‘SOFT ELECTRONIC MATTER’, MAGNETOELECTRIC COUPLING, ANDMULTIFERROISM IN COMPLEX OXIDES

By

PATRICK R. MICKEL

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

2011

c© 2011 Patrick R. Mickel

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To my father

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ACKNOWLEDGMENTS

A Ph.D. is a long road that requires committed support from numerous people, both

professionally and personally. As the saying goes, it takes a village to raise a graduate

student - and my case is no exception. It is impossible to thank everyone who has

hepled me along the way, but below are my attempts.

First and foremost, I am forever grateful for the opportunities that Art has provided

me. I came to him lost in the world of bio-physics, and he gave me the guidance and

support I needed, helping me turn my graduate career 180 around. The stimulating

projects, and positive environment that surround him have truly changed my life. Art

provided invaluable feedback and insight into every problem I presented him, and I will

forever aspire to understand physics as simply and deeply as he does.

My committee members have all helped me reach this point successfully. Amlan,

my unofficial second advisor, has provided me indispensable support. Our daily

conversations, which often end in laughter, are a cornerstone in my graduate education.

Without him, none of this work would be possible. Dr. Rinzler has also helped shape my

graduate career, and I am very grateful for his support during my lab transition. As my

teacher, Dr. Hershfield has greatly improved my understanding of quantum mechanics.

His door has always been open for questions concerning both class and research. I am

also very grateful for all the ways I have learned with Dr. Dempere. Through her class

on SEM, and the opportunities she provided me working at MAIC, I learned valuable

characterization techniques and concepts that still help me today.

I have also benefited greatly from many discussions and relationships with other

members of the University of Flordia Physics Department. First, my collaboration

with Dr. Pradeep Kumar was quite fruitful, as he taught me a great deal about

magnetoelectric coupling. The theoretical understanding of magnetoelectric coupling

in this thesis would not have been possible without him. The time I spent in Dr. Tom

Mareci’s lab was also valuable, as I learned the inner workings of Diffusion Tensor

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Imaging. Also, many students have augmented my education along the way and I list

them here in no particular order: Hyoungjeen Jeen, Sinan Selcuk, Sefaatin Tongay,

Pooja Wadhwa, Evan Donoghue, Manoj Srivastava, Andrei Kamalov, Ritesh Das,

Gueenta Singh-Bhalla, Siddhartha Ghosh, Greg Boyd, Rajiv Misra, Maureen Petterson,

Sanal Buvaev, Dan Pajerowski, Chris Pankow, Mitch McCarthy, Corey Stambaugh, and

Patrick Hearin.

I would also like to acknowledge the staffs of machine shop and electric shop.

In particular the cryogenic staffs, Greg and John, provided our labs an incredible

advantage in research through their constant supply of liquid He and N2 all year around

24/7. Also, thanks to Jay Hornton (a really nice guy) for looking after all the pumps and

chillers. Without the help of these hard workers, my work would have taken exponentially

longer to complete.

Prior to my time at the University of Florida, many people were instrumental

in nurturing my scientific career. Dr. Daniel Fleisch, my first mentor, was incredibly

generous with his time as he introduced me to science and research. My undergraduate

professors at the University of Notre Dame also provided an important chapter in my

education. Their doors were always open, even outside of office hours, as they instilled

confidence and knowledge in me. Dr. Kathy Newman, Dr. Zoltan Neda, and Dr. Mitchell

Wayne were particularly generous and patient.

Last but certainly not least, I would like to thank my family and future wife Kristina.

They support me in everything I do, and have made me who I am. Without them I am

nothing.

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TABLE OF CONTENTS

page

ACKNOWLEDGMENTS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

LIST OF FIGURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

ABSTRACT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

CHAPTER

1 Basic Physics Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.1 Mixed Valence Manganites . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141.1.2 Structure and Energy Diagrams . . . . . . . . . . . . . . . . . . . . 151.1.3 Effects of Doping and Cation Substitution . . . . . . . . . . . . . . 181.1.4 La1−xCaxMnO3, Pr1−xCaxMnO3, and (La1−yPry )1−xCaxMnO3 . . . 21

1.2 Multiferroics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281.2.2 Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . . . . . . . 291.2.3 Ferroelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341.2.4 Magnetoelectric Multiferroics . . . . . . . . . . . . . . . . . . . . . 43

1.3 Magnetoelectric Coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481.3.2 Maxwell Equations vs. Magnetoelectric Coupling . . . . . . . . . . 501.3.3 Free Energy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

2 Experimental Techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

2.1 Sample Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.1.1 Growth Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . 542.1.2 Structural and Compositional Characterizations . . . . . . . . . . . 55

2.2 Temperature and Magnetic Field Control . . . . . . . . . . . . . . . . . . . 562.3 Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 572.4 Capacitance Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 58

2.4.1 Capacitance Bridge and Stick . . . . . . . . . . . . . . . . . . . . . 582.4.2 Dielectric Electrodes . . . . . . . . . . . . . . . . . . . . . . . . . . 602.4.3 Interdigital Capacitance . . . . . . . . . . . . . . . . . . . . . . . . 632.4.4 Bandwidth Temperature Sweeps . . . . . . . . . . . . . . . . . . . 65

2.5 Ferroelectric Measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 672.5.1 Sawyer-Tower Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . 672.5.2 Precision LC: Ferroelectric Tester . . . . . . . . . . . . . . . . . . . 692.5.3 Remanent Polarization . . . . . . . . . . . . . . . . . . . . . . . . . 70

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3 ‘Soft Electronic Matter’ in LPCMO . . . . . . . . . . . . . . . . . . . . . . . . . 73

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 733.2 Transport Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.2.1 Resistance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 743.2.2 Complex Capacitance . . . . . . . . . . . . . . . . . . . . . . . . . 74

3.3 Competing Dielectric Phases . . . . . . . . . . . . . . . . . . . . . . . . . 793.3.1 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 803.3.2 Temperature Dependence of Model Parameters . . . . . . . . . . . 81

3.4 ‘Soft Electronic Matter’ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 853.4.1 Polarons and Detailed Balance . . . . . . . . . . . . . . . . . . . . 853.4.2 Testing Detailed Balance Constraints . . . . . . . . . . . . . . . . . 873.4.3 Lattice Relaxation Rates . . . . . . . . . . . . . . . . . . . . . . . . 903.4.4 Charge Density Waves . . . . . . . . . . . . . . . . . . . . . . . . . 91

3.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93

4 Strain Mediated Magnetoelectric Coupling in (La1−yPry )1−xCaxMnO3 . . . . . 95

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 954.2 Dielectric Constant Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

4.2.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 964.2.2 Modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97

4.3 Activation Energy Tuning . . . . . . . . . . . . . . . . . . . . . . . . . . . 994.3.1 Experimental Results . . . . . . . . . . . . . . . . . . . . . . . . . . 994.3.2 Comparison of Magnetoelectric Couplings . . . . . . . . . . . . . . 101

4.4 Film Thickness Study . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1014.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

5 Multiferroism in BiMnO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1055.2 Characterizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

5.2.1 Structural Characterization . . . . . . . . . . . . . . . . . . . . . . . 1065.2.2 Magnetic Characterization . . . . . . . . . . . . . . . . . . . . . . . 1075.2.3 Resistive Characterization . . . . . . . . . . . . . . . . . . . . . . . 1095.2.4 Ferroelectric Characterization . . . . . . . . . . . . . . . . . . . . . 1095.2.5 Dielectric Characterization . . . . . . . . . . . . . . . . . . . . . . . 112

5.3 Nature of Ferroelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.3.1 Relaxor Review . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1155.3.2 Comparison . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.3.3 Pulse Sequencing . . . . . . . . . . . . . . . . . . . . . . . . . . . 1185.3.4 Island Growth . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

5.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

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6 Tuning Ferroelectricity in BiMnO3 . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.2 Strain: External and Island Edges . . . . . . . . . . . . . . . . . . . . . . 123

6.2.1 External Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1236.2.2 Electrode “Lensing” . . . . . . . . . . . . . . . . . . . . . . . . . . . 1256.2.3 Island Edge Strain Gradients . . . . . . . . . . . . . . . . . . . . . 125

6.3 Magnetoelectric Coupling in BiMnO3 . . . . . . . . . . . . . . . . . . . . . 1266.3.1 Remanent Polarization Tuning . . . . . . . . . . . . . . . . . . . . . 1266.3.2 Reorientation Time-Scales . . . . . . . . . . . . . . . . . . . . . . . 1276.3.3 Connection to Lattice Transition . . . . . . . . . . . . . . . . . . . . 128

6.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

REFERENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131

BIOGRAPHICAL SKETCH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137

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LIST OF FIGURES

Figure page

1-1 Cubic ABO3 Perovskite Manganite Structure . . . . . . . . . . . . . . . . . . . 16

1-2 Orbital Energy Levels: Crystal Field Splitting and Jahn-Teller Distortions . . . . 16

1-3 Cubic and Jahn-Teller Distrotion of MnO6 Octahedra . . . . . . . . . . . . . . . 18

1-4 3d Orbitals: eg and t2g . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1-5 Polaron Depiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

1-6 La1−xCaxMnO3 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

1-7 Pr1−xCaxMnO3 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

1-8 (La1−yPry )1−xCaxMnO3 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . 25

1-9 Dark-Field Electron Diffraction Image of Phase Separation . . . . . . . . . . . 26

1-10 Magnetic Force Microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

1-11 Multiferroic Coupling Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

1-12 Types of Magnetic Ordering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31

1-13 M-H Loops for Different Magnetic Orderings . . . . . . . . . . . . . . . . . . . . 32

1-14 Stoner Band Theory of Ferromagnetism . . . . . . . . . . . . . . . . . . . . . . 35

1-15 Polarization vs. Electric Field Loops for Different Electric Orderings . . . . . . . 36

1-16 Ferroelectric Bananas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

1-17 Ferroelectric Unit Cell . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

1-18 Ferroelectric Energy Diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

1-19 Antisymmetric Dzyaloshinskii-Moriya Interaction . . . . . . . . . . . . . . . . . 42

1-20 Composite Multiferroic Geometries . . . . . . . . . . . . . . . . . . . . . . . . . 45

1-21 Bi 6s Lone Pair . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

1-22 Magnetoelectric Revival . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

1-23 Magnetoelectric Multiferroic Venn Diagram . . . . . . . . . . . . . . . . . . . . 50

2-1 PLD Schematic and Image . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

2-2 PPMS Sample Chamber Schematic . . . . . . . . . . . . . . . . . . . . . . . . 57

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2-3 Four-Terminal Resistance Geometry . . . . . . . . . . . . . . . . . . . . . . . . 58

2-4 HP4284 Circuitry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

2-5 Dielectric Electrode Circuits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

2-6 Maxwell-Wagner Circuit Simulation . . . . . . . . . . . . . . . . . . . . . . . . . 63

2-7 Interdigital Capacitance Fundamentals . . . . . . . . . . . . . . . . . . . . . . . 64

2-8 Interdigital Capacitance Fabrication . . . . . . . . . . . . . . . . . . . . . . . . 65

2-9 Multi-Frequency Consistency Check . . . . . . . . . . . . . . . . . . . . . . . . 67

2-10 Sawyer-Tower Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 68

2-11 Precision LC Circuitry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

2-12 Remanent Polarization Pulse Sequence . . . . . . . . . . . . . . . . . . . . . . 72

3-1 DC Resistance vs. Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . 75

3-2 Complex Capacitance vs. Frequency . . . . . . . . . . . . . . . . . . . . . . . . 76

3-3 Cole-Cole Plot . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

3-4 Logarithmic Parametric Slope . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

3-5 Competing Dielectric Phase Ansatz . . . . . . . . . . . . . . . . . . . . . . . . 79

3-6 Circuit Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

3-7 Model Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82

3-8 β vs. Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

3-9 ramp vs. Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

3-10 Arrheniuis Plot of Relaxation Time-Scales . . . . . . . . . . . . . . . . . . . . . 84

3-11 Polaron Depiction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85

3-12 Detailed Balance 3 State Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

3-13 ramp vs. Temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88

3-14 Energy/Population Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89

3-15 Thickness Dependence of Dielectric Constatns . . . . . . . . . . . . . . . . . . 91

3-16 Energy/Population Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

3-17 Charge Density Wave Schematic . . . . . . . . . . . . . . . . . . . . . . . . . . 93

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4-1 Magnetic Tuning of Dielectric Constants . . . . . . . . . . . . . . . . . . . . . . 97

4-2 Magnetic Tuning of Activation Energies . . . . . . . . . . . . . . . . . . . . . . 100

4-3 Modeling Results for Multiple Thickness Films . . . . . . . . . . . . . . . . . . . 102

4-4 Strain Dependence of Magnetoelectric Coupling . . . . . . . . . . . . . . . . . 103

5-1 Monoclinic Unit Cell of BiMnO3 . . . . . . . . . . . . . . . . . . . . . . . . . . . 106

5-2 BiMnO3 Θ− 2Θ Scans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108

5-3 Magnetic Characterization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

5-4 Remanent Hysteresis Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

5-5 Frequency Dependence of Imaginary Capacitance . . . . . . . . . . . . . . . . 113

5-6 Arrhenius Plot of Relaxtion Time-Scales . . . . . . . . . . . . . . . . . . . . . . 114

5-7 Temperature Dependence of Real Capacitance . . . . . . . . . . . . . . . . . . 115

5-8 Dielectric Prediction of Ferroelectric TC . . . . . . . . . . . . . . . . . . . . . . 118

5-9 Dual Pulse Sequence Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120

6-1 External Strain Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6-2 Strain Tuning of Ferroelectricity . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

6-3 Electrode “Lensing” . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 126

6-4 Island Strain . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127

6-5 Magnetoelectric Coupling in BiMnO3 . . . . . . . . . . . . . . . . . . . . . . . . 128

6-6 Pulse Sequence in Magnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . 129

6-7 Correlation of Couplings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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

‘SOFT ELECTRONIC MATTER’, MAGNETOELECTRIC COUPLING, ANDMULTIFERROISM IN COMPLEX OXIDES

By

Patrick R. Mickel

August 2011

Chair: Arthur F. HebardMajor: Physics

This thesis has focused on the electronic and magnetic properties of thin-film

oxide crystals. Oxides are home to some of the richest physics in condensed matter,

producing complex features in response to changes in temperature, electric/magnetic

fields, and strain. Three of these features have gained particular prominence, and

are among the most active research topics today: phase separation, magnetoelectric

coupling, and multiferroism.

“Phase separation” describes the state of materials containing neighboring regions

with distinct electronic and magnetic properties - an important phenomenon associated

with some of the most exotic material properties known: colossal magnetoresistance,

multiferriosm, and high-temperature superconductivity. Phase separation is commonly

explained by disorder and strain, resulting in static and stationary phases. However,

It is shown here that competing dielectric phases dynamically transform into one

another over macroscopic lengths and long time periods (10−3 − 10−5 seconds, and 1

µm2), indicating the phases are far from static. These results argue for a fundamental

reinterpretation of the physics of phase separation from localized rigid structures to

wave-like thermodynamic entities.

Magnetoelectric coupling describes the induction of electric (magnetic) polarization

via magnetic (electric) fields, and has myriad applications from sensors to data storage.

Lattice strain is commonly proposed as a mediating mechanism, but these conjectures

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have remained primarily phenomenological. However, this thesis introduces a first

principles strain-based microscopic model that describes the measured magnetoelectric

coupling of competing dielectric phases. The results of this model accurately reproduce

the effects of magnetic fields on the capacitive properties of both dielectric phases,

as well as predict additional results seen in the literature. These results provides a

direct experimental check of strain’s role in magnetoelectric coupling of single phases,

and marks an important step forward in understanding the mechanisms producing

magnetoelectric coupling.

Multiferroics are materials that display two separate orderings, typically spontaneous

magnetization (ferromagnetism) and electric polarization (ferroelectricity). These

materials promise technological revolutions and are arguably the most intensely

researched subject in materials science today. This thesis describes the single-phase

multiferroic, BiMnO3, providing important evidence for the growing debate concerning its

ferroelectric nature. It is shown that BiMnO3 displays relaxor ferroelectricity, and that its

remanent polarization is highly tunable, decreasing by as much as 10% in 7 T magnetic

fields, and increasing by almost 50% under small externally induced strains.

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CHAPTER 1BASIC PHYSICS REVIEW

1.1 Mixed Valence Manganites

1.1.1 Introduction

The basic electric and thermal properties of manganites were first investigated

as early as the 1950’s [1], where a large magnetoresistance was discovered near

the ferromagnetic Curie temperature, TC , producing moderate interest in the physics

community. In 1994, however, research in manganites surged in response to a

new-found phenomena: colossal magnetoresistance, where magnetic fields were

found to induce a decrease in resistance by more than a factor of 103 [2]. The initial

dream was to someday replace the ubiquitous giant-magnetoresistance (GMR) effect,

which had quickly become the standard in the magnetic information storage industry

for read-heads. GMR is based on spin-valve mechanisms and results in a few tens of

percent change in resistance, so colossal magnetoresistance provided a huge potential

for improved device performance.

As research progressed, however, the vision of manganites changed radically from

what seemed a straightforward application in the magnetic-information-storage industry,

to a colossal challenge to condensed-matter physics. Soon after the reemergence of

manganites, it was realized that their initial theoretical description (i.e. double exchange,

see section 1.1.3) was incapable of quantitatively reproducing the new-found colossal

properties, and that these systems were much more complex than originally thought.

Manganites are now thought of as a quintessential complex-oxide system where

the simultaneous interplay of spin, charge, orbital, and lattice degrees of freedom

spawn some of the most complicated and exotic material properties and physics in

condensed-matter today. These properties include: insulator-to-metal transitions,

charge and orbital ordering, ferro and anti-ferromagnetic ordering, charge density

waves, and multiferroism. Additionally, manganites are extrememly sensitive to external

14

perturbations, with temperature, pressure, light, electric and magnetic fields capable of

drastically altering the balance of phases.

1.1.2 Structure and Energy Diagrams

Manganites crystallize in multiple configurations, including the cubic perovskite,

double-perovskite, and hexagonal structures as well as the general Ruddlesden-Popper

series: An+1BnO3n+1. The manganites studied in this thesis, however, are exclusively

“cubic” perovskites with the B site occupied by Mn atoms and the A site occupied

by a variety of ions: La, Ca, and Pr in the mixed valence manganites, and also Bi in

the following multiferroic discussion. Figure 1-1 shows the basic cubic structure of

the manganite unit cell. The A site atoms occupy the corners of the unit cell and act

primarily as charge reservoirs for doping and space fillers for structural integrity. The Mn

atoms occupy the center of MnO6 oxygen octahedra, the network of which is the primary

mediator of electrical conductivity and magnetic structure within the crystal. The A site

atoms indirectly control the electric and magnetic properties by influencing the valence

and bond angles of the Mn atoms (which necessarily affects its magnetic moment).

Specifically, the electronic and magnetic properties of the crystal are controlled by

the 3d electrons of the Mn atom. Therefore, it is useful to consider the energies of these

orbitals. In isolation, the Mn 3d electrons share a five fold degeneracy: the dxy , dyz , dxz ,

dz2, and dx2−y2 orbitals. However, when the Mn atoms are brought near neighboring

ions, the ambient electric fields can alter the energy levels of the electronic orbitals. This

effect is called “crystal field splitting,” [3] and in cubic perovskites it results in splitting the

5 degenerate 3d orbitals into two groups of energy levels: 3 low-energy t2g orbitals and 2

high-energy eg orbitals. Figure 1-2 illustrates this degeneracy splitting. The energy level

splitting can be understood in terms of Coulomb interactions between the O2p electrons

which lie along the x, y, and z axis of the MnO6 octahedra. The Coulomb potential is

largest for orbitals along these axes, raising the energy of the dz2, and dx2−y2 orbitals

(eg), and lowering the energy of the dxy , dyz , and dxz off axis orbitals (t2g).

15

Figure 1-1. In the cubic perovskite manganite structure, the B site Mn atom (red) isencaged in an O octahetron (black). A site atoms (blue) constitute the cubicshell that supports the octahedron structurally. Alternatively, the structuremay be viewed in terms of MnO2 planes that are separated by AO planes,where the A site atoms lie in the same plane as the apical O atoms of eachoctahedron.

Figure 1-2. The 3d orbital energy levels display to modifications: crystal field splittingand Jahn-Teller distortions. On the left are the 5 degenerage 3d electronorbitals for an atom in isolation. When the atom is placed in the presence ofambient electric fields from ions within a crystal structure the degeneracy isbroken (middle section). And when one eg orbital is occupied by a singleelectron, a spontaneous Jahn-Teller deformation lowers the energy of thesytem by ∆EJT by elongating the z-axis of the unit cell and lowering theCoulomb interaction of the dz2 orbital which is aligned with the neighboringO2p orbital. See Figs. 1-4 and 1-3.

16

For certain valence states of the Mn atom, coulomb repulsions also drive a second

splitting of the orbital energy levels. Odd numbered valence states, such as Mn3+, result

in a singly occupied eg orbital, either dz2 or dx2−y2 - both of which are directly aligned

with the neighboring O2p orbitals. In this configuration, the system may lower its energy

by ∆EJT by spontaneously distorting and elongating along the z-axis to decrease the

Coulomb interaction of the dz2 and O2p electrons, thereby decreasing the potential

energy of the orbital (see Figs. 1-2 and 1-3). Because the overall Coloumbic potential

does not increase, the overall volume of the unit cell remains constant, resulting in a

contraction in the x-y plane which increases the Coulomb interaction of the dx2−y2 and

O2p orbital, raising its potential energy. The t2g orbitals are also effected, with the yz

and xz levels now lower than the xy orbitals, again because of the proximity of the O2p

electrons. This spontaneous energy lowering distortion is called a Jahn-Teller distortion

[4], and is depicted in Figs. 1-2 and 1-3. We note here that this energy gain through

spontaneous distortion is not available in Mn4+ systems because all eg orbitals are

empty and so there are no occupied electron states that decrease in energy, since

the total energy of the t2g levels is constant. This type of distortion, that is dependent

on the presence of an electron is called a polaron. A Polaron is a quasi-particle that

encompasses an electron and the induced lattice distortion surrounding it, see Fig. 1-5.

The crystal field and Jahn-Teller energetic adjustments are important because they

have a direct effect on the magnetic and electric transport properties of manganites.

The lowering of the t2g energy levels results in a localized 3/2 spin which can be treated

as a classical core spin that is tied to the lattice, providing a basis for all the magnetic

orderings present. The formation of the Jahn-Teller polaron and lowering of the eg

energy level modifies the electronic conduction by localizing the eg electrons in a “self”

potential well, resulting in a Mott like insulating state - since the conventional band

picture dictates that LaMnO3 (the prototypical Mn3+ manganite) with it’s singly occupied

eg state should be conducting.

17

B)A)

Figure 1-3. A) The oxygen octahedra of the cubic perovskite structure is shown. B) TheJahn-Teller distorsion is shown. The eg orbital is occupied by a singleelectron allowing it to become energetically favorable for the unit cell todistort from a cubic MnO6 octahedra to an octahedra that is elongated alongthe z-axis (Jahn-Teller distorted). This reduces the orbital overlap andresulting Coulomb potential energy between the Mn dz2 and O2p orbitals.See Fig. 1-4 for 3d orbital orientations, and Fig. 1-2 for the orbital energylevel modifications.

1.1.3 Effects of Doping and Cation Substitution

As mentioned above, the A site atoms indirectly control the electronic and magnetic

properties of the crystal by tunning the valence of the Mn atoms. A site atoms bond

ionically to O atoms, donating their electrons, thus obviating the further ionization of

the Mn atoms by the oxygen octahedra. By populating the A site with a combination of

tri- and di-valent atoms, a mixture of Mn3+ and Mn4+ sites can be created, which can

increase the conductivity by allowing the Mn3+ Jahn-Teller polarons to hop to cubic

undistorted/unoccupied Mn4+ cites. While the conductivity does increase, hopping

between Mn3+ and Mn4+ cites is still an insulating mechanism, due to the inherent

energy barrier involved. However, a balance of Mn3+ and Mn4+ sites can also facilitate a

special form of metallic conduction which is particularly important in manganites: double

exchange.

18

t2g:

eg:

Figure 1-4. The 5 3d orbitals are split into two groups: eg(2) and t2g(3). The 2 eg orbitalsare oriented toward the O atoms on the x, y, and z axes, while the 3 t2gorbitals are not. The Jahn-Teller distortion (see Figs. 1-2 and 1-3) causes anelongation of the unit cell along the z-axis and a contraction in the x-y planemoving O2p orbitals further away and closer, respectively. The decreasedoverlap along the z-axis lowers the energy of the dz2, dxz , and dyz orbitals,and raises the energy of the dx2−y2 and dxy orbitals.

Figure 1-5. A polaron is a quasi-particle that is defined by an electron and the “cloud” ofdistortions it induces in the surrounding lattice sites.

19

Double exchange was first proposed by Zener in 1951 [5] and was later reformulated

by Anderson and Hasegawa in 1955 [6], and consists of the simultaneous transfer of

an electron from a Mn3+ site to an O2− site and the transfer of an electron from an

O2− site to a Mn4+ site. In the charge transfer process, the hopping electrons/polarons

are coupled magnetically to the 3/2 core spin of the t2g orbitals through a large Hund

coupling (> 1eV) that energetically requires that the eg electrons have the same spin

orientation as the core spin at the new location. This results in an effective hopping

matrix between the two Mn atoms of the (simplified) form [6]:

ti ,j = t0i ,jcos(θi ,j/2) (1–1)

where θi ,j is the angle between the core spin and the spin of the eg electron (which

should have approximately the same orientation as the core spin at its initial site). This

hopping process also mediates the paramagnetic-ferromagnetic transition, because as

each eg electron/polaron aligns magnetically with the core spin at the new lattice site it

also induces a slight rotation of the core spin so that effectively, as hopping continues

the orientation of the core spins are gradually rotated to align with the core spins of the

neighboring lattice sites. Once the temperature is low enough to limit thermal fluctuation

of the core spins, this process results in the ferromagnetic alignment of all the spins in

the crystal, and induces an almost simultaneous ferromagnetic and insulator-to-metal

transition [7]. An additional consequence of the large Hund coupling is that at low

temperatures half-doped manganites are half metals with near 100% spin polarization of

the conduction electrons, making them prime candidates for spintronic applications [8].

Double exchange provides a basic description of the magnetoresistance observed

in manganites, near TC the induced alignment of core spins by external fields facilitates

hopping thereby increasing the conductivity resulting in the observed negative

magnetoresistance. This description is only qualitative, however, failing to reproduce

20

the magnitude of the effect. To accurately describe manganites, polaronic, Coulombic,

and exchange interactions must all be accounted for [9–11].

In addition to the valence of the A site atoms, their size also has a strong effect on

the electric and magnetic properties of the crystal. Varying the ionic radii of the A site

atoms can induce additional distortions of the MnO6 octahedra by allowing the O-Mn-O

bonds to buckle away from 180o . This effect is commonly quantified using the “tolerance

factor”:

f =< rA > +rO√2(rMn + rO)

(1–2)

where rA, rO , rMn are the ionic radii of the A site atom, O atom, and Mn atom respectively.

As the tolerance factor decreases from 1, the space group and crystal structure can vary

greatly, from cubic to rhombohedral to orthorhombic. Various tolerance factors can also

promote different orbital and charge ordering which in turn support different magnetic

structures: ferromagnetic, and antiferromagnetic (A, C, CE, and G type. See Ref.

[11]). Additionally, buckling the O-Mn-O bond angle encumbers double exchange by

lowering the hopping integral, preventing the alignment of core spins and delaying the

ferromagnetic and insulator-to-metal transitions to lower temperatures or eliminating

them altogether.

1.1.4 La1−xCaxMnO3, Pr1−xCaxMnO3, and (La1−yPry )1−xCaxMnO3

The manganite discussed in this chapter is the mixed valence manganite of the

composition (La1−yPry )1−xCaxMnO3 (LPCMO). LPCMO is a mixture of two parent

compounds, namely La1−xCaxMnO3 (LCMO) and Pr1−xCaxMnO3 (PCMO), each

of which have complex phase diagrams driven by valence and ionic radii changes

produced by cation substitution (discussed in section 1.1.3). LPCMO is composed

of an incommensurate (inhomogeneous) mixture of LCMO and PCMO, therefore to

understand its properties, it is necessary to review each parent compound.

The two limiting compounds in LCMO’s phase diagram (see Fig. 1-6) are the Mn3+

LaMnO3 and the Mn4+ CaMnO3, however, the properties of LCMO are considerably

21

Figure 1-6. The La1−xCaxMnO3 phase diagram shows the effect of Ca doping, whichintroduces Mn4+ into the completely Mn3+ system LaMnO3. Intermediatedopings produce physics not seen in either parent compound (LaMnO3 andCaMnO3):. At intermediate dopings below 50% a ferromagnetic metallicphase is formed at low temperatures, and intermediate dopings above 50%a charge-ordered insulator with a spatial modulation of the chargedistribution on Mn sites is formed (Mn4+ Mn3+ Mn4+ Mn3+... ). Here, CAFstands for canted antiferromagnet, FI for ferromagnetic insulator, and CO forcharge ordered. The CAF and FI could have spatial inhomogeneity with bothferro- and antiferro-magnetic states present. Figure adapted from Ref.[12, 13].

different than a simple interpolation between the properties of the limiting compounds.

LaMnO3 and CaMnO3 are both paramagnetic insulators at high-temperature and

transition to canted-antiferromagnetic insulators at low temperature. The high-temperature

paramagnetic insulating state remains at all compositions, however, at intermediate

mixings LCMO develops entirely new low-temperature electromagnetic phases. As Ca

doping increases to between 5% and 20% the lower-temperature phase transitions

to a ferromagnetic insulator. Then as Ca increases further to between 20% and 50%

the low-temperature phase becomes a ferromagnetic metal, with the development

22

Figure 1-7. The Pr1−xCaxMnO3 phase diagram shows the effects of Ca doping. Cadoping introduces Mn4+ into the completely Mn3+ system PrMnO3, and alsointroduces structural distortions because the ionic radii of Ca is significantlysmaller than Pr. Like La1−xCaxMnO3, Pr1−xCaxMnO3 develops chargeordering at intermediate dopings, but unlike La1−xCaxMnO3, remainsinsulating for all dopings. The PI, PM, and CI denote the paramagneticinsulating, paramagnetic metallic, and canted insulating states, respectively.The FI and FM denote the ferromagnetic insulating and ferromagneticmetallic states, respectively. TC and TN denote the ferromagnetic Curie andantiferromagnetic Neel temperatures, respectively. Figure reproduced fromRef. [14]

of an insulator-to-metal transition mediated by double exchange. Above 50% Ca

doping results in a low-temperature antiferromagnetic charge-ordered insulating phase

where there is a spatial modulation of the Mn charge distribution (Mn4+ Mn3+ Mn4+

Mn3+...). Above 7/8ths Ca doping the system transitions back to the original canted

antiferromagnetic phase. It should be noted that Ca and La have comparable ionic radii,

so the rich physics embodied in LCMO’s phase diagram are primarily the result of the

balance of the mixed valences, Mn3+ and Mn4+.

23

On the contrary, in PCMO, Pr and Ca have not only different valences but

considerably different ionic radii as well. Therefore, as the Ca doping is increased

the tolerance factor of the crystal decreases from 1, promoting distortions of the unit

cell which decrease the Mn-O-Mn bond angle (see section 1.1.3). These distortions

cause a larger alternating tilting of the MnO6 octahedra which reduces the one-electron

bandwidth thereby hindering double exchange [14]. As a result, PCMO’s conduction is

insulating over its entire phase diagram (see Fig. 1-7). However, there are still complex

low-temperature phases that arise with increasing Ca doping. At 15% Ca doping a low

temperature ferromagnetic insulating phase develops, and above 30% Ca doping there

are charge-ordered antiferromagnetic and canted-antiferromagnetic phases. While

PCMO is naturally insulating over its phase diagram, it is important to note that the

application of magnetic fields iss able to “melt” the charge-ordered insulating phases

and induce a low-temperature insulator-to-metal transition [15].

Naturally, the phase diagram of LPCMO is even more complex than the phase

diagrams of its two parent compounds. In this thesis, we focus on the stoichiometry with

an equal mixture of LCMO and PCMO: (La1−yPry )1−xCaxMnO3, with y = 0.5 and x =

0.33. A simplified phase diagram is shown in Fig. 1-8. At low temperatures and for the

composition x = 0.33, LCMO is a ferromagnetic metal and PCMO is a charge-ordered

insulator. Combining these compounds at equal ratios (y = 0.5) results in a coexistence

and competition between these two dissimilar phases. This coexistence has been

termed “phase-separation” and has been shown to occur over quasi-macroscopic length

scales approaching 1 µm. Figures 1-9 and 1-10 show the most convincing evidence of

phase separation in manganites. Figure 1-9 is a dark-field electron-diffraction image

taken at a second-order Bragg reflection peak. The bright spots are the result of the

constructive interference of the spatial modulation of the charge ordering of Mn3+ and

Mn4+, whereas the dark regions are charge-disordered regions which are believed to be

the ferromagnetic metallic phase (regardless of what phase they represent, the image

24

Figure 1-8. The (La,Pr,Ca)MnO3 phase diagram shows a combination of the phasediagrams of the parent compounds (La,Ca)MnO3 and (Pr,Ca)MnO3. The redline denotes the x = 0.33 Ca doping concentration, and the grey box denotesa range of compositions which exhibit phase-separation, however, this workfocuses exclusively on the center of this region at y = 0.5. In the phaseseparated region, the charge-ordered insulating phase of PCMO competeswith the paramagnetic-insulating phase of LCMO at intermediatetemperatures. At low temperatures the ferromagnetic metallic phase ofLCMO competes with the charge-ordered phase of PCMO. Illustrationprovided by Dr.Amlan Biswas.

still demonstrates phase separation between charge ordered and charge disordered

phases on µm length scales). Figure 1-10 is a magnetic-force-microscopy (MFM) image

which provides direct evidence of the percolation and the evolution of ferromagnetic

metallic phase as temperature is swept through the insulator-to-metal transition.

For the composition of LPCMO with y = 0.5 and x = 0.33, at high temperatures

the entire crystal is in the paramagnetic insulating (PMI) phase. Then at intermediate

temperatures phase separation occurs as a portion of the sample becomes charge-ordered

insulating (COI) near 240 K. Finally at low temperatures (below ≈ 115 K for 30 nm films)

the ferromagnetic metallic (FMM) phase is formed, which supplants the PMI phase and

competes with the COI phase. The competition between these three phases (PMI, COI,

25

Figure 1-9. Dark-field images for La5/8−yPryCa3/8MnO3 are obtained by using a super-lattice peak caused by charge order (CO) Panel a shows the coexistence ofcharge-ordered (insulating) and charge-disordered (FM metallic) domains at20 K for y = 0.375. The charge-disordered domain (dark area) is highlightedwith dotted lines for clarity. The curved dark lines present in CO regions areantiphase boundaries, frequently observed in dark-field images for thecommensurate CO states of La0.5Ca0.5MnO3. Panels b and c, obtained fromthe same area for y = 0.4 at 17 K and 120 K, respectively, show thedevelopment of nanoscale charge- disordered domains at T ≈ TC . Thecurved lines in a, b and c signify the presence of anti-phase boundaries ofthe CO domains. Figure and caption reproduced from Ref. [16]

and FMM) has been the subject of intense experimental and theoretical investigation

since their discovery. The percolative onset of the FMM phase has received particular

attention, producing colossal changes in resistance [2] and capacitance [18] by inducing

an early insulator-to-metal transition through its magnetic field dependent stabilization

at higher and higher temperatures. In this thesis, however, we will show that the

competition between the PMI and COI phases is also of fundamental interest, as it

provides a unique perspective into the basic nature of phase separation itself in complex

oxides. By measuring the frequency dependence of the complex capacitance of thin

26

Figure 1-10. A magnetic force microscopy (MFM) image of phase separation is shown.The temperature-dependent MFM image sequence (A) for cooling and (C)for warming, and the resistivity (B) of the La0.33Pr0.34Ca0.33MnO3 thin filmover a thermal cycle. The blue series corresponds to cooling, the red seriesto warming. The center of each image is aligned horizontally with thetemperature scale from (B) at the time of image capture. Scanned areasare 6 µm by 6µm for all cooling images and 7.5 µm by 7.5 µm for allwarming images. All cooling images were taken at one area of the sample,and all warming images were taken at another area. Figure and captionreproduced from Ref. [17]

LPCMO films, we show that the competition between these dielectric phases (PMI, COI)

provides the first evidence for “electronically soft phases.”

27

1.2 Multiferroics

1.2.1 Introduction

Multiferroics - defined as materials possesing at least two ferroic orderings [19] -

have quickly become one of the most widely researched topics in condensed matter

physics today, both for their potential applications and for their complex physical origins.

Ferromagnetism, ferroelectricity, and ferroelasticity are the classic ferroic orders,

however, contemporary focus has placed little emphasis on ferroelastic properties, and

the magnetic ferroic requirements have been broadened to include antiferromagnetism

and ferrotoroidic orderings. The first attempts to combine multiple ferroic properties

into one material started in the 1960’s by Smolenskii and Venevtsev [20, 21]. Initially,

these results inspired moderate interest in the physics community, but multiferroics have

recently undergone an intense renaissance [22–26].

The multiferroic renaissance has been fueled by multiple factors. First in 2000,

a seminal paper highlighted the curious (and inconvenient) lack of overlap between

ferromagnetic and ferroelectric materials, resulting in a dearth of single-phase

multiferroics [27]. This paper in effect issued a grand challenge to materials development

which - thanks to recent advancements in both theoretical and experimental tools - has

been aggressively (and sucessfully) pursued. Experimentally, thin film crystal growth

has progressed significantly since the initial multiferroic interest of the 1960’s, with the

advent of strain engineering through epitaxial lattice mismatch and new high-pressure

growth techniques [28, 29]. Additionally, new experimental techniques for observing

electric and magnetic domains have developed [30]. Theoretically, improvements in

first-principles and density functional theory (DFT) computational techniques have

provided insight into relevant microscopic mechanisms promoting ferromagnetism,

ferroelectricity, and their couplings. Most important, however, is the growing intersection

of experiment and theory in multiferroics - where the newfound attainability of high

quality samples creates synergy through the direct feedback between both disciplines.

28

Finally, the multiferroic renaissance has been motivated by a broad realization of

potential applications for multiferroic materials. Ferromagnets are ubiquitous in the

transformer and information storage industries, and the sensing and actuation industries

rely heavily on ferroelectrics. With the strong trend toward device miniaturization,

the potential of combining multiple functionalities into a single material has made

single-phase multiferroics highly desirable. Multiferroics can also display strong

couplings in their ferroic orders, providing a large design space which will inevitably

lead to higher efficiencies and increased capabilities (see Fig. 1-11). Coupling between

elastic and ferroic properties is widely observed in the form of piezoelectricity and

piezomagnetism, where strain can induce electric or magnetic polarization (and vice

versa). However, the most interesting coupling is between the electric and magnetic

orderings themselves: magnetoelectric coupling - for a complete discussion of

magnetoelectric coupling see Sec. 1.3

This section will cover the basic physics of multiferroics, beginning with the two most

popular ferroic orderings: ferromagnetism and ferroelectricity. This leads to a discussion

of “d0-ness,” and their seemingly incompatible mechanisms. Finally we discuss novel

approaches to combine ferromagnetism and ferroelectricity in a single material.

1.2.2 Ferromagnetism

Magnetism is one of the most ancient physical phenomenon know to man, and was

first discussed scientifically in Greece more than 2500 years ago. Since then, multiple

types of magnetic ordering have been identified: diamagnetism, paramagnetism,

ferromagnetism, ferrimagnetism, antiferromagnetism, and canted antiferromagnetism

(see Fig. 1-12). Diamagnetic systems exhibit magnetizations (coherent orientations

of internal magnetic moments) which can be induced under external fields, with

the magnetization linearly proportional to the magnitude of applied field and aligned

anti-parallel to the field. Paramagnetic systems exhibit induced magnetizations under

external fields, with the magnetization linearly proportional to the magnitude of applied

29

Figure 1-11. The ferroic orders of multiferoics can be controlled by externalperturbations. The electric field E, magnetic field H, and stress σ controlthe electric polarization P, magnetization M, and strain ε, respectively. In aferroic material P, M, and ε are spontaneously formed to produceferromagnetism, ferroelectricity, or ferroelasticity, respectively. In amultiferroic, the coexistence of at least two ferroic forms of ordering leads toadditional interactions. In a magetoelectric multiferroic, a magnetic fieldmay control P or an electric field may control M (green arrows). Figurereproduced from Ref. [24]

field and aligned parallel to the field. For these systems, once the external field is

removed the magnetic moments re-randomize canceling the macroscopic magnetization

(see Fig. 1-12a). In ferroic magnetic systems, however, there is an inherent coupling

between spins that promotes a coherent alignment of the magnetic moments even in the

absence of an external magnetic field, often resulting in a spontaneous magnetization

(see Fig. 1-12b). In antiferromagnetic systems, however, this coupling promotes an

anti-parallel alignment of magnetic moments resulting in zero magnetization (see Fig.

1-12c). Ferrimagnetism is a combination of ferromagnetism and antiferromagnetism,

where sublattices of spins exhibit ferromagnetic coupling internally but antiferromagnetic

coupling to neighboring sublattices (see Fig. 1-12e). By definition these sublattices have

unequal magnetizations (otherwise the system would be antiferromagnetic), resulting

in an over all weak ferromagnetic behavior. Canted antiferromagnetics are frustrated

antiferromagnets where it is energitically favorable for the alignment of the sublattices to

30

Figure 1-12. A sampling of magnetic order is shown. A) Paramagnetism: Magneticmoments are randomized for no net magnetization in zero external field. B)Antiferromagnetism: Magnetic moments are ordered in sublattices whichare anti-aligned with each other resulting in no net magnetization. C)Ferromagnetism: Magnetic moments are aligned parallel producing a largenet magnetization in zero external field. D) Canted-antiferromagnetism:Magnetic moments are ordered in sublattices which are only partiallyanti-aligned, producing a net magnetization (to the right here). E)Ferrimagnetism: Magnetic moments are ordered in sublattices which areanti-aligned, but unequal, resulting in a net magnetization.

skew from the 180o anti-parallel alignment, resulting in a small net magnetization (see

Fig. 1-12d).

At high temperatures, ferroic magnetic systems are typically paramagnetic before

undergoing a time-reversal invariance breaking transition at lower temperatures (the

Curie temperature, TC , for ferromagnetics; the Neel temperature, TN , for antiferromagnetics)

with the onset of magnetic coupling and the manifestation of spontaneous ordering.

Experimentally, in ferromagnets this transition is observed by the opening of magnetization

vs. magnetic field (M-H) hysteresis loops, see Fig. 1-13. Initially the system orders into

local domains which cancel globally to zero macroscopic magnetization. When a field of

sufficient strength is applied, the domains align and remain aligned after the removal of

31

M M

H

M

H H

HC

MS

T > TC T TC T < TC

Paramagnetic (PM) PM-FM Transition Ferromagnetic (FM)

M

H

Diamagnetic

Any T

A) D)C)B)

M M

H

M

H H

HC

MS

T > TC T TC T < TC

Paramagnetic (PM) PM-FM Transition Ferromagnetic (FM)

M

H

Diamagnetic

Any T

M M

H

M

H H

HC

MS

T > TC T TC T < TC

Paramagnetic (PM) PM-FM Transition Ferromagnetic (FM)

M

H

Diamagnetic

Any T

A) D)C)B)

Figure 1-13. M-H loops are shown for different magnetic orderings. A) Diamagnetism:M-H loop is closed, with a linearly induced magnetization that opposes theapplied field. B) Paramagnetism: M-H loop is closed, with a linearlyinduced magnetization aligned with the applied field. C) PM-FM transition:Near TC M-H loops begin to open with the onset of spontaneousmagnetization. D) Ferromagnetism: M-H loops are open, as there isspontaneous magnetization at zero external field (MS). MS can bereorientated under a “coercive” field, HC . Ferrimagnets andcanted-antiferromagnets have M-H loops similar to but smaller thanferromagnets.

the external field resulting in a large remanent spontaneous magnetization, MS (see Fig.

1-13). Ferrimagnets and canted antiferromagnets display reduced magnetic hysteresis

loops, however, pure antiferromagnets (with zero spontaneous magnetization) have no

magnetic hysteresis. Thus, ferromagnets are the most technologically relevant magnetic

materials, and accordingly they have received the most attention in research. Two

phenomenological theories have successfully reproduced many of the properties of

ferromagnetism: the Curie-Weiss local-moment theory, and the Stoner band theory of

ferromagnetism.

32

In 1907, Weiss developed a theory proposing that there existed some internal

“molecular field” which intrinsically aligned the individual magnetic moments of a

ferromagnet (which is now interpreted as an exchange interaction). At high temperatures

the thermal fluctuations were thought to be larger than the alignment energy of the

“molecular field,” resulting in randomized orientations of the magnetic moments and

the observed paramagnetic behavior. Below the Curie temperature, TC , the magnetic

alignment energy dominated, producing a coherent reorientation of the magnetic

moments and creating spontaneous magnetization. The Weiss local-moment theory

predicts the temperature dependence of the magnetic susceptibility for magnetic

materials according to the Curie-Weiss law:

χ =C

(T − TC)γ (1–3)

where C is the Curie constant, T and TC (the Curie temperature) are measured in

Kelvins, and γ is a critical exponent. The Curie-Weiss law accurately captures the

the high temperature behavior of ferro-, ferri-, and anti-ferromagnets, as well as the

divergence in susceptibility near TC in ferromagnets. However, the theory has two

shortcomings, namely: the theory requires the dipole moment at each site be equal in

both the paramagnetic phase and the ferromagnetic phase, and the theory also requires

the moments at each site to correspond to an integer number of electrons - neither of

which is observed experimentally. These contradictions, however, are resolved by the

Stoner band theory of ferromagnetism.

The Stoner band theory of ferromagnets is also derived from an exchange

interaction between magnetic moments. The exchange energy is minimized when

all magnetic moments are aligned parallel, however, opposing this energy gain is the

band energy required for electrons to occupy energy states higher than the nominally

degenerate anti-parallel states. The increased band energy is the primary obstacle to

magnetic order in most materials. For ferromagnetic transition metals, such as Fe, Ni,

33

and Co, the Fermi energy lies in a region of overlap between 3d and 4s orbital bands

causing the valence electrons to partially occupy both bands. The 4s bands have a low

density of states over a large energy range, meaning that to further populate this band

the electrons would have to reach to much higher energy states, as low levels fill quickly.

This large energy cost renders the energy gain from exchange coupling insignificant.

The 3d band, however, has a narrow but large density of states near the Fermi level -

meaning the cost of populating higher energy levels is much lower, and the energy gain

from exchange coupling becomes relevant again.

The exchange interaction can be thought of as shifting the 3d minority spin band

up in energy, see Fig. 1-14. The magnitude of the shift is uniform for all wavevectors,

resulting in a rigid displacement between minority and majority spin carriers. When the

Fermi level lies within the 3d band this results in an increased population of majority

spin carriers, and a spontaneous magnetization: M = µB(n↑-n↓), where n↑ and n↓ are the

majority and minority populations, respectively (see red and blue areas in Fig. 1-14),

and µB is the Bohr magneton. By incorporating Fermi statistics, this model succinctly

resolves the observation that magnetic moments do not correspond to integer numbers

of electrons, as well as the potential change of magnetic moments as energies change

during phase transitions. The model also explains the trend of ferromagnetism in

transition metals: in later transition metals the Fermi level rises above the 3d bands

causing both spin bands to be occupied equally, canceling the net magnetization.

Hence, for transition metal ions, ferromagnetism requires a partially occupied 3d band.

1.2.3 Ferroelectricity

Ferroelectricity was first discovered in 1920, in the Rochelle salt compound

(KNa(C4H4O6)•4H2O) [31], and since then numerous similarities to ferromagnetism

have been documented. Analogous to the energy gain from exchange coupling in

ferromagnetism, ferroelectricity is commonly driven by an energy gain associated

with the hybridization of ionic orbitals. Like in magnetism, there are multiple electronic

34

Figure 1-14. The Stoner Band Theory of Ferromagnetism is depicted. Exchangeinteractions raise the energy of anti-aligned spins, shifting the band ofminority spin-down carriers (blue). This results in an increased populationof majority spin-up carriers (red), and a net magnetization. 3d bandsproduce large magnetizations due to their large density of states, D(E),which produce a large population difference between spins for the smallshift induced by exchange interactions. The 4s band (green) has a lowdensity of states and does not contribute to the magnetization.

orderings: dielectric, paraelectric, ferroelectric, ferrielectric, antiferroelectric, and canted

antiferroelectric. The discussion of these orderings is almost identical to their magnetic

counterparts, with the simple replacement of magnetic dipole moments with electric

dipoles. Dielectric and paraelectric systems display induced polarizations under external

fields, with the dipoles re-randomizing once the field is removed. One small difference

is that the distinction between dielectric and paraelectric is a linearly and non-linearly

induced polarization in external field, respectively - as opposed to linearly induced with

anti-parallel and parallel alignment for diamagnetic and paramagnetic, respectively.

In ferroic electric systems - just as in ferroic magnetic systems - there is an inherent

coupling that promotes a coherent alignment of the dipole moments even in the absence

35

P P

E

P

E E

EC

PS

T > TC T TC T < TC

Paraelectric (PE) PE-FE Transition Ferroelectric (FE)P

E

Dielectric

Any T

A) D)C)B)

P P

E

P

E E

EC

PS

T > TC T TC T < TC

Paraelectric (PE) PE-FE Transition Ferroelectric (FE)P

E

Dielectric

Any T

A) D)C)B)

Figure 1-15. Polarization vs. electric field loops are shown for different electric orderings.Dielectric: P-E loop is closed, with a linearly induced polarization thataligned with the applied field. Paraelectric: P-E loop is closed, with anon-linearly induced polarization aligned with the applied field. PE-FEtransition: Near TC P-E loops begin to open with the onset of spontaneouspolarization. Ferroelectric: P-E loops are open, as there is spontaneouspolarization at zero external field (PS). PS can be reorientated under a“coercive” field, EC . Ferrielectrics and canted-antiferroelectrics have P-Eloops similar to but smaller than ferroelectrics.

of an external field, often resulting in spontaneous polarization. The definitions for

ferroelectric, ferrielectric, antiferroelectric, and canted antiferroelectric are directly

analogous to magnetic ferroic systems, see section 1.2.2 and Fig. 1-12.

The thermodynamic properties of ferroelecctrics are also analogous to magnetic

systems. Ferroelectrics are paraelectric at high temperatures before undergoing

a transition at lower temperatures (also called the Curie temperature, TC ) with the

breaking of spatial inversion symmetry and the onset of long range order among electric

dipoles. The transition can be both first and second order, as both displacive and

order/disorder transitions have been observed (discussed below). As in magnetics, the

36

Figure 1-16. Ferroelectric Bananas. A) Charge versus voltage loop typical for a lossydielectric, in this case the skin of a banana B) electroded using silver paste.The hysteresis loop for a truly ferroelectric material such as Ba2NaNb5O15C) is shown in D) ferroelectric hysteresis curve for ceramic barium sodiumniobate. Figure and caption reproduced from Ref. [32].

most common technique for observing ferroelectric transitions is hysteresis loops, here

polarization vs. electric field (P-E loops), which open near TC . Ferroelectric dipoles

also initially order into local domains, which can be aligned under a strong electric field

resulting in a remanent spontaneous polarization, PS , when the field is removed, see

Fig. 1-15d. The dipole moments in antiferroelectrics are anti-aligned resulting in zero

remanent polarization, while ferrielectrics and canted antiferroelectrics display weak

but open P-E loops. Unlike ferromagnetism, however, ferroelectric hysteresis loops are

constructed from transport measurements, making them susceptible to multiple potential

artifacts such as leakage and dielectric loss. To drive home this point, recently one

researcher humorously demonstrated that transport measurements on a banana could

produce open P-E hysteresis loops similar to those reported in the literature, despite the

obvious absence of inherent ferroelectricty [32].

37

Early work on ferroelectrics centered around Rochelle salt which was useful

for identifying basic properties, however, its complex structure and large number of

atoms per unit cell prevented theoretical progress. Today, perovskite oxides with the

cubic ABO3 structure are the most widely studied ferroelectrics, and their simplified

structure has facilitated a theoretical understanding of fundamental ferroelectric

mechanisms. Below the Curie temperature, perovskite ferroelectrics undergo a

symmetry lowering distortion caused by the off-center shift of their B site cations,

which induces a spontaneous dipole moment (see Fig. 1-17). Ferroelectricity is the

result of a delicate balance between short-range repulsions which favor non-polar cubic

states, and long-range Coloumb forces which stabilize ferroelectric distortions. Density

functional theory has provided a significant contribution to the understanding of this

balance, as it has been clearly demonstrated that the off-center shifts are the result of

the hybridization of B site 3d orbitals with O 2p orbitals, which is essential to weaken

short-range repulsions and lowers the energy of the distorted ferroelectric state (see

Figs. 1-18 and 1-17).

The energy gain associated with this hybridization can also be described analytically

[26, 33]. Upon distortion, the hybridization matrix element tpd modifies to tpd(1 + gu)

where u is the distortion and g is the coupling constant. The first order terms in the

hybridization energy cancel, with the second order approximation producing an energy

gain:

δE ≈ −(tpd(1 + gu))2/∆− (tpd(1− gu))2/∆ + 2t2pd/∆ = −2t2pd(gu)2/∆ (1–4)

where ∆ is the charge transfer gap. This energy gain is depicted in Fig. 1-18a, where

the two O 2p electrons occupy a lower energy hybridized bonding state. However,

the energy gain associated with this hybridization is dependent on the valance of

the 3d orbital. If the 3d orbital contains an electron, then one electron is forced to

occupy the higher energy, anti-bonding hybridized state - lowering the energy gain and

38

Figure 1-17. A ferroelectric unit cell, and its distortions, are shown. The off-center shift ofthe B site cation (hashed to solid red sphere) is a ferroelectric distortionwhich induces a dipole moment (solid black arrow). The hybridization of theO2p orbitals with the 3d orbitals of the B site cation, denoted here by grey Oatoms and connections, provides an energy gain which stabilizes theferroelectric distortion. In perovskites the distortion commonly takes placealong the < 111 > body diagonals, shown here by the green arrows.

destabilizing the ferroelectric distortion. Accordingly, this distortion is called the second

order Jahn-Teller effect: second order because the linear terms cancel, and Jahn-Teller

because the energy gain from the distortion is dependent on the valence of the B site

atom (see section 1.1.2). The distortion’s stability is dependent on the overall energy

gain from hybridization, however, this can be negated by an elastic energy cost. This

point makes the choice of the A site atom particularly important, as its own size and

bonding with O atoms (either covalent or ionic) can tune the elastic properties of the

lattice and therefore the ferroelectricity as well.

As shown in Fig. 1-17, the B site atom can hybridize with three oxygen atoms at

once, with its displacement oriented along the < 111 > body diagonals. Distorting

in either direction along the < 111 > axis and hybridizing with either set of O atoms

39

b)

A) B)

Figure 1-18. Ferroelectric Energy Diagrams. A) This hybridization energy diagramshows the energy gain from hybridizing O2p and B site 3d orbitals when the3d orbital is empty. When the 3d orbital is occupied (shown here withdashed arrows), its electron(s) must occupy the anti-bonding hybridizationstate, lowering the energy gain. B) This potential energy diagram shows thedouble well associated with hybridizing with both sets oxygen along the< 111 > body diagonal.

results in an identical energy gain, leading to a double well potential (see Fig. 1-18b).

This double well results in the characteristic “switchable” ferroelectric states, as opposite

distortions invert the induced dipole moment (and therefore the bulk polarization vector).

In the order/disorder interpretation, the B site cations are displaced along the body

diagonals, producing (microscopic) spontaneous dipole moments at every temperature.

At high temperatures, all eight dipole orientations are stable - which when averaged

across the sample results in zero net polarization. Then near TC the dipoles adopt either

the same orientation (rhombohedral symmetry) or two or three preferred orientations

(tetragoal or orthorhombic symmetry). The order/disorder model therefore predicts

a second order transition. Alternatively, the soft-mode model predicts a first order

transition. In the soft-mode interpretation, B site displacements are only stable below

TC . At higher temperatures, phonon modes provide a restoring force that eliminates the

40

displacement. According to the model, as temperature is reduced the frequency of the

phonon mode “softens,” decreasing to zero at TC resulting in a static ferroelectric lattice

deformation that extends throughout the crystal, and a first order volume change.

When ferroelectricity occurs via an off-center shift of the B site cation, as described

above, it is deemed ‘proper’ ferroelectricity. However, as long as inversion symmetry is

broken, there are multiple ‘improper’ mechanisms which can produce ferroelectricity.

The renaissance of multiferroics has led to the discovery of multiple new ‘improper’

ferroelectric mechanisms: interfacial effects, charge-frustration, bonding distributions,

lone-pairs, and magnetic interactions/orderings. In short-period superlattices it has

been shown that interfacial effects can break inversion symmetry through cooperative

rotations of oxygen octahedra, which produce a spontaneous electric dipole moment

[34]. Charge-frustration based ferroelectricity has been demonstrated in charge-ordered

systems with triangular lattices which introduce geometric frustration, here the

lattice is centrosymmetric, and it is the asymmetric charge distribution that breaks

inversion symmetry [29]. It was also shown that the coexistence of bond-centered and

site-centered charge orders in half-doped Pr1−xCaxMnO3 leads to a non-centrosymmetric

charge distribution and a net electric polarization [35]. Additionally, A site cations with

lone-pairs have been shown to induce ferroelectricity even when their corresponding

B site cations have partially filled 3d orbitals (see Section 1.2.2). However, despite

these promising avenues, magnetic ferroelectric mechanisms have received the most

attention.

Magnetic mechanisms began receiving attention after the popular report of a

spin-flop transition in TbMnO3 where a magnetic field of ≈ 5T was shown to rotate

the ferroelectric polarization 90o from the a-axis to the c-axis, as well as increase the

dielectric constant as much as 500% [28]. Interestingly, these phenomena were linked

to the magnetic frustration. It has been shown that inhomogeneous magnetic order

allows for third-order free-energy terms of the form PM∂M. In cubic crystals this results

41

Figure 1-19. This figure shows the effects of the antisymmetric DzyaloshinskiiMoriyainteraction. The interaction HDM = D12· [S1×S2]. The Dzyaloshinskii vectorD12 is proportional to spin-orbit coupling constant λ, and depends on theposition of the oxygen ion (open circle) between two magnetic transitionmetal ions (filled circles), D12 ∝ λx×r12. Weak ferromagnetism inantiferromagnets (for example, LaCu2O4 layers) results from the alternatingDzyaloshinskii vector, whereas (weak) ferroelectricity can be induced by theexchange striction in a magnetic spiral state, which pushes negativeoxygen ions in one direction transverse to the spin chain formed by positivetransition metal ions. Figure and caption reproduced from Ref. [22]

in polarization of the form [36]:

P = [(M · ∂)M−M(∂ ·M)]. (1–5)

Magnetic frustration induces spatial variations in magnetization, thereby inducing a

polarization through ∂M. Ferromagnetic interactions (J > 0) between neighboring spins,

with antiferromagnetic interactions (J’ < 0) between next nearest neighbors can induce

spiral magnetic states of the form:

Sn = S [cos(qxxn)x+ sin(qyxn)y], (1–6)

42

where q is a wavevector determined by the ratio of ferromagnetic and antiferromagnetic

interactions. In addition to breaking time-reversal symmetry, this ordering simultaneously

breaks inversion symmetry because the change of the sign of all coordinates inverts

the direction of the rotation of spins in the spiral. According to Eq. 1–5, this produces

a polarization orthogonal to both q and z : P || z × q. A likely microscopic mechanism

for this ferroelectric polarization is the anti-symmetric Dzyaloshinskii-Moriya (DM)

interaction. The DM interaction is a relativistic correction to superexchange, and can

be written: Dn,n+1· Sn× Sn+1, where Dn,n+1 is the Dzyaloshinskii vector - which is

proportional to x× rn,n+1, where rn,n+1 is a unit vector along the line connecting the

magnetic ions, and x is the displacement of the oxygen ion from this line. In spiral

magnets, the product Sn× Sn+1 in the DM interaction has the same sign for all pairs and

uniformly pushes negative oxygen ions in one direction perpendicular to the spin chain

composed of positive magnetic ions, thus creating polarization perpendicular to the

chain.

1.2.4 Magnetoelectric Multiferroics

Magnetoelectric multiferroics are materials that are simultaneously ferromagnetic

and ferroelectric - and also display coupling between the ferroic orders (see section

1.3). Accordingly, for a material to display both orderings it is subject to the physical,

structural, and electrical constraints required for both ferroic properties. These

constraints include: symmetry, electrical conduction, and orbital chemistry. Ferroelectric

polarization requires a low symmetry structure which breaks inversion symmetry, and

ferromagnetic polarization requires a low symmetry structure which breaks time-reversal

symmetry. There are 31 point groups which allow spontaneous ferroelectric polarization,

and there are also 31 point groups which allow spontaneous ferromagnetic polarization.

Of these two sets of point groups, 13 overlap - allowing both electric and magnetic

spontaneous polarizations [19]. With the initial number of Shubnikov point groups

at 122, reducing to 13 point groups appears to be a strong limitation. However,

43

considering that there are multiple materials within these 13 point groups which are

not magnetoelectric multiferroics, clearly additional limitations play an important role.

The electrical constraint is that ferroelectric materials must be insulating, as free charge

carriers would immediately screen the spontaneous electric polarizaiton rendering the

ferroelectricity undetectable. This is an important point because empirically almost all

ferromagnets are metallic. Despite these important factors, however, one constraint has

proved to be the most dominant: orbital chemistry.

As discussed in Sections 1.2.2 and 1.2.3 orbital chemistry plays a vital role in the

primary mechanisms for both ferromagnetism and ferroelectricity. In the Stoner band

theory of ferromagnetism, a partially full 3d orbital with exchange interactions between

spins shifts the population balance of spin-up and spin-down electrons resulting in

a net magnetization. Ferroelectricity, however, typically relies on the hybridization of

empty 3d orbitals with O2p orbitals producing an off-center shift of the B site cation which

breaks inversion symmetry and induces a net spontaneous electric dipole moment.

Therefore, the fundamental mechanisms for the two ferroic orderings seem to be

mutually exclusive, requiring both empty 3d orbitals for ferroelectricity (termed d0-ness)

and paritally full 3d orbitals for ferromagnetism. While it has been demonstrated that

ferroelectricity can also be established via ‘improper’ mechanisms (see Section 1.2.3),

it is certainly true that these conflicting processes strongly limit the coexistence of

ferromagnetism and ferroelectricity making single-phase magnetoelectric multiferroics

exceptionally rare.

The first attempts to bypass this mutual exclusion involved constructing elaborate

materials which included separate structural units to produce the individual ferroic

properties. These attempts centered around materials with BO3 groups, such as

GdFe3(BO3)4 and Ni3B7O13I, and were successful in producing multiferroic properties

[37]. However, due to the isolation of the ferromagnetic and ferroelectric components,

coupling between the ferroic orders was extremely limited. With the recent improvement

44

B)A)

Figure 1-20. Composite Multiferroic Geometries. A) Composite multiferroic can begrown in a horizontal laminated structure of epitaxial layers grownsuccessively. b) Composite multiferroic can be also grown in a verticalcolumnar structure with self organizing columns of one ferroic materialinside a parent matrix of another.

in thin film growth techniques, modern efforts have focused on nanoscale heterostructures.

Both horizontal heterostructures composed of alternating layers of ferromagnetic and

ferroelectric compounds, and vertical heterostructures composed of self-assembled

nano-pillars inside parent matrices have been investigated [38, 39]. When the

ferromagnetic and ferroelectric compounds are also piezoelectric and piezomagnetic (or

electrostrictive and magnetostrictive) this geometry produces efficient magnetoelectric

coupling mediated by strain. Here, applied electric (magnetic) fields induce a mechanical

strain in the piezoelectric (piezomagnetic) layers which, through the epitaxial growth

conditions, additionally strains the piezomagnetic (piezoelectric) layers producing

a magnetic (electric) polarization. The composite approach has proven successful

for several niche technological applications such as microwave applications and

high-resolution magnetic field sensors, however, it fails to encompass the full vision

of magnetoelectric multiferroics.

45

Composites fall short in terms of both technology and physics. Technologically,

stacking/mixing ferroic layers and coupling them via strain is incapable of producing the

properties needed for emerging technologies such as spintronics and tunnel junctions.

True magnetoelectric multiferroics are prime candidates for spintronic applications

because they are magnetic insulators in which the band gap may be tuned by the

orientations of the ferroelectric polarization. This is useful because when the band gap

is large, only majority spins can tunnel through the barrier creating a spin-filter which

passes spin-polarized currents. When the gap is reduced, the spin-polarization of the

current decreases and the spin-filter can essentially be switched “off.” Alternatively,

composites cannot emulate this functionality because the ferroelectric components

are not magnetic and spins decohere traveling through these layers. Physically, the

composite shortcut offers little insight into fundamental principles of the compatibility of

ferromagnetism and ferroelectricity. Multiferroics present condensed matter physics a

challenging puzzle, the solution of which could teach us valuable lessons about two of

the most technologically relevant material properties known. Additionally, isolating the

interaction between ferroic properties to the interfaces limits the ability to learn about

inherent couplings between them. Thus, the study of single-phase magnetoelectric

multiferroics is important, both technologically and physically.

Several single-phase magnetoelectric multiferroics have recently been discovered

thanks to ‘improper’ ferroelectric mechanisms (see Section 1.2.3). Frustrated magnets

where the spatial variation in magnetization breaks inversion symmetry have emerged

as a ferroic hotbed. Here, because the ferroelectricity is caused by the magnetic

ordering, it is particularly sensitive to magnetic fields leading these systems to have

displayed the largest magnetoelectric coupling observed in single phases to date.

Unfortunately, the coupling is unidirectional as electric fields have no effect on magnetic

properties, limiting their potential application. However, another approach has also

proved promising: Bi based multiferroics. In Bi based multiferroics, the Bi ion contains

46

Figure 1-21. The isosurface (at a value of 0.75) of the valence ELF of monoclinicBiMnO3 projected within a unit cell is shown. Blue corresponds to almostno electron localization, and white corresponds to complete localization.The projection on one of the cell faces is of the valence ELF, color coded asin the bar by the side of the figure. The view of the crystal is nearly downthe b axis. Figure and caption reproduced from Ref. [40].

6s electrons which do not bond and instead form a stereochemically active “lone-pair.”

The lone pair is extremely polarizable, and helps induce ferroelectric distortions in

the unit cell. BiMnO3 and BiFeO3 are two successful examples of the Bi lone-pair

approach, as both have displayed ferromagnetic and ferroelectric properties, as

well as magnetoelectric coupling. BiFeO3 is popular because its properties exist at

room temperature, however, its limitation is that its magnetization is the result of a

canted-antiferromagnetic state and is quite weak. On the other hand, BiMnO3 displays

true ferromagnetism and ferroelectricity at lower temperatures. Accordingly, BiMnO3 is a

model system to study in order to understand the coexistence of ferroic properties, and

it has been deemed the ‘hydrogen atom’ of multiferroics. The multiferroic portion of this

thesis focuses on the properties of BiMnO3, see Chapters 5 and 6.

47

1.3 Magnetoelectric Coupling

1.3.1 Introduction

Magnetoelectric coupling has also undergone an intense revival [41], as shown

in Fig. 1-22, where the number of publications citing ‘magnetoelectric’ as a keyword

is shown to have increased exponentially over the past 20 years. Initially, interest in

magnetoelectric coupling started modestly, with the first experimental observations

and theoretical predictions dating as far back as the 1800’s. In 1888, Rontgen found

that when a dielectric was placed in motion in an external electric field it became

magnetized - soon followed by the observation of the reverse effect (polarization of

a moving dielectric in a magnetic field) [42, 43]. Then in 1894 Curie provided the

first theoretical description of the potential for static magnetoelectric coupling on the

basis of symmetry. Much later, it was realized that magnetoelectric coupling was

only possible in materials which break time-reversal symmetry, such as: materials in

motion, materials in the presence of magnetic fields, or materials with intrinsic magnetic

ordering. Finally, in the late 1950’s a linear magnetoelectric effect based on the violation

of time-reversal symmetry was predicted by Dzyaloshinskii in a specific static material,

Cr2O3, which was followed shorty by the experimental observation of its electric field

induced magnetization. These findings galvanized the physics community briefly,

however, a general weakness of the coupling, a dearth of systems displaying it, and

a limited understanding of the microscopic mechanisms led to a decreased interest in

magnetoelectric phenomena.

After a 20 year lull, the field has been reignited in response the recent developments

in multiferroic research. Magnetoelectric coupling of as many as 5 orders of magnitude

larger than that observed in Cr2O3 has been achieved in composites of piezoelectric

and piezomagnetic materials, with strain inducing polarizations in each component.

Single-phase multiferroics have also shown great potential, where fundamental

limitations on the magnitude of magnetoelectric coupling have been shown to be

48

Figure 1-22. This figure shows the publications per year with ‘magetoelectric’ as akeyword according to the Web of Science. Figure and caption reproducedfrom Ref. [41]

loosened in these systems (see Section 1.3.3). However, as shown by the venn

diagram of Fig. 1-23, magnetoelectric coupling is not limited to ferroic materials,

and this magnetoelectric ‘momentum’ has also spilled over into additional materials

research areas. In particular, the effect of magnetic fields on the electric properties of

correlated electron systems has become an extremely active area of research. Spin

offs of magnetoelectric couplings termed ‘magnetocapacitance’ and ‘magnetodielectric’

have become ubiquitous, exemplified by report of colossal magnetoresistance and

magnetocapacitance in mixed valence manganites [18]. Magnetocapacitance was even

shown to be possible non-magnetic media as long as it is inhomogeneous [44]. Thus,

the search for magnetic field induced electric polarization has been reinvigorated, and

has expanded to multiple new landscapes.

Finally, it should be noted that the current revival is also due to a long list of

potential technological applicaitons for magnetoelectric coupling. Although many

of these concepts were conceived following the initial research surge, the recent

progress has made the realization of their potential tantalizingly close. In particular,

magnetoelectric coupling could one day lead to the writing and reading of magnetic data

with electric fields, a capability that would decrease the power usage and increase the

49

Figure 1-23. This figure shows the relationship between multiferroic and magnetoelectricmaterials. Ferromagnets (ferroelectrics) form a subset of magnetically(electrically) polarizable materials such as paramagnets andantiferromagnets (paraelectrics and antiferroelectrics). The intersection(red hatching) represents materials that are multiferroic. Magnetoelectriccoupling (blue hatching) is an independent phenomenon that can, but neednot, arise in and of the materials that are both magnetically and electricallypolarizable. In practice, it is likely to arise in all such materials, eitherdirectly or indirectly via strain. Figure and caption reproduced from Ref.[25]

speed of nearly every device involving memory. Additional devices proposed include

high-resolution magnetic field sensors, electrically tunable microwave applications such

as filters, oscillators and phase-shifters, and spintronic applications such as spin-wave

generation, amplification, and frequency conversion.

1.3.2 Maxwell Equations vs. Magnetoelectric Coupling

Classical electromagnetism is one of the most successful and influential branches

of physics in history. In 1865, Maxwell unified the theories of electric and magnetic

fields in four concise equations which now famously bear his name. These equations

predicted and explained an inherent coupling between electricity and magnetism - where

one changing field can induce the other - and successfully describe almost all physical

50

phenomena ruled by one of the four fundamental physical forces: the electromagnetic

interaction. Maxwell’s equations elegantly describe the nature of electric and magnetic

fields in vacuum (the propagation of light), and can even account for electric and

magnetic phenomena in polarizable media as well by introducing constitutive equations

(the electric displacement and magnetizing fields). Therefore, it is quite surprisingly that

the fundamental coupling between electricity and magnetism described by Maxwell’s

equations has little to offer in terms of magnetoelectric coupling.

The crux of the matter is that the coupling in Maxwell’s equations only deals with the

electric and magnetic fields and not the electric and magnetic polarization themselves.

In order to describe and understand magnetoelectric coupling, it is necessary to

understand how the polarization is induced. In solids, magnetic polarization is related

to the spins of electrons in partially filled orbitals and electric polarization results from

the covalently driven shifts of negative and positive ions. The constitutive equations of

classical electromagnetism are not equipped to describe either of these processes.

Quantum mechanics, on the other hand, has all the tools necessary to describe

the manifestations of polarization and their coupling. On the most fundamental level

the exchange interaction combines the electrostatic and magnetic interaction in solids.

The exchange interaction is strictly a quantum mechanical phenomenon which is a

geometric consequence of the symmetrization requirement, and therefore has no

classical counterpart. It determines the extent of the overlap of wavefunctions in real

space, and is also dependent on relative spin states. This directly affects the cooperative

alignment of electron spins and the charge distributions of electronic bonds, thereby

controlling the magnetization and charge separation (electric polarization).

Multiple microscopic mechanisms have been shown to modify these exchange

interactions and thereby indirectly affect magnetoelectric properties: single-ion

anisotropy (∝ (Szi )2), symmetric superexchange (∝ ri ,j(Sxi Syj + Syi Sxj )), antisymmetric

superexchange (∝ ri ,j(Sxi Syj −Syi Sxj )), dipolar interactions (∝ ~mi ~mi/r3i ,j−3( ~mi ~ri ,j)( ~mi ~ri ,j)/r 5i ,j),

51

and Zeeman energies (∝ BgiSi). Strain/stress is also an efficient mechanism to

modulate exchange interactions, because by changing crystal lattice constants the

overlap of electronic wavefunctions is directly affected. This strain can be applied

mechanically or via external fields, as described by Maxwell’s stress tensor.

1.3.3 Free Energy

Theoretically, ME coupling is typically described using Ginzburg-Landau theory,

where a free energy analysis provides a unified temperature and field dependence

of the thermodynamics and stability of the constituent electronic phases. The free

energy is commonly written phenomenologically as a series expansion in powers of the

magnetic and electric fields, ~H and ~E respectively:

− F (~E , ~H) = 12ε0εi ,jEiEj +

1

2µ0µi ,jHiHj + αi ,jEiHj +

βi ,j ,k2EiHjHk +

γi ,j ,k2HiEjEk + · · · (1–7)

where ε0 and µ0 are the permittivity and permeability of free space, εi ,j and µi ,j are the

relative permittivity and permeability, and α and β and γ are the phenomenological

magnetoelectric coupling constants. In this notation, polarizations can easily be

calculated as a function of applied fields,

Pi = (− ∂F

∂Ei) = ε0εi ,jEj + αi ,jHj +

βi ,j ,k2HjHk + · · · (1–8)

µ0Mi = (− ∂F

∂Hi) = µ0µi ,jHj + αi ,jEj +

βi ,j ,k2EjEk + · · · (1–9)

The free energy formalism is convenient because it also serves as an infrastructure

for testing specific microscopic coupling mechanisms, and multiple mechanisms have

been shown to produce ME coupling (see section 1.3.2). These mechanisms can then

be mapped onto the phenomenological expansion for systematic comparisons.

Analysis of the free energy has also provided important insight and guidelines to

researchers. In particular, it was shown that enforcing a stability condition on εi ,j and µi ,j

by requiring the sum of the first three terms in Eq. 1–7 to be greater than zero (ignoring

higher order coupling), the linear magnetoelectric coupling constant, αi ,j , is bounded by

52

the geometric mean of the diagonalized tensors εi ,j and µi ,j such that:

α2i ,j ≤ ε0µ0εi ,jµi ,j . (1–10)

This result is important because researchers knew to only investigate magnetoelectric

coupling in materials with large permittivities and permeabilities. In fact, this is the

reason that multiferroics are the most popular magnetoelectric materials - with

their typically large permittivities and permeabilities, they relax one of the strongest

constraints on magnetoelectric coupling.

53

CHAPTER 2EXPERIMENTAL TECHNIQUES

2.1 Sample Fabrication

2.1.1 Growth Methods

This thesis includes data from two types of thin film complex oxides:

(La1−yPry )1−xCaxMnO3 (LPCMO) and BiMnO3 (BMO). Both films were grown via pulsed

laser deposition (PLD) in Dr. Biswas’ lab. The growth is epitaxial (lattice matched),

where the structure of a thin (0.5 mm) single crystal substrate acts as a template for the

crystal structure of the film. The substrates used in this thesis were NdGaO3 and SrTiO3

for LPCMO and BMO respectively.

The PLD system is composed of a KrF excimer laser (248 nm), a vacuum chamber,

a target material, and a substrate heater. The laser is pulsed on and off, ablating the

target and creating a stoichiometric plume of the material’s elements that extends to just

above the substrate, resulting in the deposition of less than one monolayer per pulse.

The heater supplies the thermal energy necessary to allow the elements to shift into the

most energetically favorable configuration, resulting in the crystalline structure.

Careful optimization of multiple growth parameters was required to obtain

high-quality, stoichiometric, epitaxial, and crystalline samples. This optimization was

primarily performed by Dr. Tara Dhakal and Hyoungjeen Jeen of Dr. Biswas’ research

group for LPCMO and BMO respectively. For the LPCMO films the substrate was heated

to 820 oC in a vacuum of 10−6 Torr, then a partial pressure of oxygen of 450 mTorr was

applied during film growth with a laser pulse frequency of 5 Hz and laser energy of 480

mJ, with a pre-ablation period priming the target before a shutter guarding the sample is

removed. For the BMO films the substrate was heated to 632 oC (a lower temperature

than for LPCMO was necessary to avoid Bi evaporation) under a vacuum of 10−6, then

a partial pressure of oxygen of 37 mTorr was applied during film growth with laser pulse

frequency of 5 Hz and laser energy of 480 mJ. Additionally, the BMO films required a

54

B)A)

Figure 2-1. A) A schematic of PLD chamber is shown B) An image illustrating the plumecreated by the laser ablating the target during film growth (white) is shown.The tip of the outer edge of the plume very nearly coincides with the heater(red/orange, 820C) where the sample is mounted on the substrate, resultingin slow controlled growth. TheiImages were provided by Dr. Biswas.

non-stoichiometric target of Bi2.4MnO3 as well as an quenching oxygen pressure of 680

Torr during cooling. Film thicknesses ranged 30-150nm for LPCMO and 30-60 nm for

BMO.

For the LPCMO samples, a 10-15 nm capping dielectric layer of AlOx was grown

using rf magnetron sputtering of an alumnina target in an ultra-high vacuum chamber

with a base pressure of 10−9 Torr. Both LPCMO and BMO films also had top electrodes

deposited by thermal evaporation in a vacuum of 10−6 of high-quality metals held in

tungsten boats. The LPCMO electrodes were Al, deposited through a shadow-mask

to attain designed sizes and shapes. The BMO electrodes (Cr/Au) were an interdigital

array requiring multiple processing steps which are described in section 2.4.3.

2.1.2 Structural and Compositional Characterizations

The structure and composition of each film was confirmed using various X-ray

diffraction techniques. For LPCMO standard Θ − 2Θ X-ray diffraction measurements

were performed determining the samples were epitaxial and of a single chemical

phase [45]. For BMO, Θ − 2Θ X-ray diffraction and Auger electron spectroscopy

55

were used to analyze crystal structure and composition respectively. The Θ − 2Θmeasurements showed the BMO is free of impurities and grows with a (111) orientation

as expected for the perovskite structure. The Auger spectroscopy found the Bi, Mn, and

O concentrations to be 23.3% 24.1% and 52.6% respectively, close to the expected

1:1:3 ratio for BMO with a slight oxygen deficiency [46].

2.2 Temperature and Magnetic Field Control

In order to study our samples as a function of temperature and magnetic field,

the majority of the data presented in this thesis were measured inside a Quantum

Design cryostat, QD-6000 Physical Properties Measurement System (PPMS). The

PPMS consists of a sample chamber evacuated to a pressure of ≈ 1 Torr by an external

mechanical pump, surrounded by liquid nitrogen, liquid helium, and vacumm chambers,

as well as a 7 T superconducting magnet. A schematic of the cryostat is provided in Fig

2-2.

The PPMS provides temperature control using two heaters, three thermometers,

and a flow control valve separating the liquid He and a mechanical pump. The

mechanical pump pulls cold He gas over a cooling annulus that surrounds the sample at

set rates, providing a temperature range of 1.7 K < T < 350 K. Temperatures below 4.2

K (the boiling temperature of He) are achieved by filling the sample chamber with liquid

He and inducing evaporation via the mechanical pump. The magnetic field is generated

by current in a superconducting coil surrounding the sample chamber, and can range

from -7 T < H < 7 T, with 0.01% accuracy inside the chamber.

The PPMS connects to a local computer via GPIB cables and is controlled by either

Quantum Design’s software interface, Multiview, or by virtual instruments in Labview

programs. The samples are connected to a removable puck which locks into the sample

chamber providing thermal coupling and electrical connection to co-axial cables which

lead to an external ‘break-out box’ where instruments may be attached. When the

56

A) B)

Figure 2-2. A) A schematic of the PPMS is shown, with its insulation layers - N2,vacuum, and He. B) This schematic shows the temperature and magneticfield control design. Figures are reproduced from the PPMS manual and aQuantum-Design brochure.

capacitance stick (discussed below) is used, the electrical connections are made directly

to the stick.

2.3 Resistance

In resistance measurements one has the choice of either sourcing voltage or

current across a sample and then sensing the induced current or voltage. In principle

both of these procedures require only two contacts, with both sourcing and sensing

occurring at the positive and negative terminals. However, if there is a large resistance

associated with the contacts to the sample then there will be a large voltage drop across

the contacts, and in two terminal measurements this leads to artifacts in the measured

resistance. In a four terminal geometry, however, this problem is circumvented by

sourcing and sensing at separate locations. Because the input impedance of the

sensing voltmeter is typically large enough that there is nominally no current in its leads,

57

Figure 2-3. The four-point measurement of resistance is made between voltage senseconnections 2 and 3. The current is supplied via source connections 1 and4. The figure was provided by wikipedia.

the measured voltage drop between the sensing leads is due purely to the current

through the sample. This geometry is illustrated in 2-3. The resistance measurements

discussed in this thesis were made using the four-terminal geometry and a Keithley 220

Current Source and Keithley 2182 Nanovoltmeter. Contacts were made using gold wires

with either silver paint, carbon paint, or pressed indium.

2.4 Capacitance Measurements

2.4.1 Capacitance Bridge and Stick

The capacitance measurements in this thesis were made using a Hewlett Packard

LCR meter, the HP4284. The HP4284 has an internal voltage oscillator that is used to

excite the device under test (DUT), and has a wide range of operational frequencies,

f: 8610 frequencies over the bandwidth 20 Hz < f < 1 MHz. The excitation voltage can

range from ± 20 V, and an external bias can be applied up to ± 40 V. The impedance

measurement is made by measuring the induced current out of the DUT at a point just

below a virtual ground that is created by an “Auto Balancing Bridge,” which undoubtedly

involves operational-amplifiers, but the inner workings of which are not disclosed.

The amplitude and phase of the current with respect to the applied voltage provides a

complex impedance which can be reported in multiple formats which assume various

model circuit configurations. In our work the parallel model (a capacitor in parallel to a

58

Figure 2-4. The schematic shows the circuit design of the HP4284 capacitance bridge,including the 4 terminal geometry, virtual ground, and auto-balancing-bridge.The figure was reproduced from the HP4284 manual.

resistor) is most appropriate, although any chosen output can easily be transformed into

any other format (CS-RS , Z-Θ, etc). Finally, all of our capacitance measurements have

been made inside a custom built capacitance probe, which is designed to be electrically

isolated from its surroundings, with a copper can acting as a Faraday cage to cancel

ambient electric fields that can be sources of noise.

The simplest sample geometry for capacitance measurements is the parallel plate

capacitor where two electrodes of area A enclose an insulating medium with dielectric

constant ε, and thickness d, which stores induced polarization (or, equivalently charge)

according to

C =ε0εA

d(2–1)

where C is the capacitance and ε0 is the permittivity of free space. In our studies,

however, this geometry is not possible because our thin films are grown on insulating

substrates, prohibiting the placement of an electrode below our films. The following

two sections describe techniques which overcome this limitation, one in which the

dielectric is used as the bottom electrode and an additional dielectric layer decloaks its

59

capacitance, and a second one in which a modified top interdigital electrode geometry is

used.

2.4.2 Dielectric Electrodes

A dielectric analysis technique was developed in Dr. Hebard’s lab (prior to this

thesis research) which demonstrated that it is possible to detect the dielectric response

of the bottom electrode in a tri-layer (metal-insulator-metal) parallel plate capacitor [18].

Analyzing the complex impedance of appropriate circuit models, it was shown that if

certain experimental constraints are met, the equipotential planes in the electrode will

be parallel to the electrode-dielectric layer interface, resulting in sensitivity to the c-axis

capacitance of the bottom electrode.

They began by modeling the entire structure as a resistance in series to a lossy and

leaky capacitor, where the series resistance is the a-b plane resistance of the LPCMO

film (see Fig 2-5 a)). The term lossy here refers to a complex capacitor (C ∗ = C1 − iC2)that dissipates energy from dipole reorientations, but does not pass dc current. Thus,

the addition of a shunting resistance in parallel results in a lossy, leaky capacitor that

does pass dc. In our structures the shunting resistance was found to be extremely large

R0 > 1010Ω, whereas at maximum RS ≈ 107Ω, meaning that 99.9% of a dc voltage is

applied across C ∗ and that RS can be ignored.

However, when the voltage is ac there are multiple current paths available (R0 and

R2 = 1/ωC2, see Fig. 2-5 b)). Since we are interested in the dielectric properties, we

choose our frequency range so that R2 << R0 ensuring the current passes through R2.

By measuring at frequencies above this lower bound we ensure that we are sensitive

to C ∗, but as frequency increases the voltage drop across RS (which is frequency

independent) will increase and eventually distort the capacitance measurements.

Analyzing the complex impedance of the circuit, however, provides a set of impedance

constraints that determine when the voltage drop across RS is negligble.

60

A) B)

C) D)

Figure 2-5. A) The circuit equivalent of the two-terminal measurement configurationwhere RS is the series resistance of the LPCMO sample and the parallelcombination of a complex (lossy) capacitor C ∗(ω) with a resistor R0represents the impedance of the LPCMO in series with the aluminum oxidecapacitor is shown. In the two-terminal configuration, the longitudinal voltagedrop across Rs cannot be distinguished from the perpendicular voltage dropacross the parallel combination of C ∗(ω) and R0. B) The decomposition ofC ∗(ω) = C1(ω) iC2(ω) into a parallel combination of C1(ω) and R2(ω) =1/ωC2(ω) is shown. C), The circuit equivalent for the capacitance CP(ω) andconductance 1/RP(ω) reported by the capacitance bridge is shown. D) TheMaxwell-Wagner circuit equivalent for the LPCMO impedance in series withthe Al/AlOx capacitor is shown. The LPCMO manganite film impedance isrepresented as a lossy capacitor C ∗M(ω) shunted by a resistor RM . There isno shunting resistor across CAlOx because the measured lower bound on R0is 1010, well above the highest impedance of the other circuit elements. Thefigure is reproduced from Ref. [18]

61

With the capacitance bridge set in parallel mode, the measured complex impedance

is reported in terms of a parallel capacitance, CP , and resistance, RP (see Fig. 2-5 c)).

Using simple circuit analysis it can be shown that the CP and RP of the model circuit can

be written in terms of the model circuit components as,

CP = C1(R22 (ω)

(R2(ω) + RS)2 + (ωRSR2(ω)C1(ω))2) (2–2)

RP =(R2(ω) + RS)

2 + (ωRSR2(ω)C1(ω))2

(R2(ω) + RS) + ω2RSR22 (ω)C21 (ω))

(2–3)

. If RS is small enough that,

RS << min 1

ωC1(ω),1

ωC2(ω),C2

ωC 21 (ω) (2–4)

then RP ≈ R2 and CP ≈ C1. The constraint can then be rewritten in terms of our

measurement parameters making it directly testable.

RS << min 1

ωCP(ω),1

ωC2(ω),C2

ωC 2P(ω). (2–5)

Once it has been determined that RS has not corrupted the capacitance

measurements, complex plane analysis can be used to ensure that the measured signal

is not a convolution of the electrode and the capping dielectric layer’s capacitances,

but that the measurement of each is separated into isolated regimes of the frequency

spectrum. The complex capacitance, C ∗, is now interpreted in terms of the classic

Maxwell-Wagner circuit, shown in Fig. 2-5 d). The Maxwell-Wagner circuit is commonly

used to account for the effects of contacts in dielectric measurements and is composed

of two leaky capacitors in series - here the manganite capacitance and the AlOx

capacitance. Fig. 2-6 b) shows a parametric plot of the complex impedance of a model

Maxwell-Wagner circuit with frequency the implicit variable. If CM << CAlOx then the

impedances of each capacitor will be isolated in separate frequency ranges as shown in

Fig. 2-6, and high-frequency measurements are guaranteed to be sensitive to only the

62

Figure 2-6. A Maxwell-Wagner circuit simulation is shown. This is a parametric plot ofthe complex imepedance (with frequency the implicit variable) of a simulatedMaxwell-Wagner circuit. Note the three time-scales, one characteristic foreach series component, and one marking the cross-over in the dominantresponse.

dielectric response of the electrode. Because the transverse voltage drop is known to be

negligible, this means the measurement is sensitive to the c-axis capacitance.

2.4.3 Interdigital Capacitance

Interdigital capacitance is a well established dielectric technique enabling

polarization measurements while only connecting to a single side of a dielectric. This

geometry has proved useful for a myriad applications including microwave integrated

devices, optical and surface acoustic wave devices, optically controlled microwave

devices, thin-film acoustic/electronic transducerss, chemical and biological sensors, and

finally dielectric studies on thin films. The geometry consists of interwoven electrode

”fingers” which create fringing fields that penetrate the dielectric and create equipotential

planes between the positive and negative electrodes, thereby creating an effective

capacitance (see Fig. 2-7). We use this technique to study thin film BMO, which is

grown on insulating substrates making the placement of a bottom electrode challenging,

63

A)

B)

Figure 2-7. A) A z-axis view of an interdigital electrode array is shown λ is the spacingbetween electrode fingers of the same voltage. B) An x-axis view of aninterdigical array is shown. The alternating voltage of the fingers results inequipotential planes separating them. Figure reproduced from [47]

and the transverse resistance of which is too large to utilize dielectric-electrode

analysis described in section 2.4.2. Our interdigital electrodes were fabricated using

photolithography and thermal evaporation.

A 3:1 mixture of Shipley 1813 and thinner was spin coated at 4000 rpm for 45

seconds, depositing a 1.5 µm photoresist polymer layer on our BMO samples. The

samples were then pre-baked at 115 C for 45 seconds. Next, the sample was

positioned underneath a photolithography mask with a pattern similar to that shown

in Fig. 2-7 using a Karl Suss MA-6 Contact Mask Aligner. Once aligned the portions

of the sample visible through mask were exposed to UV light for 14.5 seconds, and

subsequently dissolved in AZ300 MIF developer and rinsed in deionized water. Finally

the samples were post-baked at 115 C for 1.5 minutes to harden them in preparation for

further processing. At this point in the process the samples now have open areas in the

shape of the electrodes, which are separated by a “snake” of photoresist that weaves

in-between the electrode fingers.

64

B)A)

Figure 2-8. A) An optical image of gold electrodes is shown. B) A schematic showingrelevant length-scales is shown.

Next, the electrodes were deposited using thermal evaporation. The chamber is

pumped down to a base pressure of 10−6 Torr, and then a large current (≈ 100 Amp)

is passed through a tungsten boat which holds a small amount of high purity metal

until the metal is so hot that it evaporates. The rate of deposition is monitored with

a Infinicon Quartz crystal monitor, and is stabilized near 5 A/s. The electrodes have

two metal layers which are grown sequentially without breaking vacuum, an ultra-thin

layer of Chromium (≈ 2 nm) followed by thin layer of gold (≈ 50 nm). The electrodes

are deposited without shadow-masks, allowing the metal to deposit over the entire

sample. However, the photoresist is sufficiently thick (≈ 1.5 µm) that the metal deposited

on it does not connect to the metal deposited on developed portions. This allows the

metal deposited on the photoresist to be removed smoothly by sonicating the sample

in acetone and dissolving the polymer. Fig. 2-8 shows the final result of our interdigital

electrode fabrication process.

2.4.4 Bandwidth Temperature Sweeps

Preceding the research presented in this thesis, the accepted measurement

procedure of the lab for the frequency and temperature dependence of manganite

capacitance was to measure a single frequency while temperature is lowered, warm

65

the system up, increment the frequency and repeat. Manganites are extremely

hysteretic systems that exist in glassy states for extended temperature ranges, and

have had reports of “blocking temperatures” representing multiple alternative states the

system can be “driven” into. So it was natural to tend toward as smooth and controlled

measurements as possible, however, the time consumed for this procedure prohibited a

more detailed investigation.

The measurement procedure was modified to eliminate the time spent warming the

system with each frequency by continuously performing complete frequency sweeps as

the temperature is lowered, thereby requiring only one temperature sweep for an entire

bandwidth of frequencies. Measuring multiple frequencies during a single temperature

sweep was thought likely to perturb the system. However, frequency values taken from a

temperature sweep where multiple frequencies were measured were compared to single

frequency temperature sweeps spaced across the bandwidth, and showed no such

perturbation. Fig. 2-9 shows the comparison of multiple/single-frequency temperatures

sweeps for frequencies of 500 Hz and 20 kHz, where the two results are identical except

for a slight temperature shift due to different thermal drifts for different temperature

sweep rates.

For reasonable temperature sweep rates of 0.5 K/min it is possible to measure

up to 200 frequencies over the bandwidth 20 Hz < F < 1 MHz, with each frequency

measured every ≈ 1K. Because the frequencies are measured sequentially as the

temperature is lowered, however, each frequency is measured at a different temperature

- which complicates their analysis. To make iso-temperature analysis of the frequency

bandwidth possible, the temperature dependence of each frequency was interpolated

onto a standard 1K step temperature grid in Matlab. Using the interpolated data

set features of the dielectric relaxation spectrum can be identified as a function of

temperature.

66

0 100 200 300102

103

104

105

Loss

Temperature

Single- Continuous-

20 kHz

0 100 200 300

101

102

103

104

Cap

acita

nce

(pF)

Temperature (K)

Single- Continuous-

500 Hz

0 100 200 300102

103

104

Loss

(nS

)

Temperature (K)

Single- Continuous-

500 Hz

0 100 200 300

10-1

100

101

102

103

Single- Continuous-

Cap

acita

nce

(pF)

Temperature (K)

20 kHzA) B)

C) D)

Figure 2-9. The capacitance and loss data for multi-frequency (black) andsingle-frequency (red) temperature sweeps at 500 Hz and 20 kHz arecompared. A) The capacitance is measured at 500 Hz. B) The capacitanceis measured at 20 kHz. C) The loss is measured at 500 Hz. D) The loss ismeasured at 20 kHz.

2.5 Ferroelectric Measurements

2.5.1 Sawyer-Tower Circuit

The Sawyer-Tower circuit is arguably the most widely accepted ferroelectric

characterization technique, and remains the standard to which all other measurement

methods are compared. The Sawyer-Tower circuit is quite simple and consists of

a “Device Under Test” (DUT) placed in series to a high-precision “sense” capacitor

(see Fig. 2-10). The design relies on the fact that capacitors in series maintain equal

charges, so that whatever polarization is induced on the surface of the DUT is also

67

Figure 2-10. The Sawyer-Tower circuit is the standard ferroelectric measurementtechnique. It includes an ac voltage source and a sense capacitor in seriesto a “device under test” (DUT). A voltage measurement on the sensecapacitor, the capacitance of which is well defined, determines thetransfered charge from the DUT. “Back Voltage” is discussed in the text.

induced on the “sense” capacitor. The total transferred charge can then easily be

determined by a simple measurement of the voltage across the “sense” capacitor. The

Sawyer-Tower circuit is typically used with ac voltages supplied by function generators,

with the output read by oscilloscopes.

Despite its simplicity, the Sawyer-Tower circuit has multiple caveats which, if

ignored, can result in systematic artifacts. The first concern is the voltage on the sense

capacitor. Capacitors in series act as voltage dividers, so if the sense capacitor is not

sufficiently large this can decrease the magnitude of the electric field across the DUT.

Also, back voltage is a concern. Back voltage occurs when the applied voltage returns

to zero, and the charge collected by the sense capacitor induces a voltage on the DUT

which is opposite to the maximum applied voltage (see Fig. 2-10). The duration of back

voltage is fleeting, however, its magnitude can be severe enough to reprogram the

domain state of the sample.

68

The second concern when using the Sawyer-Tower circuit is that of parasitic

capacitance. The accuracy of the Sawyer-Tower circuit is tied directly to the known

precision of the sense capacitor, so parasitic capacitances in parallel can compromise

the confidence of the calculations of induced charge. Parasitic capacitances come from

multiple sources including measurement circuitry, and sample cables each of which can

add as much as 10pF.

The final and most important concern when using the Sawyer-Tower circuit is that it

provides no information about the origin of the charge induced on the sense capacitor.

The charge could be due to leakage in the DUT, to the capacitance of DUT, or to the

reversal of ferroelectric domains in the DUT. While the shape of hysteresis curves

can suggest which sources are responsible, ultimately the Sawyer-Tower circuit is not

equipped to discriminate the sources of charge contributions.

2.5.2 Precision LC: Ferroelectric Tester

The ferroelectric measurements in this thesis are made using a Radiant

Technologies Precision LC, Ferroelctric Tester. The Precision LC’s design is essentially

an evolution of the Sawyer-Tower circuit, with the addition of a virtual ground preceding

the sense capacitor (see 2-11). A virtual ground is a negative feedback system that

guarantees that a specific point in a circuit maintains a desired potential, in this case

V=0. The feedback is generated by connecting the specified location to the inverting

input and a ground to the non-inverting input of an op-amp. If the specified location

develops a positive (negative) potential, then the result of the inverting input is to create

a negative (positive) voltage at the output. Because this output is connected to the

specified location this inversion provides feedback that shifts the potential until the two

inputs are equal, and in this case grounded.

The addition of a virtual ground to the circuit design elegantly solves two of the

caveats described in the ”Sawyer-Tower Circuit” section above. By guaranteeing that

the potential just below the DUT is maintained at ground, the issue of back voltage is

69

0 2 4 6 8 100

2

4

6

8

10

Figure 2-11. Precision LC circuitry is reproduced from the Radiant manual.

completely circumvented. Similarly, parasitic capacitances are also greatly reduced

because the return path is always grounded, therefore both sides of the parasitic

capacitances are at ground and no charge can accumulate. Charge only accumulates

on the integrator, which is on the other side of the virtual ground.

2.5.3 Remanent Polarization

In ferroelectrics, the most important physical quantity is typically the remanent

polarization. The remanent polarization is composed of domains of spontaneous

dipoles, the orientations of which are dependent on the voltage history of the sample.

The Sawyer-Tower circuit and Precision LC are sensitive to the remanent polarization

because once a critical voltage (the coercive voltage) is reached the spontaneous

dipoles will flip and align themselves with the electric field. This results in a transfer of

charge which is detected by a voltage across the sense capacitor. This detection is

ambiguous, however, as it is difficult to definitively state the charge is due to domain

reversals and not an alternative source of charge such as resistive leakage or the

displacement current from capacitor charging. The total polarization measured in a

70

typical PE loop can be approximated to be composed as,

Ptot ≈ remanent + capacitance + diodes + resistors. (2–6)

The current explosion of interest in multiferroics has fallen victim to this subtlety on

many occasions, and this point was recently driven home sarcastically in a paper

entitled ”Ferroelectrics go bananas” where the author literally measured a banana to

show that its resistive leakage leads to polarization-vs-electric-field (PE) loops which

resembles those claiming to be ferroelectric in the multiferroic literature. Unequivocally

determining the presence of remanent polarization is therefore crucial for studying

ferroelectrics and identifying multiferroics. By implementing specific pulse sequences

and voltage waveforms the Precision LC is capable of resolving this issue, determining

the contributions of each component to the total transferred charge.

The equation for the total polarization (2–6) may be rewritten in terms of only two

components: remanent polarization, and non-remanent polarization,

Ptot ≈ remanent + non-remanent. (2–7)

The principle that allows these components to be resolved is that the remanent

polarization is sensitive to previously applied voltages, and the non-remanent

polarization is not. By applying presetting pulses that exceed the coercive voltage,

the orientation of the polarization of the ferroelectric domains, and thus their contribution

to hysteresis loops, can be controlled. Figure 2-12 shows two hysteresis waveforms

which utilize this principle. Presetting pulses align the ferroelectric domains along the

direction of the subsequent electric field (for both the positive and negative portions of

the hysteresis voltage waveform), guaranteeing that any charge transferred during the

hysteresis measurement cannot be from ferroelectric domain reversals - the domains

are already oriented along the field. Figure 2-12b illustrates the opposite scenario, in this

waveform presetting pulses align the domains anti-parallel to the following electric fields,

71

A)

B)

Figure 2-12. In the remanent polarization pulse sequence, presetting pulses (blue)precede hysteresis-loop voltage sections (red), guaranteeing either none A)or all B) of the ferroelectric dipoles are available to flip and contribute to themeasured transfer charge.

guaranteeing that all of the domains are available to flip during the hysteresis voltage

sweeps. The non-remanent polarization is the same in both hysteresis measurements,

however, only b) has contributions from ferroelectric domain reversals. Therefore,

subtracting the hysteresis loop of waveform a) from the hysteresis loop of waveform

b) results in a purely remanent hysteresis loop. This measurement procedure is used

extensively in the discussion of multiferroics in this thesis.

72

CHAPTER 3‘SOFT ELECTRONIC MATTER’ IN LPCMO

3.1 Introduction

This chapter will present strong experimental evidence for the existence of

‘electronically soft’ phases in mixed-valence manganites, providing important context

to the debate of the fundamental mechanisms driving phase separation/competition

in complex oxides. Measuring the frequency dependence of the complex

capacitance, we identify signatures of phase separation in the dielectric response of

(La1−yPry )1−xCaxMnO3 thin films, and present an analysis enabling the simultaneous

characterization of both dielectric phases, thereby enabling a spatial and temporal

description of the dynamic competition between these phases over a broad temperature

range. We find that the thermodynamic constraints imposed by detailed balance

strongly support the notion of an ‘electronically soft’ material, as we observe continuous

conversions between micron size dielectric phases with comparable free energies

competing on time scales that are long compared with electron-phonon scattering times.

Phase separation and phase competition are associated with many of the most

exotic material properties that complex oxides have to offer and are found ubiquitously

in high-temperature superconductors [48, 49], spinels [50], multiferroics [51, 52], and

mixed-valence manganites [16, 17]. Accordingly, understanding the fundamental

mechanisms of phase separation/competition is a strong priority for physicists, and

is necessary for the technological implementation of these next generation materials.

Importantly, the findings described in this chapter support a recent theory that predicted

‘electronically soft’ phases composed of ‘charge-density-waves’, challenging the static

disorder and strain based explanations of phase separation. This theory has broad

implications for complex oxides with coexisting and competing phases [53–55], however,

evidence for ‘electronically soft’ phases has yet to be provided.

73

3.2 Transport Properties

3.2.1 Resistance

As discussed in Sec. 1.1.4, at high temperatures LPCMO is composed of two

dielectric phases: the paramagnetic insulating phases (PMI), and the charge-ordered

insulating (COI) phases. Accordingly, the resistance is expected to have insulating

temperature dependence. Figure 3-1 displays the temperature dependence of the

a-b plane resistance R(T ) of a 30 nm-thick pulsed laser deposited LPCMO film (see

Sec 2.1.1). With decreasing temperature, T , the resistance increases smoothly until

an insulator-to-metal transition at T = 115 K, and then decreases as an expanding

ferromagnetic metallic (FMM) phase forms a percolating conducting network at the

expense of the insulating dielectric phases [16, 17]. In bulk LPCMO samples there is

also a signature kink in R(T ) in the range 200-220 K, which signals the temperature

where the COI phase becomes well established [56, 57].

The COI resistance kink is systematically absent in thin films, and this has been

interpreted as the suppression of the COI phase. However, evidence for delocalized

charge-density-waves (CDWs) associated with the COI phase of manganites has

been verified by multiple experimental techniques [55, 58, 59]. The absence of a

charge-ordering feature in the resistance of LPCMO films (see Fig. 3-1) does not mean

that the COI phase is not present in thin films; rather the COI transition is smeared by

the inherent disorder and strain of thin films [58], analogous to similar behavior in CDW

systems doped with large impurity densities [60, 61]. In the following sections, further

evidence for the CDW nature of the COI phase will be presented.

3.2.2 Complex Capacitance

Advantageously our complex capacitance measurements demonstrate an increased

sensitivity to dielectric phases compared to dc resistance measurements. The dielectric

measurements are made using the trilayer configuration discussed in Sec. 2.4.2 in

which the manganite film serves as the base electrode. If the series resistance of the

74

100 150 200 250 300104

105

106

107

R (

)

Temperature (K)

TIM

Missing COI "kink"

Figure 3-1. The a-b plane resistance (measured upon cooling) shows a pronouncedpeak at the insulator-to-metal transition, TIM ≈ 115 K, but lacks a COIassociated anomaly seen in bulk manganites in the temperature range200-250 K [56, 57, 62].

electrode is not too large and if the leakage across the AlOx dielectric is negligible,

the equipotential planes will be parallel to the film surface, thereby placing the c-axis

capacitance of the electrode in series with the insulating dielectric layer, effectively

decloaking the smaller electrode capacitance [18]. Using this technique, we measure

the complex capacitance of 30 nm thick (La1−yPry )0.67Ca0.33MnO3 films (capped by

AlOx dielectric layers) over the bandwidth 20 Hz to 200 kHz, and the temperature range

100 K < T < 300 K using an HP4284 capacitance bridge in parallel mode (where

the impedance is reported in terms of a model comprising a resistor in parallel with a

capacitor).

The complex capacitance was sequentially sampled at 185 frequencies spaced

evenly on a logarithmic scale across our bandwidth as the temperature was lowered

at a rate of 0.1 K/min, thus guaranteeing a complete frequency sweep every 0.25 K.

The capacitances of individual frequencies were then interpolated onto a standard

75

102 103 104 105 10610-11

10-10

10-9

(rad/s)

C''

B)10-12

10-11

10-10

10-9

10-8

A)

C'

T = 200K

Figure 3-2. The frequency dependence of the complex capacitance is shown toqualitatively match typical dielectric relaxations. A) The real capacitanceshows a low frequency plateau, and decreases logarithmically at highfrequency. B) The imaginary capacitance shows a loss peak with logarithmicdecrease on both sides.

temperature grid with steps of 1 K for each frequency, allowing each dielectric spectrum

to be analyzed at constant temperature. As a check, the interpolated capacitance values

from the multiple-frequency temperature sweep were compared to single-frequency

temperature sweeps at several representative frequencies across the bandwidth, and

were found to be identical. Warming runs were also performed with no qualitative

change in model parameters other than a hysteretic shift in temperature. See Sec. 2.4.4

for details.

76

10-12 10-11 10-10 10-9 10-810-11

10-10

10-9

10-8

C'' (

F)

C' (F)

Figure 3-3. In the Cole-Cole representation of the complex capacitance, the imaginarycapacitance is plotted parametrically as a function of the real capacitancewith the measuring frequency, ω, varied as the implicit parameter.

At first glance our complex capacitance data (where C = C ′ − iC ′′) displays all of

the qualitative features of a dielectric relaxation (see 3-2): in the real component there

is a low frequency plateau with minimal dispersion, followed by a logarithmic decrease

at high frequencies, while the imaginary component displays a clear loss peak with

logarithmic decreases on either side. However, when the dielectric data is analyzed

quantitatively, it is not consistent with dielectric theories describing single phases.

A common convention for analyzing complex capacitance data utilizes the

Cole-Cole representation. In Cole-Cole plots, the imaginary capacitance is plotted

parametrically as a function of the real capacitance, with the measurement frequency

varied as the implicit parameter, allowing for a simultaneous analysis of both

components. Figure 3-3 shows a Cole-Cole plot of our complex capacitance data

at 200 K. To compare our data to standard dielectric theories, we further calculate

the logarithmic parametric slope, (∂(ln C ′′)/∂ω)/(∂(ln C ′)/∂ω), at each point of Fig.

3-3. Figure 3-4 shows the calculated logarithimc parametric slope as a function of

77

103 104 105 1060.0

0.4

0.8

1.2

C'/C

''(C

''/C

')

UDR

Cole-Cole

(rad/s)

T = 200K

Figure 3-4. The logarithmic parametric-slope, (C ′/C ′′)(∂C ′′/∂C ′) (green), is shown todiffer from the Cole-Cole [64] dielectric response and ‘Universal’ dielectricresponse [63].

frequency, compared to a Cole-Cole dielectric response function, and ‘Universal’

dielectric response (UDR). Cole-Cole dielectric functions have the form:

ε(ω) = ε∞ +ε0 − εinf

1 + (iωτ)1−α, (3–1)

where τ is a characteristic relaxation time-scale, α is a time-scale broadening, and

the difference between the zero frequency and infinite frequency dielectric constants,

ε0 − εinf , determines the amplitude of the relaxation. UDR refers to the ubiquitous trend

in dielectrics that at high frequency both components share a fractional power-law

dependence in frequency: ∝ ωn−1 [63]. UDR results in a logarithmic parametric slope of

1, as both components are changing with frequency at the same rate.

As seen in Fig. 3-4, our data are inconsistent with both dielectric theories (only

slopes greater than zero are shown for clarity). The Cole-Cole response increases

monotonically toward the UDR slope of 1, while our data displays a high-frequency

non-monotonic anamoly, and then appears to saturate for an extended bandwidth.

78

103 104 105 1060.0

0.4

0.8

1.2

C'/C

''(C

''/C

')

(rad/s)

PMI

COI

Figure 3-5. The logarithmic parametric-slope, (C ′/C ′′)(∂C ′′/∂C ′) (green), is shown todiffer from the Cole-Cole [64] dielectric response, providing a signature ofmultiple phases.

The following sections will demonstrate that both of these features are the result of the

first order competition between the PMI and COI dielectric phases [65]. Furthermore,

characterizing the temperature dependence of their competition reveals a highly

correlated collective transport mode of the COI phase domains, similar to the ‘coherent

creep’ preceding ‘sliding’ in CDW systems [66].

3.3 Competing Dielectric Phases

It appears that there are two dominant time-scales in the logarithmic parametric

slope of the dielectric response. The first time-scale is centered at the loss peak, where

the logarithmic parametric slope is zero (locally C ′′ does not change with frequency

there), and the second time-scale is centered at the high-frequency non-monotonic

anomaly. Knowing that there are two competing dielectric phases at high temperatures

in bulk samples motivated a simple hypothesis: one dielectric phase’s relaxation is

centered at the loss peak, and the other dielectric phase’s relaxation is centered at the

high frequency anomaly. This hypothesis is illustrated in Fig. 3-5.

79

3.3.1 Modeling

Figure 3-6 displays a circuit model that was developed which takes this hypothesis

into account. The circuit is composed of three parallel components all in series to

fractional areas of the AlOx dielectric layer: aCOI , aPMI , and aR (the fractional area of the

FMM phase, which acts as a resistive short at low temperatures), with the constraint∑ai = 1. The series resistance is an artifact of measuring the dielectric response

of an electrode in a MIM structure, and is discussed in detail in Sec. 2.4.2. Placing

the capacitances of each phase in parallel requires that the domains of each phase

span the film thickness, thus obviating a series configuration. Our film thickness of

30 nm, however, satisfies this requirement for the phase domains of manganites, the

length-scales of which are known to be on the order of microns [17]. Above TIM , aR ≈ 0,and the circuit is dominated by CPMI and CCOI in our frequency range, so that the

total complex dielectric response may be approximated by the superimposition of two

Cole-Cole dielectric functions,

C ∝ ε(ω) = ε∞ +APMI

1 + (iωτPMI )1−α+

ACOI1 + (iωτCOI )1−β

, (3–2)

where the amplitudes Ai are the product of the fractional area ai and dielectric constant

εi of each phase, i.e., Ai = aiεi , and ε∞ is the sum of the infinite frequency response of

each dielectric phase.

We model our complex capacitance data with Eq. 3–2 at fixed temperatures in 1 K

steps between 100 K and 350 K by varying ω = 2πf over 185 frequencies. The fits are

produced by simultaneously minimizing the difference between the measured complex

capacitance and both the real and imaginary parts of Eq. (2). In the low-frequency limit,

ε(0) ≈ APMI + ACOI , allowing a fitting variable to be eliminated by reparameterizing the

dielectric amplitudes in terms of their ratio, ramp = ACOI/APMI , and the measured ε(0),

i.e., APMI = ε(0)/(1 + ramp),and ACOI = ramp ε(0)/(1 + ramp). As τPMI is determined from

80

aCOIaPMI

RmetRS

CPMI & COI

CAlOx

~

aR

Figure 3-6. In our circuit model, three parallel components, CPMI , CCOI , and Rmet , are inseries to fractional areas (aPMI , aCOI , and aR respectively) of the AlOx layer.Above the insulator to metal transition temperature, TIM , aR ≈ 0.

the loss peak frequency (see Fig. 3-7), five free variables are determined by the fits: ε∞,

ramp, α, β, and τCOI .

Figure 3-7 shows a typical fit to Eq. 3–2 (green curve) where the average relative

error is less than 10−3 for each temperature. Also shown are the individual relaxations

of each phase, the low frequency PMI phase (red) and the high frequency COI phase

(blue) (their identifications are discussed below). Importantly, Fig. 3-7 explains the

high-frequency non-monotonic anomaly in the logarithmic parametric slope. At low

frequencies, the PMI phase dominates both channels, and at high frequencies the COI

dominates in the real channel, but not the imaginary channel. This mixing of relaxation

components at high frequencies results in the logarithmic parametric-slope behaviour

seen in Figs. 3-4 and 3-5, and thus are a signature of phase separation between two

separate and readily identifiable dielectrics.

3.3.2 Temperature Dependence of Model Parameters

The temperature dependence of the model parameters provides a wealth of

physical knowledge, allowing us to identify the individual dielectric phases and

determine the microscopic mechanisms that manifest their material properties.

Dielectric broadening provides a measure of the correlations among relaxors, and

81

102 103 104 105 106

10-12

10-11

10-10

10-9

COI

PMI

(rad/s)

C''

B)10-13

10-12

10-11

10-10

10-9

10-8

A)

Data PMI Model COI

C'

T = 200K

Figure 3-7. A) The measured real capacitance (black) is compared with fits to Eq. 3–2(green) at T = 200 K. B) the imaginary capacitance is compared. Theindividual capacitances of the PMI (red dash) and COI (blue dot) phases arealso shown for both components.

it is known that the COI phase is a highly correlated and ordered phase. Thus, it is

expected that the broadening of the COI phase should increase as the phase forms.

The temperature dependence of β confirms this expectation (see Fig. 3-8), matching

the known temperature dependence of the COI phase - where COI nanoclusters are

reported in a related material to appear near 280 K with the phase fully developed below

240 K [62] - while α remains relatively constant, thereby identifying the high-frequency

response as the COI dielectric phase.

The ratio of dielectric amplitudes, ramp = aCOI εCOI/aPMI εPMI , corroborates this

identification. Since it is known from the resistance measurements that the COI phase

82

100 150 200 250 300

0.00

0.05

0.10

0.15 (COI)

Temperature (K)

&

(PMI)

Figure 3-8. The temperature dependence of β (blue) is shown to match the temperaturedependence of the correlations of COI phase, where COI nanoclusters arereported in a related material to appear near 280 K with the phase fullydeveloped below 240 K [62], identifying the high-frequency relaxation as theCOI phase. α (red) is shown to display limited temperature dependence inthis range.

is not the dominant phase (absence of COI kink), we would expect that the area of the

COI phase would be a small minority. As shown in Fig. 3-9, ramp, is on the order of a few

percent. As a caveat, this argument requires that the dielectric constants of each phase

are comparable, but this is proved to be accurate in the following sections. Furthermore,

the ratio demonstrates that the COI area is increasing in the temperature range where

the dielectric broadening tells us the COI phase is forming. Thus, the model provides a

consistent identification of the PMI and COI phases.

Figure 3-10 shows Arrhenius plots of τPMI and τCOI over the temperature range

100 K < T < 350 K. Surprisingly, over the linear regions, the activation energies of

each phase are nearly equal, with EA(PMI) = 117.9 ± 0.2 meV and EA(COI) = 118.6 ±0.3 meV. These values are consistent with small polarons, the known conduction and

polarization mechanism in manganites [67]. A polaron is a quasi-particle that includes

an electron and the lattice distortion caused by its presence (see Fig. 3-11). Polarons

absorb thermal fluctuations (phonons) allowing their charge, in this case electrons, to

83

100 150 200 250 300 3500.0000

0.0125

0.0250

Temperature (K)

r amp

Figure 3-9. The temperature dependence of ramp is shown to match the temperaturedependence of the COI phase, where COI nanoclusters are reported in arelated material to appear near 280 K with the phase fully developed below240 K [62], and identifying the high-frequency relaxation as the COI phase -consistent with the dielectric broadening data (β).

2 4 6 8 10

10-6

10-5

10-4

10-3

10-2

PMI

COI

1000/T (K-1)

PM

I&

CO

I

Figure 3-10. The Arrhenius plots are shown for τCOI and τPMI , with EA ≈ 118 meV in thelinear region for both dielectric phases, which identifies the polarizationmechanism as small polarons. The nearly identical EA’s suggest thephases share a single energy barrier.

hop to new spatial locations, where the local lattice site subsequently relaxes to the

(lower energy) distorted state. In manganites, this process is adiabatic where electrons

hop quickly and the lattice relaxes slowly around the relocated electron with a time-scale

at least as long as the electron hopping time-scale [67].

84

Figure 3-11. A polaron is a quasi-particle that includes an electron and the surrounding“cloud” of lattice distortions induced by its presence.

3.4 ‘Soft Electronic Matter’

3.4.1 Polarons and Detailed Balance

The strikingly similar activation energies of the two phases suggests the relaxations

are coupled, possibly sharing a common energy barrier. Crossing this energy barrier

would result in the two phases converting into each other. In our system, however,

each dielectric also polarizes independently without converting into the other phase.

Therefore, the phases would have to be connected through a common excited state

from which relaxations can occur to either phase. This common state is consistent

with adiabatic polaron hopping in manganites [67], where the lattice relaxes slowly in

response to fast electronic hopping.

We model this process in our samples by the three state system shown in Fig. 3-12.

The electrons of the polarons of both dielectric phases absorb thermal fluctuations that

activate them over their hopping barriers to an equivalent “excited” state: a relocated

electron surrounded by a lattice site that has yet to relax. The new lattice site has some

initial distortion (either PMI or COI), but as it accommodates the new electron it can

transform/relax into distortions that correspond to either dielectric phase. The electronic

hopping happens at characteristic rates which we measure directly from loss peak

positions in the complex capacitance (γPMI = 1/τPMI , and γCOI = 1/τCOI ). The lattice site

relaxation, however, occurs at unknown rates, γE and γ′E , for the PMI and COI phases

85

COI

PMI

E

E

COI

PMI

(excited)

'

Figure 3-12. An energy level schematic of a three state system describing the polaronhopping process and detailed balance is presented. Polarons of bothdielectric phases absorb thermal fluctuations and hop to an excited state attheir characteristic rates, γPMI and γCOI . The excited state is equivalent forboth phases, corresponding to an electron surrounded by an undistortedlattice. The lattice can then relax into distortion states that correspond topolarons of either dielectric phase.

respectively. This process effectively results in two channels, one in which polarization

is manifested independently in each phase by polarons relocating without altering their

distortions, and one in which polarons relocate as well as transform their distortion state.

Since the equilibrium populations of each phase are constant in time, the rate

equations for the three state model in Fig. 3-12 are given by,

∂

∂t

nPMI

nE

nCOI

=

−γP γE 0

γP −(γE + γ′E) γC

0 γ′E −γC

nPMI

nE

nCOI

=

0

0

0

, (3–3)

where nPMI , nCOI , and nE are the populations of each state. Solving this system of

equations at equilibrium (the rightmost equality) results in a detailed balance equation of

the form,

nCOI (γCOIγE) = nPMI (γPMIγ′E) , (3–4)

86

where (γCOIγE ) and (γPMIγ′E ) are the effective transition probabilities of each phase.

Although the populations of each phase are time-independent at equilibrium, they still

have inherent temperature and energy dependences governed by Boltzmann statistics.

The populations of each phase may be written in terms of their ground-state population

and an exponential factor,

nPMI = n0PMIe

−EPMI /kT

nCOI = n0COIe

−ECOI /kT(3–5)

where EPMI and ECOI are the configuration energies of each phase. We stress here the

distinction of EA(i) and Ei with (i = PMI ,COI ). Ei is the configuration energy of the

phase, and EA(i) is the energy barrier to hopping, or equivalently the energy difference

between the current polaron state and the excited energy state: EA(i) = Eexcited − Ei with

(i = PMI ,COI ). The detailed balance equation may then be rewritten as,

(n0COI/n0PMI )e

−∆E/kT = (τCOI/τPMI )(γ′E/γE) , (3–6)

where ∆E = ECOI − EPMI is the difference in configuration energy between phases.

3.4.2 Testing Detailed Balance Constraints

By making the physically reasonable Ansatz that the ratio of populations is equal to

the ratio of fractional areas (i.e., volumes for constant thickness),

(n0COI/n0PMI )e

−∆E/kBT = aCOI/aPMI , (3–7)

our circuit model provides a direct test of the detailed balance constraint of Eq. 3–6.

This Ansatz (Eq. 3–7) combined with the detailed balance result (Eq. 3–6) allows us

to eliminate the exponential factor and obtain the result that the equilibrium ratio of

fractional areas can be written in terms of the product of two ratios of transition rates,

i.e.,

aCOI/aPMI = (γ′E/γE)(τCOI/τPMI ) = (γ

′E/γE)(γPMI/γCOI ) . (3–8)

87

100 150 200 250 300 3500.000

0.025

0.050

0.075 r ramp

Temperature (K)

r amp &

r

Figure 3-13. The temperature dependence of ramp is shown to mirror that of rτ intemperature (with a constant offset factor), confirming our use of thedetailed balance equation.

We are able to verify this equation constrains our system because our model

determines the variables on either side to within a constant factor: rτ =τCOI/τPMI and

ramp = (aCOI/aPMI )(εCOI/εPMI ). Since detailed balance does in fact constrain our system,

then the independently determined ratios should have a constant offset factor over the

measured temperature range of, rτ/ramp = (εPMI/εCOI ) (assuming that γ′E/γE ≈ 1,which is discussed below). Figure 3-13 shows the temperature dependence of the

independently determined ratios, rτ and ramp. The two ratios follow a similar trend with a

ratio of ratios,

rτ/ramp = (τCOI/τPMI )(aPMI/aCOI )(εPMI/εCOI ) ≈ 2.4 (3–9)

which is constant within ±10% in the ranges 100 < T < 250 and 270 < T < 350,

with a small deviation near 260K which we ascribe to a modification to the ratio γE/γ′E

caused by the increase of correlations in the COI phase. Therefore, the temperature

dependence of the ratios differs only by a constant factor, thus confirming that detailed

balance constrains our system.

The combination of Eq. 3–6 and Eq. 3–7 together with the result that γ′E/γE ≈ 1leads to the relation aCOIγCOI ≈ aPMIγPMI which with the normalization, aCOI + aPMI = 1,

88

ECOI

T < 135K

ECOIEPMI

135K < T< 235K

ECOIEPMI

EPMI

T > 235K

Figure 3-14. A schematic depiction of the temperature evolution of detailed balance andthe energy barriers separating the PMI (red) and COI (blue) dielectricphases is shown. At high temperatures ECOI decreasing while EPMI isconstant, causing its population to increase. At intermediate temperatures,the energies of both phases are equal leading to balanced populations.Then at low temperatures, EPMI increases while ECOI is constant, whichincreases the population of the COI phase.

gives the particularly simple relations,

aCOI ≈ γPMI/(γCOI + γPMI ),

aPMI ≈ γCOI/(γCOI + γPMI ),(3–10)

for the fractional areas occupied by each phase.

The above equations in combination with the schematics and data of Figs. 3-13

and 3-14 present a physically intuitive picture of the evolution of competing phases

with decreasing temperature. In the higher T region (T > 235 K) where ECOI is at its

maximum value, the PMI phase is trapped in a deeper well (∆E = ECOI − EPMI > 0),

implying that γPMI << γCOI (Eq. 3–6) and aCOI << aPMI (Eq. 3–7 and 3–10). As T is

lowered, ∆E decreases faster than 1/T and the COI phase becomes more prominent

with a smaller hopping rate, and its population increases proportional to e−∆E/kT

89

(polarons remain in the state longer because of a deeper potential well). In this limit,

Eq. 3–10 shows aCOI ≈ γPMI/γCOI and aPMI ≈ 1 − γPMI/γCOI , confirming that the

fractional area occupied by the COI phase is substantially smaller than that of the

PMI phase. Then at intermediate temperatures, the two phases are at equal energies

(∆E = 0) over the surprisingly large temperature range of ∼ 100 K, making their

populations temperature independent. Within this region the energy barrier to escape

to the excited state is the same for each phase (118 meV) (Fig. 3-10. Finally, at low

temperatures the PMI phase destabilizes and ∆E becomes negative as EPMI increases

relative to ECOI . As a result the COI’s population increases proportional to e−∆E/kT as

seen in Fig. 3-13.

3.4.3 Lattice Relaxation Rates

Thus far, we have assumed that the ratio of relaxation rates from the excited state

was constant in temperature, and approximately equal to 1. In this section we will

discuss both of these claims. We gain additional insight into the lattice relaxation rates

by substituting Eq. 3–8 into Eq. 3–9 to obtain the result,

εPMI/εCOI = 2.4(γ′E/γE) . (3–11)

The implication of this result becomes apparent by recalling that in our modeling of

the total dielectric response as a superposition of two Cole-Cole dielectric functions

(Eq.3–2), we assumed the dielectric constants εi of the COI and PMI phases to be

temperature independent with all of the temperature dependence in the numerators

of the respective dielectric functions subsumed into the fractional areas ai = Ai/εi .

Accordingly, Eq. 3–11 then tells us that the ratio of lattice relaxation rates γ′E/γE is a

temperature-independent constant which for simplicity we assume is unity (discussed

further below). With this choice equations 3–7 and 3–8 can be combined to give

(n0COI/n0PMI )e

−∆E/kT = aCOI/aPMI = (τCOI/τPMI ) = (γPMI/γCOI ) , (3–12)

90

30 60 90 120 1500

15

30

45

Film Thickness (nm)

PM

I &

CO

I PMI COI

Figure 3-15. The dielectric constants, calculated from the dielectric amplitudes andareas of each phase, εi = Ai/ai , are shown to saturate near their bulkvalues once substrate strain is relaxed. The agreement with bulk datavalidates the areas calculated assuming equal lattice relaxation rates,γE ≈ γ′E .

hence confirming that rτ = τCOI/τPMI does, in fact, represent the ratio of areas.

Knowledge of the ratio of fractional areas aCOI/aPMI and the ratio of dielectric

amplitudes ACOI/APMI together with the respective constraining normalizations, aCOI +

aPMI = 1 and ACOI + APMI = ε(0), allow an experimental determination of the respective

dielectric constants εCOI = ACOI/aCOI and εPMI = APMI/aPMI . Figure 3-15 shows the

dielectric constants determined in this manner for four films with thickness ranging from

30 nm to 150 nm. The dielectric constants of each phase increase and saturate near

their known bulk values [68–70] as the substrate strain relaxes. In manganites grown

on NGO, the film is relaxed at a thickness of d ≈ 100 nm [15, 18] in agreement with the

saturation of the data in the figure. The agreement of the data with these expectations

tends to validate our assumption that γ′E/γE ≈ 1.3.4.4 Charge Density Waves

Interestingly, our data also present strong evidence for the existence of

charge-density waves (CDWs) in LPCMO, describing a collective and delocalized

91

10-6 10-5 10-4 10-3 10-2

0.0

0.5

1.0

(s)

G(

Figure 3-16. The normalized distributions of hopping-rate time-scales are shown to benarrow for both phases, suggesting temporally coherent hopping (PMIdotted red, and COI solid blue).

propagation of the charge-density distribution of the COI phase. The constraints of the

parallel model require that the hopping mechanism is correlated over sufficiently long

length scales that regions equal to at least the film thickness hop together collectively,

so that as the phases convert the entire phase boundaries progress simultaneously in a

‘creep’ like manner. ‘Creep’ is typically a random phenomenon, however, transforming

our dielectric broadening to a distribution of time-scales [71] (shown in the inset of

Fig. 4) we find a narrow distribution of hopping rates suggesting an ordered process

similar to the ‘temporally coherent creep’ found in the CDW system NbSe3 [66].

The exact nature of the order is ambiguous, with two likely scenarios. The first

possibility is the coherent propagation of phase domains, where as the phase boundary

‘creeps’ forward the regions behind synchronously hop, guaranteeing the continuity of

the phase. This scenario is depicted in the schematic of Fig. 3-17, where the boundaries

of the Mn3+/Mn4+ charge ordered phase (blue) coherently propagate with successive

phase boundary hops/creeping, supplanting and converting into the charge-disordered

paramagnetic insulating phase (red). The second scenario is a ‘breathing’ mode in

which the area of different phase domains cooperatively increase and decrease at a

characteristic frequency (with total area conserved). Both scenarios demonstrate the

92

C)

B)

A)

Figure 3-17. Panels A), B), and C) represent successive snapsots in time of the phasecompetition between the COI (blue) and PMI (red) phases. The boundariesof the Mn3+/Mn4+ charge ordered phase (COI, blue) propagate withcoherent phase boundary hops, supplanting and converting into thecharge-disordered paramagnetic insulating phase (PMI, red) as a functionof space and time. The solid black line represents the oscillation of chargedensity.

collective and delocalized nature of the COI phase in which its entire charge distribution

moves collectively and coherently in dynamic competition with the PMI dielectric phase.

3.5 Summary

In summary, we have presented a dielectric characterization of the competition

between the COI and PMI dielectric phases of (La1−yPry )1−xCaxMnO3, identifying

signatures of phase separation and providing temperature dependent time-scales,

dielectric broadenings, and population fractions of each phase. More importantly, we

demonstrate that the constraints imposed by detailed balance describe an ‘electronically

soft’ coexistence and competition between dielectric phases, highlighted by continuous

conversions between phases with comparable energies on large length and time scales

as well as a collective and delocalized nature of the charge-density distribution of

93

the COI phase. Our findings provide important context concerning the fundamental

mechanisms driving phase separation, and strongly support the ‘electronically soft’,

delocalized and ordered thermodynamic phase interpretation (Ref. [72]) over the

disorder and strain based explanations which result in static phases.

94

CHAPTER 4STRAIN MEDIATED MAGNETOELECTRIC COUPLING IN (LA1−YPRY )1−XCAXMNO3

4.1 Introduction

Magnetoelectric (ME) coupling, the induction of electric (magnetic) polarization

by external magnetic (electric) fields, has recently become one of the most active

research topics due to both its complex physical origins and its potential technological

applications [22, 33, 42, 43]. Theoretically, ME coupling is typically described using

Ginzburg-Landau theory, where a free energy analysis provides a unified temperature

and field dependence of the thermodynamics and stability of the constituent electronic

phases. The free energy description also provides an infrastructure for testing

specific microscopic coupling mechanisms, and multiple mechanisms have been

shown to produce ME coupling: single ion anisotropy, symmetric and antisymmetric

superexchange, dipolar effects, Zeeman energy, and etc [73]. However, these couplings

are found to be systematically small [41].

Experimentally, large magnetoelectric coupling has been achieved in composites by

combining piezoelectric and magnetostrictive materials which couple through strain. The

coupling in composites can be multiple orders of magnitude larger than in single phase

materials, establishing strain coupling as the most powerful known magnetoelectric

mechanism and sparking a revival in magnetoelectric research [41, 74]. However, the

theoretical description of the large composite coupling has provided little insight into

the thermodynamics or microscopic magnetoelectric mechanisms, as it is estimated by

the simple phenomenological multiplication of bulk properties and is limited strictly to

the interfaces [41, 75]. Outside of composites, only indirect evidence of strain-mediated

magnetoelectric coupling has been demonstrated. For example, anomalies in the

dielectric constant have been shown to accompany structural transitions resulting from

antiferrodistortion at magnetic ordering temperatures [76].

95

In this chapter we present the first successful strain-based microscopic modeling

of magnetoelectric coupling data in single phases. Measuring the frequency

dependence of the complex capacitance, we show that the dielectric phases of thin film

(La1−xPrx )1−yCayMnO3 - paramagnetic insulating (PMI) and charge-ordered insulating

(COI) - each display two distinct magnetoelectric couplings: magnetic field tuning of both

the dielectric constants and activation energies. Using electric, magnetic, and elastic

terms in a free-energy expansion, we correctly predict the power and sign of magnetic

field induced modifications of the dielectric constant of each phase. Importantly, this

provides the first experimental verification of the widely conjectured strain origin of the

biquadratic (M2P2) magnetoelectric term in the free energy. Furthermore, through a

film thickness study we demonstrate that both magnetoelectric couplings are strongly

sensitive to the strain state of the lattice, providing valuable information for future strain

engineering studies.

4.2 Dielectric Constant Tuning

4.2.1 Experimental Results

The first type of magnetoelectric coupling present in our samples is the direct tuning

of dielectric constants with magnetic field. Figure 4-1 shows the dielectric constants

for both phases as a function of magnetic field at a fixed temperature T = 200K , with

εPMI decreasing quadratically, and εCOI increasing quadratically with increasing field in

both the 30nm and 150nm films. The dielectric constants are calculated by dividing the

dielectric amplitudes returned from the model introduced in Chap. 3, APMI = aPMI εPMI

and ACOI = aCOI εCOI , by the respective fractional areas, aPMI and aCOI , which are

determined using a detailed balance equation that was found to constrain our system,

aPMI (1/τPMI ) = aCOI (1/τCOI ), and the known constraint aPMI + aPMI = 1 (see Sec.

3.4.2). Below we describe a nearest neighbor mean-field model that reproduces all

of the qualitative features of the magnetic tuning of the dielectric constants (quadratic

field dependence, and sign of the change for each phase) using strain as the coupling

96

12.50

12.52

12.54

12.56

12.58

12.60

-2 0 2 4 6

5.25

5.50

5.75

6.00

6.25

6.50

6.75

7.00

CO

I

H (T)

PM

I

0H2

30nm

45.1

45.2

45.3

45.4

45.5

45.6

45.7

-1 0 1 2 3 432

34

36

38

40

42

0H2

C

OI

H (T)

150nm

P

MI

A)

B)

Figure 4-1. A) The magnetic field dependent dielectric constants for the 30 nm areshown. B) The magnetic field dependent dielectric constants for the 150 nmare shown. The dielectric constants of both phases are shown to tunequadratically with magnetic field. These qualitative features are reproducedby a nearest-neighbor mean field model discussed in the text. The dottedlines are fits to, ε = ε0 − φH2 (Eq. 4–6).

mechanism. Although there are two dielectric phases present, it is important to note that

their coexistence is coincidental and that our model assumes no interaction between

them: the strain mediation occurs within the individual phases.

4.2.2 Modeling

Our model considers nearest-neighbor interactions of a lattice with a local

magnetic moment and electric dipole at each site, both of which respond linearly to

their respective fields: mi = mS + µH and Pi = χE , where mS is the spontaneous

magnetization, with nearest-neighbor (i 6= j) potential energies, −J ~mi · ~mj , and −λ~Pi · ~Pj .

The magnetoelectric coupling arises by considering the spatial dependence of their

97

individual coupling coefficients (J,λ),

J(~r + ~δ) ≈ J(~r) +∇J(~r) · ~δ

λ(~r + ~δ) ≈ λ(~r) +∇λ(~r) · ~δ ,

(4–1)

and introducing an elastic term into the free energy,

F = − ~mS · ~H − J ~mi · ~mj − λ~Pi · ~Pj +1

2kδ2. (4–2)

The energy of the system is then minimized by modifying the lattice spacing by an

amount,

δ =1

k[∇J ~mi · ~mj +∇λ~Pi · ~Pj ], (4–3)

where ∇J and ∇λ are spatial gradients. When this result is substituted back into the free

energy, it results in a correctional term in the polarization that is quadratic in magnetic

field,

P = −∂F

∂E= (χ0 − φH2)E +O(E 3), (4–4)

where χ0 is the susceptibility in zero field,

φ =2χ2µ2

k∇λ∇J, (4–5)

and the terms of order E 3 are not observed. The dielectric constant can then be written

as,

ε = ε0 − φH2, (4–6)

where ε0 is the dielectric constant in zero magnetic field. Fig. 4-1 shows the fits to

Eq. 4–6 for both phases (solid lines). The magnetic field ranges were chosen to

guarantee each dielectric phase is well established, i.e. the Arrhenius plot is linear

over a sufficiently large enough range to provide a well defined EA.

With χ2, µ2, and k all positive, the sign of φ is determined solely by ∇λ and ∇J. The

functional form of λ is known from the classical result for the field of an electric dipole,

98

with ∇λ ∝ −r−4, however, the spatial dependence of J is not explicitly known for each

phase. To investigate J, we consider the relation J ≈ TC ,N , where TC and TN are the

ferromagnetic (Curie) and antiferromagnetic (Neel) transition temperatures of the PMI

and COI phases, respectively. The sign of ∇J can then be determined from the change

in the transition temperatures under the application of pressure, that is ∇J ≈ ∆TC ,N/∆x .

In compressive experiments the ferromagnetic transition of the PMI dielectric phase is

shifted to higher temperatures (∆TPMI > 0), and the antiferromagnetic charge-ordering

transition of the COI dielectric phase is shifted to lower temperatures (∆TCOI < 0)

[77]. The change in x is negative in compressive experiments (∆x < 0), meaning that

∇JPMI ≈ ∆TPMI/∆x < 0, and ∇JCOI ≈ ∆TCOI/∆x > 0, which combined with ∇λ < 0

results in φPMI > 0 and φCOI < 0, thereby predicting the experimentally observed sign of

the magnetoelectric coupling for both phases.

4.3 Activation Energy Tuning

4.3.1 Experimental Results

Figure 4-2 displays the second type of magnetoelectric coupling present in our

samples. The Arrhenius plots of the time-scales for the relaxation of both dielectric

phases are shown for field-cooled temperature sweeps in fields of 0, 2, 4, and 6 T.

The large linear regions demonstrate the process is activated, with activation energies

determined from fits to, τ = τ0eEa/kBT (solid black lines). The low temperature deviations

from linearity are due to the onset of the well known low-temperature insulator-to-metal

transition in each phase. At zero field, EA ≈ 118meV, consistent with small polarons,

the known polarization mechanism in manganites [67, 69]. As shown, the EA’s of both

dielectric phases are sensitive to magnetic fields. The inset of Fig. 3 shows that the EA’s

are found to decrease with magnetic field according to the equation,

EA(H) = E0 − ηH, (4–7)

99

3 4 5 6 7 8

10-5

10-4

10-3

10-2

10-1

PMI COI6T

4T

2T

0T

6T

2T

4T

0T

PMI &

C

OI

1000/T (K)

0 2 4 6 8

60

80

100

120

H (T)

EA

(meV

)

Figure 4-2. The magnetic field dependent EA’s are shown for the 30 nm film. The largelinear regions of the relaxation time-scales demonstrate the hopping isactivated, solid black lines are fits to the Arrhenius equation. Deviations fromlinearity at low temperatures indicate the onset of the FMM phase. Inset: EAis shown to depend linearly on H for both phases. The solid lines are fits to,EA(H) = E0 − ηH.

where η is defined as a phenomenological magnetoelectric coupling coefficient. When

polarons hop they carry a distortion of the lattice with them, so if the barrier to polaronic

hopping (EA) is modified, it is likely that the lattice strain has been modified by the

magnetic field so that distortions are easier/harder to accommodate. In support of this

interpretation, we note that polaronic barriers in the COI are 50% more sensitive to fields

(ηCOI/ηPMI ≈ 1.5), which is consistent because its larger inherent distortion suggests its

lattice coupling is stronger than the PMI phase.

The linear order of the magnetoelectric coupling in EA is also worth noting. A

non-zero spontaneous magnetization results in linear field dependence in energy, similar

to the first term in Eq. 4–2. This suggests the presence of spontaneous magnetization in

the polarons of each phase, consistent with previous reports of magnetic polarons well

above TC in magnetoresistive perovskites [78]. The relative magnitude of the coupling

of each phase is also consistent with this explanation, as the magnetically ordered COI

100

polarons should have larger local spontaneous moments at each site, even though they

cancel globally (as a result of antiferromagnetic alignment).

4.3.2 Comparison of Magnetoelectric Couplings

We find it important to clarify the difference between the two magnetoelectric

couplings presented in this chapter. In the second magnetoelectric coupling

presented, magnetic fields tune the activation energies, which as a result modulate

the characteristic time-scales of the dielectric relaxation process - resulting in faster

relaxations. If the capacitance were analyzed at a single measurement frequency

greater than the initial relaxation rate, this would appear as an increase in the dielectric

constant. However, this is only the result of the frequency time-scale now capturing

more of the dielectric relaxation process - before the magnetic field shortened the

relaxation time-scale the dielectric relaxation did not fully polarize in the measurement

time-scale.

The first magnetoelectric coupling presented in this chapter, the direct tuning

of dielectric constants by magnetic field, is fundamentally different than the second

magnetoelectric coupling presented. This magnetoelectric coupling is measured as a

result of fitting across the entire dielectric relaxation frequency spectrum and is therefore

not a time-scale effect. This magnetoelectric coupling represents a change in the

magnitude of the polarization of the dielectric, and not simply a change in the rate at

which it polarizes.

4.4 Film Thickness Study

Epitaxial films use the crystal structure of the underlying substrate as a seed from

which to grow, however, the lattice constants and crystal structures of the film and

substrate are not always identical. This can lead to large compressive or tensile strains

in the film layers near the substrate-film interface, and our substrates (NdGaO3) induce a

slight tensile strain of +0.5% [79]. As the film thickness (d) increases, however, the strain

101

B)102 103 104 105 106 107

10-14

10-13

10-12

10-11

10-10

10-9

10-8

C' (

F) (rad/s)

Measured Model PMI COI

1/COI

1/PMI

102 103 104 105 106 107

10-12

10-11

10-10

10-9

(rad/sec)

C'' (

F)

102 103 104 105 106 10710-14

10-13

10-12

10-11

10-10

10-9

10-8

1/COI

Measured Model PMI COI

C

' (F)

(rad/s)

1/PMI

102 103 104 105 106 107

10-12

10-11

10-10

10-9

(rad/s)

C'' (

F)

A)

Figure 4-3. A) The modeling results are shown for the 30 nm film. B) The modelingresults are shown for the 150 nm film. The measured real capacitance(black) is compared to the parallel model results (green). The individualcapacitances of the PMI (red dash) and COI (blue dot) phases are alsoshown. Insets: Imaginary capacitance, where loss peak positions show thecharacteristic relaxation time-scales.

relaxes as the lattice constants of the film approach their bulk values, and only the layers

near the interface remain strained.

In manganites, thickness-dependent studies have shown that for d > 125 nm the

lattice constants have almost completely relaxed to their bulk values [15]. Therefore,

varying film thickness in this range provides a direct probe of strain in thin films. To

investigate the role of strain in the magnetoelectric coupling of our samples, we have

measured LPCMO films of four thicknesses: 30, 60, 90, and 150 nm. Figure 4-3 shows

the modeling results for the thinnest and thickest films in this range.

102

30 60 90 120 1500.0

0.2

0.4

0.6

B)

PMI COI

A)

PMI| &

C

OI

Film Thickness (nm)

6

9

12

PMI &

C

OI

Figure 4-4. A) The activation energy coupling constant, η, is shown to increase andsaturate with film thickness. B) The dielectric constant coupling constant, φ,is also shown to increase and saturate with film thickness. Bothmagnetoelectric couplings correlate with the relaxation of substrate inducedmismatch strain

Both magnetoelectric coupling coefficients are found to depend strongly on

the strain state of the lattice. Fig. 4-4 shows that the magnetoelectric coupling

coefficients correlate with strain relaxation, increasing and saturating near d ≈ 125

nm. This provides further evidence that both magnetoelectric couplings are mediated

through strain. According to our model, the magnetoelectric coupling results from the

modification of lattice constants. Therefore, as the mismatch strain relaxes, the ability of

the electric and magnetic properties to couple would be amplified as the lattice softened

and was free to respond elastically and modify its spacings, which is consistent with our

observations.

4.5 Summary

In this chapter, we have reported two distinct magnetoelectric couplings in each of

the dielectric phases of LPCMO: magnetic field tuning of both the activation energies

and the dielectric constants of each phase. We provided strong evidence that the

magnetoelectric coupling is strain mediated, including a film thickness study that

shows the coupling coefficients depend on the strain state of the lattice. Importantly,

103

our findings demonstrate the first successful strain-based microscopic modeling of

magnetoelectric coupling data of individual phases.

104

CHAPTER 5MULTIFERROISM IN BIMNO3

5.1 Introduction

BiMnO3 is perhaps the most fundamental multiferroic, and has been referred to

as the “hydrogen atom” of multiferroics [80]. In BiMnO3, as in BiFeO3, the 6s2 lone

pair on the Bi ion leads to its displacement from the centrosymmetric position at the

A-site of a perovskite unit cell. However, in BiMnO3, the resultant distortion modifies the

superexchange integrals and leads to a ferromagnetic interaction between the Mn ions

at the B-site in BiMnO3 [81, 82]. In bulk form, BiMnO3 has been observed to be both

ferromagnetic and ferroelectric [83], while in thin film few reports have demonstrated

similar magnetic properties or low enough leakages to allow a clear ferroelectric

measurement [84? ]. A possible reason for the low resistivities of BiMnO3 thin films

is the substrate induced strain, which exacerbates the growth of a highly distorted

perovskite structure, making growth optimization critically important.

This chapter will discuss the growth of BiMnO3 on SrTiO3 substrates, and its

multiferroic properties. The results of structural, magnetic, resistive, ferroelectric,

and dielectric characterizations are presented and discussed. BiMnO3 is shown to

be multiferroic, demonstrating both magnetic and ferroelectric polarizations at low

temperature. Finally, the relaxor nature of the ferroelectricity is discussed, and the

ferroelectric polarization is proposed to likely be located near island edges.

5.2 Characterizations

Arguably the most difficult aspect of studying BiMnO3 experimentally (especially

in thin film) is the difficulty of attaining consistently high quality samples. Therefore,

prior to presenting the ferroelectric investigation, it is worthwhile to briefly discuss the

structural, stoichiometric, and magnetic characterizations performed by collaborators

and used to optimize the film growth. In particular, the film growth is complicated by, (1)

the volatility of Bi which has a very high vapor pressure at the film growth temperature,

105

Figure 5-1. The monoclinic unit cell of BiMnO3 is shown. The figure is reproduced fromRef. [46]

and (2) the metastability of the BiMnO3 structure itself. The stoichiometric unit cell of

BiMnO3 is the cubic perovskite (see Fig. 1-1), however, it is highly distorted leading to a

larger monoclinic unit cell (see Fig. 5-1). The films were characterized structurally using

Θ − 2Θ scans and Atomic Force Microscopy, stoichiometrically with Auger Electron

Microscopy, and magnetically with a Superconducting-Quantum-Interference-Device

(SQUID).

5.2.1 Structural Characterization

Figure 5-2 shows the Θ − 2Θ x-ray diffraction data for a 60-nm-thick BiMnO3 thin

film (sample type 1). The inset shows that the BiMnO3 grows with a 111 orientation

as expected from the structure of BiMnO3. A small peak corresponding to Mn2O3

impurities is also visible in the semi-log plot [Fig. 5-2 (b)] (the integrated intensity ratio

of the Mn2O3 peak to the BiMnO3 111 peak is 0.025). To remove these impurities, the

post-deposition cooling rate of the substrate was increased to about 40 C/min in an O2

atmosphere of 680 Torr. Figure 5-2b also shows the x-ray data for a 60 nm film grown

using the new cooling rate (sample type 2), which confirms that the impurity peaks have

106

been successfully removed using the modified growth conditions. The stoichiometry

of the samples was investigated with Auger electron spectroscopy measurements at

300 K in ultra high vacuum (UHV) conditions. Derivative Auger electron spectroscopy

surface spectra were taken using a 5 keV primary electron beam from kinetic energies

of 50 to 1500 eV at incident angles from 30 to 60. Depth profiling was performed by

taking surface spectra with the parameters given above followed by an in situ repeated 3

keV Ar-ion sputtering. Surface spectra of the BiMnO3 films displayed three manganese

(Mn) peaks located at 548, 595, and 645 eV, two bismuth (Bi) peaks at 106 and 254

eV and one oxygen (O) peak at 518 eV together with residue carbon (C) peak at 273

eV with concentrations less than 1%. After 6 s of Ar sputtering on the surface, the C

peak disappeared and Bi, Mn, and O concentrations are found to be 23.3, 24.1, and

52.6%, respectively, with about a 2% error. These concentrations imply that the BiMnO3

stoichiometry is consistent with the measured BiMnO3 x-ray peaks from the Θ − 2Θmeasurements. Moreover, the sensitivity factor for oxygen is based on an MgO matrix,

and because there is no matrix parameter in the atomic percentage calculations, this

could account for the slightly lower than stoichiometric oxygen concentrations.

5.2.2 Magnetic Characterization

The magnetic properties of BiMnO3 are closely related to its unique crystal

structure. BiMnO3 is similar to the compound LaMnO3 (LMO) but due to the 6s lone

pair, the Bi ion moves away from the centrosymmetric position at the B-site of a

perovskite structure. LMO is an A-type antiferromagnet due to antiferromagnetically

stacked ferromagnetic layers [85]. In BiMnO3, the distortion caused by the Bi ion

leads to an ferromagnetic interaction between the layers [81, 82]. Hence, BiMnO3 has

an overall magnetic moment that has been measured to be as high as 3.6 µB /Mn in

polycrystalline samples (where µB is the Bohr magneton); this is close to maximum

possible magnetization of 4 µB /Mn [86]. In thin films, the magnetic moment is reduced

quite likely due to the substrate induced strain. The TC in thin films is also lower

107

B)

A)

Figure 5-2. A) The Θ2Θ x-ray diffraction pattern of a 60-nm-thick BiMnO3 thin film isshown. The inset shows the BiMnO3 (111) peak in detail. B) A semilog plotshowing a small amount of Mn2O3 impurity phase and a small BiMnO3(20-3)peak (higher intensity line) and the impurity free film grown using the rapidquenching technique (lower intensity line) are shown. This Figure andcaption is reproduced from reference [46].

than the value of about 105 K obtained in polycrystalline samples [83, 86]. Figure

5-3 shows the M-T and M-H curves of 60 nm-thick BiMnO3 thin films (sample types

1 and 2). The magnetic field was applied in the plane of the film for the magnetic

measurements. The M-T plot reveals a TC of about 85-65 K for both sample types.

Saturation magnetizations of about 1.0 and 1.1 µB /Mn were obtained at 10 K in a field

of 50 kOe for sample types 1 and 2, respectively. The inset of Fig. 5-3 (b) shows the

hysteresis in the M-H plot at different temperatures for sample type 2. A coercive field of

108

about 270 Oe is observed at 10 K, which drops to about 30 Oe at 50 K. The hysteresis

of the M-H curves and the magnetic moment become negligible at about 80 K, which

is close to the estimated TC , confirming that the observed magnetization is associated

with magnetic ordering that happens at a temperature lower than the corresponding

TC of bulk BiMnO3 [83]. The reduced magnetic moment of our thin films compared to

bulk BiMnO3 is not due to the presence of the nonmagnetic impurities because both the

sample types 1 and 2 have similar saturation magnetizations and coercive fields.

Finally, the inset of Fig. 5-3 shows the surface morphology of sample type 2.

Both type 1 and 2 thin films show three-dimensional (3-D) island growth mode with

an r.m.s roughness of about 10 nm. It has been shown that 3-D island growth leads

to nonuniform strain resulting in high values of strains at the island edges [87, 88].

Because the crystal structure of BiMnO3 is closely related to that of antiferromagnetic

LaMnO3, the nonuniform strain distribution could be responsible for both the reduced

values of TC and the saturation magnetization.

5.2.3 Resistive Characterization

One of the most important qualities of a potentially ferroelectric sample is a large

resistivity, as the free charge carriers in low resistance materials would screen the

ferroelectric polarization, rendering it undetectable. In fact, this has been the primary

limitation in many thin film BiMnO3 multiferroic studies. Our optimized thin films have

a room temperature resistivity of about 10 Ω-cm, which is lower than values reported

by other groups [25, 89] However, by 140 K (below 140 K the resistance is too high to

measure with our instrumentation), the resistivity increases to about 1 MΩ-cm, and it

was possible to make direct polarization versus electric field (P-E) measurements at

temperatures below ≈ 100 K.

5.2.4 Ferroelectric Characterization

Ferroelectric measurements typically utilize a capacitor structure, the simplest

of which is a tri-layer metal-insulator-metal structure. However, due to the insulating

109

A)

B)

Figure 5-3. A) The magnetization vs temperature (M-T) plot for two 60-nm-thick BiMnO3thin films in an in-plane field of 500 Oe are shown: sample 1 (circles) andsample 2 (triangles). The full symbols and open symbols are the zero fieldcooled and field cooled data respectively. The inset shows a 5x5 µm atomicforce microscope image of the surface of sample 2. B) The magnetization vsmagnetic field (M-H) plot for sample 1 (circle) and sample 2 (triangle) at 10 Kare shown. The inset shows the reduction in the hysteresis of the M-H datafor sample 2 as the temperature is increased. This Figure and caption isreproduced from reference [46].

110

-100 -50 0 50 100-30

-20

-10

0

10

20

30

Rem

anen

t Pol

ariz

atio

n (

C/c

m2 )

E (kV/cm)

100K

100K

5K 0.00

0.25

0.50

0.75

1.000 50 100

M RP

Temperature

RP

& M

Figure 5-4. Remanent polarization hysteresis loops are shown. The ferroelectrictransition is shown to span ≈ 100 K, opening near 100 K with a maximumremanent polarization of ≈ 23 µ C/cm2 at 5 K.

substrate (STO), we have implemented an interdigital capacitance geometry. The

capacitor is composed of alternating V+/V− electrodes uniformly spaced on the film

surface, which leads to equipotential planes intersecting the film between each pair of

electrodes, resulting in a capacitance between the projected areas of each electrode

within the film (see Sec. 2.4.3). The projected areas were calculated analytically using

conformal mapping and equating the capacitor thickness to half the electrode spatial

wavelength [47]. The polarization is then calculated by dividing the transferred charge

by this projected area. The polarization in a typical hysteresis loop is calculated by

integrating the total transferred charge during application of a bipolar triangular voltage

waveform, with contributions from leakage current, capacitance, and ferroelectric

domain switching. However, as described in detail in Sec. 2.5.2, we measure remanent

hysteresis loops, where the contributions from leakage and capacitance have been

eliminated through specific pulsing sequences.

111

The ferroelectric transition was found to span more than 100 K, with the remanent

polarization (RP) vs. electric field loops opening slowly near 100 K and the RP

increasing to 23 µC/cm2 at 5 K. Incorporating the remanent polarization into typical

hysteresis loops (which include the total polarization, see Sec. 2.5.3), this broad

transition would result in the slow “square-to-slim-loop” transition in temperature that is

typical in relaxor ferroelectrics[90]. Interestingly, the ferroelectric transition is found to

coincide with the ferromagnetic transition. The inset of Fig. 5-4a shows that both the

ferroelectric polarization and magnetization increase from zero simultaneously, signaling

their respective transitions and suggesting a strong coupling between the orderings (see

Chap. 6 for a complete discussion).

The ferroelectric polarization is also found to be highly tunable, and is modulated

by both magnetic fields (magnetoelectric coupling) and external strain (see sections 6.2

and 6.3, respectively). The strain coupling is much stronger than the ME coupling,

lowering the coercive field and increasing the RP by as much as 50% at modest

strains of less than 10−2%. The strain coupling is strongest at low temperatures, and

disappears above T ≈ 50 K. The strain coupling is also found to be extremely sensitive

to the orientation of the strain (discussed in Chap.

5.2.5 Dielectric Characterization

To gain further insight into the ferroelectric transition, we have also completed a

dielectric characterization. The complex capacitance was measured as a function of

temperature at 200 frequencies over the bandwidth 20 Hz - 1 MHz, from 5 K< T <300

K. The characterization reveals two dielectric relaxations both of which display strong

temperature and frequency dependence.

Analyzing the frequency dependence of the complex capacitance shows two distinct

loss peaks centered at 1/τ1 and 1/τ2, indicating dielectric relaxations (see Fig. 5-5). The

112

101 102 103 104 105 106

10-12

10-11

10-10

1/ 2

C'' (

F)

Frequency (Hz)

1/ 1

T = 135K

Figure 5-5. The frequency dependence of the imaginary capacitance, C′′, displays twoloss peaks, shown here at 135 K over 200 frequencies between 20 Hz and 1MHz. The solid and dashed lines are fits to complex Cole-Cole dielectricfunctions.

relaxations are modeled by Cole-Cole dielectric functions of the form,

ε(ω) = ε∞ +ε0 − εinf

1 + (iωτ)1−α, (5–1)

where τ is a characteristic relaxation time-scale, α is a time-scale broadening, and

the difference between the zero and infinite frequency dielectric constants, ε0 − εinf ,

determines the amplitude of the relaxation. Simultaneous fits to both the imaginary and

real (not shown) capacitances are performed across the entire measurement bandwidth,

and the Cole-Cole functions of each relaxation are shown as the solid and dotted lines in

Fig. 5-5.

Plotting the temperature dependence of the relaxation time-scales provided by the

Cole-Cole fits in Arrhenius format reveals that the polarization mechanism is activated

(see Fig. 5-6). Fitting the time-scales to the Arrhenius equation, τi = τ0,ieEA/kBT , over the

large linear regions provides a well defined activation energy, EA, and pre-exponential

113

6 9 1210-6

10-5

10-4

10-3

10-2 1

2

1 &

2

1000/T (K-1)

EA1 = 205meVEA2 = 189meV

Figure 5-6. Arrhenius plots of τ1 and τ1 are shown. Both relaxation time-scales arethermally activated, with activation energies of 205 meV and 189 meV,respectively.

factor for each loss peak. The activation energies are 205 ± 5 meV and 189 ± 4 meV,

with pre-factors of ≈ 2.9× 10−11± 0.1× 10−11 s and ≈ 3.610−13± 0.1× 10−13 s for τ1 and

τ2, respectively. We note that the pre-factors are quite small for a dielectric relaxation,

and are in the range of phonon frequencies.

Figure 5-7 shows the temperature dependence of the real component of the

complex capacitance for a selection of frequencies: 20 Hz, 200 Hz, and 2 kHz. At

high temperatures, there is evidence of a broad maximum, however, because of

the increased leakage the temperature range is limited and the peak is not directly

observed. The “maximum” is frequency dependent, however, occurring at higher

temperatures for higher frequencies. At low temperature, the dispersion disappears

and the real capacitance diverges (for reasons which are discussed below). The lack of

dispersion signifies that the capacitance is representative of the static dielectric constant

at our measurement frequencies.

114

0 50 100 150

0.1

0.2

0.3

0.4

0.5

C' (

nF)

Temperature (K)

20 Hz 200 Hz 2000 Hz

Figure 5-7. The temperature dependence of the real capacitance is shown for selectfrequencies. The high temperature range is limited due to increased leakagethere.

5.3 Nature of Ferroelectricity

5.3.1 Relaxor Review

Relaxor ferroelectricity can be understood by considering a representative model

called superparaelectricity [90], which is directly analogous to superparamagnetism,

except that in superparaelectricity the magnetic domains are replaced by ferroelectric

domains. Superparamagnets are typically composed of nano-sized magnetic

particles each of which act as their own ferromagnetic domain. In an applied field the

magnetic moments align resulting in a magnetization which is much larger than typical

paramagnets. However, once the field is removed the individual domains randomize

canceling the macroscopic magnetization over an activated time-scale known as the

Neel relaxation time. Below a certain temperature, the Neel relaxation time exceeds the

measurement time-scale and the system is said to be “blocked”, and behaves similar

to a regular ferromagnet. Relaxor ferroelectrics behave almost identically, with dipoles

originating from “polar-nano-regions” or PNRs. PNRs exhibit large induced polarizations

115

which reorient and randomize upon the removal of the electric fields. PNRs appear at

a high temperature transition, known as the Burns temperature (TB)[91], and grow in

size slowly with decreasing temperature until the freezing temperature. Then, at TF ,

the PNRs grow rapidly causing the fluctuation time-scales to diverge as a result of

the coalescing and coupling of neighboring PNRs[92, 93]. Besides their spontaneous

ferroelectric dipole moments, these regions have larger intrinsic dielectric constants as

well, leading to an increase in the overall dielectric constant as their % area increase at

TF .

In normal ferroelectrics, the ferroelectric transition is typically caused by a soft

long-wavelength phonon-mode that decreases to zero frequency at TC , resulting

in a static lattice deformation that extends throughout the crystal[94]. However,

relaxor ferroelectrics are inherently disordered by the PNRs which critically damp

the zone-center phonon modes and prohibit their propagation[95, 96]. As a result the

transition is very broad (diffuse), and the corresponding slow “slim to square loop”

transition in the hysteresis as the temperature is lowered is a signature of relaxor

behavior. The PNRs act as local “frozen phonon modes” that vibrate out of phase.

These vibrations are thermally activated and correspond to the flipping of PNR dipole

moments. PNRs contribute to the dielectric response via two distinct mechanisms: the

thermally activated reorientation of their dipole moments, and the displacement of their

boundaries[97]. The PNRs are also believed to cause the broad peak in ε′ as a function

of temperature, which is a classic signature of relaxor ferroelectricity[93].

5.3.2 Comparison

The first comparison to note between our data and the expected trends of relaxor

ferroelectricity is the diffuse (slow in temperature) ferroelectric transition. Figure 5-4

shows that the ferroelectric transition spans more than 100 K. If these remanent

polarization loops were incorporated into total polarization hysteresis loops (where

charge contributions from leakage and capacitance were included), it would result in a

116

slow “slim to square loop” hysteresis loop transition in temperature, which as discussed

is one of the primary signatures of relaxor ferroelectricity.

The dielectric data presented here also display all of the hallmarks of relaxor

ferroelectricity. First we note the presence of two relaxations is consistent with the two

dielectric mechanisms discussed above (PNR reorientation and boundary fluctuations).

The specific frequency dependence of the dielectric relaxations is also consistent with

PNRs. Phonon modes in typical dielectrics (ferroelectrics included) contribute to the

dielectric function according to the Lorentzian oscillator model:

ε = ε∞ +∑

j

Ajω2j

ω2j − ω2 − iγjω , (5–2)

where where ωj , γj , and Aj are the frequency, width, and dimensionless oscillator

strength of the jth phonon mode [98]. Phonon modes that are critically damped,

however, are instead represented by Cole-Cole dielectric functions instead (see Eq.

5–1) [64]. As shown in Fig. 5-5, the relaxations are both successfully modeled by

Cole-Cole functions.

The temperature dependence of the dielectric relaxations is also in agreement

with relaxor ferroelectricity. The relaxation time-scales are activated as expected

for PNR reorientation time-scales in the superparaelectric model, and fitting the

temperature dependence of the relaxation time-scales (provided by the Cole-Cole

fits) to the Arrhenius equation, τi = τ0,ieEA/kBT , results in quite small pre-exponential

factors of τ0,1 ≈ 3 × 10−11 and τ0,2 ≈ 3.5 × 10−13, clearly establishing their phonon

origin of the PNRs (see Fig. 5-8. The real capacitance, shown in Fig. 5-7, is also

consistent with PNRs and relaxor ferroelectricity. At high temperatures we see evidence

of the tail end of the characteristic frequency dependent maximum (we are limited to

lower temperatures due to leakage) - a hallmark of relaxor ferroelectricity - and at low

temperatures we see a frequency independent divergence of the dielectric constant as

is expected with the rapid growth of the PNRs at the freezing temperature.

117

0 4 8 1210-14

10-12

10-10

10-8

10-6

10-4

10-2

100

Phonon Frequencies

1

2

1 &

21000/T (K-1)

Loop Time-scale

TC

Figure 5-8. The relaxation time-scales are extrapolated using the Arrhenius equation. Atlow temperatures the time-scale of the PNR dipole reorientations is shown tointersect with the RP loop time-scale almost exactly at TC = 100K . At hightemperatures the pre-exponential factors are shown to be in the range ofphonon frequencies.

The most convincing evidence for relaxor ferroelectricity, however, is the accurate

prediction of the ferroelectric TC from the dielectric data. At high temperatures the

PNRs randomize quickly so that once the ferroelectric measurements are finished the

remanent polarization has completely randomized. As temperature is lowered, however,

the randomization is less and less complete. Therefore, one would expect that just

as the ferroelectric hysteresis loops are opening that the measurement time-scale is

just equal to the PNR reorientation time-scale. As shown in Fig. 5-8, extrapolating the

activated time-scales via the Arrhenius equation, we see that the temperature at which

the PNR reorientation time-scale, τ1, and the RP loop measurement time-scale are

equal corresponds exactly to the temperature where the RP loops begin to open (TC ),

confirming the PNRs as the source of ferroelectricity.

5.3.3 Pulse Sequencing

As a final check of relaxor ferroelectricity, we study a simple two pulse waveform

depicted schematically in the inset of Fig. 5-9. The first voltage pulse poles the domains,

and the second voltage is applied after a variable delay time. After the second pulse is

118

applied a charge transfer measurement is conducted (green point in the inset of Fig.

5-9) to determine if any ferroelectric domains have reoriented during the delay time. The

charge transfer due to resistive leakage and capacitive charging should be the same

for each pulse independent of delay time, therefore any change in charge transfer with

delay time is attributed to reoriented ferroelectric domains. It is worth noting that this

procedure is distinct from the remanent polarization measurement procedure discussed

in Sec. 2.5.3. In the remanent polarization measurement, there are presetting pulses

which pole the domains, however, the charge transfer is only measured during the

following triangular waveform. Here the charge transfer is measured only as a result of

the pulse itself.

In a normal ferroelectric, the second pulse of our two pulse waveform should have

no effect as there should be minimal domain reorientations. In a relaxor ferroelectric,

however, the charge transfered during the second pulse should increase as the delay

time between pulses increases allowing time for more reorientations. The inset of Fig.

5-9 shows that the charge transfered by the second pulse increases logarithmically

with delay time, once again confirming the presence of rapidly disordering ferroelectric

domains, i.e. relaxor ferroelectricity.

5.3.4 Island Growth

Recent electron and neutron diffraction data have cast doubt over the purported

non-centrosymmetry of the BiMnO3 crystal structure [99], and centrosymmetric

structures have also been predicted using density functional theory calculations

[100]. This point is of fundamental importantance because a non-centrosymmetric

crystal structure is essential for ferroelectricity. Therefore, although BiMnO3 was first

thought a prime example of multiferroicity, there has been a growing debate concerning

whether it is an intrinsic ferroelectric. If the crystal structure of BiMnO3 is indeed

centrosymmetric, then the possible reasons for the ferroelectric behavior of BiMnO3 thin

films are: 1 structural distortions due to oxygen vacancies [99, 101], a centrosymmetric

119

0.0 0.5 1.0 1.5 2.0 2.5 3.04

6

8

10

Delay (s)

P (

C/c

m2 )

T = 90K

DelayV

time

0T

Measurement

Figure 5-9. The switched polarization is shown to increase with delay time betweenpulses, confirming the quick reorientation of PNRs. Inset: Schematic of thesimple pulse sequence.

to noncentrosymmetric transition below TC that is, below 100 K [24, 99], and substrate

induced strain [102].

Although, the Auger electron spectroscopy measurements on our thin films reveal

an oxygen deficiency that could lead to the ferroelectric behavior, we cannot rule out

the role of substrate induced strain. If the film is uniformly strained, the lattice mismatch,

which is -0.77% (compressive), is not enough to break the centrosymmetry as shown by

Hatt et al [102]. However, it has been shown that compressive lattice mismatch strain

could lead to a nonunifrom strain distribution in the thin film due to island formation

and the strain at the island edges could far exceed the average lattice mismatch strain

[87, 88]. The growth morphology of our thin films ( see the inset of Fig. 5-3) suggests

that such nonuniform strain distribution is also a possible mechanism for the appearance

of ferroelectricity. It is also worth noting that this interpretation is consistent with relaxor

ferroelectricity. The individual island edges would serve as PNRs, and the inherent

120

disorder of island growth explains the dampening of the phonon modes discussed

above.

5.4 Summary

In summary, impurity free multiferroic thin films of BiMnO3 (111) on SrTiO3 (001)

substrates have been grown with the desired structure and stoichiometry. The results

of structural, magnetic, resistive, ferroelectric, and dielectric characterizations were

also presented. The films are shown to be ferromagnetic with a TC of is 85-65 K and

a saturation magnetization of about 1 µB /Mn at 10 K. Also, the films demonstrated a

sufficiently high resistivity at low temperatures to allow the clear measurement of P-E

loops, and a ferroelectric remnant polarization of 23 µC/cm2 was measured at 5 K.

Finally, correlations between the dielectric and ferroelectric properties show that BiMnO3

is a relaxor ferroelectric, and it suspected that the PNR dipoles are located near island

edges.

As a caveat, it is appropriate to also mention an additional potential source of the

ferroelectric polarization presented in this chapter. Bismuth doped SrTiO3 is a well

known relaxor ferroelectric, and it is possible that during the pulsed laser deposition

some Bi ions may be implanted in the top layers of the substrate. However, we have

also grown BiMnO3 on LSAT and NGO substrates, which show small but non-zero

ferroelectric polarizations. Furthermore, the reduced polarization values are expected as

compressive strain decreases the ferroelectric polarization (see Sec. 6.2.1) and LSAT

and NGO substrates induce much larger compressive strains than STO. Therefore the

thin film BiMnO3 researched in this thesis is believed to display ferroelectric polarization

(potentially limited to island edges only).

121

CHAPTER 6TUNING FERROELECTRICITY IN BIMNO3

6.1 Introduction

Ferromagnetic and ferroelectric materials are ubiquitous in modern technology.

The vast majority of data storage technologies use magnetic materials, and the sensor

and actuator industries rely heavily on ferroelectric components. Multiferroics are

exciting because they offer the potential of one multi-functional material performing more

than one task. Furthermore, multiferroics are prime candidates for the technological

implementation of magnetoelectric coupling, offering an additional dimension to the

design phase-space and undoubtedly facilitating efficiency and progress. In particular,

magnetoelectric coupling could one day lead to the writing and reading of magnetic data

with electric fields, a capability that would decrease the power usage and increase the

speed of nearly every device involving memory. Additional devices proposed include

high-resolution magnetic field sensors, electrically tunable microwave applications such

as filters, oscillators and phase-shifters, and spintronic applications such as spin-wave

generation, amplification, and frequency conversion. However, the technological

implementation of multiferroic materials hinges on their ability to couple to external

perturbations.

In this chapter we will demonstrate that BiMnO3 shows multi-functional

characteristics, displaying strong couplings to both external strains and magnetic

fields. External strains of less than 10−2% change in lattice constants are shown

to increase the ferroelectric polarization by almost 50%. The strain coupling is also

shown to be anisotropic, sensitive only for specific induced distortions. Magnetic

fields of 7 T are found to decrease the ferroelectric polarization by 10%. While the

magnetoelectric coupling is small compared to the strain coupling, it is important to note

that while commonly discussed theoretically, direct observation of tuning of ferroelectric

polarization by magnetic fields is rare. Interestingly, all these properties appear to

122

be connected as the couplings are shown to correlate with each other through a low

temperature transition in the lattice.

6.2 Strain: External and Island Edges

6.2.1 External Strain

Stress is applied to the thin films directly using an external three-point beam

bending technique. In this technique the film is supported on opposite edges while

an external force is applied in the center of the film. The result is a bending of the film

similar to a classical beam, where the strain is quantified as,

ε = ∆L/L0 = xt/L20, (6–1)

where ∆L is the change in length at the surface of the beam due to a small displacement

(x) of the beam center, and L0 and t are the original length and thickness of the beam,

respectively. This strain can be either compressive or tensile depending on whether the

force is applied to the film surface or to the back-side of the substrate, as shown in Fig.

6-1, and is nominally uniaxial. The stress is applied by turning a 60 turns/inch screw in

contact with the sample surface with the aid of a worm gear providing spatial resolution

of ≈ 1 µm displacements.

The remanent polarization was found to depend strongly on externally applied

stress, and increases of almost 50% were observed at very modest strains of less than

10−2% change in lattice constants (see Fig. 6-2). The effect was found to be odd in

strain, with the sign of the change in polarization depending on whether compressive

or tensile strain was applied. The top inset of Fig. 6-2 shows that the coupling is linear

in strain at low strains, with the coupling diminishing at high strains. The bottom inset

of Fig. 6-2 show that the strain coupling displays strong temperature dependence,

appearing below approximately 50 K and diverging at low temperautres.

123

Figure 6-1. The three-point beam bending technique for applying compressive andtensile strain to a thin film is shown. When the film is on the opposite side ofthe applied stress (orange) the strain is tensile. When the film is on the sameside as the applied stress (purple), the strain is compressive. The areabetween the orange and purple lines represents the substrate. The dashedbox is the undistorted beam/substrate. Figure reproduced from [45].

-80 0 80-40

-30

-20

-10

0

10

20

30

40

Rem

anen

t Pol

ariz

atio

n (

C/c

m2 )

T = 5K

E (kV/cm)

5KS = 10-2%

0

5

100 25 50 75

Temperature

RP

2 4 6 8

0

5

10

10-3%)

R

P

Figure 6-2. An external strain of less than 10−2 is shown to increase the FE polarizationby ≈ 50% (blue curve). Insets: (upper) The strain coupling is shown toincrease as a function of tensile strain. (lower) The strain coupling is shownto decrease as a function of temperature, disappearing near ≈ 50 K.

124

6.2.2 Electrode “Lensing”

Advantageously, the interdigital capacitance geometry facilitates a “lensing”

experiment where the polarization may be probed as a function of the relative

orientations of the induced strain and applied electric field (see Fig. 6-3). Applying strain

along the (100) and (010) axes produced no change in remanent polarization regardless

of electric field orientation. Straining along (110) and (1-10), however, produces the

giant changes shown in Fig. 6-2. The increase is only for electric fields parallel to the

strain, with perpendicular fields inducing no change in remanent polarization (see Fig.

6-3). The insensitivity of the perpendicular fields rules out a rotation of the polarization,

suggesting that the strain induces ferroelectric polarization by stabilizing the monoclinic

ferroelectric distortion of BiMnO3 (see Fig. 6-3).

6.2.3 Island Edge Strain Gradients

With a SrTiO3 substrate, the lattice mismatch strain for BiMnO3 is approximately

-0.77% (compressive), which according to density functional theory calculations is not

sufficient to break centrosymmetry and induce ferroelectricity [102]. Furthermore, the

externally applied strain is not sufficiently large either. Therefore another factor that is

sensitive to strain must be present: island edge strain gradients.

Compressive lattice mismatch strain can lead to a non-unifrom strain distribution in

thin film due to island formation, and the strain at the island edges can far exceed the

average lattice mismatch strain (see Fig. 6-4) [87, 88]. The growth morphology of our

thin films (see Fig. 5-3) suggests that such non-uniform strain distribution is a definite

possible mechanism for the appearance of ferroelectricity. Additionally, this mechanism

explains the high sensitivity to external strain. As shown in the inset of Fig. 5-3, the

strain gradients diverge near the island edge. Therefore, it is likely that our externally

applied strain converts the regions near the island edges just short of the critical strain

necessary to produce ferroelectric polarization.

125

A)

B)

Figure 6-3. A) The schematic shows the orientations of the electric fields and inducedstrains. The remanent polarization is insensitive to < 100 > and < 010 >strains, however, < 110 > and < 1− 10 > strains induce/stabilize themonoclinic ferroelectric distortion of BiMnO3. Only electrodes with electricfields parallel to the induced tensile strains detect changes (increases),meaning the RP does not rotate, but that new ferroelectric dipoles arecreated as the PNRs grow. B) Arrows indicate the monoclinic distortions ofthe unit cell induced by straining along < 110 > and < 1− 10 >. Bi is blue, Ois red, and Mn is yellow.

6.3 Magnetoelectric Coupling in BiMnO3

6.3.1 Remanent Polarization Tuning

In addition to strain tuning, the ferroelectric polarization is also modulated by

external magnetic fields. This magnetoelectric coupling is shown in Fig. 6-5, where the

remanent polarization is shown to decrease by approximately 10% in a field of 7 T. The

coercive field is unchanged, with only the magnitude of the polarization effected, and the

coupling is isotropic - equivalent for all angles of applied fields.

The magnetoelectric coupling is shown to be linear in field (upper inset of Fig. 6-5),

with a maximum linear magnetoelectric coupling coefficient of, α = 0.1 µC/cm2T near

126

C)

B)

D)

A)

Figure 6-4. A) An AFM image shows Ge islands grown on a Si substrate. B) Across-section TEM image shows a Ge island on a 6 monolayer thick Gewetting layer on a Si substrate. C) The boundary of Lings mound and acontour diagram shows the calculated strain εxx in the system. D) Thevariation of surface strain εs is shown along the system surface. The figure isreproduced from reference [88].

65 K (see Sec. 1.3.3. The magnetoelectric coupling is also found to decrease with

temperature, disappearing completely below T ≈ 50K (see lower inset of Fig. 6-5).

6.3.2 Reorientation Time-Scales

To investigate the magnetoelectric coupling further, we also checked the magnetic

field dependence of our dual pulse waveform introduced in Sec. 5.3.3 (where an intitial

pulse poles the ferroelectric domains, and after a variable delay time a second pulse

checks for domains which have reoriented). Figure 6-6 shows the result of varying

the delay time between pulses for both 0 T and 7 T fields. As seen, for all delay times,

the magnetic field data have larger measured switching polarizations. Therefore, we

conclude that the magnetic field decreases the reorientation time-scale so that more

PNRs are available to flip during the second pulse. The dielectric data also support this

127

-100 -50 0 50 100

-5

0

5

= 0.1 C/cm2T

H = 7T

E (kV/cm)

65KT = 65K0 2 4 6 8

5.45.65.86.0

RP

H (T)

Figure 6-5. A magnetic field of 7 T is shown to decrease the ferroelectric polarization byapproximately 10%. Inset: The magnetoelectric coupling is shown to belinear in field. The decrease in remanent polarization is believed to becaused by an increased reorientation rate of the PNRs (see Sec. 6.3.2).

interpretation as a decrease in EA of a few percent is seen at 7 T in both relaxations (not

shown).

This mechanism explains the negative sign of the magnetoelectric coupling,

however, the temperature dependence of the coupling and how magnetic fields increase

the reorientation rate are still not understood. A possible explanation for this is that the

ferroelectricity is linked to or partially caused by the spatial variation of the magnetization

(see Sec. 1.2.3). In this scenario magnetic fields would force magnetization vectors

to align and would eliminate a portion of the ferroelectric regions. The reduction in

size of the ferroelectric domains would then make them more susceptible to thermal

fluctuations, thereby reducing the reorientation time-scales of the PNRs.

6.3.3 Connection to Lattice Transition

Interestingly, all of the material properties of BiMnO3 discussed thus far

demonstrate a strong correlation. Five simultaneous phenomena demonstrate the

strong connection between the dielectric, ferroelectric, ferromagnetic, structural, and

magnetoelectric properties. Near T ≈ 50K : the remanent polarization increases rapidly,

128

0.0 0.5 1.0 1.5 2.0 2.5 3.0

4

6

8

10

Delay (s)

P (

C/c

m2 )

T = 90K

DelayV

time

7T0T

Figure 6-6. The dual pulse sequence shows that for every delay time the magnetic fielddata have larger switching polarization, indicating that the reorientationtime-scales are decreased by the 7 T field.

C)B)

0 50 100

0.00

0.25

0.50

0.75

1.00

RP M

Temperature (K)

Nor

mal

ized

RP

& M

0 50 100 150

0.1

0.2

0.3

0.4

0.5

C' (

nF)

Temperature (K)

20 Hz 200 Hz 2000 Hz

0 25 50 75 100

0.0

0.2

0.4

0.6

0.8

1.0

Temperature (K)

Nor

mal

ized

R

P

A)

Figure 6-7. The magnetic, electric, and lattice properties are all shown to correlate. Near50 K, five simultaneous phenomena occur as the temperature is lowered. A)The remanent polarization (filled circles) and magnetization (open circles)increases rapidly. B) The strain coupling (blue) appears and themagnetoelectric coupling (red) disappears. C) The static dielectric constant,ε0, diverges. This provides evidence that the magnetoelectric coupling islinked to the strain state of the lattice (see text).

the magnetization increases rapidly, the strain coupling appears, the magnetoelectric

coupling disappears, and ε0 diverges. Figure 6-7 displays all of these simultaneous

phenomena.

129

The rapid increase in the remanent polarization tells us the PNR vibrations have

slowed down significantly (and are growing), indicating a stiffening of the lattice (see Fig.

6-7 (a)). This is also suggested by the appearance of the strain coupling (see Fig. 6-7

(b)), indicating that the lattice is now rigid enough to hold the induced distortions. The

cause of the change in the lattice is not clear, however, the coincidence of the increase

in magnetization suggests ferrodistortion is likely (see Fig 6-7 (a)). The increase in

dielectric constant indicates PNRs are likely increasing in size (see Fig. 6-7 (c)). These

phenomena suggest that 50 K is the freezing temperature known in relaxor ferroelectrics

(where fluctuation time-scales diverge, and the PNRs coalesce and couple). The

simultaneous disappearance of magnetoelectric coupling suggests its mechanism is

tied to the flexibility of the lattice (see Fig. 6-7 (b)), consistent with a recent theoretical

prediction of giant magnetoelectric coupling induced by ’structural softness’ [103].

6.4 Summary

In this chapter we demonstrated that the ferroelectric polarization in BiMnO3 may

be modulated in two distinct manners: external strain coupling, and magnetoelectric

coupling. The external strain tuning showed an increase of more than 50% in the

ferroelectric polarization when the strain was oriented to induce monoclinic distortions.

The magnetoelectric coupling manifested a 10% decrease in ferroelectric polarization

in fields of 7 T. Furthermore, these couplings provided a window into the mechanism

driving the multiferroic behavior: The low levels of strain required to increase the

ferroelectricity (less than 10−2%) suggest the large strain gradient regions near island

edges allow regions with strain just below the critical strain for ferroelectricity to be

converted. Also, the magnetic field was shown to alter the reorientation time-scales of

the PNRs providing insight into the relaxor nature of the ferroelectricity.

130

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BIOGRAPHICAL SKETCH

Patrick R. Mickel was born in 1982 to two loving parents, Stan and Karen Mickel.

He was born into a family with two older brothers, Andy and Jeremy, where his parents

fostered all of their talents without pushing in preset directions (evidenced by their

diverse choices: art, writing, and science). As a child, his affinity for tools shined

through, as he was constantly building toys. He enjoyed some success - a full-scale

half-pipe for roller-blading and skateboarding, and a make-shift BB gun - as well as

some failures, such as a pressurized squirt-gun and battery powered moped (much to

his teasing friends’ delight). Although he excelled in math growing up, it was not until

late into high school that he discovered his love for physics.

During the second semester of his senior year, he enrolled in an astronomy course

at Wittenberg University, taught by the physics professor Dr. Daniel Fleisch - who

was famous on campus for his talent and charisma. Patrick would stay after class,

sometimes for hours, talking with Dr. Fleisch and learning about all different areas

of science. He was hooked. He started college at the University of Notre Dame the

following fall listed as a tentative theology/philosophy major, but quickly changed

course and majored in physics. During his undergraduate years, he sought a diverse

experience and volunteered for research in many different areas: high energy physics,

optics, statistical physics, and condensed matter.

His senior year he decided to apply for graduate school and pursue a Ph.D. in

physics (not surprising for the son of two academics). Then, under the guidance of a

few trusted physics professors, he accepted a fellowship from the University of Florida.

While initially focused on biophysics (Nuclear Magnetic Resonance diffusion tensor

imgaing), during his second year, he took a course with Dr. Arthur Hebard and quickly

realized how great an opportunity it would be to work with him. The next semester he

joined Art’s lab, and began the research that has culminated in this dissertation. Finally,

he graduate with his Ph.D. from the University of Florida in August of 2011.

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