of wavenumbers. Wavenumbers are found by taking the
reciprocal of the wavelength (in centimeters) and thus they have dimensionality cm−1.
1 cm−1 corresponds to an energy of 0.124 meV and a frequency of 30 GHz. The majority
of the infrared and Raman plots in this dissertation employing wavenumbers will have
a twin axis of the appropriate scale in the units of electron volts. All other measured
quantities will be reported in the SI units.
16
CHAPTER 2MULTIFERROICS AND MAGNETOELECTRICS
2.1 Fundamentals
2.1.1 Definitions and Applications
Multiferroics are defined as materials that simultaneously possess multiple ferroic
properties (order parameters). The primary ferroic order parameters under consideration
are ferromagnetism, ferroelectricity, and ferroelasticity; however, the contemporary
definition of multiferroics has come to also include antiferromagnetism, ferrimagnetism,
and antiferroelectricity as well. Basic science interest and potential technological
applications arise because the simultaneously existing orders are often coupled to
one another. For a full appreciation of multiferroic physics, let us consider Figure 2-1.
Ferroelectricity is defined by spontaneously ordered electric dipoles that are switchable
by an applied electric field. Ferroelastic materials exhibit a spontaneous deformation
that is controllable by an externally applied strain. Ferromagnetic materials display a
spontaneous magnetization that is switchable by an applied magnetic field. (It is worthy
of note that the aformementioned ferroic orders only manifest themselves in a finite region
of temperature space and whose onset is signified by a Curie or Neel temperature.) In
terms of coupling, one could imagine applying an external electric field and subsequently
harness magnetically ordered dipoles or control a spontaneous deformation. Such coupling
is widely referred to as magnetoelectric coupling. The most direct form of magnetoelectric
coupling in multiferroic materials occurs when the spontaneous onset of one ferroic
order at a critical temperature promotes a second order to crop up at that very same
temperature. A common example of this, which will be discussed in Section 2.1.3, is
magnetically driven ferroelectricity.
Magnetoelectric materials, by definition, also exhibit the magnetoelectric coupling
effect; however, they are different from multiferroic materials in the following subtle
way: In the absence of external perturbations, magnetoelectric materials only possess
17
Figure 2-1. In multiferroics, spontaneous order parameters can be altered by applyingexternal perturbations (e.g., external fields of conjugate orders). Whenferroelectricity, ferromagnetism, and ferroelasticity coexist, a spontaneouselectric polarization, P , can be tuned by not only an external electric field E,but also an external magnetic field H , or an applied strain σ. Figurereproduced from Ref. [9].
one ferroic order. Nevertheless, upon applying an external perturbation of the sole
ferroic order (e.g., electric field for a ferroelectric), one can bring about a second ferroic
order (e.g., ferromagnetism). Magnetoelectrics are thus a close cousin to multiferroic
materials, and consequently, the following discussion of applications will be relevant to
both magnetoelectrics and multiferroics.
The great technological interest in multiferroic materials revolves around ferroelectric
ferromagnets and is driven by a demand for spintronic (utilizing electrons spin and charge)
devices. Before applications can be fully appreciated, it is worthwhile to consider the
18
implementation of ferroelectrics and ferromagnets alone in devices. Ferroelectrics are
ubiquitous in the actuator and sensing industries, specifically being used as tunable
capacitors and for non-volatile random-access memory (Ferroelectric RAM). Ferromagnets
are almost exclusively utilized in the design of transformers, as well as for the recording
and storage of data. The properties of multiferroics can potentially be exploited in
many ways. At the mundane level, the hope is to eventually make some of our current
magnetic technologies electric-field controllable, for electric fields produce less heat
dissipation and they are easier to generate. Furthermore, one can imagine novel devices
that would ensue (e.g., tunneling magnetoresistance sensors, magnetic read-head electronic
write-head multistate memory devices, and magnetic field sensors where the magnetic field
is determined by monitoring a material′s electronic properties).
The current push towards miniaturizing technological devices correlates well with the
design of materials that can simultaneously perform multiple tasks. The implementation
of multiferroic materials in spintronic devices would certainly advance the progression
predicted by Moore′s law, for such devices would not be as susceptible to heating effects
when the overall dimensionality is reduced.
2.1.2 History and Renaissance
In the 19th century, physicist Pierre Curie first postulated a coupling between
magnetic and electric orders in a single homogeneous material. Despite this early
postulation, it was not until 1960 that the magnetoelectric effect was for the first
time experimentally observed in the compound Cr2O3.[1, 10, 11] Subsequent work on
the magnetoelectric effect in the 1960s and 1970s were spearheaded by the groups of
Smolenskii[1] and Venevtsev[2] (operating in the former Soviet Union). At the time,
the work stirred moderate interest in the physics community; however, theoretical
understanding of the magnetoelectric effect was sparse and attempts to improve the
coupling strength for device applications proved bleak.
19
A renewal of interest in multiferroics1 and magnetoelectrics following the first years
of the 21st century has been driven by multiple factors. One such factor is the theoretical
work of N. Hill (now N. Spaldin)[3] published in 2000 that illuminated the seemingly
contradicting nature of multiple ferroic orders (c.f. Section 2.1.3), and thus explaining
the scarcity of multiferroic and magnetoelectric materials existing at the time. The newly
founded knowledge ushered in a “materials by design” approach in which the amounts and
nature of certain material constituents were tailored to meet the suspected requirements
of multiple orders.[6] This “materials by design” approach had the serendipitous benefit of
promoting synergy between theoretical and experimental groups, thus spreading interest in
the field.
Another equally important factor fueling the renewed interest in multiferroic materials
stems from the advancement of instrumentation. In the pioneering work of Smolenskii
and Venevtsev[1, 2] 40 years prior, the tools were not developed yet to capitalize fully
on the wealth of knowledge and discoveries intrinsic to multiferroic materials. The
biggest strides in the realm of fabrication techniques have occurred in the growth of thin
films. Specifically, the introduction of strain via the lattice mismatch with a substrate
and new growth techniques employing pressurized environments have greatly benefited
the production of new pristine materials. Moreover, the advent of new experimental
techniques to probe the microscopic phenomena associated with ferroic orders (e.g.,
mapping domain boundaries) are now widespread. On the theoretical side, faster
and more powerful computing capabilities have paved the way for first-principle and
density functional calculations to shed light on the microscopic interactions and coupling
mechanisms active in the materials.
Breakthroughs themselves, such as new and unexpected mechanisms for ferroelectricity
(c.f. Section 2.1.3) in manganese-oxide peroskite compounds, have stirred curiosity and
1 The term “multiferroic” was not coined until 1994 by H. Schmid.[12]
20
stimulated even greater vigor in the field.[4] Many unanswered questions, such as those
pertaining to mechanisms of coupling, still remain however.
2.1.3 Contradicting Requirements
The synthesis of new multiferroics has proven to be a formidable task, considering
that ferroic orders favor material properties that generally contradict one another. For
simplicity, let us consider ferroelectric ferromagnetic multiferroics and examine the
electrical properties, crystal symmetry, and electron configurations indicative of the two
orders. Ferroelectric materials, by nature, must be electrically insulating; otherwise, when
electric fields are applied, charges would flow rather than become polarized. In contrast,
many ferromagnets are metallic, although metallicity is not a strict requirement.
Concerning symmetry requirements, ferroelectrics mandate a noncentrosymmetric
structure because the electric dipoles must reverse direction upon reversing the field
polarity. In contrast, ferromagnets necessitate a structure that breaks time reversal-symmetry.
There exist 31 point groups that satisfy the symmetry requirement of ferroelectricity.
Ironically, 31 point groups also satisfy the symmetry requirement of ferromagnetism. Of
the original 122 Shubnikov-Heesch point groups, 13 overlap, permitting simultaneously
ferromagnetism and ferroelectricity.[12]
As noted above, loopholes exist for both electrical properties and symmetry
requirements; however, the prevailing concepts about electron configurations supporting
ferroelectricty and ferromagnetism are truly incompatible. It was empirically noted in
the 1940s by Matthais that ferroelectricity is favored by empty d orbitals. Upon closer
examination of our classical ferroelectrics (e.g., BaTiO3 and PbTiO3), it was discovered
that the switchable off-centering between cation and anion, which defines ferroelectricity,
is a result of a hybridization between the filled 2p orbitals of the oxygens and the empty
3d orbitals of the transition metal ions (titanium in this example). On the other hand,
21
ferromagnetism requires unpaired electrons in the 3d shell of a transition metal.2 This
incompatibility amounts to a competition between Hund and Pauli. Hund′s rules promote
unpaired spins, while those of Pauli would like to have all spins paired in the same orbital.
When localized electrons are involved, this competition tends to lean towards magnetism,
and the ferroelectricity is forced to arise from “improper” means.3
The “improper” or unusual mechanisms of creating ferroelectricity were first reported
in TbMnO3,[13] CuFeO2,[14] and Ni2V3O8.[15] A 2007 article by Cheong and Mostovoy[4]
defined improper ferroelectricity as a spontaneous polarization, which is a by-product
of a more complex lattice distortion towards a noncentrosymmetric state driven by an
alternate ordering phenomena. Table 2-1 details the mechanisms of proper and improper
ferroelectricity. In what follows, the improper mechanism of magnetic ordering will be
discussed, for it is a direct manifestation of magnetoelectric coupling. In addition, electric
dipoles induced by the ordering of magnetic dipoles remain highly susceptible to applied
magnetic fields, thus prompting tremendous interest in this class of materials.
Magnetic frustration leading to a spiral spin order can induce an electric polarization
stemming from the free energy term, —P——M—∂—M—, in the following way. In a
cubic crystal, the magnetically driven electric polarization takes the form
P = [(M · ∇)M−M(∇ ·M)]. (2–1)
Consider a chain of spins in which nearest neighbors and next-nearest neighbors are
coupled ferromagnetically and antiferromagnetically respectively. This frustrated ordering
2 Magnetism arising from p and f electron physics is excluded in this discussion.
3 Ferroelectricity necessitates a distortion of the crystal lattice at its onset. Properferroelectric order occurs when the distortion is driven by the mechanism of ferroelectricityitself. Improper ferroelectric order occurs when the lattice distortion is just a by-productof a more complex ordering phenomena.
22
Table 2-1. Mechanisms of inducing ferroelectricity
Type Mechanism of inversion symmetry breaking MaterialsProper Covalent bonding between 3d0 transition metal BaTiO3
(Ti) and oxygenPolarization of 6s2 lone pair of Bi or Pb BiMnO3, BiFeO3
Pb(Fe2/3W1/3)O3
Improper Structural transition K2SeO4, Cs2CdI4‘Geometric ferroelectrics’ hexagonal RMnO3
Charge ordering LuFe2O4
‘Electronic ferroelectrics’Magnetic ordering Orthorhombic RMnO3
‘Magnetic ferroelectrics’ RMn2O3, CoCr2O4
Table adopted from Ref. [4].
has been shown to stabilize in a spiral magnetic state (Figure 2-2) of the form:
Sn = S[e1 cosQxn + e2 sinQxn]. (2–2)
In the above equation S describes the spin-density-wave ordering, e1 and e2 are orthogonal
unit vectors, and Q is the wavevector given by cos (Q/2) = −J ′/(4J). Traditional
magnetic ordering is invariant under spatial inversion (-xn goes to xn), however, changing
the sign of all coordinates in the aforementioned spiral configuration inverts the direction
of the rotation of spins in the spiral, thus breaking inversion symmetry. A spontaneous
polarization arises in the form:
P ∝ γ(e12 × Js), (2–3)
where Js is the spin current from two coupled noncollinear spins (Js ∝ S1 × S2) and e12
is the vector connecting the two magnetic ions. Sergienko et al.[16] postulated that the
inverse Dzyaloshinskii-Moriya (DM) interaction was a likely mechanism for the induced
polarization. Put briefly, in spiral magnets the DM interaction acts to move negative
oxygen ions in the direction normal to the spin chain formed by the positive magnetic
ions, thus inducing an electric polarization normal to the chain.[16]
23
Figure 2-2. A spiral spin configuration that has been known to break inversion symmetry.The frustrated spin chain has ferromagnetic nearest-neighbour andantiferromagnetic next-nearest-neighbour interactions J and J’.
2.2 Experimental Realization
2.2.1 Infrared Activity in Multiferroics and Magnetoelectrics
This section is intended to provide a brief review of the infrared signatures used to
probe multiferroic and magnetoelectric materials. Infrared activity unambiguously related
to the onset of ferroelectric order, magnetic order, and magnetoelectric coupling as well as
a novel excitation denoting multiferroism are discussed in this section.
When ferroelectricity arises in a centrosymmetric crystal, a symmetry lowering
transition occurs in which the inversion center is removed. Such a transition alters the
distribution of infrared-active Brillouin-zone-center phonon modes. The compounds
YMnO3[17] and TbMn2O5[18] both show the clear appearance of new absorption
peaks upon entering the ferroelectric state. Even in materials already possessing
noncentrosymmetric structures, the onset of ferroelectricity corresponds to a physical
displacement that often manifests itself in a phonon mode that softens with temperature
(cf. EuTiO3[19] and KH2PO4[20]).
Infrared spectroscopy is also sensitive to onset and nature of magnetic ordering. The
spin-wave dispersion branches associated with long range magnetic order can be detected
near the Brillouin zone center by a coupling to the ac magnetic field of the light. More
24
specifically, magnons (quantized excitations of spin waves) give rise to absorptions in the
infrared, typically in the range 100 µeV-10 meV.[21] The dynamics of magnon absorptions
as a function of magnetic field can help to distinguish between antiferromagnetic and
ferromagnetic order. Specifically, antiferromagnetic magnons are typically degenerate
in zero applied field. Moreover, when a magnetic field is applied,4 antiferromagnetic
magnons split into two branches of opposing dispersion.[23] Ferromagnetic magnons are
non-degenerate even in zero applied field and have a zero field intercept at zero energy.[24]
Magnetoelectric coupling in the infrared is commonly observed via the magnetodielectric
effect (a change in the dielectric constant induced by the onset of spontaneous magnetization).
The dielectric constant, which is sensitive to the electric nature of a material, can be
estimated from IR measurements through causality relations and oscillator modeling (cf.
Section 3.2.1 and Section 3.2.2). TbMnO3[25] and Cu2OSeO3[26] both exhibit an abrupt
change in the infrared-extracted dielectric constant coinciding with the onset of magnetic
order. Moreover, because a sudden change in the infrared-extracted dielectric constant
of a material usually stems from a number of strong infrared phonon anomalies in the
temperature range of interest, our discussion is extended to include materials that show
such phenomena. CoO,[27] MnO,[28] and MnF2[29] all exhibit a renormalization of their
respective phonon parameters below a magnetic ordering temperature.5
2.2.2 Electromagnons
The coupling between electric and magnetic orders in multiferroic and magnetoelectric
materials has recently manifested itself in a new fundamental infrared excitation coined
“electromagnon.” In 2006, Pimenov et al.[30] first observed electromagnon excitations
4 Subtleties in splitting behavior depends on the orientation of the external field withrespect to the easy axis of magnetization.[22]
5 In all three materials, the phonon renormalization is suspected to occur from magneticordering alone; however, only MnF2 does not exhibit a structural transition in the vicinityof the magnetic transition temperature.
25
in the dielectric constants of multiferroic TbMnO3 and GdMnO3 at an energy of
∼3 meV (See Figure 2-3). Electromagnons have subsequently been identified in numerous
compounds.[31–34] Electromagnons were originally postulated to arise from the interaction
of a spin wave with the ac electric field of the light; however, the current view of an
electromagnon is much more nuanced. Conversely, a large body of literature (Raman,[35]
neutron,[36] and infrared[37]) indicates that electromagnons are actually a hybrid
magnon-phonon resonance mode. With this modified definition in mind, and considering
electromagnons are observed in the dielectric constant of a material, the resonance mode
must gain spectral weight from a dipole-active excitation, the main candidates being
domain relaxations, phonons, and electronic transitions. In other words, the optical f-sum
rule (oscillator strength sum rule), given by
∫ ∞
0
σ1(ω)dω =πne2
2m, (2–4)
states that the area underneath the optical conductivity is a constant regardless of
temperature. Therefore, when an electromagnon appears, the spectral weight of another
existing feature must decrease accordingly. This method of spectral weight transfer has
become the primary method for distinguishing electromagnons from their seemingly
identical counterpart, traditional magnons, which do not contribute to the optical
conductivity.[38]
Additional fruitfulness in the study of electromagnons relates to the ease with which
the low frequency excitation can induce considerable changes in a materials dielectric
constant and therefore index of refraction (√ε=n) from dc up to terahertz frequencies.
The ability to tune easily the refractive index of a material allows for the design of a new
generation of optical switches and optoelectronic devices.
26
Figure 2-3. Spectra of the electromagnons in TbMnO3 and GdMnO3 as manifested in thereal (ε1) and imaginary (ε2) parts of the dielectric function. The spectra areobtained with the ac electric field of the light oriented parallel to the a axis ofboth materials. The electromagnon excitation has characteristic frequencies of23 ± 3 cm−1 and 20 ± 3 cm−1 in GdMnO3 and TbMnO3, respectively. Figurereproduced from Ref.[30].
27
CHAPTER 3EXPERIMENTAL METHODOLOGY
This chapter will cover specific spectroscopy instrumentation and analytical
techniques utilized to acquire and process data in this dissertation. The principles of
a Fourier transform interferometer will be discussed along with sources and detectors.
The intricacies of a two-color photomixing system used to probe the terahertz region
will also be illuminated. Finally, two methods to estimate complex response functions
(e.g., complex dielectric function) from the measured reflectance or transmittance will
be discussed. Appendix B provides details on a combined reflection and transmission
approach.
3.1 Instrumentation
3.1.1 Fourier Transform Interferometry
The principal element of a Fourier transform interferometer is a Michelson interferometer.1
As shown schematically in Figure 3-1, a Michelson interferometer utilizes a beamsplitter
(BS) to partition equal amounts of the incoming radiation towards mirrors M1 (fixed) and
M2 (movable). The movable mirror scans back and forth through the zero path difference
position (ZPD) and the light is recombined at the BS; a portion of the recombined light
travels towards the detector. In Figure 3-1 the ZPD position occurs when both mirrors are
a distance “d” away from the BS. Infrared radiation is typically modulated by the movable
mirror in the kHz frequency range.
The principles of an interferometer are best developed by first considering a
monochromatic source of frequency ν (in wavenumbers) and a displacement of the movable
mirror denoted x . (The optical path difference is given by ∆ = 2x because the light travels
1 For a detailed introduction to this technique, see for example Ref. [39] and Ref. [40]
28
BS
Source
M1
M2
DET
d
d
x
Figure 3-1. A schematic of a Michelson interferometer, which constitutes the principleelement of a Fourier transform interferometer.
out and back the distance x .) The intensity at the detector is given by,
Idet(x) =1
2Isource[1 + cos (2πνx)]. (3–1)
When x=0 (the ZPD position), light from the two arms of the interferometer interfere
constructively and the intensity at the detector is maximum. As x deviates from zero, the
intensity varies sinusoidally, reaching a minimum when x is such that the argument of the
cosine term is an odd integer multiple of π. Because we are concerned with the dynamics
of an interferometer, let us only consider the second term (oscillatory term); namely,
I ′det(x) =1
2Isource cos (2πνx). (3–2)
29
The 12prefactor stems from the realization that an ideal beamsplitter transmits (reflects)
one half of the incident light. Therefore, only one half of the light from the source will
impinge on the detector, with the other half directed back towards the source.2 In reality
losses are incurred (e.g., unwanted absorption in beamsplitters and light reflected from
mirrors that do not produce phase shifts equal to 180, which in principle are frequency
dependent). In lieu of this let us replace the prefactor 12Isource by a term of lesser value,
denoted K(ν). The signal at the detector becomes
S(x) = K(ν) cos (2πνx). (3–3)
At this point we can extend our formalism to the practical case of a broadband source by
integrating in frequency space from zero to infinity,
S(x) =
∫ ∞
0
K(ν) cos (2πνx)dν. (3–4)
From the signal as a function of retardation, we can obtain the signal as a function of
frequency by applying a Fourier Transform:
K(ν) =
∫ ∞
−∞
S(x) cos (2πνx)dx. (3–5)
(Note that we have extended our frequency interval to be symmetric in x , thus facilitating
the computation.)
In practice displacing the mirror an infinite distance on either side of the ZPD is
unachievable. The limits of integration become plus and minus the maximum distance
that the mirror moves, denoted as xmax. The corresponding optical path difference (OPD)
is ∆max = 2xmax (light travels out and back). The OPD is inversely proportional to the
2 In practice interferometers are slightly misaligned on purpose so that the light leavingthe interferometer that is not directed at the detector does not modulate the source.
30
spacing between statistically significant data points in the spectrum (resolution3 ) by the
relation ν=1/2xmax.
For computational handling of the interferogram one must convert the continuous
intensity, measured as a function of OPD, to a discrete sum of voltages at specific values
of x via an analogue to digital converter (ADC) board. The method of extracting discrete
values from a continuous response function is termed sampling. Sampling motivates
a discussion of the Nyquist criterion. According to Nyquist′s theorem, the sampling
frequency must be equal to or larger than twice the bandpass of the measurement
(fsamp ≥2fmax). If the Nyquist theorem is not satisfied, then aliasing occurs where the
outlying frequencies are folded back into the assumed frequency domain resulting in
the creation of unwanted spectral artifacts. Therefore, critical sampling (at the Nyquist
frequency) necessitates the need for efficient analogue filters to cut signal beyond the
sampling range.
A second issue, apropos sampling, relates to the averaging of multiple interferograms
in order to improve the signal-to-noise ratio of the measurement. To be collectively
averaged, all interferograms must be sampled (or digitized) on the same grid of data
points (i.e., same values of retardation, x ). This feat is accomplished by utilizing a second
reference interferometer. The details of the reference interferometer vary for different
instruments, so at this time we will restrict the discussion to a Bruker 113v FTIR. Here
the reference interferometer is separate from the main interferometer but utilizes the same
moving mirror. A white light source and a HeNe laser source (νHeNe=15798 cm−1) are
passed through the reference interferometer, resulting in a second interferogram and a
sine wave at their respective detectors. The intensity of the three sources (IR through the
3 Other parameters such as the source aperture and the focal length of mirrors in theinterferometer also limit the resolution, but further details are beyond the scope of thisdiscussion.
31
main interferometer and WL and HeNe through the reference interferometer) from the
two interferometers as a function of time (optical distance is determined by the mirror
velocity) are depicted in Figure 3-2. The interferogram peak from the WL source always
precedes the main interferogram peak from the infrared source. Simply put, when the WL
interferogram peak occurs, a threshold is exceeded that signals the ADC board to being
sampling. Thereafter, every zero crossing (or multiple of) of the HeNe sine wave triggers
the sampling of the main infrared interferogram. The unwavering position of the WL peak
with respect to the main interferogram peak combined with a stable laser source equates
to repeated interferograms sampled at the same retardation.
Before the Fourier transform can be computed, the interferogram must be modified
to account for several factors. First, the interferogram must be truncated by a piecewise
function that is real and positive in the interval -xmax to xmax and zero everywhere else.
(In practice the truncation function is not symmetric about x=0 because only half of the
interferogram lineshape is collected.) A simple boxcar function that takes the form
f(x) =
1 |x| ≤ xmax
0 |x| > xmax
(3–6)
satisfies this criterion. However, this boxcar function proves to be insufficient for another
reason; its Fourier transform is the sinc function which has “side lobes” or “feet” that
drop off 22% below zero on either side of the principal maximum. To remedy, the boxcar
function is then replaced by an apodization function, which serves to suppress the side
lobes but has the drawback of broadening lines, thus decreasing the resolution of the
measurement. A commonly used apodization was derived by Norton and Beer and consists
of specialized sums of polynomial functions:
W (x) =xmax∑
m=0
αm(1− x2)m, (3–7)
32
Intensity
Time
Vtrig
Figure 3-2. The sampling methodology of a Bruker 113v FTIR. When the white lightinterferogram (gold line) reaches the threshold Vtrig sampling commences. Thesampling of the main interferogram (blue line) occurs at every zero crossing(dotted black line) of the HeNe laser interferogram (red line). The relativescaling of intensity in this plot has not meaning.
subject to the constraint thatxmax∑
m=0
αm = 1.0. (3–8)
Another apodization function is the Happ-Genzel function which takes the form
W (x) = 0.54 + 0.46 cosπx
xmax
. (3–9)
The Happ-Genzel function has a corresponding Fourier transform given by,
F.T.[W (x)] =sin 2πνxmax
2π
[
1.08
ν+
0.46
xmax/ω − ν− 0.46
xmax/2 + ν
]
. (3–10)
33
Second, a phase correction must ensue to account for numerous factors that act to
distort the interferogram: off-axis rays in the interferometer, erroneous determination
of the ZPD position, and frequency dependent noise coming from electrical processing
components. To compensate for the error, we can introduce a phase term on the cosine
leading to the equation
S(x) =
∫ ∞
0
K(ν) cos (2πνx−Oν)dν, (3–11)
with
Oν = A+Bν + Cν2 +Dν3. (3–12)
When higher than first order terms exist, the interferogram is said to be “chirped” and
the phase has to be calculated and subsequently removed from the spectrum. The phase is
usually calculated directly from the interferogram,
Oν = arctanIm(K(ν))
Re(K(ν)). (3–13)
Two methodologies of phase correction are double sided, where the phase is calculated for
all points in the interferogram, and single sided, where the phase is calculated for half the
interferogram and then mirrored to form the other half.
3.1.1.1 Detectors
Infrared detectors are grouped into two categories: photon detectors and thermal
detectors. Photon detectors rely on photo-excited charge carriers changing the electrical
properties of the sensing chip. The sensing chips are typically semiconductors with
a band gap in the vicinity of the low energy side of the frequency band that is being
measured. The frequency response of a photon detector is not constant. Light incident on
a photon detector must be electrically or mechanically chopped to allow for electron hole
recombination. Time constants for recombination vary but are generally on the order of
nanoseconds. Typical photon detectors utilized in the infrared include HgCdTe, InGaAs,
InAs, and PbSe.
34
On the other hand, thermal detectors sense the heat of the impinging infrared
radiation. The temperature change caused by heating alters the sensing materials
properties (e.g., thermoelectric voltage or electrical conductivity). Thermal detectors
typically operate at low temperatures to avoid spurious fluctuations in the thermal
properties of the environment. One such thermal detector utilized for the majority of the
infrared measurements in this dissertation is a liquid-helium-cooled (LHC) bolometer.
LHC bolometers consist of a semiconducting sensing chip (typically germanium or silicon
doped with gallium or phosphorus) cooled at or below 4.2 K. Thermal detectors also need
time for relaxation of the sensing chip′s properties. The time constant of a bolometer
detector, given by the ratio of the heat capacity of the chip to its thermal conductance
with the bath, is around 1 ms. We therefore chop the light at a frequency equivalent to a
few multiples of the time constant.
3.1.1.2 Sources
Sources in the infrared can also be partitioned into two categories: continuous wave
(CW) sources and intermittent sources. The most common CW sources of infrared
radiation are high-temperature blackbodies, which continuously emit photons. The work
in this dissertation employed multiple blackbody sources to achieve spectral coverage
from the far infrared to the ultraviolet (10–50000 cm−1 or 2 meV–8 eV). A high pressure
mercury arc lamp with an effective4 operating temperature of 2000 K covered the
far-infrared (10–400 cm−1 or 2–50 meV). A globar lamp, which consists of a silicon carbide
rod heated to 1200 K, was used to cover the mid-infrared range (400–5000 cm−1 or
50–625 meV). The near infrared and visible ranges (5000–50000 cm−1 of 0.6–8 eV) are
spanned by tungsten based lamps and various arc lamps (deuterium and xenon).
4 Radiation from the mercury plasma is supplemented by radiation from the glowing hotquartz casing of the arc lamp.
35
Intermittent sources are less advantageous than CW sources for broadband spectroscopy
(e.g., one has to factor in the repetition rate of the source when choosing a chopping
frequency for the detector); however, the drawbacks can be compensated for by higher
intensities, widely tunable photon energies, and micrometer sized focal spots. One
such intermittent source utilized throughout this dissertation is synchrotron radiation.
Synchrotron radiation is released by electrons as they change the direction of their
acceleration vector (via bending magnets) while traveling around a circular storage ring
at relativistic speeds. Synchrotron radiation exhibits an intrinsic frequency at the height
of its broad spectral distribution, which is best described by a modified Bessel function.5
Electrons do not continuously circulate the storage ring of a synchrotron, but rather travel
in bunches (like carriages on a carousel wheel) thus creating the intermittent nature of the
source.
A discussion of polarization effects arising from synchrotron radiation is of interest
since most optical measurements in this dissertation necessitated linearly polarized light.
Synchrotron storage rings are typically the size of a modern day sports arena and oriented
parallel to the earth′s surface (hereafter horizontal). Because an accelerating electron
emits radiation, with the electric field of the radiation in the direction of the electron′s
acceleration vector, the intrinsic polarization of the synchrotron light is horizontal.
However, from the perspective of an observer either above or below the storage ring,
there are also circularly polarized components noticeable that would otherwise cancel out
when observation point is in the horizontal plane of the ring. When the observation point
is not entirely in the horizontal plane (the case for measurements in this dissertation),
the dominant circularly polarized portion of the radiation leads to a vertical component
5 The broad frequency distribution of synchrotron radiation stems from the Fouriertransform of the short pulses of electric field in the time domain from the electron bunchescirculating the ring.
36
to the overall polarization of the light. Moreover, for a fixed observation point not in
the horizontal plane, the ratio of the intensities of horizontal to vertical components of
the light is not constant but rather depends on the wavelength of the light. Briefly, the
wavelength dependence stems from a diffraction effect occurring at the very beginning of
the beamline in which the solid angle capturing the synchrotron radiation becomes smaller
as wavelengths increase (frequency decreases). Therefore, the first part of the beam′s
profile to be “trimmed” off will be the top and bottom, which constitute the circularly
polarized light that gives rise to the small vertical component. Since very long wavelengths
were measured in this dissertation, the horizontal polarization of the light below 100 cm−1
for the specific synchrotron beamline utilized, U4IR (NSLS, Brookhaven), is shown as a
function of increasing wavelength in Figure 3-3.
In lieu of this unusual polarization dependence, a wire grid polarizer was placed
before the sample to force the polarization of the light to become completely horizontal.
To change the geometry of the polarization, one must rotate the sample (as opposed to the
common method of rotating the polarizer when blackbody sources are utilized).
3.1.2 Terahertz Two-color Photomixing System
A terahertz (THz) two-color photomixing system at the University of Wollongong,
Australia, was used to measure lower photon energies (lower frequencies) than those
attainable by the sources and techniques of conventional Fourier transform infrared
spectroscopy. The THz two-color photomixing system was interfaced to a superconducting
magnet, thus representing a novel and unique experimental setup. It merits a detailed
explanation.
A THz two-color photomixing system consists of two tunable near-infrared laser
diodes adjusted to have a difference frequency on the order of 1 THz. The lasers are
coupled in either free space or via an optical fiber, and focused onto a photomixer. The
photomixer consists of a low-temperature grown (LTG) GaAs wafer onto which a pattern
of six interdigitated electrode fingers are deposited by lithographic techniques. The
37
20 40 60 80 100 120Frequency (cm−1 )
0.0
0.2
0.4
0.6
0.8
1.0
I polarized−h
orizon
tal/I
unpolarized
2 4 6 8 10 12 14
Energy (meV)
Figure 3-3. The horizontal polarization percentage of radiation below 120 cm−1 atbeamline U4IR of the NSLS.
wafer is supported by a GaAs substrate. An image of the interdigitated pattern and
corresponding antenna geometry is shown in Figure 3-4. Electron-hole pairs generated in
the LTG GaAs by photon absorption are accelerated towards the electrodes which have a
voltage bias across them in the manner seen in Figure 3-4. Due to the high charge-carrier
mobility and sub-picosecond recombination time of the LTG GaAs, conduction band
bending at the metal-semiconductor interface is negligible, meaning the electrical
properties of the photomixer may mostly be represented by a photoconductance that
is a function of the absorbed optical power.[41] Therefore, the dominant photocurrents
in the photomixer′s fingers and hence antennas are generated at the sum and difference
38
Figure 3-4. Image of the two-color photomixer utilized at the University of Wollongong,Australia. Top panel (a) depicts the geometry of the gold antenna pattern.Bottom panel (b) is an enlarged view of the separation between the twoantennas.
frequencies of the near-infrared pump lasers. By use of an optical filter, the difference
frequency is selected out and coupled into free space, producing CW THz radiation.
3.2 Analytical techniques
3.2.1 Lorentz Model
This section derives the formalism of the Lorentz oscillator model. The Lorentz
oscillator model is applicable to describe the optical behavior of insulators, and it is thus
utilized multiple times in this dissertation.
The model is based on a classical harmonic oscillator that is simultaneously damped
and driven. In the model, electrons are viewed as being bound to atoms with tiny springs,
39
and the damping force arises from various scattering mechanisms (e.g., phonons and
electron-electron interactions). The oscillatory driving force on the electron comes from
the ac electric field of the light. The force on an electron is written as
mx = −mω20x−mγx + Fext. (3–14)
The force from the driving field is
Fext = −eE0e−iωteiq·r. (3–15)
At this point two assumptions must be made. First, we ignore the spatial dependence
of the field, eiq·r, by restricting ourselves to wavelengths of light that are much greater
than the radius of an atom (λ a0). Second, the microscopic field Eloc is set equal to
the microscopic field E. In general Eloc 6= E; however, solving for the exact relationship is
an extremely formidable task, and the aforementioned assumption turns out to be quite
robust in preserving the essential features necessary to describe the optical properties of a
material.
With x = x0e(−iωt) our relation becomes
−mω2x0 − iωmγx0 +mω20x = eE. (3–16)
The factorized expression if given by
x0 =−e/m
ω20 − ω2 − iωγ
E. (3–17)
The displacement of the electron with respect to the driving field can be grouped into
three characteristic regions. As ω → 0, the displacement is approximately in-phase with
the driving field. At the resonance frequency, ω = ω0, the displacement is 90 out-of-phase
with the driving field. In the limit ω → ∞ the displacement is 180 out-of-phase with the
driving field.
40
Assuming the displacement x is sufficiently small so that a linear relationship between
dipole moment and electric field is accurate, we can write that
p = −ex0 − αpE. (3–18)
The atomic polarizibility is then given by
α =e2/m
ω20 − ω2 − iωγ
. (3–19)
The complex dielectric function for a system of n electrons takes the form
ε = 1 + 4πnαp = 1 +4πn e2
m
ω20 − ω2 − iωγ
. (3–20)
Let the plasma frequency of bound electrons be defined as ωp =√
4πne2
mand consider that
a number (j) of Lorentzian oscillators are needed to model accurately most insulating
materials. The Lorentz dielectric function then becomes,
ε = ε∞ +∞∑
j=1
ω2pj
ω2j − ω2 − iωγj
. (3–21)
Note that we have replaced 1 with the value ε∞. ε∞ encapsulates all the absorption
processes at frequencies higher than those measured in a typical experiment. If all the
absorption processes are measured and modeled by Lorentzian oscillators (typically
needing data up to x-ray wavelengths) then ε∞ takes the value of unity.
3.2.1.1 Fitting procedures
Optical measurements in this dissertation were modeled with the dielectric function
given in Eq. (3–21). Each individual Lorentz oscillator is defined by a resonance frequency
ω0, a line width γ, and a strength ωp. A series of j Lorentz oscillators were initially fit to
the experimental spectra by sight, and then further optimized by a least squares fitting
routine developed by Bevington.[42] Figure 4-7, Figure 5-6, and Figure 6-4 show results
of the fitting routine and optimization. With an accurate representation of the complex
dielectric function from the Lorentz model, it is then only a matter of algebra to arrive
41
at expressions for other frequency-dependent complex response functions (e.g., optical
conductivity and refractive index).
3.2.2 Kramers Kronig
Kramers-Kronig analysis is a second method to estimate the complex dielectric
function (and other complex response functions) from the measured reflectance or
transmittance. In short, Fourier transform infrared spectroscopy (FTIR) measures
power reflectance and transmittance; therefore, no information about the phase of the
light is obtained. The phase is the imaginary part of either response, and it is necessary to
calculate the complex (real plus imaginary) dielectric function. Kramers-Kronig relations,
via causality, provide a method to estimate the imaginary part of a response function with
knowledge of the real part and vice versa.
To develop the Kramers-Kronig formalism let us begin by defining a linear response
function. Consider the relation
X(t) =
∫ ∞
−∞
G(t− t′)f(t′dt′), (3–22)
where X(t) represents the response of a system (e.g., conductivity), f(t) is the external
stimulus (e.g., a driving electric field), and G(t) is the response function (e.g., susceptibility).
It should be noted that we are neglecting the spatial dependence, r, in this discussion by
restricting ourselves to wavelengths much longer than the length scales of the microscopic
interactions giving rise to such responses. In Eq. (3–22) we see that the response is
effected by a previously applied stimulus, similar to the way the instantaneous speed
of a free falling object depends on the duration of the free fall. On the other hand, the
principle of causality states that the response function, G(t), cannot depend on t when t is
a future time:
G(t− t′) ≡ 0, t < t′. (3–23)
At this point it is intuitive to consider the Fourier transform of G(t) into the frequency
domain and map its contribution in the complex plane using both real and imaginary
42
ω0
Im[ω]
Re[ω]
41
23
Figure 3-5. As described in the text, the Kramers-Kronig integral only necessitatesintegration in the upper half of the complex plane. Segments 1 and 3 areprincipal value integrals and segment 2 can be evaluated analytically.
frequencies ω = ω1 + iω2,
G(ω) =
∫ ∞
−∞
G(t− t′)eiω1(t−t′)e−ω2(t−t′)dt. (3–24)
The term eiω1(t−t′) is bounded for all frequencies in the complex plane. The term eiω2(t−t′)
is only bounded when its argument is negative; therefore, when ω2 is negative the term
(t − t′) must also be negative. Recalling our causality relation, it follows that G = 0
when ω2 is negative, thus restricting our integration of Eq. (3–24) to the upper half of the
complex plane (See Figure 3-5). Next we use Cauchy′s residue theorem for the integration
of the analytic function G(ω) in the upper half plane, namely,
∮
G(ω)
ω − ω0dω = 0. (3–25)
The integral around the closed contour can be broken into four segments as shown in
Fig. 3-5. The contribution of segment 4 is disregarded because G(ω) → zero as ω →
43
infinity. Segments 1 and 3 are principal value integrals (denoted by P) along the real
axis, and segment 2 can be evaluated analytically for the results iπG(ω0). The result of
Cauchy′s theorem then becomes,
iπG(ω0) = P∫ ∞
−∞
G(ω)
ω − ω0dω. (3–26)
Now we can split G(ω) into its real and imaginary components to obtain,
Re[G(ω0)] =1
πP∫ ∞
−∞
Im[G(ω)]
ω − ω0dω. (3–27)
Im[G(ω0)] =−1
πP∫ ∞
−∞
Re[G(ω)]
ω − ω0dω. (3–28)
We have now proved that the real and imaginary parts of the response function G(ω) are
not independent but rather connected via Kramers-Kronig relations.
The Kramers-Kronig relations of interest in this dissertation connect the reflectance,
r, and the phase, φ. The complex reflectance is given by
r = |r|eiφ. (3–29)
where the power reflectance, |r2|, is the typical measured quantity. Taking the natural
logarithm of Eq. (3–29), we can separate the real and imaginary parts of the complex
reflectance. Deriving the Kramers Kronig relations for a logarithmic function is a
challenging task because | ln r(ω)| → ∞ as |ω| → ∞. For a complete derivation
the interested reader is referred to a very informative discussion of the Kramers-Kronig
relations written by Fredrick Wooten.[43] The resulting Kramers-Kronig relation becomes
ln r(ω) =2
π
∫ ∞
0
ωφ(ω)− ω0φ(ω0)
ω2 − ω20
dω. (3–30)
φ(ω) =−2ω
π
∫ ∞
0
ln |r(ω)| − ln |r(ω0)|ω2 − ω2
0
dω. (3–31)
In practice, since we only measured a finite interval of frequencies, extrapolations must
be employed to extend the measured reflectance towards the limits of integration in
44
Eq. (3–30). Below our lowest measurable frequency, the calculated reflectance from
Lorentz oscillator modeling of our data is used to extended towards zero frequency. At
high frequencies, two avenues open for use: either power law extrapolations appropriate
for the suggested behavior of optical constants are used, or x-ray reflectance spectra are
estimated from the constituent atoms and bridged to the measured range with a single
Lorentz oscillator.6
3.2.3 Sum Rules
Sum rules can be derived from Kramers-Kronig relations and are important tools
for analyzing optical data. One particular sum rule states that the spectral weight (area
under the real part of the optical conductivity) is conserved. This rule essentially states
the conservation of charge and is expressed as
∫ ∞
0
σ1(ω)dω =ω2p
8=
π
2
ne2
m. (3–32)
This sum rule is often applied to understand changes in the optical conductivity of a
material with respect to temperature. A well-known example is the loss of spectral weight
at low frequency when a superconducting gap opens. The missing spectral weight in the
superconducting state is pushed into a delta function at zero frequency, which represents
the response of the condensate.
Infrared-active phonons often exhibit dynamical behavior in the optical conductivity
with respect to temperature. At low temperatures the phonons typically resemble sharp
Lorentzian oscillators. As temperature increases, the linewidth of a typical phonon
broadens. One can often extend the above sum rule to a single phonon. If the phonon is
not coupled to another phonon, transition, or broad continuum, then its area should be
conserved with temperature. This methodology has been recently extended to identify the
formation of electromagnons, which have to gain spectral weight at the expense of existing
6 For details on the latter approach see Section 5.3.4
45
dipole active transitions. Electromagnons are usually observed to borrow spectral weight
from a nearby phonon.
46
CHAPTER 4MAGNETODIELECTRIC COUPLING IN Cu2OSeO3
4.1 Overview
The magnetodielectric effect refers to a change in the dielectric constant induced by
an external magnetic field or by the onset of spontaneous magnetization.
Here we report our infrared studies on a single crystal of Cu2OSeO3, a piezoelectric
with a ferrimagnetic transition temperature of Tc ∼ 60 K.[44] A pervious study on
Cu2OSeO3 by Bos et al.[44] reported a magnetodielectric effect (anomalous jump in
dielectric constant) at the ferrimagnetic transition temperature, as observed through
dielectric capacitance measurements. Although we found no drastic anomalies across
Tc, a thorough inspection of the data combined with some modeling lead us to a
magnetodielectric effect as well. Recently Gnezdilov et al.[45] have presented a Raman
study of Cu2OSeO3 prepared in the same way as our crystal. They observed the abrupt
appearance of 3 new lines in the spectra upon cooling below Tc, and an additional 2 lines
that appeared below 20 K. Gnezdilov et al. also gave a detailed description of the crystal
and magnetic symmetry of this compound.
Crystal Structure: Effenberger and Pertlik[46] solved the crystal structure using
single-crystal X-ray diffraction. The compact crystal structure consists of three basic
building blocks, square pyramidal CuO5, trigonal bipyramidal CuO5, and a lone pair
containing tetrahedral SeO3 unit. The oxygen atoms in the unit cell are shared amongst
the three building blocks. The square pyramidal CuO5 units exist in a 3-to-1 ratio to
the trigonal bipyramidal CuO5 units within the conventional unit cell. This ratio will
be important subsequently when explaining the magnetic structure. All copper ions
Reprinted with permission from K. H. Miller, X. S. Xu, H. Berger, E. S. Knowles, D.J. Arenas, M. W. Meisel, and D. B. Tanner, Phys. Rev. B 82, 144107 (2010).
47
possess a +2 oxidation state. More detailed descriptions of the crystal structure are found
elsewhere.[44, 47]
The material crystallizes in the P213 cubic space group and has been shown to
remain metrically cubic with no abnormal change in the lattice constant through the
Curie temperature and down to 10 K.[44] The onset of magnetic order does have the
effect of reducing the crystal symmetry to R3. Full cubic symmetry would require all
copper ions to feel the same Coulomb interaction from nearest neighbor copper spins. The
proceeding explanation is the case for ferromagnetism and antiferromagnetism but not
for ferrimagnetism, which is why a reduction from cubic symmetry must accompany this
ordering.
4.2 Experimental Procedures
Single crystals of Cu2OSeO3 were grown by a standard chemical vapor phase
method. Mixtures of high purity CuO (Alfa-Aesar, 99.995%) and SeO2 (Alfa-Aesar,
99.999%) powder in molar ratio 2:1 were sealed in quartz tubes with electronic grade HCl
as the transport gas for the crystal growth. The ampoules were then placed horizontally
into a tubular two-zone furnace and heated slowly by 50 C/h to 600 C. The optimum
temperatures at the source and deposition zones for the growth of single crystals have
been found to be 610 C and 500 C, respectively. After six weeks, many dark green,
indeed almost black, Cu2OSeO3 crystals with a maximum size of 8 × 6 × 3 mm3 were
obtained. X-ray powder diffraction (XRD) analysis was conducted on a Rigaku X-Ray
diffractometer with Cu Kα radiation (λ = 1.5418 A). An electron microprobe was used for
chemical analysis of all solid samples.
The temperature dependent (5–300 K) reflectance and transmittance measurements
employed a Bruker 113v Fourier Transform interferometer in conjunction with a helium
Crystal growth was carried out by Helmuth Berger at the EPFL in Lausanne,Switzerland
48
cooled silicon bolometer detector in the spectral range 30–700 cm−1 and with a nitrogen
cooled MCT detector from 700–5,000 cm−1 . Room temperature measurements from
5,000-40,000 cm−1 were obtained with a Zeiss microscope photometer. After measuring
the bulk reflectance over the entire spectral range, a crystal was polished to a thickness
of 194 µm for transmittance measurements. All measurements were performed using
unpolarized light at near-normal incidence, with the electric field of the light in the 〈111〉
crystal plane. The cubic nature of the material forbids anisotropy in the optical spectra.
Magnetic measurements were performed in a commercial SQUID magnetometer (Quantum
Design MPMS-XL7) on a single crystal sample mounted with the [111] axis parallel to the
applied field. After cooling the sample in zero field to 50 K, magnetization was measured
in an applied field of 10 G while warming to 70 K. The dc susceptibility was calculated
in the low-field limit as χ(T ) = M(T )/H . In addition, the isothermal magnetization as a
function of applied field was measured at a temperature of 2 K, while sweeping the field
from zero to 2 kG and back to zero.
4.3 Results and Analysis
4.3.1 Magnetism
Recent studies have measured the magnetic susceptibility of powdered samples
Cu2OSeO3, finding ordering temperatures of T inflectionc = 55 K[48] and T onset
c =
60 K.[44] Because anomalies in the infrared spectrum at the transition temperature
are important, an accurate determination of Tc for the single crystal of interest was
desired. The measured dc susceptibility as a function of temperature, χ(T ), is shown in
Fig. 4-1. Taking the transition temperature to be where the susceptibility turns upward,
T onsetc = 60 K is found, consistent with the observations of Bos et al.[44] At 2 K (Fig. 4-1
inset), well within the ordered state, the magnetization saturates in a field of 800 G at
1.0 NµB, which is half of the expected saturation value for a S = 1/2 system, indicating
a ferrimagnetic ordering in a three-up and one-down configuration. No coercive field was
49
χ
µ
!" #$%&
'( ) *
Figure 4-1. The dc susceptibility, χ(T ), of Cu2OSeO3 near the ordering temperature(Tc = 60 K) in an applied field of 10 G as measured while warming afterzero-field cooling to 50 K. The inset shows the isothermal magnetization,M(H), as a function of applied field at 2 K, where the field was swept up to2 kG before being reduced to zero. In all instances, the lines connecting thedata points are guides for the eyes. The schematic shows the orientation of thesingle crystal with respect to the field applied parallel to the [111] direction.
measured; however, an inflection point with some slight hysteresis was observed near
400 G, which is also consistent with the findings of Bos et al.[44]
4.3.2 Reflectance and Transmittance Spectra
The temperature-dependent reflectance spectrum of Cu2OSeO3 between 30 and 1,000
cm−1 (4–120 meV) is shown in Fig. 6-2. A strong sharpening of many phonon modes
is observed with decreasing temperature. It should be noted that there are no drastic
anomalies in the far-infrared spectrum, such as the presence of new modes or the splitting
of existing modes. Because infrared spectroscopy is extremely sensitive to changes in
dipole moments and force constants, the absence of these anomalies gives strong support
to the assertion of no lattice distortions at Tc as initially determined by X-ray diffraction
measurements.[44]
50
0 200 400 600 800 1000Frequency (cm−1 )
0.0
0.2
0.4
0.6
0.8
1.0
Reflectance
20 40 60 80 100 120Energy (meV)
20K70K150K200K300K
Figure 4-2. Temperature-dependent reflectance spectrum of Cu2OSeO3.
The top panel of Fig. 4-3 shows the 300 K reflectance up to 40,000 cm−1. The
onset of electronic absorption is indicated by the upturn around 26,000 cm−1 (3.2 eV).
The transmittance spectra, as depicted in Figure. 5-4 and Figure. 4-5, are in good
agreement with the reflectance measurements. At frequencies below the strong phonon
modes (< 80 cm−1), the crystal transmits. The periodic oscillations in this range are
the Fabry-Perot interference fringes due to multiple internal reflections. Transmission
gaps open between the infrared phonon modes. These regions become increasingly more
evident as temperature is lowered and the modes sharpen. The low-frequency transmission
spectrum exhibits a weak phonon with a resonance frequency of 68 cm−1 that first appears
as a small structure around 120 K and strengthens with decreasing temperature. The
mode has a small oscillator strength, thus explaining why it was not observed in reflection.
The weak phonon shows no response to fields of 10 T applied perpendicular to the
crystal surface (inset of Figure. 5-4). In contrast, the high frequency spectra (Figure 4-5)
possesses two sharp dips, 1530 and 2054 cm−1, which are too high to be single phonon
peaks. Consequently, we are not able to make an assignment of these features.
51
Frequency (cm−1 )
0.1
0.2
0.3
0.4
0.5
0.6
Reflectance
1 2 3 4 5Energy (eV)
a300K
0 10000 20000 30000 40000Frequency (cm−1 )
0
0.4
0.8
1.2
1.6
2.0
σ1(103 Ohm−1cm
−1) b
Figure 4-3. The upper panel (a) shows the 300 K broadband reflectance out to 40,000cm−1. The lower panel (b) depicts the optical conductivity at roomtemperature over the same frequency region.
4.3.3 Kramers-Kronig Analysis and Optical Properties
Kramers-Kronig analysis can be used to estimate the real and imaginary parts
of the dielectric function from the bulk reflectance R(ω).[43] Before calculating the
Kramers-Kronig integral, the low frequency data were extrapolated as a constant for
ω → 0 as befits an insulator. At high frequencies the reflectance was assumed to be
constant up to 1 × 107 cm−1, after which R ∼ (ω)−4 was assumed as the appropriate
behavior for free carriers. The optical properties were derived from the measured
52
0 20 40 60 80 100Frequency (cm−1 )
0.0
0.2
0.4
0.6
0.8
1.0
Transm
ission
0 2 4 6 8 10 12Energy (meV)
50K75K100K120K300K
0 2 4 6 8 10Field (T)
67.5
68.0
68.5
Frequency (cm
−1)
Figure 4-4. Temperature dependent transmission spectrum of 194 µm thick Cu2OSeO3
single crystal below 100 cm−1. This region is highlighted by a weak phonon(∼68 cm−1) that begins to appear around 120 K. The inset shows themagnetic field dependence of the weak low-energy absorption with a resolutionof 0.25 cm−1. It seems to have no response to magnetic field.
reflectance and the Kramers-Kronig-derived phase shift on reflection; in particular, we
estimated the real part of the optical conductivity, σ1(ω).
Figure 4-3b shows the optical conductivity at room temperature out to 40,000
cm−1. The low absorption in the infrared is consistent with the insulating nature of the
compound. Our temperature-dependent measurements ceased at 5,000 cm−1. Accordingly,
the dynamics of interband transitions with temperature in the region of high photon
energy do not contribute to our overall findings. Figure 4-6 depicts σ1(ω) at 20 K (below
Tc) and 70 K (above Tc) from 30 to 1,000 cm−1. The principal effect is a sharpening of
most modes, yielding a larger conductivity at the resonant frequency.
The phonon modes in reflectance appear as Lorentzian lines in the optical conductivity,
making them intuitive for modeling as harmonic oscillators. The vanishingly small static
limit of σ1(ω) and the low background level of conductivity throughout the infrared
regions is further evidence of the insulating nature of the compound.
53
101 102 103 104
Frequency (cm−1 )
0.0
0.2
0.4
0.6
0.8
1.0
Transm
ission
300K
Figure 4-5. Broadband transmission on the thinned piece of Cu2OSeO3 crystal at 300 Kmeasured out to the 40,000 cm−1.
4.3.4 Oscillator-Model Fits
The Drude-Lorentz model was used to fit the reflectance and obtain a second estimate
of the complex dielectric function in the infrared range. The model consists of a damped
oscillator for each putative phonon in the spectrum plus a high frequency permittivity ε∞
that describes the contribution of all electronic excitations. The model has the following
mathematical form:
ε =
∞∑
j=1
Sjω2j
ωj2 − ω2 − iωγj
+ ε∞ , (4–1)
where Sj, ωj, and γj represent the oscillator strength, center frequency, and linewidth of
the jth damped oscillator. The complex dielectric function provided by the Drude-Lorentz
model is used to calculate the reflectivity, which agrees well with our original measured
quantity. Figure 4-7 compares the calculated reflectivity and the measured reflectivity at
70 K. Similar quality fits were obtained at all other temperatures.
54
0 200 400 600 800 1000Frequency (cm−1 )
0
50
100
150
200
250
σ1(Ohm−1cm
−1)
20 40 60 80 100 120Energy (meV)
20K70K
Figure 4-6. Far Infrared optical conductivity Cu2OSeO3 at 20 K(red) 70 K(black).
4.4 Discussion
4.4.1 Magnetodielectric Effect
Equipped with oscillator parameters to describe each of the infrared phonons at all
measured temperatures, one is now in position to closely monitor the subtle dynamics
of the phonon structures across the transition temperature. Despite the lack of drastic
changes in the phonon spectrum at Tc, it is worthwhile examining whether a combination
of many small anomalies in the phonon dynamics might sum to give an overall effect. At
this point it is logical to examine the static dielectric constant because it is the sum of
parameters that describe the dielectric nature of the compound (the side to which the
infrared is most sensitive). Taking the zero frequency limit of the Drude-Lorentz formula,
one arrives at the following simple expression for the static dielectric constant:
εo =
∞∑
j=1
Sj + ε∞ . (4–2)
55
0 200 400 600 800 1000Frequency (cm−1 )
0.0
0.2
0.4
0.6
0.8
1.0
Reflectance
70 K
20 40 60 80 100 120Energy (meV)
Figure 4-7. Experimental reflectance (blue points) and calculated reflectance (red line)from the Drude-Lorentz model of the Cu2OSeO3 70 K dielectric function.
The calculated static dielectric constant at temperatures near Tc, is shown in Fig. 4-8.
There is an anomalous jump at 60 K. It should be noted that this magnetodielectric effect
in Cu2OSeO3 was previously observed by Bos et al.[44] through dielectric capacitance
measurements. While the value of the dielectric constant measured here is considerably
smaller than that in the previous report, the direction and magnitude of the anomaly at
Tc are in good agreement. It is also worthwhile to note that the systematic changes in
reflectance near Tc over the mid infrared band (1,000–5,000 cm−1) corresponded to changes
in the mid infrared dielectric function that were one order of magnitude less than our
magnetodielectric effect, thus identifying phonons as the main contributor to this effect.
4.4.2 Anomalous Phonons
The observed phonons in Cu2OSeO3 may be divided into two categories: conventional
and anomalous. Conventional phonons show a slight hardening of their frequencies when
cooled to low temperatures, accompanied by a significant reduction in linewidth and at
most modest changes in oscillator strength. Moreover, the temperature variation is smooth
with no sudden changes in slope or value. Anomalous phonons violate one or more of
56
Table 4-1. Oscillator parameters for Cu2OSeO3 (at 20 K) and their corresponding assignments for a few strong modes in thefar infrared.
Index Osc Strength Center Freq Linewidth Anomalous AssignmentS ω (cm−1) γ (cm−1) Type A, B, C, or No
1 0.003 67.7 1.1 No -2 0.501 84.9 0.6 No Vibration of CuO5 units3 0.337 91.2 0.6 No -4 0.233 100.7 0.5 No -5 0.073 126.1 0.8 No -6 0.134 147.5 1.2 Type A -7 0.544 184.8 1.5 Type B -8 0.071 204.8 2.4 Type C -9 0.521 212.0 2.9 No SeO3 vibrating against CuO5
10 0.038 273.9 5.5 Type C -11 0.123 293.2 7.6 Type C -12 0.042 311.7 6.2 Type B -13 0.107 335.8 3.6 Type C Internal in-plane vibration of CuO5 units14 0.188 385.5 2.0 No Internal bending mode of SeO3 units15 0.063 399.4 3.6 No -16 0.095 437.4 5.9 Type C Internal out-of-plane vibration of CuO5 units17 0.007 454.1 4.3 No -18 0.445 504.5 7.4 No -19 0.224 537.9 6.5 Type C -20 0.091 551.3 3.8 Type C -21 0.094 592.5 3.4 Type C -22 0.291 717.1 15.8 Type B Antisymmetric stretch of SeO3 units23 0.024 753.1 11.6 No -24 0.054 781.7 13.9 Type A Antisymmetric stretch of SeO3 units25 0.017 813.8 6.6 No -26 0.016 831.3 5.1 No Radial breathing mode of SeO3 units
57
0 20 40 60 80 100Temperature (K)
8.0
8.1
8.2
8.3
8.4
8.5
8.6
Dielectric Constant ε o
Figure 4-8. The static dielectric constant as calculated from the Drude-Lorentz model attemperatures between 20 and 100 K, including Tc.
these expectations. We have identified 13 anomalous phonons in our spectra, which show
26 total modes. A numbering scheme has been used which corresponds to the sequence
of appearance of oscillators in the infrared spectrum starting with #1 (lowest frequency
mode) and ending with #26 (highest frequency mode)
Figure 4-9 displays the temperature dependence of the three oscillator parameters
for 3 of the 13 anomalous phonons (#13, #19, and #20) in the infrared spectrum. The
oscillator strength, represented by Sj, was calculated from the spectral weight and center
frequency of each phonon using the relation Sj = (ωpj/ωoj)2. The behavior of Sj directly
results from changes in ωp. The oscillator parameters for one conventional phonon (#2)
are shown for comparison. It was observed that anomalous behavior usually could be
found in all three oscillator parameters.
4.4.3 Assignment of Phonon Modes
Because the magnetodielectric effect is observed through lattice dynamics, we make
a partial analysis of the phonon spectrum. The number of phonon modes expected in
Cu2OSeO3 can be found by space group analysis. Using the SMODES program,[49] we
58
0 100 200 300Temperature (K)
84.6
84.7
84.8
Tc
#2
334
335
336#13
536
537
538#19
547
549
551#20
0 100 200 300
Cente
r Fr
equency
+ (cm,1 )a
0 100 200 300Temperature (K)
0.80
0.85
0.90
0.95
1.00
1.05
Norm
aliz
ed O
scill
ato
r Str
ength
, Sj(T
)/Sj(20K)
Tc
b #2#13#19#20
0 100 200 300Temperature (K)
0
5
10
15
Line W
idth
- (cm,1 )
c
Tc
#2#13#19#20
Figure 4-9. The (a) center frequency, (b) normalized oscillator strength, and (c) linewidthof four oscillators as a function of temperature. Oscillator #2 is a typicalconventional phonon, whereas the other three oscillators show anomalousbehavior as temperature is lowered across Tc.
arrive at the following distribution of modes:
Γoptical = 14A(R) + 14E∗(R) + 41T (R,IR) , (4–3)
where (R) and (IR) denote Raman active and infrared active modes respectively. We
therefore have the potential of 41 total infrared active modes, all of which possess
threefold degeneracy as indicated by their irreducible representation. However, only
26 modes in the infrared spectrum are detected. The oscillator parameters (at 20 K) of the
26 experimentally observed modes are listed in Table 6-1. Based on our lattice dynamical
calculation of the position of the 41 predicted modes and the experimental linewidth of the
26 observed modes, we suspect that weaker phonons are buried within stronger phonons
and merge in the spectrum. The Lorentz analysis for merged phonons would result in an
average of oscillator parameters weighted by oscillator strength.
59
More specifically, the lattice dynamical calculation employed herein are based on
a real-space summation of screened coulomb interactions involving a spherical cut-off
boundary. [50] Frequency, mode intensity, as well as displacement pattern were calculated
based on the structure and valence as reported by Bos et al. [44] The center frequency,
mode intensity, and specific nature of atomic vibration of all calculated modes are listed
in Table 4-2. We have indicated groups of the calculated modes where a potential merging
may have occurred in the experimental spectra. We were also able to assign eight strong
modes in the infrared spectrum by comparing the calculated and experimental spectra.
Table 4-2 details the mode assignments made using the adopted numbering
scheme and center frequency of each oscillator for identification. In addition, we have
distinguished between oscillators exhibiting anomalous behavior in one parameter (type
A), two parameters (type B), and all three parameters (type C). It should be noted that
oscillators #13 and #16, which exhibit anomalous behavior at the transition temperature
in all three oscillator parameters (#13 is shown in Fig. 4-9), are associated with vibrations
of oxygen around the central copper, the ion responsible for magnetic ordering.
4.5 Summary
Far-infrared measurements of single crystal reflectance from Cu2OSeO3 reveal no
drastic anomalies in the phonon spectrum at the ferrimagnetic transition temperature.
However, a closer inspection of the dynamics of the phonon spectrum, as modeled through
Drude-Lorentz fitting, uncover an anomalous jump in the dielectric constant near Tc. It is
observed that 13 of the 28 total far-infrared phonons contribute to this magnetodielectric
effect. A few strong far-infrared phonons have been assigned to motion of the CuO5 and
SeO3 units via a lattice dynamical calculation. It is noteworthy that 2 of the 13 modes
exhibiting anomalous behavior across Tc have been assigned to the motions of oxygen
around the central copper, the ion responsible for magnetic ordering. A weak phonon that
was not resolved in reflectance is observed below 120 K in the transmission spectrum. A
magnetic origin for this structure has yet to be ruled out.
60
Our infrared results agree with the Raman studies of Gnezdilov et al.[45] in that we
also observed a number of phonon modes that exhibit anomalies in their strength, center
frequency, or linewidth, but we differ on other issues. For example, we do not observe any
new modes below the magnetic transition whereas Genzdilov et al.[45] do detect some
additional (rather weak and broad) features. They report three new modes appearing in
the Raman spectra below the transition temperature at frequencies of ∼261, 270, and
420 cm−1. The only structure we observe in these three spectral regions is at 270 cm−1,
where 1 of the 13 reported anomalous phonons is present. The absence of a typical
mode at 420 cm−1 gives further support to the claim by Genzdilov et al. that the new
line appearing in their spectra at this frequency “unambiguously has magnetic origin.”
Genzdilov et al.[45] also report two new lines originating below 20 K in the Raman
spectra at ∼86 and 203 cm−1. Our infrared studies reveal a strong rather typical mode
at ∼86 cm−1 and a weak anomalous mode at 203 cm−1. We have assigned the typical
mode at ∼86 cm−1 to the collective vibration of the edge sharing CuO5 units. If any new
infrared features possessed the same relative intensities as reported for the Raman spectra,
we would have observed them clearly.
61
Table 4-2. Center frequency and intensity of the 41 predicted infrared active modes in Cu2OSeO3.
Center Freq Dipole Moment Experimentally Observed Atomic Motionω (cm−1) (Arb. units) Exp. index Description87.1? 0.02 - General motion of CuO5 units89.9? 0.05 - Vibration of CuO5 units92.5 0.16 2 Vibration of CuO5 units104.5 0.02 - General motion of CuO5 units112.7 0.08 - Vibration of CuO5 trigonal bipyramidal vs CuO5 square pyramidal114.3 0.13 - Vibration of CuO5 trigonal bipyramidal vs CuO5 square pyramidal127.9? 0.05 - General motion of CuO5 units130.8? 0.04 - General motion of CuO5 units143.9 0.11 - General motion of CuO5 units159.7? 0.03 - General motion of CuO5 units173.8? 0.03 - General motion of CuO5 units213.6 0.15 9 SeO3 units vibrating against CuO5 units232.2 0.12 - General motion of SeO3 and CuO5 units277.9? 0.09 - Vibration of SeO3 units289.2? 0.01 - General motion of SeO3 units305.6 0.20 - General motion of SeO3 units317.8 0.23 13 Internal in-plane vibration of CuO5 units374.3 0.57 - Internal in-plane vibration of CuO5 units386.7 1.85 14 Internal bending mode of SeO3 units401.3 0.35 - General motion of CuO5 units408.9 0.71 - General motion of CuO5 units412.0 0.80 - General motion of CuO5 units423.8 0.70 - Internal out-of-plane vibration of CuO5 units435.3 0.73 16 Internal out-of-plane vibration of CuO5 units444.4 0.90 - General motion of Oxygen atoms450.2† 0.22 - General motion of Oxygen atoms
The term “general motion” refers to modes where the atoms in motion are known, but specific motion is unclear. Theannotation ? indicates modes that are possibly not resolved from merging due to weak dipole moments on adjacent phonons.
The annotation † indicates phonons that are possibly not resolved due to merging (broad experimental linewidths).
62
Table 4-2. Continued.Center Freq Dipole Moment Experimentally Observed Atomic Motionω (cm−1) (Arb. units) Exp. index Description453.3† 0.63 - General motion of Oxygen atoms469.0? 0.07 - General motion of Oxygen atoms485.6? 0.06 - General motion of Oxygen atoms492.1 0.24 - General motion of Oxygen atoms498.7 0.43 - General motion of Oxygen atoms512.6 0.68 - Internal vibration of SeO3 units517.7 0.49 - General motion of Oxygen atoms525.5 0.17 - General motion of Oxygen atoms557.4 0.46 - General motion of Oxygen atoms716.2 1.30 22 Antisymmetric stretching of SeO3 units750.7 0.56 - General motion of SeO3 units782.4† 0.83 24 Antisymmetric stretching of SeO3 units790.0† 0.57 - Antisymmetric stretching of SeO3 units803.3 0.34 - General motion of SeO3 units830.7 0.30 26 Radial breathing mode of SeO3 units
63
CHAPTER 5OPTICAL PROPERTIES OF MULTIFERROIC FeTe2O5Br
5.1 Motivation
A technological shortcoming of magnetically driven ferroelectrics has been the small
value of induced spontaneous electric polarization observed. Much larger spontaneous
polarizations have been predicted in materials where ferroelectricity arises from collinear
magnetic ordering of spins.[51, 52] In collinear magnetic ordering, moments may vary in
amplitude but do not vary in direction; therefore, one does not expect the aforementioned
spin-orbit-related interactions to be important,[53] enabling the much stronger exchange
striction interaction as the mechanism by which ferroelectricity is induced.
This chapter repots infrared studies on single crystals of the multiferroic FeTe2O5Br
compound, a magnetically driven ferroelectric with nearly-collinear incommensurate
spin order. A previous study by Pregelj et al.[54] has reported significant changes in the
magnetic and electric orders when external magnetic fields are applied along the different
crystallographic directions. Strikingly, for B‖b they observed that fields greater than 4.5 T
destroyed the electric polarization completely. In our investigation of magnetoelectric
coupling we measured transmission in the far-infrared down to 15 cm−1 (2 meV) and in
external magnetic fields up to 10 T oriented along all three crystallographic axes. The
significant magnetoelectric coupling previously shown by the application of external
magnetic fields did not surface in the far infrared to the extent we examined it. However,
we present a comprehensive report of the excitation spectrum in single crystal FeTe2O5Br.
The modeling of the infrared active phonons as well as lattice dynamical calculations
have lead us to make an interesting comparison to a previous infrared study on a similar
Reprinted with permission from K. H. Miller, X. S. Xu, H. Berger, V. Craciun,X. Xi, C. Martin, G. L. Carr, and D. B. Tanner, submitted to Phys. Rev. B, eprintarXiv:1301.5881 (2013).
64
geometrically frustrated spin-cluster oxyhalide compound, FeTe2O5Cl,[55] where an equal
number of modes was predicted but a significantly lower number was reported.
Crystal Structure: Becker et al.[56] solved the crystal structure using single-crystal
x-ray diffraction. The FeTe2O5Br compound crystallizes in a layered structure where the
individual layers comprise the bc plane in the monoclinic system (P21/c). The layers are
weakly connected via van der Waals forces and slip at an 18 angle as they stack along the
normal to the bc plane, thus defining the monoclinc a axis. A single layer is composed of
[Fe4O16]20− groups confined on top and bottom by [Te4O10Br2]
6− groups. The three groups
contain common oxygens which connect them and also serve to create charge neutrality
within the layers. The [Fe4O16]20− groups consist of four edge-sharing [FeO6] distorted
octrahedra. All iron ions possess a +3 oxidation state. More detailed descriptions of the
crystal structure are found elsewhere.[56]
The FeTe2O5Br compound exhibits nearly-collinear (7 of canting between the Fe1
and Fe2 sites) incommensurate antiferromagnetic ordering of its moments below TN =
10.6 K.[57, 58] The amplitude-modulated magnetic order is described with the wave vector
q=(1/2,0.463,0). Simultaneously, a ferroelectric polarization is induced perpendicular to q
and the Fe3+ moments.[57] The ferroelectric order is attributed to the highly polarizable
Te4+ lone pair electrons. Single-crystal x-ray diffraction measurements in the ordered state
did not detect any change in crystal symmetry from the high temperature phase; however,
clearly distinguishable changes in Fe-Te and Fe-O interatomic distances were observed
that signified exchange striction as the means by which the inversion center is removed,
thus allowing for ferroelectricity to arise.[58] Exchange striction is the mechanism by which
magnetic ions shift away from their centrosymmetric positions to maximize their exchange
interaction energies.[4, 59]
65
5.2 Experimental Procedures
Single crystals of FeTe2O5Br have been grown by standard chemical vapor phase
method. Mixtures of analytical grade purity Fe2O3, TeO2 and FeBr3 powder in molar
ratio 1:6:1 were sealed in the quartz tubes with electronic grade HBr as the transport
gas for the crystal growth. The ampoules were then placed horizontally into a tubular
two-zone furnace and heated very slowly by 50C/h to 500C. The optimum temperatures
at the source and deposition zones for the growth of single crystals have been 500C and
440C, respectively. After six weeks, many dark yellow, almost orange FeTe2O5Br crystals
with a maximum size of 8 × 12 × 1 mm3 were obtained. The powder x-ray diffraction
pattern obtained by using a Rigaku x-ray diffractometer shows the monoclinic space group
P21/c for all FeTe2O5Br crystals.
The zero-field temperature-dependent (5–300 K) reflectance and transmittance
measurements were performed on a 1.3 × 8 × 6 mm3 single crystal (crystal 1) using
a Bruker 113v Fourier transform interferometer in conjunction with a 4.2 K silicon
bolometer detector in the spectral range 25–700 cm−1 and a 300 K DTGS detector
from 700–7,000 cm−1. The crystal was cooled using a flow cryostat. Room temperature
measurements from 7,000–33,000 cm−1 were obtained with a Zeiss microscope photometer.
Appropriate polarizers were employed to span the desired spectral range.
Field-dependent transmission measurements in the far infrared at a resolution of
0.25 cm−1 were performed on a 0.3 × 8 × 5 mm3 single crystal (crystal 2) at beam
line U4IR of the National Synchrotron Light Source, Brookhaven National Laboratory.
The measurements employed a Bruker IFS 66-v/S spectrometer in conjunction with a
10 T Oxford superconducting magnet and a 1.8 K silicon bolometer detector. A free
Crystal growth was carried out by Helmuth Berger at the EPFL in Lausanne,Switzerland
66
standing wire grid polarizer was oriented along the preferential polarization direction of
the synchrotron beam.
At this point it is important to establish with confidence that we cooled our single
crystal below 10 K in both the Oxford superconducting magnet and the Janis flow
cryostat. The superconducting magnet is equipped with a variable temperature insert
that cools in liquid He vapors. This same system has been used to accurately measure
the properties of superconducting materials down to 2 K.[60] The accuracy of the cooling
capacity of the flow cryostat was corroborated by comparing ratios of Ts/Tn for the
superconductor Nb0.5Ti0.5N with those same ratios measured in the superconducting
magnet. With the superconductor fastened to the cold finger of our flow cryostat with a
copper clamp, the cooling capabilities, below 10 K, agreed to within ±1 K.
In materials where the crystal symmetry is lower than cubic, electrical excitations
strongly depend on the orientation of the electric field. For these low symmetry crystals,
the dielectric constant is a 3 × 3 tensor and will generally have off diagonal components.
For every crystal symmetry, however, there exists a set of orthogonal Cartesian axes,
called the principal dielectric axes, which diagonalize the dielectric tensor.[61] In an
optics experiment, the principal dielectric axes are found by rotating the orientation
of the sample which is placed between two polarizers that are crossed and remain
fixed. Throughout a revolution of 360, four orthogonal orientations will result in zero
transmitted light beyond the second polarizer. These four dark orientations correspond to
the electric field of the incoming light becoming aligned parallel to a principal dielectric
axis of the system.
For a monoclinic system, the crystallographic b axis is unique because it is perpendicular
to both the a and c axes. Accordingly, one principal dielectric axis of the system is fixed
along the crystallographic b axis, but the other two axes can orient themselves arbitrarily
as long as they remain orthogonal. Consequently, we performed single crystal x-ray
diffraction on crystal 1 with knowledge of how the principal dielectric axes fell in relation
67
to the crystal facets in order to determine their orientation with respect to the a and c
crystallographic directions.
X-ray diffraction measurements were collected at 300 K on a four-circle Panalytical
X’Pert MRD in a parallel beam geometry with Kα radiation. A θ-2θ scan was used to
check that the sample surface corresponded to the bc plane while pole figure measurements
were performed to find the spatial orientation of the a axis with respect to the bc plane.
5.3 Results and Analysis
5.3.1 X-Ray diffraction
The diffraction experiment showed that the crystallographic c axis also coincides with
a principal dielectric axis of the system. Furthermore, the third principal dielectric axis
makes an 18 angle with the crystallographic a axis. The XRD patterns acquired from
the sample with the surface aligned perpendicular to the diffraction plane are displayed
in Fig. 5-1. Only very narrow (h00) diffraction lines were observed, confirming that the
sample was a high quality single crystal, with the surface aligned parallel to the bc plane.
Several acquired pole figures helped identify the orientation of the a axis with respect to
the bc plane. The inset to Fig. 5-1 contains the results of our XRD investigation, namely,
a sketch of crystal 1 with the principal dielectric axes (eA, eB, and eC) superimposed on
the monoclinic crystal axes. (Note that c, a, eC , and eA all lie in a plane.)
5.3.2 Reflectance and Transmittance Spectrum
The temperature-dependent reflectance spectrum of FeTe2O5Br for electric field
oriented parallel eA, eB, and eC in the frequency interval 30–1000 cm−1 (4–120 meV) is
shown in Fig. 6-2. A strong sharpening of many modes accompanied by a hardening of
resonance frequencies is observed with decreasing temperature. No drastic deviations from
the room-temperature spectrum were observed along any of the three directions when
cooling to 5 K. Particular attention was given to the excitation spectrum slightly above
and below the 10.6 K multiferroic ordering temperature. The absence of anomalies in the
spectra upon cooling below 10.6 K gives strong support to the observation of no crystal
68
10 20 30 40 50 602θ (degree)
0
1000
2000
3000
4000
Intensity (arb. units)
(100)
(200)
(400)
(600)
(700)
(800)
Figure 5-1. The XRD pattern for the sample surface aligned perpendicular to thediffraction plane. The inset depicts the orientation of the principal dielectricaxes and crystallographic axes with respect to the facets of crystal 1.
symmetry change in the ordered state; however, a subtle contradiction exists.[57] The
high temperature P21/c space group is centrosymmetric, but ferroelectricity, for which
non centrosymmetry is a strict requirement, is observed below TN . This in turn suggests
that new infrared modes should accompany the transition. However, the small value of
polarization measured (8.5 µC/m2) leads one to suspet that a subtle transition occurs and
additional infrared modes are likely below our experimental resolution (<0.5 cm−1).
Figure 5-3 displays the optical conductivity along the three principal dielectric axes in
the frequency interval 30–33,000 cm−1. The strong anisotropy observed in the far infrared
persists at high frequencies resulting in three distinct absorption edges. The inset of
Fig. 5-3 shows the 300 K reflectance up to 33,000 cm−1 along eA, eB, and eC .
Crystal 1 exhibits transmission at frequencies above the infrared phonon modes and
below the absorption edge. The dimensions of the crystal only allowed for transmission
with light polarized along eB and eC . As shown in Fig. 5-4, a sharp dip at 1850 cm−1 as
well as a broad absorption centered at 3250 cm−1 are present along both polarizations.
Structures at these particular frequencies call to mind the absorption bands of H2O,
which in turn suggest water of hydration in the crystal structure. But the anisotropic
strengths of these absorptions can be used to rule out H2O. Therefore the structures
suggest absorptions that are intrinsic to FeTe2O5Br. The differences in energy of the high
69
Frequency (cm−1 )0.0
0.4
0.81.0
E ∥ eB
20 40 60 80 100 120Energy (meV)
5K50K100K200K300K
Frequency (cm−1 )0.0
0.4
0.8
Reflectance E ∥ eC
0 200 400 600 800 1000Frequency (cm−1 )
0.0
0.4
0.8 E ∥ eA
Figure 5-2. Temperature-dependent reflectance spectrum of FeTe2O5Br along eA, eB, andeC .
frequency transmission edge along eB and eC reinforce the anisotropy seen in the optical
conductivity at high frequencies (Fig. 5-3).
5.3.3 Field-Dependent Transmittance
Crystal 2 transmits at frequencies below the strong infrared phonons, in the gaps
in-between the phonons, and in the mid infrared between the highest phonon and the
onset of interband transitions. Particular attention was given to the transmission below
the strong infrared phonons. No appreciable change in the spectra were observed for
transmittance at 5 K measured along eB and eC with magnetic fields of 10 T applied along
all three crystallographic directions. A class of low energy magnetic excitations called
antiferromagnetic resonance modes (AFMR) are predicted to exist at finite frequencies
in zero field and shift upon the application of external fields. AFMR modes in the
FeTe2O5Br have been reported in a recent electron spin resonance study below 400 GHz
70
0 10000 20000 30000Frequency (cm−1 )
0
1000
2000
3000
4000
5000
σ1(O
hm−1cm
−1)
1 2 3 4Energy (eV)
E ∥ eCE ∥ eBE ∥ eA
0 10000 20000 30000Frequency (cm−1 )
0.0
0.2
0.4
0.6
Reflectance
E ∥ eCE ∥ eBE ∥ eA
Figure 5-3. The optical conductivity in the frequency interval 30–33,000 cm−1. The insetshows the 300 K broadband reflectance out to 33,000 cm−1 from which theoptical conductivity was extracted via Kramers-Kronig relations.
2000 4000 6000 8000 10000 12000Frequency (cm−1 )
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Transm
ittance
0.25 0.50 0.75 1.00 1.25 1.50Energy (eV)
E ∥ eCE ∥ eB
Figure 5-4. Transmittance of crystal 1 in the mid-infrared and near-infrared regions.
(∼13 cm−1).[62] However, in an external field of 10 T the modes remain slightly below our
measurable frequency range (15 cm−1).
71
103 104
Frequency (cm−1 )
0.0
0.2
0.4
0.6
0.8
1.0
Reflectance
& Transm
ittance E ∥ eC
Tmeasured*1.4Rmeasured*1.4Rsingle
Figure 5-5. Single-bounce reflectance (red) as approximated from the scaled measuredreflectance (black) and the scaled measured transmittance (blue) of crystal 1.The elevated level of measured reflectance throughout the transmission intervalis expected due to additional light reflected from the back surface of thecrystal.
5.3.4 Determination of Optical Properties
We have used the measured reflectance and transmittance spectra of crystal 1 to
extract the complex dielectric function. Because Kramers-Kronig analysis requires a
single-bounce reflectance spectrum over a broad frequency region, we must first correct
the measured reflectance spectrum in regions where the crystal transmits by using
a combined reflection and transmission analysis. An attempt to solve for the single
bounce reflectance by inverting the full reflection and transmission equations leads to a
transcendental equation with infinitely many solutions. Nevertheless, using the method
employed by Zibold et al.,[8] i.e., assuming that k is small in the region of interest, leads
to an approximation of reflection and transmission that can be solved numerically for a
single solution.
The measured reflectance (not shown) and transmittance (Fig. 5-4) in the frequency
interval 1500–10000 cm−1 were significantly lower than expected throughout this region
of low absorption (Fig. 5-3). The low measured values were attributed to the scattering
72
of light from facets existing on the back surface of crystal 1. To compensate, we assumed
a mask on the back surface of crystal 1 which lead to scaling the measured reflectance
and transmittance up by a factor of 1.4 in the interval 1500–10000 cm−1, thus explaining
the discrepancy between Fig. 5-4 and Fig. 5-5. The scaled reflectance and transmittance
yielded a single bounce reflectance which agreed well with the measured bulk reflectance in
regions where crystal 1 did not transmit, as shown in Fig. 5-5.
Kramers-Kronig analysis can be used to estimate the real and imaginary parts
of the dielectric function from the bulk reflectance R(ω).[43] Before calculating the
Kramers-Kronig integral, the low frequency data (0.1–30 cm−1) were extrapolated using
parameters from the complex dielectric function model described in Sec. III E. At high
frequencies (8 × 104–2.5 × 108 cm−1) the reflectance was approximated from known x-ray
scattering functions of the constituent atoms. The gap between our highest measurable
frequency and the x-ray reflectance data was bridged with a Lorentz oscillator at
60,000 cm−1. Subsequently, the phase shift on reflection was obtained via Kramers-Kronig
analysis, and the optical properties were calculated from the reflectance and phase.
5.3.5 Lorentz Oscillator Fits
The measured reflectance was fit with a Drude-Lorentz model to obtain a second
estimate of the complex dielectric function in the infrared range. The model assigns a
lorentzian oscillator to each distinguishable phonon in the spectrum plus a high frequency
permittivity, ε∞, to address the contribution of electronic excitations. The model has the
following mathematical form:
ε =∞∑
j=1
Sjω2j
ωj2 − ω2 − iωγj
+ ε∞ , (5–1)
where Sj, ωj, and γj signify the oscillator strength, center frequency, and the full width at
half max (FWHM) of the jth lorentzian oscillator. The Drude-Lorentz complex dielectric
function is used to calculate the reflectivity, which is compared to the original measured
73
0 200 400 600 800 1000Frequency (cm−1 )
0.0
0.2
0.4
0.6
0.8
1.0
Reflectance
E ∥ eB100K
datafit
20 40 60 80 100 120Energy (meV)
Figure 5-6. Experimental reflectance (blue points) and calculated reflectance (red line)from the Drude-Lorentz model of the FeTe2O5Br 100 K eB dielectric function.
quantity in Fig. 5-6. Similar qualities of fits were obtained for all other temperatures and
polarizations.
5.4 Discussion
5.4.1 Group theory and Lattice Dynamics
Due to the lack of anomalies upon cooling below 10.6 K, we turn our focus to
the highly anisotropic far-infrared excitation spectrum. The number of phonon modes
expected in FeTe2O5Br can be found by group-theory analysis. Using the SMODES
program,[49] we arrive at the following distribution of modes:
Γoptical = 54A(R)g + 54B(R)
g + 53A(IR)u + 52B(IR)
u , (5–2)
where (R) and (IR) denote Raman active and infrared active modes respectively. The
53Au modes are expected to lie along the unique b axis (eB) of the monoclinic system.
The 52Bu modes are expected to lie in the ac plane, and one should be able to resolve all
52 modes using the two orthogonal polarization spectra measured in this plane. However,
the merging of weaker modes near stronger modes can hinder this count. We therefore
74
Table 5-1. Oscillator parameters for experimentally observed phonons along eA (at 20 K).The motions of atoms associated with groups of phonons have been assignedwhen the pattern was apparent. The atoms are listed in descending order oftheir calculated net atomic displacement.
Idx Osc Str Ctr Freq FWHM Degen AsgnS ω (cm−1) γ (cm−1) idx in eC
1 0.207 45.5 2.0 1 Br2 1.320 79.6 2.1 5 -3 0.356 88.4 1.1 6 Br,Te,O,Fe4 0.042 98.1 1.0 8 Br,Te,O,Fe5 0.056 107.5 2.1 9 Br,Te,O,Fe6 0.048 137.1 1.9 12 Br,Te,O,Fe7 0.015 157.6 1.6 14 Te,Br,O,Fe8 0.023 166.0 2.2 15 Te,Br,O,Fe9 0.354 184.9 1.3 16 Te,O,Fe10 0.519 194.3 2.4 17 Te,O,Fe11 0.159 211.7 2.0 Te,O,Fe12 0.007 236.4 4.9 22 -13 0.084 247.8 1.7 23 -14 0.009 256.5 4.0 24 -15 0.169 291.9 1.8 27 Fe,O,Te16 0.030 302.8 3.4 Fe,O,Te17 0.033 321.6 3.6 28 Fe,O,Te18 0.006 328.0 2.4 Fe,O,Te19 0.003 348.5 2.0 29 Fe,O,Te20 0.102 370.0 2.5 -21 0.008 387.1 2.6 -22 0.030 401.3 4.1 -23 0.005 418.6 4.0 -24 0.005 426.2 2.6 30 -25 0.090 435.0 3.9 31 -26 0.119 463.6 5.4 32 O,Fe,Te27 0.004 481.0 1.4 O,Fe,Te28 0.029 493.2 4.9 33 O,Fe,Te29 0.350 575.3 7.5 35 O,Fe30 0.203 666.1 8.0 37 O,Fe31 0.257 730.5 13.9 O,Fe32 0.018 784.0 9.2 42 -
employed lattice dynamical calculations to determine the relative resonance frequency,
mode intensity, as well as displacement pattern for the 105 infrared active modes. The
calculations, which utilized the structure and valence as reported by Becker et al.,[56] were
75
based on a real-space summation of screened coulomb interactions involving a spherical
cut-off boundary.[63] The following two sections describe the method used to compare the
observed and calculated spectrum for the Au and Bu modes respectively.
5.4.2 52Bu Modes
All 52Bu modes predicted lie in the ac plane and should possess a finite intensity
along both measured polarizations in this plane unless the direction of the induced dipole
moment for a particular vibration is parallel to either eA or eC . (Small angle deviations
from the parallel case may be hard to detect.) For all other modes one must sort out
which excitations along eA and eC corresponding to the same physical vibration so that
they are not counted twice. To count the modes, we used the Drude-Lorentz parameters
for each phonon at 20 K as well as the resonant frequency and mode intensity from our
lattice dynamical calculation. With knowledge of the optical conductivity spectra along
eA and eC at 20 K, modes were grouped together if they were observed in the same small
frequency interval (on the order of a typical linewidth) of one another in the two spectra.
Next, we compared the experimental linewidths of the grouped modes to ensure that
they would overlap if plotted side-by-side. As two final criteria, we compared both the
ratio of experimental oscillator strengths with the computed oscillator strengths, and also
compared the displacement patterns generated by our calculation to make sure they were
in good agreement as to which excitations corresponded to the same physical vibrations.
All 52Bu modes predicted in the ac plane were observed in the experimental spectrum.
Lorentz oscillator parameters for all modes as well as modes present along both eA and eC
polarizations can be found in Table 5-1 and Table 5-2.
5.4.3 53Au Modes
Counting the experimentally observed Au modes is straightforward; we observe
43 of the 53 predicted modes along this direction. We believe that 10 modes were not
resolved as a result of weaker modes being buried within stronger modes and merging
together in the spectrum. In the optical conductivity, all single phonon excitations
76
77 79 81 83 85 87Frequency (cm−1 )
0
5
10
15
20σ1(O
hm−1cm
−1) 5Ka data
fit
270 275 280 285Frequency (cm−1 )
0
100
200
300
400
σ1(O
hm−1cm
−1)
b E ∥ eB 5K50K100K150K200K300K
Figure 5-7. The left panel (a) shows unaccounted for spectral weight on the high frequencyside of the asymmetric oscillator at 85 cm−1 along with a fit to a symmetricoscillator (79 cm−1). We suspect the asymmetric shoulder is the result of aburied mode. The right panel (b) depicts the appearance of a once buriedmode (∼276 cm−1) upon cooling.
should be represented by symmetric lorentzian oscillators. Figure 5-7 left panel depicts
a slightly asymmetric lorentzian oscillator with a resonance of 85 cm−1. Our attempt to
fit the resonance at 85 cm−1 with a symmetric lorentzian oscillator yields unaccounted
for spectral weight on the high frequency side of the resonance. A symmetric lorentzian
oscillator centered at 79 cm−1 and corresponding fit are also included as a reference. (Note
that the additional spectral weight between the resonances at 75 cm−1 and 85 cm−1 is
attributed to absorption between the bands and therefore not to be mistaken for a lattice
vibration). We suspect that this asymmetric shoulder is due to a weak buried mode that is
not fully resolved. To further support our interpretation of buried modes, the right panel
of Fig. 5-7 depicts the appearance of a once buried mode (∼276 cm−1) upon cooling. The
Lorentz oscillator parameters of the 52Bu modes at 20 K are found in Table 5-3.
77
5.4.4 Assignment of Modes
The assignment of a particular phonon, observed in the experimental spectrum, to
its precise atomic displacement pattern cannot be made without sacrificing objective
certainty because of the density of the phonon spectrum. Assignments to groups of
modes, however, can be made through the displacement patterns generated by our lattice
dynamical calculations. Using a simplified model of a phonon as a driven oscillator with
resonant frequency inversely proportional to mass, we proceed to list the constituent
atoms in the structure from heaviest to lightest: Te, Br, Fe, and O. Normally, the low
frequency modes involve the heaviest atoms, but in the FeTe2O5Br structure the Br atoms
have a weak interaction with the other atoms and therefore are responsible for the lowest
frequency phonons. At a certain frequency the motion of Br atoms will cease. Likewise,
at a higher frequency the vibration of atoms involving Te will cease, leaving the motion
of Fe and O to be expected in the highest frequency phonons. Accordingly, the phonon
spectrum can be divided into three clusters that is reflected in Fig. 6-2 (namely, below
180 cm−1, between 180 cm−1 and 530 cm−1, and above 530 cm−1). The clustering is also
observed in the calculated spectrum via the displacement patterns produced, in terms of
first the Br and subsequently the Te ceasing to vibrate. Within each cluster trends are
to be noted on the following principle: atoms are listed corresponding to their net atomic
displacement. This approach, in turn, has allowed us to identify another characteristic of
atomic displacement within the three clusters. The atom of focus in a cluster (e.g., Br in
the first cluster) has a net atomic displacement ever diminishing. Cessation does not arrive
abruptly. The three clusters are profiled in each of Table 5-3, Table 5-1, and Table 5-2.
Apropos the third cluster that focuses on Fe and O, an additional resource is provided
by the familiar orthoferrite family of compounds because the FeO6 octrahedral building
block that is present in FeTe2O5Br is itself a defining unit of the orthoferrite family.
Given the abundant literature on vibrational assignments made to the FeO6 ochrahedral
78
unit,[64, 65] we are consequently able to map these well recognized assignments onto our
experimental findings, thus corroborating both perspectives.
5.5 Summary
Far-infrared measurements on multiferroic single-crystal FeTe2O5Br reveal no drastic
anomalies in the excitation spectrum below the multiferroic ordering temperature.
Nonetheless, a thorough inspection of the phonon spectra has lead to the identification of
all 52 predicted modes in the degenerate ac plane and 43 of the 53 predicted modes along
the unique b axis. The absence of 10 modes along the b axis is presumably due to the
merging of weaker modes near stronger modes in the dense excitation spectrum.
The motions of individual atoms have been assigned to groups of phonons via
the displacement patterns produced through our lattice dynamical calculation. The
assignments divide the overall phonon spectrum into three clusters, which are characterized
by an atom that is common to every vibration in that cluster. We observed that as
frequency increases, vibrations involving the motion of Br atoms will progressively cease.
Likewise, at higher frequencies the vibrations involving Te atoms will cease in the same
manner, leaving the motion of Fe and O to be expected in the highest frequency phonons.
Moreover, the particular atoms involved in a vibration are listed in descending order of net
atomic displacement, which in turn makes clear a trend within each cluster (namely, the
atom of focus in a particular cluster has a net atomic displacement ever diminishing).
It remains to compare our results to a recent infrared study by Pfuner et al.[55] on
the similar FeTe2O5Cl compound where an equal number of phonons were predicted.
Our analysis of the phonon spectrum agrees with the infrared study of Pfuner et al. in
that we also believe modes resonating at equivalent energy scales can overlap, preventing
them from being resolved separately, but we differ on other issues. Pfuner et al. report
a combined 38 modes along the eB and eC directions. We observe more modes along the
unique b axis alone, as well as modes below 50 cm−1, whereas the study by Pfuner et al.
does not report measurements below this frequency.
79
We also observe drastically different and more strongly anisotropic high frequency
optical properties. Pfuner et al.[55] report an absorption edge around 4,000 cm−1 in
FeTe2O5Cl, where as we observe low absorption accompanied by substantial transmission
through the FeTe2O5Br crystal in this region. In speculating about the differences in high
frequency optical properties of the two similar compounds, we point out that spurious
changes of slope of the reflectance throughout the mid-infrared region, such as those
resulting from the contribution of back surface reflection (black line in Fig. 5-5), can
drastically change the Kramers-Kronig-derived optical properties.
Transmission measurements as well as reflectance along eA were not reported in the
study by Pfuner et al..[55] Similar to the FeTe2O5Br system reported here, the FeTe2O5Cl
system also showed no response to magnetic fields above or below the magnetic ordering
temperature.
80
Table 5-2. Oscillator parameters for experimentally observed phonons along eC (at 20 K).The motions of atoms associated with groups of phonons have been assignedwhen the pattern was apparent. The atoms are listed in descending order oftheir calculated net atomic displacement.
Idx Osc Str Ctr Freq FWHM Degen AsgnS ω (cm−1) γ (cm−1) idx in eA
1 4.275 46.7 0.2 1 Br2 1.335 53.9 0.5 Br3 0.951 65.8 1.0 -4 0.216 71.7 0.8 -5 0.435 81.2 0.9 2 -6 0.205 89.3 0.9 3 Br,Te,O,Fe7 0.061 97.0 0.8 Br,Te,O,Fe8 0.130 99.0 1.1 4 Br,Te,O,Fe9 0.533 109.2 0.8 5 Br,Te,O,Fe10 0.294 112.8 1.0 Br,Te,O,Fe11 0.369 128.5 0.8 Br,Te,O,Fe12 0.127 135.4 0.9 6 Br,Te,O,Fe13 0.411 140.6 0.9 -14 0.857 157.0 0.5 7 Te,Br,O,Fe15 0.107 166.8 1.0 8 Te,Br,O,Fe16 0.212 184.5 0.6 9 Te,O,Fe17 0.039 194.8 1.2 10 Te,O,Fe18 0.066 204.0 1.4 -19 0.014 206.3 1.1 -20 0.085 221.7 1.1 -21 0.101 225.4 0.9 -22 0.065 232.6 1.0 12 -23 0.107 248.0 1.5 13 -24 0.291 254.2 0.6 14 -25 0.239 274.4 1.1 -26 0.195 284.9 1.2 -27 0.633 294.8 1.4 15 Fe,O,Te28 0.138 319.0 1.9 17 Fe,O,Te29 0.173 349.0 2.2 19 Fe,O,Te30 0.514 426.7 2.4 24 -31 1.562 437.0 2.3 25 -32 0.088 459.3 3.4 26 O,Fe,Te33 0.006 488.5 4.3 28 O,Fe,Te34 0.067 509.6 9.6 O,Fe,Te35 0.260 573.5 5.6 29 O,Fe36 0.096 626.7 5.2 O,Fe37 0.015 662.0 3.2 30 O,Fe38 0.061 672.0 5.7 O,Fe
81
Table 5-2. Continued.Idx Osc Str Ctr Freq FWHM Degen Asgn
S ω (cm−1) γ (cm−1) idx in eA39 0.182 693.5 8.9 O,Fe40 0.172 707.4 5.2 O,Fe41 0.101 749.8 9.4 -42 0.014 780.4 6.6 32 -43 0.033 814.7 6.9 -
82
Table 5-3. Oscillator parameters for experimentally observed phonons along eB (at 20 K).The motions of atoms associated with groups of phonons have been assignedwhen the pattern was apparent. The atoms are listed in descending order oftheir calculated net atomic displacement.
Idx Osc Str Ctr Freq FWHM AsgnS ω (cm−1) γ (cm−1)
1 1.388 35.7 0.8 Br2 0.330 43.5 1.4 Br3 0.899 53.2 0.8 -4 0.289 60.5 0.9 Br,Te,O,Fe5 0.119 79.1 1.3 Br,Te,O,Fe6 0.075 85.0 1.2 Br,Te,O,Fe7 0.052 91.2 1.0 Br,Te,O,Fe8 0.138 93.3 1.3 -9 0.259 98.0 0.9 -10 1.359 102.0 1.1 -11 0.073 106.9 0.7 -12 0.170 123.7 0.7 Te,Br,O,Fe13 0.092 126.3 0.9 Te,Br,O,Fe14 0.074 131.4 1.1 Te,Br,O,Fe15 1.066 184.0 0.4 Te,O,Fe16 0.128 190.2 0.7 Te,O,Fe17 0.491 198.6 0.8 Te,O,Fe18 0.440 210.0 0.3 -19 0.023 215.3 1.3 -20 0.053 222.0 1.1 -21 0.058 240.1 1.3 -22 0.028 249.0 0.9 -23 0.274 273.9 0.9 Fe,O,Te24 0.102 275.8 1.4 Fe,O,Te25 0.661 295.5 1.3 Fe,O,Te26 0.366 317.6 1.1 Fe,O,Te27 0.101 324.3 1.5 Fe,O,Te28 0.261 341.7 1.2 Fe,O,Te29 0.808 393.7 2.6 Fe,O,Te30 0.371 418.0 4.1 -31 0.009 437.7 3.5 -32 0.209 462.8 3.2 O,Fe,Te33 0.173 478.0 3.3 O,Fe,Te34 0.012 499.0 4.6 -35 0.009 508.3 1.2 -36 0.047 595.3 6.9 O,Fe37 0.080 626.6 3.6 O,Fe38 0.323 652.0 4.9 O,Fe
83
Table 5-3. Continued.Idx Osc Str Ctr Freq FWHM Degen Asgn
S ω (cm−1) γ (cm−1) idx in eB39 0.018 682.1 8.2 O,Fe40 0.217 728.1 7.8 O,Fe41 0.027 746.2 7.4 -42 0.031 779.9 23.5 -43 0.071 836.0 9.6 -
84
CHAPTER 6PHONON ANOMALY AND MAGNETIC EXCITATIONS IN Cu3Bi(SeO3)2O2Cl
6.1 Overview
6.1.1 Crystal Structure
Cu3Bi(SeO3)2O2Cl is a geometrically-frustrated layered material possessing
magnetic order. P. Millet et al.[66] determined the room temperature crystal structure
of Cu3Bi(SeO3)2O2Cl using single crystal x-ray diffraction. The Cu3Bi(SeO3)2O2Cl
compound crystallizes in a layered structure where the individual layers stack along
the c axis of the orthorhombic cell (Pmmn). The magnetic Cu2+ ions form a hexagonal
arrangement within each plane that is reminiscent of a Kagome lattice, and thus implies
magnetic frustration. Within any one hexagon there exist two unique copper sites, Cu1
and Cu2, which possess different out-of-plane oxygen bonding. Both copper sites form
distinct [CuO4] units that are linked by Se4+ and Bi3+ ions. The Cl− atoms, which
sandwich the planes formed by the copper hexagons, position themselves along the
axis that defines the parallel stacking of the copper hexagons. An image of the crystal
structure emphasizing the hexagonal arrangement of copper ions is shown in Figure 6-1.
Additional pictures and more detailed descriptions of the crystal structure are found
elsewhere.[66, 67]
P. Millet et al.[66] also reported magnetic susceptibility measurements on a powder
sample of Cu3Bi(SeO3)2O2Cl. Near 150 K they observed a change in slope of 1/χ(T )
that resulted in two distinct and positive Weiss temperatures. To reconcile this anomaly
in 1/χ(T ), they performed linear birefringence on a single crystal and deduced that a
second-order structural transition was likely, but they were not able to determine the exact
Reprinted with permission from K. H. Miller, P. W. Stephens, C. Martin, E.Constable, R. A. Lewis, H. Berger, G. L. Carr, and D. B. Tanner, Phys. Rev. B 86,174104 (2012).
85
Figure 6-1. One unit cell is boxed in the lower left. The hexagonal arrangement of copperions (blue spheres) is emphasized.
nature of the transition. P. Millet et al. also reported ferromagnetic-like behavior below Tc
≈ 24 K.
6.1.2 Beauty of Infrared Spectroscopy
The technique of infrared spectroscopy lends itself well to the investigation of
geometrically frustrated magnetic materials. For example, the spin-driven Jahn-Teller
effect in the Cr spinel compounds CdCr2O4 and ZnCr2O4 acts to lift the degenerate
ground state arising from competing magnetic interactions via a coupling to the
lattice degrees of freedom; the ensuing lattice distortion is unambiguously observed in
the infrared as a splitting of infrared-active phonon modes.[68, 69] Furthermore, the
a.c. electric (magnetic) field of the infrared light can interact with ordered moments and
excite an electromagnon (magnon) excitation, which in turn can provide information about
the symmetry and nature of the magnetic order.[30, 70]
6.1.3 Major Findings
Here we present our infrared studies on single crystal Cu3Bi(SeO3)2O2Cl. We observe
16 new modes in the phonon spectra originating below 115 K. Strikingly, our subsequent
86
powder x-ray diffraction measurements reveal the same 300 K structure existing at 85 K.
Preliminary Raman measurements suggest that a loss of inversion symmetry is likely. In
addition, we observe new excitations in the infrared arising in the magnetically ordered
state (below 24 K) that show isotropic infrared polarization dependence but anisotropic
external magnetic field dependence. In light of the novel excitations observed in the
magnetically ordered state, we performed d.c. susceptibility measurements to examine
the anisotropic magnetic properties of Cu3Bi(SeO3)2O2Cl. The results of our magnetic
susceptibility measurements are in agreement with a recent report[67] of the magnetic
properties in the similar Cu3Bi(SeO3)2O2Br compound (Tc = 27.4 K). (This manuscript is
accompanied by supplementary material including additional figures referred to in the text
and the results of full Rietveld refinements.)
6.2 Experimental Procedures
Single crystals of Cu3Bi(SeO3)2O2Cl were grown by standard chemical vapor-phase
method. Mixtures of analytical grade purity CuO, SeO2 and BiOCl powder in molar
ratio 3:2:1 were sealed in quartz tubes with electronic grade HCl as the transport gas for
the crystal growth. The ampoules were then placed horizontally into a tubular two-zone
furnace and heated at 50C/h to 450C. The optimum temperatures at the source and
deposition zones for the growth of single crystals were found to be 480C and 400C,
respectively. After four weeks, many tabular green Cu3Bi(SeO3)2O2Cl crystals with a
maximum size of 15 × 12 × 1 mm3 were obtained, which were indentified as synthetic
francisite on the basis of x-ray powder diffraction data.
Zero-field temperature-dependent (7–300 K) reflectance and transmittance measurements
were collected on a 6.7 × 4.8 × 0.7 mm3 single crystal (crystal 1) using a Bruker 113v
Fourier transform interferometer in conjunction with a 4.2 K silicon bolometer detector
Crystal growth was carried out by Helmuth Berger at the EPFL in Lausanne,Switzerland
87
in the spectral range 25–700 cm−1 and a nitrogen cooled MCT detector from 700–5,000
cm−1. Room temperature measurements from 5,000–33,000 cm−1 were obtained with a
Zeiss microscope photometer. Magnetic field-dependent transmission measurements in
the spectral range 15–100 cm−1 were performed on a 7.5 × 3.8 × 0.2 mm3 single crystal
(crystal 2) at beam line U4IR of the National Synchrotron Light Source, Brookhaven
National Laboratory, utilizing a Bruker IFS 66-v/S spectrometer. The crystal was placed
in a 10 T Oxford superconducting magnet and the transmitted intensities were measured
using a 1.8 K silicon bolometer detector. Additional magnetic field-dependent transmission
measurements were performed on a 5.1 × 3.5 × 0.3 mm3 single crystal (crystal 3) in the
spectral range 10–45 cm−1 using a modified Polytec FIR 25 spectrometer interfaced to
a 7 T Oxford split-coil superconducting magnet at the University of Wollongong. The
transmitted intensities were subsequently converted to absorption coefficients using a
modified Beer-Lambert law that accounted for reflection losses at the surfaces. Linearly
polarized light was used in all infrared measurements and oriented along the three
principal dielectric axes of the system. For an orthorhombic system, the three principal
dielectric axes coincide with the three crystallographic axes.[61] In the following report,
the symbol “E” always refers to the polarization of the incoming light, while “H” is the
orientation of the external field (when applicable).
Powder x-ray diffraction measurements were performed at beamline X16C of the
National Synchrotron Light Source, Brookhaven National Laboratory. A small crystal was
crushed in a mortar and pestle, mixed with a small amount of Si powder (NIST Standard
Reference Material 640c) and roughly three times its volume of ground cork, and loaded
in a thin-walled glass capillary of 1 mm nominal diameter. The cork served to dilute
the sample so that it could fill the cross-section of the x-ray beam without drastically
absorbing it; measured transmission at the center of the capillary was 12%. X-rays of
nominal wavelength 0.6057 A were selected by a channel-cut Si(111) monochromator
before the sample; small drifts in the x-ray wavelength were corrected via the Si internal
88
standard. The diffracted beam was analyzed by a Ge(111) crystal and detected by a
commercial NaI(Tl) detector. Data were typically collected in steps of diffraction angle 2θ
of 0.005. For lattice parameter measurements as a function of temperature, the sample
was inside a Be heat shield in a closed cycle He refrigerator, which was rocked several
degrees at each 2θ step. Data for crystallographic refinements were collected with the
sample continuously spinning several revolutions for each point, in an Oxford Cryostream
sample cooler. Powder x-ray diffraction data were analyzed using Topas-Academic
software.[71]
Magnetic measurements were performed in a commercial superconducting quantum
interference device magnetometer on crystal 2.
Magnetic field-dependent capacitance measurements were performed at SCM2 at
the National High Magnetic Field Laboratory, Tallahassee, Florida. A single crystal of
dimension 4.2 × 4.0 × 0.2 mm3 (crystal 4) was utilized for this measurement. Aluminum
electrodes were deposited on both ab faces of cyrstal 4. The electrodes possessed an
approximate area of ∼4.8 mm2. The crystal was inserted in an 18 T magnet equipped
with a 3He insert. Measurements were recorded with a commercially available AH 2700A
Ultra-precision Capacitance Bridge.
6.3 Results and Analysis
6.3.1 Zero Field Reflectance and Transmittance Spectra
The temperature-dependent reflectance spectra of Cu3Bi(SeO3)2O2Cl along the a,
b, and c axes in the frequency interval 30–1,000 cm−1 (4–120 meV) is shown in Fig. 6-2.
Remarkably, many new phonon modes are observed in the reflectance spectra along all
three crystal directions upon cooling from 120 K to 110 K. The arrows in Fig. 6-2 indicate
the position where new modes occur. Meticulous temperature sweeps (every 1 K) between
120 K and 110 K indicate that all the new modes arise at 115 K; this trend was verified
upon cooling and warming.
89
In addition, a strong sharpening of many modes is observed with decreasing
temperature. In regions where new modes appear, the typical softening of resonance
frequencies with increasing temperature is not systematically observed. The typical
softening of resonance frequencies with increasing temperature, which is correlated to
the expansion of the lattice, can be counteracted by repulsion due to phonon mixing. An
explanation of the repulsion of phonons, which are bosonic excitations, will be presented
in the discussion section. In addition to the new phonons, we observe the softening (at
5 K to ∼30% of the 300 K value) of the lowest frequency mode along the b direction,
an occurrence which is not associated with phonon repulsion. The soft mode behavior
exhibited by this low frequency phonon is reminiscent of that occurring in a displacive
ferroelectric material.
Cu3Bi(SeO3)2O2Cl is opaque between ∼40 and 800 cm−1 due to the strong optical
phonon absorptions in this region; however, above the optical absorptions light is
observed to transmit. Figure 6-3 displays the zero field 300 K transmission spectra
of Cu3Bi(SeO3)2O2Cl along the a and b axes in the mid-infrared and near-infrared
regions. (Due to the geometry of the sample no light was observed to transmit along
the c axis.) The anisotropy, which is prominent in the infrared reflectance spectra, is
also observed in transmission at high frequencies. Sharp but weak absorption features
exist at 1950 cm−1 along the a axis and at 2000 and 2050 cm−1 along the b axis. The
aforementioned absorptions exhibit a minimal strengthening with decreasing temperature.
The downturn in transmission at 9,000 cm−1 (∼1.1 eV) indicates the onset of electronic
absorptions. The energy of the observed gap is consistent with the insulating nature of the
material. Transmission below the optical absorptions (below 40 cm−1) is the subject of
section III E.
6.3.2 Kramers-Kronig and Oscillator-model fits
Along the a and b axes we have utilized a combined reflection and transmission
analysis to extract the single bounce reflectance necessary for the Kramers-Kronig
90
Frequency (cm−1 )0.0
0.4
0.81.0
E ∥ a20 40 60 80 100 120
Energy (meV)
7K50K100K150K200K300K
Frequency (cm−1 )0.0
0.4
0.8
Reflectance E ∥ b
0 200 400 600 800 1000Frequency (cm−1 )
0.0
0.4
0.8 E ∥ c
Figure 6-2. The temperature-dependent reflectance spectra of Cu3Bi(SeO3)2O2Cl along thea, b, and c axes. Crimson arrows indicated the positions of new phononsarising at 115 K.
1000 10000Frequency (cm−1 )
0.0
0.1
0.2
0.3
0.4
0.5
Transm
ission
1.00.10.6
Energy (eV)
E ∥ aE ∥ b
Figure 6-3. Transmission along the a and b axes of crystal 1 in the mid-infrared andnear-infrared regions. The inset is a sketch of crystal 1 with the a, b, and caxes indicated.
transformation. More specifically, in the frequency region where the crystal transmits
along the a and b axes, the measured reflectance has an additional contribution from the
back surface. Using the method employed by Zibold et al.,[8] i.e., assuming that k is small
in the region of interest, we are able to extract the single bounce reflectance of the sample.
Since the c axis does not exhibit transmission, the measured reflectance was assumed to be
single bounce reflectance along this direction.
91
The real and imaginary parts of the dielectric function were estimated from the
single bounce reflectance, Rs(ω), using the Kramers-Kronig transformation.[43] Before
calculating the Kramers-Kronig integral, the low frequency (0.1–30 cm−1) data were
approximated using the dielectric function determined from the fitting procedure described
below. At high frequencies the reflectance was assumed to be constant up to 1× 107 cm−1,
after which R ∼ (ω)−4 was assumed as the appropriate behavior for free carriers. The
optical properties were obtained from the measured reflectance and the Kramers-Kronig
derived phase shift on reflection.
The single bounce reflectance was fit with a Lorentz oscillator model to obtain a
second estimate of the complex dielectric function in the infrared range. The model
assigns a Lorentzian oscillator to each phonon mode in the spectrum plus a high frequency
permittivity, ε∞, to address the contribution of electronic absorptions. The model has the
following mathematical form:
ε(ω) =∞∑
j=1
Sjω2j
ωj2 − ω2 − iωγj
+ ε∞ , (6–1)
where Sj, ωj, and γj signify the oscillator strength, center frequency, and the full width
at half max (FWHM) of the jth Lorentzian oscillator. The reflectivity is then calculated
using the Drude-Lorentz complex dielectric function. A comparison of the calculated and
measured reflectivity is shown in Figure 6-4.
6.3.3 Optical Properties
The optical properties extracted from both Kramers-Kronig analysis and our fits
of reflectance can be used to investigate further the appearance of new phonon modes
in the infrared spectra at 115 K. The upper panel of Fig. 6-5 depicts the real part of
the optical conductivity, σ1(ω), along the b axis in the frequency range 250–285 cm−1.
The lower panel of Fig. 6-5 depicts the loss function, Im(-1/ε(ω)), in the same region.
Both optical properties shown are from Kramers-Kronig analysis of the single bounce
reflectance. The peaks observed in σ1(ω) and Im(-1/ε(ω)) closely correspond respectively
92
0 200 400 600 800 1000Frequency (cm−1 )
0.0
0.2
0.4
0.6
0.8
1.0
Reflectance
E ∥ a100K
datafit
20 40 60 80 100 120Energy (meV)
Figure 6-4. The calculated reflectance from the Drude-Lorentz model (red line)superimposed on the measured reflectance (blue points) of Cu3Bi(SeO3)2O2Clalong the b direction at 100 K. Similar qualities of fits were obtained at allother measured polarizations and temperatures..
to the TO and LO phonon frequencies. The inset of the lower panel of Fig. 6-5 depicts
the plasma frequency (Ωs =√
Sjω20) associated with both phonons as a function of
temperature as obtained by our Drude-Lorentz fitting. The equation defining the plasma
frequency of ionic vibrations is similar in form to that which describes the free carrier
response in metallic systems after the electronic mass and charge are replaced with the
reduced mass of the normal mode and ionic charge. The plasma frequency is of particular
interest here because its square is proportional to the spectral weight associated with a
particular phonon. The new phonon mode that appears at ∼ 276 cm−1 below 115 K gains
spectral weight at the expense of the existing mode at ∼ 256 cm−1. The smooth shift of
spectral weight with decreasing temperature from the existing mode to the new mode is
reminiscent of a second order transition.
6.3.4 Powder X-Ray Diffraction
The changes in the infrared spectra, namely the appearance of new vibrational modes
below 115 K, suggest a reduction in lattice symmetry. To investigate the low temperature
93
250 255 260 265 270 275 280 285Frequency (cm−1 )
0.0
0.2
0.4
0.6
0.8
Loss Function Im(1/ε)
LO phonons
0
40
80
120
σ1(Ohm−1cm
−1)
E ∥ bTO phonons
31 32 33 34 35Energy (meV)
7K15K30K50K70K85K100K110K120K130K140K150K200K250K300K
0 100 200 300Temperature (K)
60
80
100
120
140
Tc=115 K
256 (cm−1 )
276 (cm−1 )
(cm−1)
Ωs
0 100 200 300Temperature (K)
254
260
266
272
278
Tc=115 K(cm−1)
ω0
Figure 6-5. The temperature dependent optical conductivity σ1(ω) (upper panel) and lossfunction Im(-1/ε(ω)) (lower panel) along the b axis in the frequency range250–285 cm−1. The insets of the upper and lower panels depict respectivelythe temperature dependence of the resonance and plasma frequencies of thetwo modes.
structure of Cu3Bi(SeO3)2O2Cl, we performed powder x-ray diffraction measurements
between 30 and 300 K. The lattice parameters as a function of temperature extracted from
Rietveld fits of the diffraction spectra are shown in Fig. 6-6. There exists some structure
in the curves of lattice parameters vs. temperature, but there is nothing that could be
regarded as conclusive evidence of a phase transition. The negative thermal expansion
observed for the a lattice parameter below 100 K is not unusual for an ionic compound
of orthorhombic symmetry where it may be energetically favorable for the unit cell to
contract in one direction while expanding in other directions. The Rietveld refinements
of the powder x-ray diffraction pattern at 295 and 85 K are shown in Figure. 6-7. Both
refinements were consistent with the published structure of P. Millet et al. in space
group Pmmn. Full details of the refinements, including bonding geometry, may also be
found in the supplementary information. We saw no evidence for splitting or broadening
of diffraction lines nor the appearance of new diffraction lines at low temperature. We
94
0 50 100 150 200 250 300Temperature (K)
0.999
1.000
1.001
1.002
1.003
1.004
Lattice parameter relative to 30 K
Tc=115 K
a
b
c
Volume
Figure 6-6. Lattice parameters and unit cell volume vs. temperature. Error bars reflectthe reproducibility of independently measured data sets at selectedtemperatures. Solid lines are drawn as guides to the eye. Lattice parameters at30 K are a = 6.3463(2) A, b = 9.6277(3) A, c = 7.2186(3) A, Volume =441.10(2) A3.
conclude that there is no direct evidence for a lowering of crystallographic symmetry
near 115 K. This is a serious puzzle because the appearance of 16 new infrared modes
below 115 K implies that the symmetry of the crystal is lower than Pmmn below that
temperature.
6.3.5 Magnetic Field-Dependent Transmission
Far-infrared transmission as a function of temperature and external magnetic field
was measured on crystal 2 and crystal 3 with light polarized along the a axis, b axis,
and at 45 to both the a and b axes. Upon cooling to 5 K an excitation was observed
at 33.1 cm−1 that only existed in the magnetically ordered state (below 24 K). The
excitation was observed in all four polarizations. Figure. 6-8 depicts the excitation, at 5 K,
in each of the four polarizations measured, as well as the evolution of the excitation as the
external field is ramped to 10 T (H ‖ c geometry). When the external magnetic field was
applied parallel to the c axis (H ‖ c) and the field was ramped to 10 T, the excitation at
95
5 10 15 20 25 30 35 402θ [degrees]
-8
0
8
Diff/ESD0
20
40
60
80
√ Intensity [Arb. units]
85K
-8
0
8
Diff/ESD0
20
40
60
80
√ Intensity [Arb. units]
295K
Figure 6-7. Rietveld fits of the powder diffraction patterns of Cu3Bi(SeO3)2O2Cl at 295 K(upper panel) and 85 K (lower panel). Red dots represent data, blue line theRietveld fit. The difference curve is normalized to the statistical uncertainty ofeach data point. The fits have χ2 values of 2.21 (295 K) and 1.70 (85 K).
33.1 cm−1 increased linearly with increasing field (inset Fig. 6-9a). No hysteresis effects
were observed upon ramping the magnetic field. To establish confidence that the observed
excitation only existed in the magnetically ordered state, the crystal was warmed up to
40 K and the external field was ramped from 0 to 10 T. The excitation was not observed.
The results are shown in Fig. 6-9a for the E‖ b+ 45 polarization.
In addition, for fields 1 T and greater applied parallel to the c axis, a second magnetic
excitation was observed which also possessed isotropic polarization dependence in the
ab plane. The excitation increased linearly from 10.5 cm−1 at 1 T to 19.3 cm−1 at
10 T. Because of the low signal to noise level it is not known whether the excitation
disappeared below 1 T or it was just unresolved. The excitation is pictured for the
E ‖ a polarization in Fig. 6-10. The spectra in Fig. 6-10 is a conjunction of data
96
30 32 34 36 38Frequency (cm−1 )
0
120
160
200
α (cm
−1)
E ∥ b+45
H ∥ c
5K
10T8T6T4T2T0T
30 32 34 36 38Frequency (cm−1 )
0
120
160
200
α (cm
−1) E ∥ b
H ∥ c
5K
10T8T6T4T2T0T
30 32 34 36 38Frequency (cm−1 )
0
40
80
120
160
α (cm
−1) E ∥ a
H ∥ c
5K
10T8T6T4T2T0T
30 32 34 36 38Frequency (cm−1 )
0
40
80
120
160
α (cm
−1) E ∥ a+45
H ∥ c
5K
10T8T6T4T2T0T
Figure 6-8. Field-dependent absorption obtained from the transmitted intensities of fourpolarizations in the ab plane. The external magnetic field was oriented parallelto the c axis for all spectra shown.
from two different interferometers on two separate crystals of Cu3Bi(SeO3)2O2Cl (see
Experimental Procedures section). The complimentary techniques on separate crystals
establish confidence in the existence of the excitation.
When the external magnetic field was applied perpendicular to the c axis (H ⊥ c)
and ramped to 10 T, the excitation at 33.1 cm−1 decreased its resonance frequency
quadratically with field (inset Fig. 6-9b). The field dependence of the excitation for the
E‖ a + 45 polarization is depicted in Fig. 6-9b. It should be noted that the details of
our experimental setup required that the external magnetic field be applied parallel to the
polarization of the incoming light in the H ⊥ c geometry. No additional modes in this
orientation of external field were detected.
Similar experiments in magnetic fields were carried out in reflectance geometry;
however, the excitations were much too weak to give rise to reflectance bands.
97
30 32 34 36 38Frequency (cm−1 )
0
150
200
250
300
α (cm
−1)
5K
40K
E ∥ b+45
H ∥ c
a.
10T
8T
6T
4T
2T
0T
0T
4T
6T
10T
0 2 4 6 8 10Field (T)
10
14
18
ω0(cm
−1)
HMM
33
34
35
22 26 30 34 38 42Frequency (cm−1 )
0
50
100
150
200
250
300
α (cm
−1)
E ∥ a+45
H⟂ c
b.
5K
0T2T4T6T8T10T
0 2 4 6 8 10Field (T)
25
30
35
ω0(cm
−1)
Figure 6-9. Experimental support (a) for claiming that the magnetic resonance observedat 33.1 cm−1 exists below the magnetic ordering temperature (Tc ∼24 K).External fields, which act to sharpen the resonance, were also applied above Tc
for further verification that the excitation was not observed. In the inset, thedashed red line denoted HMM indicates the region where the metamagnetictransition occurs in the H‖ c geometry. The isotropic magnetic excitation’sfield dependence in the H ⊥ c geometry (b) with light polarized along theE‖ a+ 45 direction.
6.3.6 Magnetic Properties
The anisotropic response to external magnetic fields of the magnetic excitation
observed at 33.1 cm−1 in transmission has inspired an investigation of the anisotropic
magnetic properties of Cu3Bi(SeO3)2O2Cl through d.c. susceptibility measurements.
The results of our d.c. susceptibility measurements are depicted in Fig. 6-11. Isothermal
magnetization measurements taken at 5 K exhibit the strong anisotropic response to the
direction of an external field previously noted in transmission measurements. As shown
in Fig. 6-11a, with H ⊥ c, the resulting magnetization vs. field loop resembles that of an
antiferromagnet. On the contrary, with H ‖ c (Fig. 6-11b), antiferromagnetic behavior is
observed from 0 to 0.1 T, followed by a metamagnetic transition from 0.1 to 0.8 T, and
then ferro or ferrimagnetic behavior persists from 0.8 to 5 T. Figure 6-11c is an enlarged
98
10 11 12 13 140
20
40
60
80
100
α (cm
−1)
5T4.5T
4T3.5T
3T2.5T
2T1.5T
1T
Frequency (cm−1 )16 18 20 22
10T9T8T
7T6T
E ∥aH ∥c
5K
Figure 6-10. The lower frequency magnetic resonance mode observed in the H‖ c geometryfor fields of 1 T and greater. The two panels result from complementarytechniques on two separate crystals of Cu3Bi(SeO3)2O2Cl. Experimentallimitations prevented a study of the mode below 1 T fields.
view of the metamagnetic transition in the H ‖ c geometry. The hysteresis observed upon
sweeping the field is likely associated with the metamagnetic transition. To verify further
the antiferromagnetic to ferro or ferrimagnetic transition occurring in the H ‖ c geometry,
magnetization was measured as a function of temperature in fields of 0.01 and 1 T. The
results are shown in Fig. 6-11d. The low temperature cancellation of oppositely aligned
moments expected for an antiferromagnet is observed at 0.01 T, whereas a strong ferro
or ferrimagnetic magnetization is observed at low temperature for the 1 T magnetization
data.
Our results and interpretation are in qualitative agreement with a recent report by
M. Pregelj et al.[67] on the similar Cu3Bi(SeO3)2O2Br compound (Tc = 27.4 K). Small
differences in the magnetization saturization value with H ‖ c is likely linked to defects
arising from different growth conditions. We normalize our susceptibility curves to moles
of Cu while M. Pregelj et al. normalize to moles of formula unit (scaling factor of three
difference).
The origin of the anomaly in 1/χ(T ), first observed in powder samples by P. Millet
et al.,[66] and subsequently observed by us along multiple axes of single crystals of
Cu3Bi(SeO3)2O2Cl (Fig. 6-12) and Cu3Bi(SeO3)2O2Br,[67] remains an unresolved issue.
99
0 50 100 150Temperature (K)
0.00
0.10
0.20
0.30
0.40
M/H (cm
3/m
ol of Cu)
1 T0.01 T
FCH ∥c
d.
-6 -4 -2 0 2 4 6Field (T)
-0.8
-0.4
0
0.4
0.8
µB/Cu
5KH ∥a
a.
-6 -4 -2 0 2 4 6Field (T)
-0.8
-0.4
0
0.4
0.8
µB/Cu
5KH ∥c
b.
-0.5 0 0.5Field (T)
-0.1
0.0
0.1
µB/Cu
c.
5KH ∥c
Figure 6-11. Isothermal magnetization measurements at 5 K, which is well within themagnetically ordered state, for H ⊥ c (a) and H ‖ c (b,c). The magnetizationas a function of temperature (d) for H ‖ c at fields above and below themetamagnetic transition as measured while warming after field cooling to5 K.
Below 150 K, the plot of H/M exhibits a slight curvature. We therefore conclude that the
data is no longer accurately described by the Curie-Weiss law below 150 K (a dashed line
fit below 150 K remains in Fig. 6-12 to exemplify the discrepancy). This behavior could be
indicative of more complex spin-correlation functions between 150 and 24 K due to the low
dimensional and frustrated nature of the spin system.
We observed the anomaly for external fields of 1 T and 0.01 T in both zero field
cooling as well as field cooling measurements.
6.3.7 Magnetic Field-Dependent Capacitance
Below 24 K and with an external magnetic field oriented parallel to the c axis, the
capacitance of Cu3Bi(SeO3)2O2Cl exhibits an abrupt transition at fields coinciding with
that of the metamagnetic transition along this orientation (0.1–0.8 T). Upon sweeping the
magnetic field, the transition displays significant hysteresis (See Fig. 6-13). Proof that the
capacitance transition is associated with the metamagnetic transition is given by both its
100
0 50 100 150 200 250Temperature (K)
0
50
100
150
200
250
300
H/M (mol of Cu/cm
3)
T=115 K
H ∥a1 T (ZFC)
0 50 100 150 200 250 300Temperature (K)
T=115 K
H ∥c1 T (FC)
Figure 6-12. Plots of H/M as a function of temperature measured with a 1 T magneticfield oriented parallel to the a axis (left panel), and also parallel to the c axis(right panel).
−2.0 −1.5 −1.0 −0.5 0.0 0.5 1.0 1.5 2.0Field (T)
4.6065
4.6070
4.6075
4.6080
4.6085
Capacitance (pF)
4.6090
300 mKH ∥c
Figure 6-13. Field-dependent capacitance of Cu3Bi(SeO3)2O2Cl at 300 mK with H ‖ c.
temperature dependence (Fig. 6-14) and its disappearance upon rotating the direction of
the externally applied magnetic field (not shown).
The exact nature of the capacitance transition at the time of compiling this
dissertation is unknown. Briefly, metamagnetic transitions are often first-order phase
transformations resulting in discontinuities of a material′s the unit cell volume.[72]. The
compounds Mn1.8Co0.2Sb,[73] Ca1.8Sr0.2RuO4,[74] and UNiGa[75] all show an abrupt
101
0 0.5 1 1.5 2 2.5Field (T)
0.9995
0.9996
0.9997
0.9998
0.9999
1.0000
1+C(B
)−C(0
)
C(0
)
H ∥c 0.370 K (up)3 K (up)6 K (up)9 K (down)15 K (up)20 K (down)25 K (up)30 K (down)
Figure 6-14. Normalized temperature-dependent capacitance of Cu3Bi(SeO3)2O2Cl withH ‖ c. Up and down refer to the direction the field was swept.
change in volume at fields near their respective metamagnetic transitions. Further
analysis and experimental investigations (e.g., field-dependent dilatometry) are needed to
determine whether the capacitance transition is magnetostrictive in nature, or rather a
direct signature of magnetoelectric coupling in this compound.
6.4 Discussion
6.4.1 Group Theory and Observed Modes
The number of optical modes in the Cu3Bi(SeO3)2O2Cl compound can be determined
by group theory analysis. Using the SMODES[49] program, we arrive at the following
distribution of modes:
Γoptical = 14B(IR)1u + 14B
(IR)2u + 11B
(IR)3u
+12A(R)g + 6B
(R)1g + 9B
(R)2g + 12B
(R)3g + 9A1u
where (R) and (IR) denote respectively Raman active and infrared active modes. The B3u,
B2u, and B1u modes are infrared active along the a, b, and c crystal axes respectively. The
9A1u modes are silent. The 12Ag modes are Raman active radial breathing modes and
102
require parallel incoming and outgoing polarizations according to the selection rules of
the Pmmn spacegroup. The B3g, B2g, and B1g modes are also Raman active, but require
crossed polarization of the light (bc, ac, and ab respectively).
At 300 K we observe all 11 of the 11 predicted B3u modes along the a axis, all 14 of
the 14 predicted B2u modes along the b axis, and 11 of the 14 predicted B1u modes along
the c axis. The discrepancy between observed and detected modes along the c axis is likely
an experimental shortcoming arising from the low signal to noise when measuring along
the thin dimension of the crystal. At 115 K we detect 8 additional modes along the a axis,
6 additional modes along the b axis, and 2 additional modes along the c axis. To study
the observed modes, the reader is referred to Table 6-1 where the modes at 7 K along all
three crystal axes are identified as well as their respective Lorentz oscillator parameters.
An asterisk after the TO frequency indicates a new mode that arises at 115 K.
In materials with an inversion center, the k = 0 optical modes as measured through
Raman and infrared reflectance are mutually exclusive. When the inversion center in a
structure is removed, the local centers of symmetry about which all the Raman modes
have zero net dipole moment will be removed, and all Raman modes will therefore
become infrared active.[76] To further investigate the 16 new modes observed at 115 K,
preliminary 300 K Raman measurements were recorded in the ab plane of crystal 1.
Raman modes observed at 172.9, 323.5, and 484.4 cm−1 closely correspond to three of
the new infrared modes observed below 115 K. If we take into account the fact that
Raman modes can slightly increase their resonance frequencies upon cooling to low
temperatures, then 300 K Raman modes at 538.3 and 583.1 cm−1 draw close redolence
to two additional new modes observed in the infrared below 115 K. We therefore turn to
a closer examination of a possible centrosymmetric to non-centrosymmetric transition in
Cu3Bi(SeO3)2O2Cl near 115 K.
103
6.4.2 Powder X-Ray Diffraction
An outstanding problem in the interpretation of our results is the fact that the
powder x-ray diffraction measurements did not find any evidence for a phase transition
accompanying the dramatic appearance of 16 infrared-active modes below 115 K. In their
2001 report, P. Millet et al.[66] sought to detect a potential structural phase transition
by optical birefringence, and they placed an upper limit of 1 on the possible rotation
of optical axis, thus indicating no distortion to monoclinic symmetry at that level. The
present powder diffraction measurements would be sensitive to a monoclinic distortion on
the order of 0.001, and none is observed. Their analysis of linear birefringence suggest
that the nature of the suspected transition is second-order, which is consistent with
what we have found in our analysis of shifts in spectral weight from existing modes to
new modes (inset lower panel Fig.6-5). In what follows, we consider lower-symmetry
non-centrosymmetric orthorhombic structures that might be a potential host for
Cu3Bi(SeO3)2O2Cl. In addition, we scrutinize the possibility of a transition to an
incommensurate lattice, and we discuss the implications of the recent neutron diffraction
report on Cu3Bi(SeO3)2O2Br.[67]
The centrosymmetric Pmmn space group has extinction class P - -n, i.e., it obeys
the condition that h + k must be even for (hk0) reflections. A continuous transition to
a non-centrosymmetric orthorhombic subgroup of the same lattice dimensions could lead
to space groups P21212, Pmm2, Pm21n, or P21mn. The two former choices would allow
additional powder x-ray diffraction peaks. We have carefully searched for them throughout
the temperature range below 115 K without success. The two latter space groups have
the same extinction class as Pmmn, and so they represent a mechanism for breaking of
inversion symmetry without producing a qualitative change in the powder diffraction
pattern. Such a distortion might not be easy to recognize from powder diffraction data
because the atoms would presumably move a small distance from their undistorted
locations, and so the Pmmn model would still give a reasonably accurate description of
104
the data in the acentric phase. One possibility is to look for unusual behavior in thermal
displacement parameters, similar to the method used to decode complicated distortions in
magnetically frustrated spinels.[77] We have not been able to detect such an effect from
the data at hand.
Neutron powder diffraction measurements on Cu3Bi(SeO3)2O2Br at 60 K were
refined in Pmmn,[66, 67] but that has only limited bearing on the present issue. First,
it is not known that data from the bromide material suggests the same loss of inversion
symmetry that we have reported here for the chloride material. Second, as noted above,
the difference between centric and acentric structures could be very difficult to see in
neutron or x-ray powder diffraction. Indeed, it is worth pointing out that the magnetic
transition studied in the bromide analog may well have occurred from a symmetry lower
than Pmmn.[66, 67]
Another possibility is that an incommensurate transition occurs in Cu3Bi(SeO3)2O2Cl
below 115 K. Incommensurate phases acquire satellite x-ray diffraction lines around
the Bragg peaks of the symmetric phase.[78] We did not observe satellite peaks, but
it is possible they exist and were too weak to be detected in the present powder x-ray
diffraction measurements. We plan to perform more sensitive measurements (e.g., single
crystal or neutron powder diffraction) to resolve this issue.
6.4.3 Phonon Repulsion
The typical dispersion of phonon resonance frequencies (i.e., softening with increasing
temperature) as dictated by the anharmonic term in the lattice potential is not observed
for a number of the new modes and existing modes in close proximity to the new modes
appearing below 115 K. According to our presumption that Raman modes become infrared
active with a loss of inversion symmetry, we examined the physics behind phonons of
similar strengths resonating at contiguous frequencies. Figure 6-15 depicts a new mode
appearing around ∼328 cm−1 along the a axis that is in close proximity to an existing
mode at ∼323 cm−1. As further emphasized in the inset of Fig. 6-15, the two modes
105
300 305 310 315 320 325 330 335 340Frequency (cm−1 )
0
20
40
60
80
100
σ1(Ohm−1cm
−1) E ∥ a
38 39 40 41 42Energy (meV)
7K15K30K50K70K85K100K110K120K130K140K150K200K250K300K
0 100 200 300Temperature (K)
318
322
326
330
334
ω0(cm−1) Tc=115 K
Figure 6-15. The real part of the optical conductivity along the a axis in the frequencyrange 300–340 cm−1. At 115 K a new mode appears around ∼328 cm−1 thatis in close proximity to an existing mode at ∼323 cm−1. The repulsion of thetwo modes as temperature is decreased is emphasized in the inset where theDrude-Lorentz resonance frequencies are plotted as a function of temperature.
strongly repel one another as temperature decreases, which seems uncharacteristic of
bosonic excitations. The repulsion is due to phonon mixing and can be understood by an
analogy to the classical system of two coupled ideal harmonic oscillators. The classical
problem amounts to solving for the eigenvalues of a matrix with off diagonal terms arising
from the coupling between the oscillators. If both oscillators are given the same initial
frequency (analogous to both phonons resonating at the same energy) and the coupling
is turned on, then the oscillators will repel one another. The repulsion increases with
increasing coupling. Applying the same concept to the case of phonon mixing, one can
see that the coupling between the phonons is increased with decreasing temperature,
an effect which is well understood via the thermal contraction of the lattice. It is also
worthwhile noting that the new phonon mode depicted in Fig. 6-5 at ∼276 cm−1 causes
a repulsion of the resonant frequency of the existing mode at ∼256 cm−1 below 115 K.
The two modes appear to be initially non-degenerate because of their separation in energy
(∼20 cm−1); however, as seen in the inset of Fig. 6-5 lower panel, there is certainly an
interaction between the two modes. The coupling is verified further by the shifts in
oscillator strengths between the two modes (lower panel Fig. 6-5 inset).
106
6.4.4 Magnetic Excitations
Magnetic excitations stimulated by infrared light can be grouped into one of two
categories: (a) traditional magnons that are excited by the a.c. magnetic field of the light,
and (b) novel electromagnons that are excited by the a.c. electric field component of the
light. Single-magnon absorptions, which at infrared frequencies are commonly observed
in antiferromagnets and thusly tabbed antiferromagnetic resonance modes (AFMR),
are magnetic dipole transitions that occur when the oscillating frequency of the light
corresponds to the k = 0 frequency of a spin wave. Electromagnons were first proposed
as strongly renormalized spin waves with dipolar momentum.[30] Since their discovery,
electromagnons have been extensively studied in rare earth manganite compounds,
namely in TbMnO3, where a large body of recent literature (Raman[35], neutron[36],
and infrared[37]) has tied the excitation to the lattice itself. The work has resulted in
a generalized hybrid magnon-phonon mode picture of electromagnons. Challenges arise
in differentiating the two aforementioned excitations because they resonate in the same
general frequency intervals. A common solution is to measure the different faces of a
crystal while rotating the polarization of the incoming light.
The geometry of the Cu3Bi(SeO3)2O2Cl crystal only allowed for transmission
measurements with the k vector of the incoming light aligned perpendicular to the ab
plane. However, a thorough polarization study within the ab plane was carried out and it
has lead to the observation of an isotropic magnetic excitation at 33.1 cm−1. These results
contradict the previously reported anisotropic nature of magnons and electromagnons.
Moreover, when external magnetic fields are applied in the H ‖ c and H ⊥ c geometries,
the 33.1 cm−1 resonance shifts to higher and to lower frequencies respectively. In what
follows we will examine the nature of the excitation as well as propose a reason for its
isotropic behavior. External field dependent spectra with H ‖ c and H ⊥ c will be
discussed separately. (The following section will focus on the 33.1 cm−1 excitation because
it is the only mode observed in zero field.)
107
6.4.4.1 Nature and isotropy
To elucidate the nature of the magnetic excitation observed in Cu3Bi(SeO3)2O2Cl at
33.1 cm−1, we compare the strength of the observed excitation to the extensive literature
on magnons and electromagnons in the infrared. Our magnetic excitation creates a peak
in α(ω) that is ∼ 30 cm−1 above the baseline. Taking the average d.c. index of refraction
to be 3 in the ab plane, we determine that our magnetic excitation corresponds to an
optical conductivity of about 0.23 Ω−1cm−1 (σ1 = c4πnα), which is roughly the same
strength as other reported single-magnon excitations.[79–81] Electromagnons, which have
been extensively studied in a number of rare-earth manganites, are observed to possess
optical conductivities of at least a factor of 10 larger.[30, 82] Although this method of
comparison does not yield objective certainty, we can further support its conclusion
by employing optical sum rule analysis on our measured reflectance data. Since an
electromagnon contributes to the dielectric constant, it must gain spectral weight from
a dipole active excitation, the main candidates being domain relaxations, phonons, or
electronic transitions. The reflectivity spectra measured at 7 K and 30 K do not show any
change associated with the magnetic excitation. Therefore, the magnetic excitation does
not gain spectral weight from the low frequency infrared-active phonon modes.
The isotropic character of the magnetic excitation observed at 33.1 cm−1 is
depicted in Fig. 6-16 where the oscillator strength of the excitation is plotted versus
the polarization angle of the light in the ab plane. (It is worthwhile noting that infrared
reflectance spectra measured on crystal 2 revealed the same strong anisotropy as depicted
in Fig.6-2, thus excluding twinning of the surface.) A recent report on TbMnO3 by
Pimenov et al.[83] details the observation of a magnon and an electromagnon, active
along perpendicular directions, resonating at the same frequency. This occurrence
gives the illusion of an isotropic magnetic excitation and is worthy of consideration
in Cu3Bi(SeO3)2O2Cl. Strongly opposing this argument are the nearly equivalent
oscillator strengths measured in any two orthogonal polarizations, as seen in Fig. 6-16.
108
E ∥b
E ∥b+45
E ∥a
E ∥a+45
E ∥b
E ∥b+45
E ∥a
E ∥a+45
12
34= Ωs
Figure 6-16. A polar plot of the oscillator strengths associated with the 33.1 cm−1 mode at0 T for all four polarizations measured in the ab plane.
Electromagnons typically have oscillator strengths that are at least one order of magnitude
greater than traditional magnons, as noted in the comparison of oscillator strengths above.
A second more plausible explanation is the occurrence of weak magnons at two
orthogonal polarizations. N. Kida et al.[82] observed a similar phenomena in DyMnO3,
namely, weak excitations arising with the a.c. magnetic field oriented along both the a
and c crystal axes. They supported their interpretation of two orthogonal magnons by
inelastic neutron scattering experiments on a similar rare-earth maganite where it was
reported that two magnon dispersions curves from orthogonal axes crossed k = 0 at the
same energy. In Cu3Bi(SeO3)2O2Cl, we suspect that magnon dispersion curves from the
[100] and [010] directions cross k = 0 at 33.1 cm−1 (∼4 meV); however, inelastic neutron
scattering measurements are needed to support our hypothesis.
6.4.4.2 H ‖ c field dependence
To analyze the H ‖ c spectra (i.e., H parallel to the easy axis), we will utilize
our d.c. susceptibility measurements as well as generalize the magnetic structure
determined for Cu3Bi(SeO3)2O2Br to Cu3Bi(SeO3)2O2Cl. At zero field, six distinct
magnetic sublattices can be identified, which would presumably lead to six distinct
109
magnon branches; however, due to the canted nature of the copper ions occupying the
4(c) sites, the exact number of branches could differ from six. At zero field, we only
resolved one magnetic excitation (33.1 cm−1). No clear signature of the metamagnetic
transition can be identified when tracking this excitation at fields between 0 and 1 T. In
a 1 T field, Cu3Bi(SeO3)2O2Cl has already undergone a metamagnetic transition where
magnetic moments on every second layer flip, resulting in ferromagnetic interlayer and
canted ferrimagnetic behavior overall.[67] (It should be noted that the c axis remains
the easy axis after the transition). Ironically, although the metamagnetic transition
effectively reduces the number of magnetic sublattices from six to three, we observe an
additional magnetic excitation appearing at fields of 1 T and greater. At this point, two
logical questions arise: First, why would the 33.1 cm−1 excitation present in the low field
antiferromagnetically-ordered state persist smoothly through the transition state and into
the high-field ferro or ferrimagnetically ordered state? Second, why are more magnetic
excitations present in the high field phase despite a reduction in the number of magnetic
sublattices?
As to the first question, magnons resulting from ferromagnetic resonance have
been observed at infrared frequencies; however, they typically resonate at much lower
frequencies because they must extrapolate linearly to zero frequency at zero applied
magnetic field. It is therefore very unlikely to observe a ferromagnetic resonance at
33.1 cm−1 in a 1 T external field. But there is an exceptional case that was first observed
by Jacobs in FeCl2 apropos metamagnetic transitions.[84] When the metamagnetic
transition occurs in FeCl2, which signals a transition from a two-sublattice antiferromagnet
to a ferromagnet, the AFMR line disappears in favor of a ferromagnetic resonance line
around the same frequency. The high frequency of the ferromagnetic resonance line in
FeCl2 is explained by large anisotropy fields in the material, and theoretical calculations
support its existence.[85] We suspect that a similar situation occurs in Cu3Bi(SeO3)2O2Cl
that explains the smooth movement of the excitation at 33.1 cm−1 from below to above
110
the metamagnetic transition. Likewise, we suspect that the magnetic excitation at 10.5
cm−1 (1 T) also originates from an antiferromagnetic resonance line, which would possess
a zero field resonance frequency of 9.5 cm−1 (slightly below our measurable range).
Apropos the second question, that is to say, the presence of more excitations above
the metamagnetic transition versus below it contradicts the reduction in magnetic
sublattices from six to three. We suspect that more excitations do exist beneath the
metamagnetic transition that are either below our measurable frequency range or that are
too weak for us to resolve (resolution 0.3 cm−1). We have thoroughly inspected the low
frequency spectra as well as the 33.1 cm−1 excitation between 0 and 1 T, and we observe
no clear signature of new resonances or the splitting of the existing 33.1 cm−1 excitation.
It is possible that the 10.5 cm−1 excitation (at 1 T) splits beneath the metamagnetic
transition; however, below 1 T experimental limitations prevented us from tracking the
excitation. Future studies involving a high-resolution, low-frequency source are needed to
investigate further.
In the interval 1–10 T we can estimate the effective g factor from the slope of
the observed resonance lines at 10.5 and 33.1 cm−1 using the formula ~ω = gµBHeff .[24]
Chosing an effective field corresponding to a plane sample with external field perpendicular
to the plane (i.e., Heff = H − 4πMs) we arrive at an equation in which the g factor can
independently be determined from both the slopes and intercepts of the two resonance
lines; however, because the material is no longer ferromagnetic below 0.8 T, we utilized
the measured slopes between 1 and 10 T. The slope of the resonance line at 10.5 cm−1
corresponds to a g factor of 2.16. The slope of the resonance line at 33.1 cm−1 varied
slightly with the polarization angle in the ab plane (see supplementary information).
The g factors and corresponding uncertainties for the resonance line at 33.1 cm−1 as
measured along the a axis, b axis, a axis+ 45, and b axis+ 45 are 0.24± 0.05, 0.44± 0.11,
0.35± 0.04, and 0.42± 0.05 respectively. The low g factor obtained for the 33.1 cm−1
resonance line can be reconciled in part by associating it with the magnetic moments on
111
the 4c sites, which are canted± 50 from c towards b.[67] The energy of a dipole is ~m · ~H ,
or mHcosθ. Assuming the magnetic moments on the 4c sites are the same as those on the
2a sites, one can expect a reduction in energy (slope) of the resonance line by cos50=0.64.
The resulting g factor estimate is increased, but not by enough to obtain the expected
value for Cu2+ (i.e., 2.0).
6.4.4.3 H ⊥ c field dependence
Again, we use our d.c. susceptibility measurements and the neutron scattering results
on Cu3Bi(SeO3)2O2Br to conclude that antiferromagnetic interactions exist within the ab
plane at all measurable fields in the H ⊥ c geometry. The theory of antiferromagnetic
resonances for uniaxial or cubic antiferromagnetic crystals was worked out by Keffer and
Kittel,[23] who showed that when the external field is oriented perpendicular to the easy
axis, the two k = 0 magnon branches correspond to frequencies ω/γ ∼= ±[2HEHA +H20 ]
1/2.
In the previous equation, γ = ge/2mc where g is the spectroscopic splitting factor, and H0,
HA, and HE represent the static, anisotropy, and exchange fields respectively. Generalizing
this theory, which was worked out for two sublattices, to a situation with more than two
sublattices, we can see that the dispersion of the excitation we observe (inset of Fig. 6-9b)
is in qualitative agreement (i.e., both are non-linear) with the lower frequency branch
predicted by the AFMR theory. (Further experiments are needed to determine HA , HE,
and g and subsequently fit the formula to the data to either validate or undermine our
generalization.) When there is no external field, the two absorptions are predicted to be
degenerate. However, in antiferromagnets with strong magnetic anisotropy, which is the
case for Cu3Bi(SeO3)2O2Cl, the degeneracy of magnon branches in zero field is lifted (e.g.,
MnO,[70] NiO,[70] and NiF2[79]). Following our previous reasoning that the resonance
observed in Cu3Bi(SeO3)2O2Cl, which moves to lower frequency with increasing field, is
the lower frequency branch of the k = 0 AFMR, we conclude that the higher frequency
branch becomes masked, thus unobservable, behind the strong phonon absorptions starting
∼40 cm−1. AFMR theory also predicts that when the lower frequency branch reaches zero
112
the external field is able to rotate the spins, at the expense of the anisotropy field, away
from the easy axis (c axis) and into the ab plane.[21] Extrapolating the lower frequency
branch towards zero, while taking into account the uncertainty in the data points, we
estimate a zero frequency crossing somewhere between 18.5 and 20.1 T. Our predicted field
range at which the spins rotate away from the easy axis is slightly higher, but remains
in rough agreement with the values predicted on the Cu3Bi(SeO3)2O2Br analogue.[67]
But unlike Cu3Bi(SeO3)2O2Br, we can not identify intermediate and hard axes because
the movement of our observed mode does not seem to depend on the orientation of the
external field within the ab plane.
6.5 Summary
The novel geometrically-frustrated layered compound Cu3Bi(SeO3)2O2Cl has been
characterized using infrared spectroscopy, powder x-ray diffraction, and d.c. magnetic
susceptibility measurements. Far-infrared reflectance measurements have revealed 16 new
infrared-active phonon modes below 115 K. The plethora of new modes observed strongly
suggest a rearrangement of atomic positions within the unit cell; however, our subsequent
powder x-ray diffraction measurements are completely consistent with the same 300 K
structure (Pmmn) existing at 85 K. Preliminary Raman spectra taken at 300 K on crystal
1 have revealed five phonon modes at frequencies close to five of the new modes observed
in the infrared below 115 K. The results suggest a loss of inversion symmetry below 115 K.
Upon further investigation we have identified two non-centrosymmetric orthorhombic
space groups (Pm21n and P21mn) that have the same allowed Bragg reflection peaks as
the 300 K Pmmn structure. Therefore we suspect that a subtle second-order transition
from Pmmn to either Pm21n or P21mn occurs near 115 K that is below the resolution of
our powder x-ray diffraction experiment. We plan to perform more sensitive measurements
(e.g., single crystal or neutron powder diffraction) to resolve the uncertainty of this issue.
In addition, an isotropic magnetic excitation is observed at 33.1 cm−1 at 5 K. We
have tentatively assigned the magnetic excitation to a magnon based on analysis of
113
previously reported oscillator strengths of magnons and electromagnons. The isotropic
behavior of the excitation within the ab plane is potentially due to the Brillouin zone
center crossing of two magnon dispersion curves along orthogonal directions, but inelastic
neutron scattering measurements are needed to investigate further the excitation’s
seemingly isotropic existence.
The resonance frequency of the 33.1 cm−1 excitation strongly depends on the
orientation of the static magnetic field. An anisotropic response to the orientation of a
static magnetic field is also seen in d.c. susceptibility measurements on Cu3Bi(SeO3)2O2Cl,
as well as neutron diffraction measurements on the similar Cu3Bi(SeO3)2O2Br compound.
When the external magnetic field is applied parallel to the c axis (H ‖ c), the resonant
frequency of the 33.1 cm−1 excitation increases linearly with increasing field. For fields
of 1 T and greater applied along the c axis an additional linearly-increasing magnetic
excitation is observed (10.5 cm−1 at 1 T).
When the external magnetic field is applied perpendicular to the c axis (H ⊥ c), the
resonant frequency of the 33.1 cm−1 excitation decreases quadratically with increasing
field. The results are in agreement with the behavior of an antiferromagnetic resonance
line in the presence of strong magnetic anisotropy.
114
Table 6-1. Oscillator parameters for the infrared observed modes of Cu3Bi(SeO3)2O2Cl (at 7 K) along all three crystal axes.The new modes arising below 115 K are indicated with an asterisk next to their corresponding TO frequencies.
a b c
Osc Str TO Freq LO Freq FWHM Osc Str TO Freq LO Freq FWHM Osc Str TO Freq LO Freq FWHMS ω (cm−1) ω (cm−1) γ (cm−1) S ω (cm−1) ω (cm−1) γ (cm−1) S ω (cm−1) ω (cm−1) γ (cm−1)3.947 52.8 57.7 1.7 26.221 36.3 55.6 1.9 0.035 53.2 53.6 1.51.291 69.9∗ 72.4 1.9 2.389 68.3 78.8 1.3 2.228 99.8 112.9 3.94.096 89.0 101.9 1.9 0.034 99.8∗ 100.2 1.0 0.055 115.1∗ 118.4 2.80.126 101.1∗ 110.7 5.7 0.468 115.2 126.2 2.4 0.059 144.8 145.8 3.00.029 137.6 138.2 1.4 0.071 128.9∗ 130.5 0.9 0.023 161.5 162.0 1.90.433 161.9 164.8 2.6 0.154 133.5 138.2 1.3 0.210 204.0 208.1 1.60.223 172.3∗ 174.1 1.9 0.744 185.8 196.2 1.2 0.035 273.8∗ 274.7 1.51.540 191.6 201.4 1.0 0.211 256.9 261.9 1.2 0.204 284.4 289.8 2.40.115 202.1 228.3 3.1 0.106 276.1∗ 279.2 2.0 0.093 337.6 340.5 2.00.085 320.0∗ 323.1 1.8 0.062 300.3 302.3 1.6 0.154 433.5 439.3 4.70.045 331.1 333.6 2.1 0.038 313.9 315.2 1.6 0.586 528.4 547.4 4.80.067 422.9 426.0 4.7 0.291 456.3 466.4 1.9 0.145 554.4 579.4 12.40.055 470.2 471.6 12.2 0.156 484.7∗ 489.9 3.8 0.136 794.9 812.7 6.30.139 542.4∗ 546.1 5.0 0.336 507.0 528.0 3.70.293 557.3 577.5 4.0 0.059 542.3 550.2 3.10.066 587.2∗ 587.4 3.9 0.045 571.4∗ 575.7 22.00.564 688.2 669.1 7.1 0.122 716.1∗ 725.9 5.30.122 703.8∗ 705.5 16.8 0.062 730.0 767.6 10.30.005 737.0∗ 774.0 19.9 0.024 811.3 815.0 16.1
0.030 825.0 834.4 7.9
115
CHAPTER 7CONCLUSIONS
This dissertation has reported comprehensive experimental results for three novel
complex oxide single crystals. The investigations were motivated by a search for new
multiferroics and large magnetoelectric coupling. The primary tool of characterization
was infrared spectroscopy; however, infrared results motivated further investigation using
complementary experimental techniques for the three materials investigated. In what
follows, an overview of the major experimental findings for each material will be discussed.
The relationship of our findings to multiferroic and magnetoelectric phenomena as well as
future aims of study will be highlighted for each material.
Chapter 4 details the experimental findings of single-crystal Cu2OSeO3. Cu2OSeO3
was previously reported to show magnetoelectric coupling via an anomalous jump in the
dielectric constant (obtained by dielectric capacitance measurements) at the ferrimagnetic
ordering temperature (Tc ∼ 60 K). In this dissertation the dielectric constant of Cu2OSeO3
as a function of temperature was obtained by an alternative method. Briefly, by fitting
the infrared reflectance spectra with a Drude-Lorentz model, the dielectric constant was
estimated by taking the zero frequency limit of the Drude-Lorentz dielectric function.
An anomaly in the infrared-obtained dielectric constant was observed, and it agreed well
in both magnitude and direction with the dielectric anomaly reported by capacitance
measurements. Furthermore, lattice dynamical calculations were carried out to further
analyze the nature of anomalous phonons that contributed to the jump in dielectric
constant. It was determined that 2 of the 13 modes exhibiting anomalous behavior near
the ferrimagnetic ordering temperature were linked to the motion of copper ions, the ions
responsible for magnetic ordering.
Since the time of the infrared study on Cu2OSeO3 reported in this dissertation, a
number of intriguing reports of exotic magnetoelectric phenomena in the title material
have surfaced. For example, Seki et al.[86] observed magnetic skyrmions in Cu2OSeO3,
116
and also deemed the material to be multiferroic. A subsequent report by White et al.[87]
detailed how the application of an external electric field oriented parallel to the [111] axis
and magnetic field oriented parallel to the [110] axis could be used to controllably rotate
the skyrmion lattice around the magnetic field axis. A future infrared study utilizing the
new findings of an externally controllable skyrmion lattice in Cu2OSeO3 (detailed above)
would be of grave interest. Specifically, measuring the infrared phonon spectra while
simultaneously applying external fields to Cu2OSeO3 could potentially shed further insight
into the microscopic atomic motions responsible for the rotation of the skyrmion lattice.
Chapter 5 reports on multiferroic single crystal FeTe2O5Br. FeTe2O5Br was previously
shown to exhibit the most direct form of magnetoelectric coupling, namely, the case in
which a ferroelectric polarization arises as a by-product of a lattice distortion driven by
the onset of complex magnetic ordering (Tn=10.6 K). The infrared results presented
in this dissertation did not contain the magnetoelectric coupling previously reported in
FeTe2O5Br despite the application of external magnetic fields (at 5 K) oriented along the
various crystal directions; however, upon orienting a single crystal of FeTe2O5Br using
x-ray diffraction, a comprehensive study of the anisotropic phonon spectra of the title
material was carried out. With the aid of lattice dynamical calculations, all 52 infrared
active modes predicted in the monoclinic ac plane were accounted for. Along the unique
b axis of the monoclinic cell, 43 of the 53 modes were observed. In addition, the lattice
dynamical calculations revealed interesting trends relating to the motion of the constituent
atoms as the far-infrared wavelengths were spanned. Furthermore, a combined reflection
and transmission analysis was utilized throughout the mid-infrared region to accurately
obtain the complex dielectric function and other complex optical properties. The results
of the aforementioned analysis shed light on the contradicting measurements of absorption
edges in FeTe2O5Br, and the related analogue, FeTe2O5Cl.
Future studies on FeTe2O5Br involve high-resolution magneto-terahertz spectroscopy
to examine the plethora of low energy magnetic excitations predicted for frustrated
117
antiferromagnetic compounds. Additionally, high-resolution infrared spectroscopy above
and below the multiferroic ordering temperature (10.6 K) are needed to observe the
expected splitting of phonon modes accompanying the centrosymmetric to non-centrosymmetric
crystal structure rearrangement mandated by the onset of ferroelectricity.
Chapter 6 details the very intriguing and novel Cu3Bi(SeO3)2O2Cl system. The
infrared results reported in this dissertation give strong evidence for a symmetry-lowering
structural transition existing below 115 K via the plethora of new phonon modes arising
along all three crystal axes of the unit cell. Surprisingly, powder x-ray diffraction results
indicate that the 300 K orthorhombic structure persists down to 30 K. Room-temperature
Raman spectra reveal phonon modes resonating at energies corresponding to the
energies of the new phonon modes in the infrared below 115 K, thus giving strong
experimental support for the loss of inversion symmetry at 115 K. In addition, magnetic
excitations are observed at 33.1 and 10.5 cm−1 in transmission geometry at 5 K. The
excitations disappear above the magnetic ordering temperature (Tc=24 K). When a
magnetic field is applied parallel to the c axis, a metamagnetic transition is triggered
in the field range 0.1–0.8 T. The metamagnetic transition signifies a change from
antiferromagnetically ordered spins to ferromagnetically ordered spins. Very recent
dielectric capacitance measurements have revealed an anomaly of electric nature coinciding
with the metamagnetic transition. It is still yet-to-be determined whether the transition is
ferroelectric in nature or rather due to magneto-strictive phenomena.
Future studies on Cu3Bi(SeO3)2O2Cl involve high resolution neutron diffraction
to investigate further the suspected structural transition at 115 K. In addition, the
ferroelectric properties of Cu3Bi(SeO3)2O2Cl will be measured to provide more insight into
the dielectric capacitance anomaly observed to coincide with the metamagnetic transition.
118
APPENDIX APRELIMINARY RESULTS ON SINGLE CRYSTAL Cu3(SeO3)2Cl2
A.1 Background and Crystal Structure
Work on single crystals of Cu3(SeO3)2Cl was on going at the time of compiling
this dissertation. Along with being a potential candidate for a new multiferroic and
magnetoelectric material, Cu3(SeO3)2Cl2 is additionally of interest because it was
previously reported to exist in two distinct crystal structures (triclinic and monoclinic).
In 2000 P. Millet et al.[88] synthesized Cu3(SeO3)2Cl2 by means of chemical transport
reaction. Single crystal x-ray diffraction was employed to determine that the material
possessed a triclinic structure with space group P 1. In 2007 Helmuth Berger[89]
synthesized Cu3(SeO3)2Cl2 also by chemical vapor transport reaction and, surprisingly,
the material was reported to crystallize in the monoclinic space group C2/m. Moreover,
the 2007 report claimed that Cu3(SeO3)2Cl2 possessed a layered structure as opposed to
the seemingly three-dimensional structure reported in 2001. It should be noted that the
crystals of Cu3(SeO3)2Cl2 reported in this dissertation were grown by Helmuth Berger.
Our initial motive to determine crystal structure solely by optical means is discussed in
Section A.3.
A.2 Magnetic Properties
Our preliminary magnetic susceptibility measurements on Cu3(SeO3)2Cl2 indicate
the onset of long range order below 40 K. The susceptibility of Cu3(SeO3)2Cl2 (not
shown) is in qualitative agreement with a previous report on the isostructural compound
Cu3(TeO3)2Br, which exhibits long range magnetic ordering at Tc=70 K.
A.3 Room-temperature Infrared and Raman Results
The study of the plate-like geometry of the Cu3(SeO3)2Cl2 crystals allowed only for
in-plane infrared measurements. We have identified the two orthogonal principal dielectric
axes in-plane (hereafter denoted 1 and 2). The 300 K infrared reflectance along both
axes is shown in Figure A-1. We observe 17 phonon modes with the light polarized
119
0 200 400 600 800 1000Frequency (cm−1 )
0
0.2
0.4
0.6Reflectance
E ∥20
0.2
0.4
0.6
0.8
E ∥1
20 40 60 80 100 120Energy (meV)
300K
Figure A-1. Infrared reflectance of Cu3(SeO3)2Cl2 at 300 K with light polarized along thetwo in-plane principal dielectric axes.
parallel to the principal axis 1, and 22 phonon modes are observed along 2. Group
theoretical calculations based on the two reported crystal structures result in the following
distribution of modes:
Γmonoclinic = 11A(R)g + 7B(R)
g + 7A(IR)u + 11B(IR)
u (A–1)
and
Γtriclinic = 39A(R)g + 36A(IR)
u (A–2)
where (R) and (IR) denote Raman active and infrared active modes respectively. We
observe many more phonon modes than predicted by the monoclinic space group and three
more than predicted by the triclinic space group. Since the plate-like nature of the crystals
resembles a layered materials (layered materials are typically thin along the layering axis),
we cannot rule out the possibility of a disordered monoclinic structure.
120
Table A-1. The partial point group symmetry character table of C2h.
C2h E C2 i σh
A g 1 1 1 1 Rz x2, y2, z2, xyB g 1 -1 1 -1 Rx, Ry xz, yz
Raman spectrosocpy was also utilized to examine the phonon spectra of the exact
crystal measured in the infrared. Focusing on the monoclinic space group, we now discuss
the Raman selection rules of the 11A g and 7B g modes. The character table of the
C2h point group is shown in Table A-1. The standard crystallographic convention is
to associate x with the a axis, y with the b axis, and z with the c axis. Following this
logic, one geometry to select the 11Ag occurs when both the incident light and scattered
light are polarized along the a axis (hereafter we adopt the notation aa). To summarize
the table, Ag modes are selected with aa, bb, cc, and ab. The Bg modes are selected
with ac and bc. The 2007 report states that the c axis is the layering axis; therefore, if
crystal is in fact monoclinic, then the in-plane face constitutes the ab plane. Since x-ray
analysis to orient the crystal is lacking, we have assumed the ab plane corresponds to
the in-plane orientation for preliminary Raman measurements. Room temperature ab
Raman spectrum is shown in Figure A-2. We observe 12 Raman active phonons in the ab
geometry, which is 1 more than predicted for the monoclinic space group. Although more
modes are observed than predicted for the monoclinic space group (same conclusion as the
infrared), a disordered monoclinic structure still cannot be ruled out. No clear selection
rules exist for the 39Ag Raman modes in the triclinic structure, and therefore they will
not be discussed. Powder x-ray measurements are expected in the near future to further
investigate the room-temperature structure of our exact Cu3(SeO3)2Cl2 crystal.
A.4 Temperature-dependent Infrared Spectra
Infrared reflectance and transmittance have been measured between 4 and 300 K.
Strikingly, as the crystal of Cu3(SeO3)2Cl2 is cooled, many new phonon modes appear in
the infrared. With light polarized parallel to the 1 direction 10 new phonons are observed.
Parallel to the 2 direction 30 new phonons arise. The temperature-dependent infrared
121
0 200 400 600 800 1000Frequency (cm−1 )
0
5
10
15
20
25
30
35
40
√ Intensity [Arb. units]
20 40 60 80 100 120Energy (meV)
(ab)
Figure A-2. Raman spectra of Cu3(SeO3)2Cl2 with incident light polarized parallel to thea axis and scattered light polarized parallel to the b axis. The ab and bageometries are identical, as is the standard case.
spectra along the two in-plane principal dielectric axes are shown in Figure A-3. Crimson
arrows denote the energies of new modes arising in the infrared.
It is also noteworthy that new infrared phonons arise in two different temperature
ranges: around 80 K and around 40 K. The new modes arising around 40 K suggest that
a symmetry-lowering structural transition accompanies the onset of long range magnetic
ordering. Figure A-4 depicts a region along the 2 direction where new modes appear in
both temperature ranges. Magneto-infrared transmission measurements in fields up to
5 T, oriented both parallel and perpendicular to the flat face of the crystal, have also been
carried out at the University of Wollongong, Australia. At 5 K the modes do not show an
observable field dependence.
Temperature-dependent powder x-ray diffraction measurements are anticipated in
the near future to determine the low temperature structure and the existence of multiple
transitions.
122
0 200 400 600 800 1000Frequency (cm−1 )
0
0.2
0.4
0.6Reflectance
E ∥20
0.2
0.4
0.6
0.8
E ∥1
20 40 60 80 100 120Energy (meV)
11K18K24K30K38K46K70K80K90K100K150K300K
Figure A-3. Temperature-dependent infrared reflectance of Cu3(SeO3)2Cl2 at 300 K withlight polarized along the two in-plane principal dielectric axes. Crimsonarrows denote the energies of all new modes arising in the infrared.
123
235 245 255 265 275Frequency (cm−1 )
0.0
0.1
0.2
0.3
0.4
0.5
0.6
Reflectance
E ∥2
30 31 32 33 34 35Energy (meV)
11K18K24K30K38K46K56K70K80K90K100K150K270K
0 100 200 300Temperature (K)
240
250
260
270
ω0(cm−1)
Figure A-4. The frequency interval 235–280 cm−1 along the 2 direction where new infraredmodes arise on either side of an existing mode at ∼256 cm−1. The resonancefrequencies of the new modes are estimated from reflectance peaks and plottedin the inset. Black markers correspond to the existing mode, red markers tonew modes around 80 K, and blue markers to new mode around 40 K.
124
APPENDIX BCALCULATING SINGLE-BOUNCE REFLECTANCE
B.1 Preface
Certain ionic insulators exhibit low absorption in the frequency range above
the infrared active phonons and below the onset of electronic transitions (typically
mid-infrared frequencies). Depending on crystal geometry and thickness, the transmittance
can take on finite values in this region, and therefore the reflectance contains an additional
contribution from the back surface of the crystal (cf. Figure 5-5). Our spectral analysis,
which involves Kramers-Kronig relations to estimate the phase shift upon reflection and
subsequently inverting reflection and phase to calculate the complex response functions,
assumes that a single layer of material is doing the reflection (single-bounce reflectance).
Briefly, for multi-layer reflection the phase estimated from Kramers-Kronig is still correct;
however, inverting reflectance and phase are non-trivial processes. An alternative option is
to estimate the single-bounce reflectance using the measured non-single-bounce reflectance
and non-single-pass transmittance. The method utilized in this dissertation is summarized
in a journal article by Zibold et al.[8] and is fully developed in this appendix. In what
follows, we will restrict ourselves to the incoherent limit of transmission (sample thickness
much greater than the wavelength of the light) which is justified by examining the
dimensions reported.
B.2 Formalism
The complex amplitude of transmittance through a slab of thickness d is given by
t =√
T (ω)eiφ(ω) = (1− r2)a(1 + a2r2 + a4r4 + · · ·), (B–1)
Ionic insulators also exhibit transmission at frequencies below the infrared-activephonons and at frequency intervals between the infrared-active phonons.
125
which can be re-written as
t =a(1− r2)
1− a2r2(B–2)
by assuming —ar—¡1 and using the geometric series. In the above equations the
amplitude attenuation coefficient is given by
a = ei2πνnd (B–3)
and the complex reflectance is given by
r =1− n
1 + n=
√
Rs(ω)eiφ(ω). (B–4)
We now compute the power transmittance, T = tt∗, for that is the quantity we actually
measure in an experiment. To do so it is useful to re-write Eq. (B–2) as
t = (r−1 − r)[(ar)−1 − (ar)]. (B–5)
For simplicity we let a = aeiφa and r = reiφr . Utilizing the double angle formula we arrive
at the following expression for the power transmittance:
T =r−2 + r2 − 2 cos 2φa
(ar)−2 + (ar)2 − 2 cos 2φr + 2 cosφa. (B–6)
Using the relation cos 2x = 1 − 2 sin2 x we can put Eq. (B–6) in the following very useful
from:
T = a2(1− r2)2 + 4a2 sin2 φa
(1− a2r2)2 + 4a2r2 sin2 φa + cosφr
. (B–7)
Following the methodology of E. E. Bell[90] we integrate Eq. (B–7) over a cycle of φa by
the standard technique: the trigonometric function of φa is replaced by a function of a
complex variable and the resulting integrand is integrated around the origin on a circle of
Eq. (B–2) is utilized to compute absorption directly from transmission whileaccounting for reflection losses at the surfaces.
126
unit radius. The values of the integrals are determined by the residues of the singularities
within the unit circle. The average value of power transmittance that results is given by:
T =a2(1− r4)− 2a2r2 cos 2φr
1− a4r4. (B–8)
We now make the critical assumption that the imaginary part of the refractive index, k, is
small. This assumption is justified by examining the argument of the exponential in the
following equation:
e−ω
ckd = e
−2π
λkd. (B–9)
As mentioned above, we are working in the incoherent limit (d >> λ) so k must be small
to keep the argument of the exponential not too big so as to produce a finite transmission.
Neglecting k allows the following approximation to the phase of reflectance [cf. Eq. (B–4)]:
r =√
Rs(ω)eiπ, (B–10)
and hence
rr∗ = r2 = Rs. (B–11)
With φr equal to π radians and defining A(ω) = aa∗ = e−4πνkd we arrive at the following
expression for the power transmittance:
T (ω) =A(ω)[1− Rs(ω)]
2
1−A2(ω)R2s(ω)
. (B–12)
At this point a relation is necessary that expresses the measured reflectance, R(ω), as a
function of both the single-bounce reflectance, Rs(ω), and measured power transmittance,
Eq. (B–12). A relation can be obtained by following the work of Tinkham[91] on thin
films. To do so, the normalized admittance y = Z0/Z = y1 − iy2 (y1 and y2 are real)
is introduced. The relation Z = 1/σd relates the complex admittance to the complex
optical conductivity. The measured power reflectance and transmittance can be expressed
127
in terms of the real and imaginary part of the admittance by
R =y21 + y22
(y1 + 2)2 + y22(B–13)
and
T (ω) =4
(y1 + 2)2 + y22. (B–14)
Following the pioneering work of Tinkham[91] we invert Eq. (B–13) and Eq. (B–14) to
solve for the y1 and y2 by way of factoring in the index of refraction of the material, which
can be written in terms of Rs(ω) (cf. Eq. (B–18)). One arrives at the following desired
relation:
R(ω) = Rs(ω)[T (ω)A(ω) + 1]. (B–15)
In practice one can now invert R(ω) and T (ω) to obtain Rs(ω) and A(ω). Specifically,
we solve for A(ω) in Eq. (B–12) and Rs(ω) in Eq. (B–15) to obtain
A(ω) =−(1 −Rs(ω))
2 +√
(1−Rs(ω))4 + 4(T (ω)Rs(ω))2
2T (ω)Rs(ω)2(B–16)
and
Rs(ω) =R(ω)
T (ω)A(ω) + 1. (B–17)
Both the measured reflectance and transmittance, R(ω) and T (ω), must be interpolated
to the same set of data points (frequency values) to carry out the numerical calculation of
arrays that ensues. A two column array with first column equal to the frequency values
determined in the previous step and second column values all initialized to unity is then
fabricated [label A(ω)]. We then compute the array Rs(ω) (Eq. (B–18)) and use this result
to re-assign values to A(ω) [Eq. (B–16)]. An iterative process ensues in which the loop is
not broken until a convergence limit is reached. Typically the convergence is signified by
reaching a lower limit of the change in A(ω) from iteration to iteration [∆A(ω) = 1× 10−6
has been used successfully in the past].
128
2000 4000 6000 8000 10000Frequency (cm−1 )
0.0
1.0
2.0
3.0
4.0
Refractive index (n)
0.25 0.50 0.75 1.00 1.25Energy (eV)
from Kramers-Kronigfrom Rs method
Figure B-1. The refractive index, n, extracted directly from the computation of Rs(ω) (reddashed line) and from Kramers-Kronig analysis of Rs(ω) (blue solid line).
Although the calculated single-bounce reflectance shown in Figure 5-5 seems to
completely eliminate the elevated level of reflectance stemming from the additional back
surface contribution, it is always assuring to further validate assumptions made. We now
examine the refractive index, n, as computed from both Kramers-Kronig analysis of the
single-bounce reflectance and also as directly extracted from the above computation by
inverting the relation
Rs(ω) ≈(n− 1)2
(n + 1)2. (B–18)
(To recall, we have neglected k.) The two calculations of n are compared in Figure B-1.
The n extracted directly from computing Rs(ω) is only meaningful in the region where the
crystal transmits (∼1600–10000 cm−1).
The procedure detailed in this appendix can be summarized as follows. The measured
reflectance and transmittance in frequency intervals where both quantities exhibit
finite values can be used to solve for the real part of the refractive index, n, under the
129
assumption that the imaginary part of the refractive index, k, is not too big. The real
part of the refractive index can be used to calculate the bulk (single-bounce) reflectance
throughout the frequency interval by inverting the approximation given in Eq. (B–18).
The bulk reflectance can then be merged with the measured reflectance in regions where
the transmission is zero. (In regions where the transmission is zero, k is not necessarily
small and typically the measured reflectance is equal to the single-bounce reflectance.)
Kramers-Kronig relations are then used to extract the real and imaginary parts of the
dielectric function (and other optical properties) in the usual manner.
130
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135
BIOGRAPHICAL SKETCH
Kevin Miller was born in Washington DC in 1985. The majority of his early years
were spent in Philadelphia. Following a tradition of the males in his family, he attended
LaSalle College High School. Subsequently, he was awarded a scholarship to Saint
Bonaventure University in Olean, New York where he majored in Physics. Since Saint
Bonaventure does not conduct graduate studies in the natural sciences, Kevin′s only
experience with academic-style research came during a research experience for undergrads
(REU) at Notre Dame in the summer of 2007 (laboratory of Prof. Jacek Furdyna).
Following his graduation from Saint Bonaventure in May 2008, Kevin decided to pursue
graduate studies in physics, and in doing so, moved far away from the cold rain and snow
of western New York. He matriculated at the University of Florida in the Fall of 2008 and
promptly join the group of Prof. David B. Tanner. In the last five years, he has met many
wonderful people and established bondings with them that will strengthen his work in
physics for years to come. Kevin Miller will be awarded a Doctor of Philosophy in Physics
in May 2013.
136