Ferroic Materials for Smart Systems: From Fundamentals to Device Applications [1 ed.] 3527344764, 9783527344765

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Ferroic Materials for Smart Systems

Ferroic Materials for Smart Systems From Fundamentals to Device Applications

Jiyan Dai

Author Prof. Jiyan Dai

The Hong Kong Polytechnic University Department of Applied Physics Hung Hom Kowloon Hong Kong

All books published by Wiley-VCH are carefully produced. Nevertheless, authors, editors, and publisher do not warrant the information contained in these books, including this book, to be free of errors. Readers are advised to keep in mind that statements, data, illustrations, procedural details or other items may inadvertently be inaccurate.

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© antoniokhr/Getty Images; (graph) Courtesy of Professor Jiyan Dai, The Hong Kong Polytechnic Univeristy

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A catalogue record for this book is available from the British Library. Bibliographic information published by the Deutsche Nationalbibliothek

The Deutsche Nationalbibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data are available on the Internet at . © 2020 Wiley-VCH Verlag GmbH & Co. KGaA, Boschstr. 12, 69469 Weinheim, Germany All rights reserved (including those of translation into other languages). No part of this book may be reproduced in any form – by photoprinting, microfilm, or any other means – nor transmitted or translated into a machine language without written permission from the publishers. Registered names, trademarks, etc. used in this book, even when not specifically marked as such, are not to be considered unprotected by law. Print ISBN: 978-3-527-34476-5 ePDF ISBN: 978-3-527-81534-0 ePub ISBN: 978-3-527-81537-1 oBook ISBN: 978-3-527-81538-8 Typesetting SPi Global, Chennai, India Printing and Binding

Printed on acid-free paper 10 9 8 7 6 5 4 3 2 1

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Contents

1

General Introduction: Smart Materials, Sensors, and Actuators 1

1.1 1.2 1.2.1 1.2.2 1.2.3 1.3 1.3.1 1.3.2 1.4 1.5

Smart System 2 Device Application of Ferroelectric Materials 5 Piezoelectric Device Applications 6 Infrared Sensor 7 Ferroelectric RAM (FeRAM) 8 Device Application of Ferromagnetic Materials 9 Spin-Transfer Torque Memory 9 Magnetic Field Sensor Based on Multiferroic Device 9 Ferroelastic Material and Device Application 10 Scope of This Book 12 References 13

2

Introduction to Ferroelectrics 15

2.1 2.1.1 2.1.2

What Is Ferroelectrics? 15 P–E Loop 15 Relationships Between Dielectric, Piezoelectric, Pyroelectric, and Ferroelectric 16 Ferroelectric–Dielectric 16 Ferroelectric–Piezoelectric 17 Ferroelectric–Pyroelectric 18 Origin of Ferroelectrics 18 Structure-Induced Phase Change from Paraelectric to Ferroelectric 18 Soft Phonon Mode 19 Theory of Ferroelectric Phase Transition 21 Landau Free Energy and Curie–Weiss Law 21 Landau Theory of First-Order Phase Transition 23 Landau Theory of a Second-Order Phase Transition 26 Ferroelectric Domains and Domain Switching 28 Domain Structure 28 Ferroelectric Switching 28 Ferroelectric Materials 29

2.1.2.1 2.1.2.2 2.1.2.3 2.2 2.2.1 2.2.2 2.3 2.3.1 2.3.2 2.3.3 2.4 2.4.1 2.4.2 2.5

vi

Contents

2.5.1 2.5.2 2.5.3 2.5.3.1 2.5.4 2.5.4.1 2.5.4.2 2.5.4.3 2.6

From BaTiO3 to SrTiO3 29 From PbTiO3 to PbZrO3 32 Antiferroelectric PbZrO3 33 Pb(Zrx Ti1−x )O3 (PZT) 35 Relaxor Ferroelectrics 36 Relaxor Ferroelectrics: PMN-xPT Single Crystal 37 Polar Nano Regions 38 Morphotropic Phase Boundary (MPB) of PMN-PT Crystal Ferroelectric Domain and Phase Field Calculation 41 References 42

3

Device Applications of Ferroelectrics 47

3.1 3.2 3.2.1 3.2.2 3.2.3 3.2.4 3.3 3.3.1 3.3.2 3.3.3 3.4 3.5 3.6

Ferroelectric Random-Access Memory 47 Ferroelectric Tunneling Non-volatile Memory 50 Tunneling Models 51 Metal–Ferroelectric–Semiconductor Tunnel Junction 55 Ferroelectric Tunneling Memristor 56 Strain Modulation to Ferroelectric Memory 57 Pyroelectric Effect and Infrared Sensor Application 58 Pyroelectric Coefficient 58 Pyroelectric Infrared Sensor 59 Pyroelectric Figures of Merit 60 Application in Microwave Device 63 Ferroelectric Photovoltaics 65 Electrocaloric Effect 67 References 68

4

Ferroelectric Characterizations 73

4.1 4.2 4.3 4.3.1 4.3.2

P–E Loop Measurement 73 Temperature-Dependent Dielectric Permittivity Measurement 76 Piezoresponse Force Microscopy (PFM) 77 Imaging Mechanism of PFM 77 Out-of-plane Polarization (OPP) and In-plane Polarization (IPP) PFM 80 Electrostatic Force in PFM 83 Perspectives of PFM Technique 84 Structural Characterization 86 Domain Imaging and Polarization Mapping by Transmission Electron Microscopy 87 Selected Area Electron Diffraction (SAED) 88 Convergent Beam Electron Diffraction (CBED) for Tetragonality Measurement 91 References 92

4.3.2.1 4.3.2.2 4.4 4.5 4.5.1 4.5.2

5

Recent Advances in Ferroelectric Research 95

5.1 5.2

Size Limit of Ferroelectricity 95 Ferroelectricity in Emerging 2D Materials 96

40

Contents

5.3 5.4 5.5 5.6

Ferroelectric Vortex 99 Molecular Ferroelectrics 104 Ferroelectricity in HfO2 and ZrO2 Fluorite Oxide Thin Films 106 Ferroic Properties in Hybrid Perovskites 114 References 117

6

123 General Introduction to Piezoelectric Effect 123 Piezoelectric Constant Measurement 124 Piezoelectric Charge Constant 125 Piezoelectric Voltage Constant 126 Dielectric Permittivity 127 Young’s Modulus (Elastic Stiffness) 127 Elastic Compliance 127 Electromechanical Coupling Factor 128 How to Measure Electromechanical Coupling Factor? 129 Equivalent Circuit 132 Characterization of Piezoelectric Resonator Based on a Resonance Technique 135 Length Extensional Mode of a Rod 135 Extensional Vibration Mode of a Long Plate 138 Thickness Shear Mode of a Thin Plate 139 Thickness Mode of a Thin Disc/Plate 140 Radial Mode in a Thin Disc 141 Mechanical Quality Factor 141 References 141

6.1 6.2 6.2.1 6.2.2 6.2.3 6.2.4 6.2.5 6.2.6 6.2.6.1 6.3 6.4 6.4.1 6.4.2 6.4.3 6.4.4 6.4.5 6.4.6

Piezoelectric Effect: Basic Theory

7

Piezoelectric Devices 143

7.1 7.1.1 7.1.2 7.1.3 7.1.3.1 7.1.3.2 7.1.3.3 7.1.4 7.1.4.1 7.1.5 7.1.6

Piezoelectric Ultrasonic Transducers 143 Structure of Ultrasonic Transducers 143 Theoretical Models of Ultrasonic Transducer (KLM Model) 145 Characterization of Ultrasonic Transducers 147 Bandwidth (BW) 147 Sensitivity of the Transducer 148 Resolution 148 Types of Ultrasonic Transducers 149 Medical Application 149 Piezoelectric Film Application in Ultrasound Transducers 149 Challenges and Trend of Developing New Advanced Ultrasound Transducers 150 Ultrasonic Motor 150 Terminologies 151 Design of USM 153 Surface Acoustics Wave Devices 154 Interdigital Electrode in SAW Device 155 Acoustic Wave 155 Piezoelectric Property Considerations for SAW Devices 157

7.2 7.2.1 7.2.2 7.3 7.3.1 7.3.2 7.3.3

vii

viii

Contents

7.3.4 7.3.5

Characterization of SAW Devices 159 Lead-Free Piezoelectric Materials 161 References 163

8

8.1 8.2 8.3 8.4 8.4.1 8.4.2 8.4.3 8.4.4 8.4.4.1 8.5 8.5.1 8.5.2 8.5.3 8.5.3.1 8.6 8.6.1 8.6.2 8.6.3 8.6.4 8.7 8.7.1 8.7.2 8.7.3

165 General Introduction to Ferromagnetics 165 Ferromagnetic Phase Transition: Landau Free-Energy Theory 168 Domain and Domain Wall 169 Magnetoresistance Effect and Device 171 Anisotropic Magnetoresistance (AMR) 171 Giant Magnetoresistance (GMR) 172 Colossal Magnetoresistance (CMR) 175 Tunneling Magnetoresistance (TMR) 176 Spin-Transfer Torque Random-Access Memory (STT-RAM) 177 Magnetostrictive Effect and Device Applications 178 Magnetostrictive Properties of Terfenol-D 180 Magnetostrictive Ultrasonic Transducer 183 Magnetoelastic Effect 184 Magnetomechanical Strain Gauge 185 Characterizations of Ferromagnetism 186 Vibrating Sample Magnetometer (VSM) 186 Superconducting Quantum Interference Device (SQUID) 187 Magnetic Force Microscopy (MFM) 188 Magneto-Optical Kerr Effect (MOKE) 189 Hall Effect 191 Ordinary Hall Effect 191 Anomalous Hall Effect 191 Spin Hall Effect 192 References 193

9

Multiferroics: Single Phase and Composites 197

9.1 9.2 9.3 9.4 9.4.1 9.5 9.6

Introduction on Multiferroic 197 Magnetoelectric Effect 199 Why Are There so Few Magnetic Ferroelectrics? 199 Single Phase Multiferroic Materials 200 Switching Mechanism in BFO Films 204 ME Composite Materials 205 Modeling the Interfacial Coupling in Multilayered ME Thin Film 207 PZT/CFO Multilayered Heterostructures 207 Ferroelectric Properties of PZT/CFO Multilayers 209 References 212

9.6.1 9.6.2

10

10.1 10.1.1 10.2

Ferromagnetics: From Material to Device

217 ME Composite Devices 217 Effect of Preload Stress 221 Memory Devices Based on Multiferroic Thin Films 223

Device Application of Multiferroics

Contents

10.3

Memory Devices Based on Multiferroic Tunneling 224 References 229

11

Ferroelasticity and Shape Memory Alloy 231

11.1 11.1.1 11.1.2 11.1.3 11.1.4 11.1.5 11.2 11.2.1 11.2.2 11.2.3

Shape Memory Alloy 231 SMA Phase Change Mechanism 232 Nonlinearity in SMA 233 One-Way and Two-Way Shape Memory Effect Superelastic Effect (SE) 235 Application Examples of SMAs 236 Ferromagnetic Shape Memory Alloys 237 Formation of Twin Variants 238 Challenges for Ni–Mn–Ga SMA 242 Device Application of MSMA 243 References 244 Index 247

235

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1 General Introduction: Smart Materials, Sensors, and Actuators The early twenty-first century has foreseen acceleration of innovations in robotics and automations as well as artificial intelligence (AI), where sensors/ transducers and smart materials play very important roles. The concept of AI has been around since the late 1950s; however, it’s only since the first decade of the twenty-first century that excitement about it has really begun to grow due to the ability of fast computation and abundant size of memory devices. A very successful demonstration of AI is Google’s AlphaGo, which is the first computer program to defeat a professional human Go player (see Figure 1.1). Another successful application of AI is the unmanned vehicles and aircrafts where large number of sensors and actuators are used. One may ask, what is the relation between the key word “ferroic materials” of this book and the mentioned robotics, automations, and AI? The answer is that these smart and intelligence systems rely on large amount of data from sensors and memories for machine learning and actuators for close-looped feedback control systems; and among these sensors, actuators, and memories, ferroic materials play very important roles. For example, the piezoelectric property of a ferroelectric material (one of the most typical ferroic materials) can be used for ultrasound sensors to detect distance of your car from a wall for auto parking system. A ferroelectric material can be used as the functional element for many kinds of sensors from pressure sensor to acceleration sensor, infrared sensor, etc. Beyond that, a ferroelectric polarization and its switching can also be used in memory devices such as ferroelectric random-access memory (FeRAM) where the ferroelectric layer acts as gate insulator in a field-effect transistor (FET) structure. Ferroelectric tunneling-based resistive random-access memory (RRAM) has also been demonstrated, and such ferroelectric-based memory has been shown to be able to perform as an artificial synapse. More interestingly, artificial neural networks (ANNs) based on these ferroic synapses can realize brain-like computing and AI functions such as image recognition. As shown in Figure 1.2, synapses with BiFeO3 (BFO) ferroelectric layer has been successfully demonstrated. This book will tell you fundamentals and characterization methods of ferroic materials, physics, and technologies behind ferroic device design and applications as well as their recent advances.

Ferroic Materials for Smart Systems: From Fundamentals to Device Applications, First Edition. Jiyan Dai. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

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1 General Introduction: Smart Materials, Sensors, and Actuators

Figure 1.1 AI beats human chess player.

Synapse Nucleus Axon

Axon Memristor

Input

Co BFO CCMO

Synapses To pre-neuron

To post-neuron

Output

Figure 1.2 A cross-bar structure of synapse and artificial neuron networks based on a cross-bar structure. Ferroelectric thin films such as BiFeO3 can be used as the junction material. Source: Adapted from Boyn et al. (2017).

1.1 Smart System A smart system, such as a self-driving car or remote-control aircraft, is a system that relies on sensors and actuators to realize instant feedback of controlled variables (CVs) such as speed, height, etc. The basic component of a smart system usually contains sensors, actuators, and control system. An intelligent smart system needs large amount of data processing and memories, while ferromagnetic and ferroelectric materials have been implemented in realizing non-volatile memristors. Beyond that, memories based on ferroelectric thin films may also find application in electronic synapses as building blocks toward building ANN.

1.1 Smart System

Proportional +

Work

Integral Sensor Derivative

Actuator

(a)

(b)

Figure 1.3 (a) Photo of a remote control copter and (b) diagram of a PID feedback control system where sensors and actuators are implemented.

As an example, a smart system of remote control copter relying on proportional–integral–derivative (PID) feedback control system is shown in Figure 1.3. PID is a three-order feedback control system that has been widely used in auto driving vehicles and auto-pilot airplanes to make the dynamic system operate smoothly or being stable during video imaging. Equation (1.1) illustrates the mechanism of PID control where three terms of proportional gain (P), integral gain (I), and derivative feedback (D) can provide instant response to cure the error (E) between the set point (speed of vehicle or height of the copter) and controlled variable (CV): t

MV (t) = PE(t) + I

∫0

E(t ′ )dt ′ − D

dCV (t) dt

(1.1)

In this feedback control system, we can find applications of ferroic materials, for example, piezoelectric-based gyroscope, surface acoustic wave device for wireless communication, and ferroelectric-based infrared detector. In gyroscope, rotation and acceleration can be sensed by measuring induced voltage generated by piezoelectric effect; a surface acoustic wave device based on piezoelectric effect is used for communication band selection; and a ferroelectric-based infrared detector can be used as an intruder sensor making the copter able to find people for rescue mission. Figure 1.4 shows the finite element modeling (FEM) simulation of three resonant motions in a Pb(Zrx, Ti1−x )O3 (PZT)-based gyroscope and photo of a fabricated gyroscope (Chang and Chen 2017). Sensors: Devices that can “sense” a change in some physical characteristics and perform an electrical input function are commonly called sensors. For example, a strain sensor converts mechanical strain into electrical signal. Actuators: Devices that perform an output function are generally called actuators. An actuator can be utilized to control some external moduli or output mechanical movement such as ultrasonic wave. For example, an atomic force microscope (AFM) uses piezoelectric actuators to realize scanning along three directions. Transducers: Both sensors and actuators are collectively known as transducers because they are used to convert energy of one kind into energy of another kind. Transducers can be used to sense a wide range of different energy forms

3

4

1 General Introduction: Smart Materials, Sensors, and Actuators Y Z

X

MEMS scanner

Horizontal scan Vertical scan

PZT on stainless steel

1 mm

Gyro sensors

Resonant in Y-direction

Resonant in Y-direction

4 mm

Resonant in X-direction

Figure 1.4 FEM simulations of three resonant motions in a PZT-based gyroscope and photo of a fabricated gyroscope. Source: Adapted from Chang and Chen (2017).

such as movement, electrical signals, thermal or magnetic energy, etc. The type of input or output transducers being used really depends on the type of signal or process being “sensed” or “controlled,” but we can define a transducer as a device that converts one physical quantity into another. A smart system needs sensors and actuators to realize the sensing functions such as distance, movement, and acceleration as well as actions. These sensors and actuators use smart materials to realize the conversion between different energies and moduli to electrical signals such as voltage, current, and capacitance. Of course, many sensor devices are made of semiconductors such as the FET, but this is not the focus of this book. “Smart material” is a very large concept, in fact, there is no stupid material (a joke), i.e. all materials are smart in some way since they all have their own properties and response to external stimuli. But in this book, we restrict the “smart materials” to those materials with “ferroic” characteristics. We focus on basic physics, materials science, structures, devices, and applications of ferroic materials for smart systems. The ferroic materials are usually classified as possessing one of the followings based on coupling of stimuli: (i) Ferroelectric, which is also piezoelectric when electromechanically coupled and pyroelectric when thermoelectrically coupled. (ii) Ferromagnetic, which is also magnetostrictive when magnetomechanically coupled. (iii) Ferroelastic, which also includes shape memory when thermomechanically coupled. Among these ferroics, we can see that strain, electric polarization and magnetization, and their interplay or coupling are involved. We call a material as ferroic material if it possesses at least one of the properties of ferroelectric, ferromagnetic and ferroelastic. If we look at the diagram shown in Figure 1.5, we can see that the coupling and interplay between electricity, mechanics, magnetism, heat, and optics result in many smart functions, such as ferroelectric, piezoelectric, pyroelectric, ferromagnetic, electromechanical, etc. One book cannot cover all of them, but those belong to ferroic materials and devices especially in the form of thin films will be

1.2 Device Application of Ferroelectric Materials

Electricity

as

to-

op

ic

tic

Py The roelectr rmo i elec city tricit y

s

ism

et

n ag

om

rm

e Th

y tricit elec cs Opto tovoltai Pho

loric toca Elas oelastic rm The

Magnetostriction ric alo c to ne ag M

Magnetism

etooptic

El

Mag n

Mechanics

e

El

ict

str

o ctr

ion

Elec troc alor

oe

ez

Pi

E Ma lectr gn om eto ag ele ne ctr tism ici ty

ptic tro-o Elec

ty

ici

tr lec

Thermo-optics Heat

Optics

Figure 1.5 Diagram showing coupling between different moduli and the clarification of smart materials.

extensively introduced in this book. Before going into details, some application examples of ferroic materials in smart systems are given in this chapter.

1.2 Device Application of Ferroelectric Materials When people talk about applications of ferroelectric materials, the first thing jumps out is most possibly the PZT (lead–zirconium–titanate with chemical formula Pb(Zrx Ti1−x )O3 ), which is known as an excellent piezoelectric material. As the most popular ferroelectric, PZT is also the most important piezoelectric material in commercial applications. Piezoelectric materials have very broad applications in many fields, from medical ultrasound imaging to ultrasonic wire bonding machine in semiconductor industry, from pressure sensors to accelerometer, etc. The market size of piezoelectric materials is more than US $1 billion now and is expected to be US $1.68 billion by 2025 (GRAND VIEW RESEARCH). Another field of application of ferroelectric materials is the infrared sensors based on their pyroelectric property, which is also one of the most important properties of a ferroelectric material. Beyond these well-known applications, another important application based on the switching of ferroelectric polarization is the non-volatile memory device such as FeRAM. Examples are given in the following and details will be introduced in the following chapters.

5

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1 General Introduction: Smart Materials, Sensors, and Actuators

1.2.1

Piezoelectric Device Applications

An example of smart system using piezoelectric material is the distance radar system in a car or a sonar system in submarines as shown in Figure 1.6, where the key sensing element is based on piezoelectric material to realize the conversion between electrical energy and acoustic energy for sending and receiving sound waves. Other application examples of piezoelectric devices include active damping system, micro-scanning system in scanning probe imaging instrument (such as AFM), force sensor, accelerometer, energy harvesting, etc. Medical ultrasound imaging system with piezoelectric material as the transducer to convert electrical and acoustic energies is another very good example of device application where the piezoelectric material plays the roles of sensing and actuating functions. Figure 1.7 shows photos of ultrasound transducers developed in our group. Knowledge in ultrasound transducer fabrication, characterization, and applications will be intensively introduced in Chapter 6. A very new application example is piezoelectric-based fingerprint ID system in mobile phone. The currently used finger identification system is based on capacitance measurement to obtain two-dimensional (2D) information of fingerprint, but it faces the problem of difficulty to identify the fingerprint when the finger is dirty or wet. Ultrasound fingerprint identification system based on piezoelectric ultrasonic transducer and imaging system can obtain a three-dimensional image

(a)

(b)

Figure 1.6 Piezoelectric materials-based sonar system for car (a) and submarine (b).

(a)

(b)

Figure 1.7 (a) Transducers and (b) B-mode image of a wire phantom acquired with PolyU-made array ultrasound transducer.

1.2 Device Application of Ferroelectric Materials

PMUT unit

Capping layer

Coupling materials + MEMS Piezoelectric layer Cavity CMOS wafer Bottom electrode

Top electrode

Figure 1.8 Illustration of concept of a ultrasonic transducer-based fingerprint ID system based on complementary metal-oxide-semiconductor micro-electro-mechanical systems (CMOS-MEMS) technology.

of fingerprint with a certain depth. This can overcome the problems of the current fingerprint identification system in most mobile phones. InvenSense, Inc. is one of the main suppliers of this solution, and Figure 1.8 is an illustration of the ultrasonic fingerprint system. 1.2.2

Infrared Sensor

An infrared sensor is usually made of a ferroelectric material, which is also pyroelectric that generates surface electric charges when exposed to heat in the form of infrared radiation. A pyroelectric-based infrared sensor can detect the temperature change but produce no response for a steady temperature since the pyroelectric sensing element can only produce polarization change-induced electric charge when the sensor is subject to temperature change. Figure 1.9a shows a photo of a real infrared detector with its internal device structure illustrated in Figure 1.9b, where the active element is made of pyroelectric materials such as LiTaO3 . Those pyroelectric materials with their polarization able to be switched are ferroelectrics. Therefore, pyroelectric sensors that are widely used as infrared detectors are important device applications for ferroic materials in a smart system. Window

Bottom electrode

Absorbing electrode layer Pyroelectric plate

Circuit board

(a)

(b)

Figure 1.9 A photo of an infrared detector (a) and illustration of its internal structure (b).

7

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1 General Introduction: Smart Materials, Sensors, and Actuators

1.2.3

Ferroelectric RAM (FeRAM)

Ferroelectric RAM (FeRAM, F-RAM, or FRAM) is a random-access memory that is similar to Dynamic Random Access Memory (DRAM) in structure but uses a ferroelectric layer instead of a dielectric layer to achieve non-volatility. FeRAM is one of a growing member of alternative non-volatile random-access memory technologies that offers the same functionality as flash memory. Advantages of FeRAM over flash memory include lower power usage, faster write performance, and much greater maximum read/write endurance (about 1010 –1014 cycles). FeRAMs have data retention of more than 10 years at +85 ∘ C (up to many decades at lower temperatures). Market disadvantages of FeRAM are much lower storage densities than flash devices and higher cost. A ferroelectric material has a nonlinear relationship between the applied electric field and the stored charge. Specifically, the ferroelectric characteristic has the form of a hysteresis loop, which is very similar in shape to the hysteresis loop of ferromagnetic materials. The dielectric constant of a ferroelectric is typically much higher than that of a linear dielectric because of the effects of electric dipoles formed in the crystal structure of the ferroelectric material. When an external electric field is applied across a dielectric, the dipoles tend to align themselves with the field direction. This alignment process is produced by small shifts in the positions of ions and shifts in the distributions of electric charges in the crystal structure. After the charges are removed, the dipoles retain their polarization state. Binary “0”s and “1”s are stored as one of the two possible electric polarizations in each data storage cell. For example, in Figure 1.10, a “1” is encoded using the negative remnant polarization “−Pr ,” and a “0” is encoded using the positive remnant polarization “+Pr ”. In this FET structure with a ferroelectric layer as gate dielectric, the two polarization states correspond to different V th , resulting in a memory window within which the ON and OFF states of the FET can be read.

Spontaneous polarization of ferroelectric layer Memory window

Gate Drain

Source

–

–

–

–

– – – – – – – –– –

N

IDS

N –––––––––

P

–––––––––

–

–

–

–

VG

Figure 1.10 Schematic diagram of field-effect transistor (FET) and the current–voltage (I–V) characteristics induced by two different polarization state.

1.3 Device Application of Ferromagnetic Materials

FeRAM remains a relatively small part of the overall semiconductor market. In 2016, worldwide semiconductor sales were US $338.93 billion (according to WSTS, SIA), with the flash memory market accounting for US $59.2 billion (according to IC insights (Cho 2018)). The 2017 annual revenue growth of Cypress semiconductor, perhaps the major FeRAM vendor, were reported to be US $2.33 billion. The much larger sales of flash memory compared with the alternative FeRAMs support a much larger research and development effort. Flash memory is produced using semiconductor linewidths of 15 nm at Renesas Electronics Corporation (2017). Flash memory can store multiple bits per cell (currently Samsung has announced the 64-layer 512-Gb in the NAND flash devices). As a result of innovations in flash cell design, the number of bits per flash cell is projected to increase to double or even to triple. As a consequence, the areal bit densities of flash memory are much higher than those of FeRAM, and thus the cost per bit of flash memory is orders of magnitude lower than that of FeRAM.

1.3 Device Application of Ferromagnetic Materials Among many successful applications of ferromagnetic-based devices, memory device based on ferromagnetic material is one of the most successful examples, especially in the thin film form. This is manifested by the very large market of magnetic hard disc in computing systems. But in most recent years, solid state memory (mainly flash memory) is superseding the magnetic hard disc. Nevertheless, ferromagnetic material also finds its application in non-volatile memories such as spin-transfer torque memory. 1.3.1

Spin-Transfer Torque Memory

Spin-transfer torque can be used to flip the active elements in magnetic random-access memory. Spin-transfer torque magnetic random-access memory (STT-RAM or STT-MRAM) has the advantages of lower power consumption and better scalability over conventional magnetoresistive random-access memory (MRAM), which uses magnetic field to flip the active elements. Spin-transfer torque technology has the potential to make possible MRAM devices combining low current requirements and reduced cost; however, the amount of current needed to reorient the magnetization at present is too high for most commercial applications, and the reduction of this current density alone is the basis for present academic research in spin electronics. Figure 1.11 is a schematic diagram of spin valve structure, while arrows indicate the magnetization direction. 1.3.2

Magnetic Field Sensor Based on Multiferroic Device

If a material possesses more than one of the ferroic properties of ferroelectric, ferromagnetic and ferroelastic, it is called multiferroics. Unfortunately, such

9

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1 General Introduction: Smart Materials, Sensors, and Actuators

Free layer

Pin layer

Low resistance state

High resistance state

Figure 1.11 Schematic diagram of spin valve structure where arrows indicate the magnetization directions.

materials are rare and are usually strong in one property but very weak in another, such as BiFeO3 , which is very strong in ferroelectrics but very weak in ferromagnetic (it is antiferromagnetic in fact). This makes multiferroic materials hard to be practically applied in devices. However, people have been trying to make composite materials such as piezoelectric with magnetostrictive materials, where the mechanical coupling between them makes the “multiferroic” meaningful for device application, for example, making very sensitive magnetic field sensor. The magnetoelectric (ME) effect is the phenomenon of inducing magnetization by an applied electric field (E) or polarization by magnetic field (H). Many efforts have been devoted to improve the limit of detection of the ME composite at low frequency range, and values of ∼10−7 Oe at 1 Hz has been reported (Wang et al. 2011). Based on the magnetic–strain–electric coupling, scientists have demonstrated dc magnetic field sensor with a detection limit of 2 × 10−5 Oe to dc magnetic field with a nonlinear ME magnetic effect (Li et al. 2017). Ferroelectric material PZT and magnetostrictive material Metglas have been implemented in the composite device (see Figure 1.12).

1.4 Ferroelastic Material and Device Application Shape memory alloys (SMAs) are the most typical ferroelastic material, which is an important member of ferroics. The shape memory characteristics originates from the phase transition between high temperature austenite phase and low temperature martensite phase, where the shape at cubic-austenite phase can be resumed from low temperature martensite phase whose lattice can be largely twisted (see Figure 1.13). The SMAs have been widely used in devices from brace of orthodontia and other medical applications, air jet, satellite antenna, etc. Research was carried out in developing systems that would optimize the chevron “immersion”

1.4 Ferroelastic Material and Device Application

ID electrodes Metglas

Piezofiber Epoxy

Kapton

VirginiaTech

Figure 1.12 Outline of ME device from Virginia Tech and schematic of the cross-section of the ME composite. Source: Wang et al. (2011). Adapted with permission of John Wiley and Sons.

He

Stress/load

ati

ng

Loading

Cooling Heating

Temperature

Figure 1.13 Phase transition between high temperature austenite phase and low temperature martensite phase, where the shape at cubic-austenite phase can be resumed from low temperature martensite phase whose lattice can be largely twisted.

into the jet flow based on the flight condition. As shown in Figure 1.14a, SMAs activated by heat were developed that would allow for full chevron immersion in jet flow during high thrust requirements (e.g. during take-off ) and not immersing it during cruise where the thrust efficiency is of greater importance (Anon n.d.).

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1 General Introduction: Smart Materials, Sensors, and Actuators

(a)

(b)

Figure 1.14 (a) Brace of orthodontia using shape memory alloys and (b) arthrodesis device developed by Karnes et al.

For broken bone rehabilitation, a SMA plate with a memory transfer temperature close to body temperature can be attached to both ends of the broken bone as shown in Figure 1.14b. From body heat, the plate will contract and retain its original shape, therefore exerting a compression force on the broken bone at the place of fracture. After the bone has healed, the plate continues exerting the compressive force and aids in strengthening during rehabilitation (Garlock et al. 2017).

1.5 Scope of This Book In Chapters 2–5, fundamentals of ferroelectrics, applications of ferroelectric materials, recent advances, and advanced measurement and testing techniques in ferroelectrics will be introduced. In particular, device applications of ferroelectric materials in thin film form will be introduced including FeRAM, ferroelectric tunneling-based resistive switching, etc. The recent advances include ferroelectricity in emerging materials such as 2D materials and high-k gate dielectric material HfO2 , while the advanced characterization technologies include the piezoresponse force microscopy (by imaging and switching ferroelectric domains) and Cs-corrected transmission electron microscopy (TEM) where atomic level ionic displacement can be identified. As the most important property application of ferroelectric materials, fundamentals of piezoelectric physics and engineering considerations for device design and fabrication are introduced in Chapters 6 and 7. In Chapter 8, starting with a brief introduction on origin of ferromagnetism and its analogy to ferroelectrics, device applications, particularly for magnetostrictive devices, are introduced. Chapters 9 and 10 will introduce the multiferroics of materials possessing both ferromagnetic and ferroelectric orders including single phase and composite materials. In particular, devices based on the integration of ferroelectric and ferromagnetic materials such as multiferroic memory device and ME coupling device for sensor applications will be introduced. In Chapter 11, ferroelastic materials represented by SMA and magnetic SMAs as well as their device applications will be introduced.

References

References Boyn, S., Grollier, J., Lecerf, G. et al. (2017). Learning through ferroelectric domain dynamics in solid-state synapses. Nature Communications 8: 1–7. Chang, C.-Y. and Chen, T.-L. (2017). Design, fabrication, and modeling of a novel dual-axis control input PZT gyroscope. Sensors 17 (11): 2505. Cho, J. (2018). Amid contradictory forecast: IC insights: ‘Memory chips will grow at annual rate of 5% only on average by 2022’. Seoul, Korea: BusinessKorea. Garlock, A., Karnes, W.M., Fonte, M. et al. (2017). Arthrodesis devices for generating and applying compression within joints. US 2017/0296241 A1, Available at: https://patents.google.com/patent/US20170296241A1/en. Li, M., Dong, C., Zhou, H. et al. (2017). Highly sensitive DC magnetic field sensor based on nonlinear ME effect. IEEE Sensors Letters 1 (6): 1–4. Renesas Electronics Corporation (2017). Renesas electronics achieves large-scale memory operation in fin-shaped MONOS flash memory for industry’s first high-performance, highly reliable MCUs in 16/14nm process nodes and beyond. Wang, Y., Gray, D., Berry, D. et al. (2011). An extremely low equivalent magnetic noise magnetoelectric sensor. Advanced Materials 23 (35): 4111–4114. Available at: https://doi.org/10.1002/adma.201100773.

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2 Introduction to Ferroelectrics 2.1 What Is Ferroelectrics? Ferroelectric can be defined as switchable spontaneous polarization at temperatures below the Curie point (T c ). The spontaneous polarization and its switching make the ferroelectric material exhibit polarization–electric field hysteresis loop (called P–E loop) as shown in Figure 2.1, where the Pr is called remnant polarization, Ps is the saturated polarization, and Ec is the coercive field, a field at which the polarization direction can be switched. A very important characteristic of ferroelectric materials is the presence of domain structure where each domain has its own polarization direction. The formation of domain structure is to align dipoles in different directions in order to minimize electrostatic and elastic energy, and it is the domain switching and domain wall movement that produce hysteresis and saturation nonlinearities. As shown in the inset of the figure, the polarization switching is accompanied by domain wall movement and new domain formation. It is worthy to note that, except with presence of charged defects, the polarizations are usually aligned head-to-tail. However, in some nanostructured ferroelectrics, or under a special strained situation, charged domain walls, topological domains such as vertex type of domains can be formed. Details will be introduced in Chapter 5. 2.1.1

P–E Loop

People usually use the term of P–E loop to prove the ferroelectricity. But, strictly speaking, it should be called D–E loop since the so-called polarization is actually obtained by measuring the electric displacement D through capacitance measurement. Nevertheless, it is approximately correct, since for ferroelectric material, the electric polarization is nearly equal to the electric displacement. For a dielectric material, the presence of an electric field E makes the electron cloud slightly shifted inducing a local electric dipole moment that is called electric displacement, while for a ferroelectric material, its polarization density P (spontaneous polarization or induced by electric field) will dominate the electric displacement D that can be defined as D = 𝜖0 E + P

(2.1)

Ferroic Materials for Smart Systems: From Fundamentals to Device Applications, First Edition. Jiyan Dai. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

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2 Introduction to Ferroelectrics

P

Figure 2.1 Typical P–E loop from a ferroelectric crystal. Insets illustrate domain structures for the corresponding polarizations. As the field is increased, the polarization of domains with an unfavorable direction of polarization will start to switch in the direction of the field.

Ps Pr

–Ec

+Ec

E

–Pr

where 𝜖 0 is the vacuum permittivity or called permittivity of free space and P depends linearly on the electric field in a homogeneous and isotropic material, i.e. P = 𝜖0 𝜒 e E

(2.2)

where 𝜒 e is electric susceptibility of the material. Therefore, we have D = 𝜖0 (1 + 𝜒e )E = 𝜖E

(2.3)

where the permittivity 𝜖 = 𝜖0 𝜖r

(2.4)

and relative permittivity 𝜖r = 1 + 𝜒e

(2.5)

For a ferroelectric material, its electric displacement is contributed mainly by ionic displacement, which is much larger than the contribution from vacuum dielectric; therefore, the polarization P is approximately equal to D. Practically, P–E loop is measured by a Sawyer–Tower method that will be introduced in Chapter 3. 2.1.2 Relationships Between Dielectric, Piezoelectric, Pyroelectric, and Ferroelectric 2.1.2.1

Ferroelectric–Dielectric

Why does a ferroelectric material have a very large dielectric constant? In a large category, a ferroelectric belongs to dielectrics. There are three primary contributions to electric polarization: electronic, ionic, and dipole reorientation; and the polarization is expressed quantitatively as the sum of electric dipoles per unit volume (C/m2 ).

2.1 What Is Ferroelectrics?

Electronic

Ionic

Dipole reorientation

E=0

E>0

Center of negative charge Electric field

Figure 2.2 Schematic diagram showing the origin of the electric polarization.

For all dielectric materials, the electron clouds deform under electric field, forming electric dipoles. Electric polarization from this electron clouds deformation is usually much smaller compared with ionic displacement. In ionic crystals, when an electric field is applied, cations and anions are attracted to the cathode and anode, respectively. While for a ferroelectric, electric polarization reorientation (rotation or reversing) will result in a significant change of electric displacement (see Figure 2.2). This diagram also illustrates that a ferroelectric falls into a larger category called dielectrics. Let’s look at their difference with an example by comparing refractive index of a typical dielectric material SiO2 and a typical ferroelectric material BaTiO 3 . By following relationship of refractive index with dielectric constant √ n = 𝜖r , with dielectric constant of about 3.6 for SiO2 , its refractive index is about 1.4–1.5. However, for a ferroelectric material, its dielectric constant is usually very large due to the reorientation of electric polarization, a few hundred for BaTiO3 , for example, but we cannot say that its refractive index is also very large √ based on the relation of n = 𝜖r . In fact, the refractive index of a material is an optical constant that only works in optical frequency regime in which electron cloud can respond to the optical frequency of electric field (say 1014 Hz), but the ferroelectric polarization √ cannot follow the optical frequency to rotate or switch, i.e. this equation n = 𝜖r does not work for ferroelectrics in optical frequency. 2.1.2.2

Ferroelectric–Piezoelectric

Many materials such as AlN, GaN, and ZnO exhibit electric polarization originated from their crystal structure symmetry breaking, but their polarizations cannot be switched by external electric field. These electrically polarized materials are piezoelectric but not ferroelectrics. Figure 2.3 shows the relationships between the piezoelectric and ferroelectric materials as well as pyroelectric and dielectric materials. A ferroelectric is also piezoelectric since the electric polarization change induced by mechanical strain will generate electric charges and voltage at the surface of a ferroelectric plate, i.e. piezoelectricity.

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Figure 2.3 Relationships of the piezoelectric, ferroelectric, pyroelectric, and dielectric. Ferroelectric

Pyroelectric Piezoelectric Dielectric

2.1.2.3

Ferroelectric–Pyroelectric

As illustrated in Figure 2.3, a ferroelectric is also a pyroelectric, but a pyroelectric material may not be a ferroelectric. Pyroelectricity is a phenomenon of temperature dependence of the spontaneous polarization. As the temperature of a crystal is changed, electric charges corresponding to the change of the spontaneous polarization appear on the surface of the sample. Among many pyroelectric materials, only those whose spontaneous polarization can be reversed by an electric field belong to ferroelectrics.

2.2 Origin of Ferroelectrics 2.2.1 Structure-Induced Phase Change from Paraelectric to Ferroelectric The origin of ferroelectric comes from space inversion symmetry breaking, i.e. the structure symmetry breaking results in spontaneous polarization of a crystal and generates the ferroelectricity. A simple understanding is that there must be a relative shift of cations and anions, i.e. non-coincidence of positive and negative charge centers, in the crystal unit cell. This will not happen in a crystal of cubic structure with central symmetry. To generate electric polarization in a perovskite structure, lattice distortion along either ⟨100⟩, ⟨110⟩, or ⟨111⟩ is a must, plus relative shift of cations and anions. This only happens when the crystal is below its Curie temperature. The following describes the formation of ferroelectricity from the crystal symmetry point of view: • Crystals can be classified into 32 point groups according to their crystallographic symmetry, and these point groups can be divided into two classes, one with a center of symmetry and the other without, as indicated in Table 2.1. • Among these 32 point groups, 21 of them do not have a center of symmetry. Except the point group 432, in crystals belonging to 20 of these non-centrosymmetric point groups, positive and negative charges are generated on the crystal surfaces when a stress is applied; so these crystals are known as piezoelectrics.

2.2 Origin of Ferroelectrics

Table 2.1 Crystallographic classification according to crystal centrosymmetry. Symmetry

Crystal system

Polarity

Cubic Hexagonal Tetragonal Rhombohedral Orthorhombic Monclinic Triclinic Centro (11) m3m m3 6/mmm 6/m 4/mmm 4/m mmm 2/m 3m 3 32 point group Non-polar 422 622 432 of (22) 23 32 222 6 4 42m 43m 6m2 crystallographic Non-centro (22) 2 symmetry Polar (10) 6mm 6 4mm 4 3m 3 mm2 1 m Inside

are piezoelectrics (20)

Inside

are pyroelectrics (10)

Source: Adapted from Uchino (2009).

Ti

Ti

O

Ti

Ps

Ps

Ba T > Tc

T < Tc

Figure 2.4 Illustration of crystal structure symmetry breaking-induced ferroelectricity in BaTiO3 .

• Among these 20 point groups, 10 of them have a unique polar direction, and these 10 polar classes are pyroelectrics. Among these pyroelectrics, only those whose polar can be reversed by external electric field are ferroelectrics. BaTiO3 is an example of a typical ferroelectric crystal with perovskite structure as illustrated in Figure 2.4. Above the Curie temperature T c , the paraelectric cubic structure is centrosymmetric, whereas in the tetragonal phase below T c , it is energetically favorable for the O2− ions to be shifted slightly below face centers, and Ti4+ ions are shifted upward from the unit cell center. The relative change in positions of the Ti4+ and O2− ions produces a spontaneous polarization Ps as well as the non-centrosymmetric structure. More ferroelectric phases of BaTiO3 will be introduced later. 2.2.2

Soft Phonon Mode

Ionic displacement can be expected through lattice vibrations at a finite temperature. Figure 2.5 shows some of the possible lattice vibrations in a perovskite-like crystal, where Panel (a) shows an initial cubic (symmetrical) structure. Panel (b) is a symmetrically elongated one. Panel (c) has coherently shifted center cations exhibiting ferroelectricity. If the shift is anti-polarized, it results in antiferroelectricity (AFE) that will be introduced later.

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2 Introduction to Ferroelectrics

Elongation direction

(a)

(b)

(c)

Figure 2.5 Starting from the original cubic structure (a), if (b) is stabilized, only oxygen octahedra are distorted without generating dipole moments (acoustic mode). When (c) is stabilized, dipole moments are generated (optical mode). The final stabilized state (c) corresponds to ferroelectric state.

Since only oxygen octahedra are distorted in Figure 2.5b without generating electric polarization, its vibration is a kind of phonon mode. However, if the particular mode Figure 2.5c becomes stabilized, with decreasing temperature, the vibration frequency decreases (soft phonon mode), and finally at a certain phase transition temperature (Curie temperature), this frequency becomes zero, i.e. the electric polarization state is stabilized. Since the soft modes in ferroelectrics lead to electric polarization, they are optically active and can be detected by means of optical spectroscopy in the spectra of dielectric permittivity (real and imaginary parts). Therefore, the vibration shown in Figure 2.5c is also called optical mode. Spectroscopic studies of the soft phonon modes provide a very powerful tool for investigating the ferroelectric transitions. Figure 2.6 shows Raman spectra of BaTiO3 phase transition, where the 305 cm−1 peak reduces its sharpness as the in situ temperature increases and becomes indistinct at 150 ∘ C, indicating that tetragonal phase is transformed to cubic phase. This trend is consistent with the results that the peak around 310 cm−1 appeared below Curie temperature (T c ) and vanished above T c (Hayashi et al. 2013). The dielectric permittivity in the soft mode is governed by three laws (Cochran 1960; Shirane et al. 1970; Luspin et al. 1980): (a) Static dielectric permittivity 𝜖 r (0) produced by the soft mode obeys the Curie–Weiss law: 𝜖r (0) ∼ (T − Tc )−1

(2.6)

(b) The eigen frequency 𝜔(T) of the soft mode follows the Cochran behavior: 1

𝜔(T) ∼ (T − T0 ) 2 where T 0 is the soft mode condensation temperature.

(2.7)

Intensity (a.u.)

2.3 Theory of Ferroelectric Phase Transition

200 °C 150 °C

100 °C 50 °C 25 °C

200

300

400 500 Raman shift (cm–1)

600

700

800

Figure 2.6 In situ Raman spectra of BaTiO3 particles measured at different temperatures: 25, 50, 100, 150, and 200 ∘ C. Source: Hayashi et al. (2013). Adapted with permission of Elsevier.

(c) The static dielectric constant and the soft mode frequency are connected via the Lyddane–Sachs–Teller (LST) relation: 𝜔2 (T) 𝜖∞ (T) = T 2 𝜖r (0) 𝜔L

(2.8)

where 𝜖 ∞ is the high-frequency dielectric constant and 𝜔T and 𝜔L are the transverse and longitudinal frequencies of the corresponding vibrations, respectively.

2.3 Theory of Ferroelectric Phase Transition 2.3.1

Landau Free Energy and Curie–Weiss Law

Ferroelectric phase transition can be well described by thermodynamic theory. Landau addressed this problem by considering free energy in the form of expansion of ferroelectric polarization P, from which fundamental properties of the ferroelectric material such as polarization and dielectric permittivity can be derived. For simplicity, we only consider one-dimensional case, so the expression of Landau free energy as a function of polarization and temperature T is F = F0 (T) + F(P, T) F(P, T) =

1 2 1 4 1 6 𝛼P + 𝛽P + 𝛾P + … 2 4 6

(2.9) (2.10)

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where F 0 does not depend on the order parameter P and it describes the temperature dependence of free energy of high temperature phase near the phase transition (Graz University of Technology n.d.; Uchino 2009). The coefficients 𝛼, 𝛽, and 𝛾 are temperature dependent in general, while their signs are different for the first- and second-order phase transitions. The reason to have only even number of powers is that when the polarization changes signs, the free energy of the crystal will not change. Since this phenomenological formula describes the free energy across the paraelectric to ferroelectric phase transition, it should be applicable to all temperature ranges. In our calculation, however, the coefficient 𝛼 is the only coefficient that is assumed to be temperature dependent. The fraction numbers are for convenience when we differentiate F in respect of P. To define the relation of 𝛼 and T, the free energy for certain polarization should be considered. As in paraelectric state (i.e. P = 0), the free energy should be zero (F(P, T) = 0) at any temperature above its Curie point. In ferroelectric state, there are two situations for free energy, either F(P, T) > 0 or F(P, T) < 0. For free energy greater than zero, the paraelectric state should be realized because it always tends to the minimum energy state. Therefore, the free energy for a certain polarization must be lower than zero in order to stabilize the ferroelectric state. Thus, the coefficient 𝛼 of the P2 term must be negative in ferroelectric state, while it is positive passing through zero at temperature T 0 . According to this concept, a relation of 𝛼 and T is formed as the following: 𝛼 = 𝛼0 (T − T0 )

(2.11)

where 𝛼 0 is a temperature-independent constant and T 0 the Curie–Weiss temperature, which is equal to or lower than the actual transition temperature T c (Curie temperature). By differentiating the free energy, electric field E related to the equilibrium polarization can be expressed as 𝜕F (2.12) = 𝛼P + 𝛽P3 + 𝛾P5 𝜕P By considering the existence of polarization without external electric field, i.e. E = 0, we can get E=

P(𝛼 + 𝛽P2 + 𝛾P4 ) = 0

(2.13)

This equation has two possible solutions, among them the trivial solution P = 0 that corresponds to a paraelectric state is not our concern for ferroelectric state. √ −𝛽± 𝛽 2 −4𝛼𝛾

Another finite solution P2 = corresponds to a ferroelectric state. 2𝛾 From Landau free energy, the temperature-dependent dielectric constant (relative permittivity) of a ferroelectric material at ferroelectric and paraelectric phases (below and above Curie temperature) can be derived. For the first order of phase transition from paraelectric phase to ferroelectric phase, the temperature-dependent Landau free energy, dielectric constant, and spontaneous polarization are illustrated in Figure 2.7. One can see that the characteristic of the first-order ferroelectric phase transition is that the order parameter Ps drops to zero abruptly and the dielectric constant reaches a finite

2.3 Theory of Ferroelectric Phase Transition

T > T1 T c < T < T1

1/ϵr

Ps

T = Tc

ϵr

T0 < T < Tc T < T0

(a)

(b)

T0

Tc T1

Figure 2.7 (a) Free energy, (b) dielectric constant and spontaneous polarization, of first-order phase transition.

peak value continuously; while 1/𝜖 r is linear at temperatures lower or higher than Curie point. The meaning of T 0 and T 1 will be shown in the following derivations. 2.3.2

Landau Theory of First-Order Phase Transition

For second-order phase transition, the order parameter grows continuously from zero at the phase transition so the first two terms of the power series will dominate. If the free energy is expanded to the sixth order in the order parameter, the system will undergo a first-order phase transition with 𝛼 > 0, 𝛽 < 0, and 𝛾 > 0 (Wang 2010). 1 2 1 4 1 6 (2.14) 𝛼P + 𝛽P + 𝛾P 2 4 6 When we try to find the existent polarization P without the external electric field E, the free energy is minimized. F(P, T) =

E=

𝜕F = 0 = 𝛼P + 𝛽P3 + 𝛾P5 𝜕P

P=0

or

𝛼 + 𝛽P2 + 𝛾P4 = 0

So, nonzero solution of P is √ −𝛽 ± 𝛽 2 − 4𝛼𝛾 2 P = 2𝛾 Here the “−” sign is meaningless since it will result in P2 < 0. So, √ −𝛽 + 𝛽 2 − 4𝛼𝛾 2 P = 2𝛾 Since 𝛼 = 𝛼 0 (T − T 0 ), the solutions of P are √ √ −𝛽 + 𝛽 2 − 4𝛼0 (T − T0 )𝛾 P=± for T < T0 2𝛾

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√ −𝛽 +

P = 0, ± P=0

√ 𝛽 2 − 4𝛼0 (T − T0 )𝛾 2𝛾

for T0 < T < T1

for T > T1

The temperature T 1 is where the two nonzero solutions become unstable. For T > T 1 , there is only one free energy minimum. For T 1 > T > T 0 , there are three potential √ minima. P = 0 is the lowest free energy solution when T 1 > T > T c , while P=±

√ −𝛽+ 𝛽 2 −4𝛼0 (T−Tc )𝛾 2𝛾

exhibits the lowest free energy when T c > T > T 0 . For

T < T 0 , there are double potential minima of the free energy that correspond to stable spontaneous polarization. The difference between the Curie–Weiss temperature T 0 , the Curie temperature T c , and the ferroelectric limit temperature T 1 can be verified based on the potential minima obtained in the first-order phase transition. As the potential minima are obtained from 𝜕F = E = 𝛼P + 𝛽P3 + 𝛾P5 = 0 𝜕P which is valid for any temperature below or above Curie temperature, there are three possible minima including P = 0 (i.e. F = 0). At T = T c , the free energy at nonzero polarization must be equal to that of the paraelectric state, i.e. F(P, T)|T=Tc = F(P, T)|P=0 = 0 So, F(P, T) =

1 2 1 4 1 6 𝛼P + 𝛽P + 𝛾P = 0 2 4 6

then we get 𝛼 + 𝛽P2 + 𝛾P4 = 0 1 1 𝛼 + 𝛽P2 + 𝛾P4 = 0 2 3 By eliminating the P4 term from these two equations, 1 2 𝛽P = 0 2 4𝛼 P2 = − 𝛽

2𝛼 +

into the equations, Putting P2 = − 4𝛼 𝛽

( ) ( )2 4𝛼 4𝛼 𝛼+𝛽 − =0 +𝛾 − 𝛽 𝛽

or

( )2 1 4𝛼 𝛼− 𝛾 − =0 3 𝛽

2.3 Theory of Ferroelectric Phase Transition

Taking 𝛼 = 𝛼 0 (T − T 0 ) = 𝛼 0 (T c − T 0 ), 𝛼 = 𝛼0 (Tc − T0 ) =

3 𝛽2 16 𝛾

the Curie temperature is thus calculated as Tc = T0 +

3 𝛽2 16 𝛼0 𝛾

This indicates that T c is little higher than T 0 . Meanwhile, when T = T 1 , the free energy has only one solution at P = 0, i.e. 𝛽 2 − 4𝛼0 (T1 − T0 )𝛾 = 0 T1 = T0 +

𝛽2 4𝛼0 𝛾

For this equation, we can identify that T 1 is higher than T c . As a result, we can conclude that T0 < Tc < T1 To identify the dielectric constant, we first have to calculate the relative permittivity 𝜖 r which is the response of the system to an electric field. 𝜖r =

dP dE

In free energy equation, the field has been included to calculate 𝜖 r 𝜕F =E 𝜕P E = 𝛼0 (T − Tc )P + 𝛽P3 + 𝛾P5 dE = [𝛼0 (T − Tc ) + 3𝛽P2 + 5𝛾P4 ]dP √ , the susceptibility in the vicinity of the At the critical temperatures P = 0, −𝛽 2𝛾 phase transition can be obtained. at Tc , at T1 ,

dP || 1 = dE ||P=0 𝛼0 (T − Tc ) dP || 1 𝜖r = √ −𝛽 = | dE |P= 𝛼0 (T − T1 ) ,

𝜖r =

2𝛾

Based on the equations earlier, we can verify that in case of 𝛽 < 0, the permittivity shows a maximum and a discontinuity of the spontaneous polarization appears at T c .

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2.3.3

Landau Theory of a Second-Order Phase Transition

At a second-order phase transition, the order parameter increases continuously from zero starting at the critical temperature of the phase transition (i.e. T c = T 0 ). It is assumed that 𝛽 > 0, so that the free energy has a minimum for finite values of the order parameter. When 𝛼 > 0, there is only one minimum at P = 0. When 𝛼 < 0, there are two minima with P ≠ 0 (Figure 2.8). For the ferroelectric materials following the second-order phase transition, it can be derived as 1 1 F(P, T) = 𝛼P2 + 𝛽P4 where 𝛼0 > 0, 𝛽 > 0 2 4 It is assumed that 𝛽 > 0 so that the free energy has a minimum for finite values of the order parameter. Similar to the previous case, polarization can be found when E = 0. 𝜕F = 0 = 𝛼P + 𝛽P3 = 𝛼0 (T − Tc )P + 𝛽P3 𝜕P So, P=0

or

𝛼0 (T − Tc ) + 𝛽P2 = 0

When T > T c , there is only one minimum at P = 0. When T < T c there are two minima with P ≠ 0, i.e. for T > Tc √ 𝛼0 (Tc − T) P=± 𝛽

P=0

for T < Tc

For dielectric constant, 𝜕F =E 𝜕P dE = [𝛼0 (T − Tc ) + 3𝛽P2 ]dP T > Tc T = Tc Ps 1/ϵr

ϵr

T < Tc

(a)

(b)

Tc

Figure 2.8 (a) Free energy, (b) dielectric constant and spontaneous polarization, of second-order phase transition.

2.3 Theory of Ferroelectric Phase Transition

dP || 1 C = = for T > Tc dE ||P=0 𝛼0 (T − Tc ) (T − Tc ) dP || C 1 𝜖r = = = dE ||P2 = 𝛼0 (Tc − T) 2𝛼0 (Tc − T) 2(Tc − T) 𝛽 𝜖r =

for T < Tc

where C = 𝛼1 is Curie constant. 0 Now we can see that the dielectric permittivity can be derived from second-order differentiation of Landau free energy with boundary conditions of E = 0 (without electric field). The temperature-dependent dielectric permittivity for ferroelectric materials following the first-order phase transition can thus be derived. In both cases, 1∕𝜖r linearly depends on temperature. The classification of order of phase transition comes from thermodynamic free energy according to Ehrenfest (Jaeger 1998; Blundell and Blundell 2009), where it can be labeled as the lowest derivation (order parameter) of the free energy that shows discontinuity as a function of other thermodynamic variables during phase transition. For instance, if a discontinuity is exhibited in the first derivative of Gibbs free energy, it is classified as first-order phase transition. If the discontinues order parameter is the second order of differentiation of Gibbs free energy, the corresponding phase transition is classified as second-order phase transition. Although Ehrenfest classification is clear to understand, it is not a complete method as it is not suitable if the derivative of free energy diverges. Therefore, a modern classification is made. Similar to the Ehrenfest classes, two broad categories are divided by taking the latent heat into account (Maris and Kadanoff 1978). For example, various solid/liquid/gas transitions usually involve a discontinuous change in density, which is the first derivative of free energy with respect to pressure. Therefore, they are classified as first-order phase transition based on Ehrenfest classification. In the view point of modern classification, during the water to ice phase transition, latent heat is added while the temperature of the system remains unchanged. The specific discontinuity with respect to energy and temperature of this phase transition thus assorts to the first-order differentiation of Gibbs free energy, and it belongs to first-order phase transition associated with latent heat. While for β-brass (an alloy of copper and zinc, in equal amounts) phase transition, there is no latent heat. However, when the temperature is lower than transition temperature, the probability for each copper atom to have more zinc nearest neighbors increases, which would result in a completely random arrangement of copper and zinc atoms. To say, the material undergoes a continuous phase transition in which the specific heat has a singularity and it belongs to second-order phase transition. For the ferroelectric phase transition, for example, BaTiO3 , there is latent heat during paraelectric to ferroelectric phase transition, and the spontaneous polarization abruptly (discontinuously) changes to zero, which is the first-order differentiation of Landau free energy. Therefore, this phase transition belongs to first-order phase transition. While for the second-order phase transition, the Ps continuously changes to zero, but the dielectric permittivity, which is the second-order of differentiation of Landau free energy, is discontinuous at T c .

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Most ferroelectric phase transitions are first order, very few (such as triglycine sulfate [TGS]) are second-order transition.

2.4 Ferroelectric Domains and Domain Switching 2.4.1

Domain Structure

In order to reduce electrostatic and elastic energy, a ferroelectric material usually presents a domain structure where every single domain possesses a polarization direction. Depending on the structure, the domain polarizations are formed into different angles. For example, in a perovskite ferroelectric material BaTiO3 with polarizations along ⟨100⟩ and ⟨110⟩ for a pseudocubic unite cell, the domains are aligned with 180∘ or 90∘ as shown in Figure 2.9a. While if the polarizations are along ⟨111⟩ directions, 71∘ and 109∘ domains will be formed (angles between the ⟨111⟩ directions). However, for a hexagonal structure such as LiNbO3 with polarizations along ⟨0001⟩, only 180∘ domains can be formed. As shown in the figure, the polarizations in adjacent domains are usually arranged head-to-tail in order not to form charged domain walls. But some charged domain walls can also be formed in some crystals if defects exist. There are also vortex types of domains formed in some ferroelectric materials, for example, YMnO3 , or in nanostructures and thin films. Figure 2.9b is a vortex domain structure in YMnO3 . These vortex domains have been theoretically predicted and experimentally observed and will be introduced in Chapter 5. 2.4.2

Ferroelectric Switching

In a ferroelectric material, polarization switching can be realized by applying electric field and/or mechanical stress. It should be noted that an electric field can induce both 180∘ and 90∘ polarization switching, whereas a mechanical stress can only induce 90∘ switching of polarization. This can be understood by considering an ellipsoid representing a polarized unit cell as illustrated in Figure 2.10. One can see that an electric field can force the ellipsoid from out-of-plane to in-plane 90° boundary

180° boundary

β+ α–

γ–

α+

(a)

(b)

γ+ β–

Figure 2.9 (a) Ferroelectric domain structure in BaTiO3 and (b) cloverleaf domain patterns and the enlarged single vortex domain in the ferroelectric YMnO3 . Source: Adapted from Choi et al. (2010) and Reprinted with permission of Springer Nature: Zhang et al. (2013).

2.5 Ferroelectric Materials E-field direction

E-field direction

or

Adding electric field

180° switching

90° switching Stress

Adding mechanical stress

or 90° switching

Figure 2.10 Illustration of polarization switching by electric field and mechanical stress.

as well as up-to-down, i.e. 90∘ and 180∘ polarization rotations, respectively. However, a mechanical stress along the long axis can only press the ellipsoid into an in-plane ellipsoid. In a ferroelectric crystal, its polarization switching is accompanied by domain switching as well as domain wall movement. This process will induce strain in the crystal, where the non-180∘ switching results in large piezoelectric strain.

2.5 Ferroelectric Materials The first ever ferroelectric material is Rochelle salt discovered in 1675 by an apothecary, Pierre Seignette, of La Rochelle in France. But this Rochelle salt is not suitable for engineering application due to its poor property and stability. The most typical and applicable ferroelectric material is BaTiO3 , which has been widely used in electronic components as capacitors due to its very large dielectric constant and very low dielectric loss. Nevertheless, the most popular and useful ferroelectric material is Pb(Zrx Ti1−x )O3 (PZT) due to its very large piezoelectric constant and remnant polarization. The applications of PZT in sensors and actuators are the implementation of its excellent piezoelectric property and applications in infrared sensors utilizing its pyroelectric property; while some small-scale application of resistive random access memory (ReRAM) implement its ferroelectric property. Relaxor ferroelectrics such as Pb(Mg1/3 Nb2/3 )O3 –xPbTiO3 (PMN-xPT) emerged in recent decades have attracted great attention due to their outstanding ferroelectric and piezoelectric properties. 2.5.1

From BaTiO3 to SrTiO3

BaTiO3 is a model ferroelectric material, when temperature decreases, it goes through a series of phase transitions from cubic to tetragonal, orthorhombic, and then rhombohedral. The spontaneous polarization directions corresponding to these non-centrosymmetric structures are ⟨100⟩, ⟨110⟩, and ⟨111⟩, respectively, indexed by pseudocubic structure. The so-called pseudocubic means we still use

29

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2 Introduction to Ferroelectrics a′

a′

Rhombohedral (a = b = c; α = β = γ ≠ 90°)

–90 °C

b′

Orthorhombic (a = b = c; α ≠ 90°, β = γ = 90°)

130 °C c

0 °C

Tetragonal (a = b ≠ c; α = β = γ = 90°)

Ba

Ti

b Cubic (a = b = c; α = β = γ = 90°)

a

Figure 2.11 Illustration of BaTiO3 structures at different temperatures showing different polarization directions at different phases.

cubic structure to index the atomic planes and directions for convenience, but in fact, the crystal structure is not cubic any more due to its distortion. You may be wondering, the so-called orthorhombic structure in Figure 2.11 should be a monoclinic structure when the cubic unit cell is stretched along [110] direction. It seems to be right, but since there is relative shift of cations and anions, the real Bravais unit cell is actually a larger orthorhombic unit cell with the shadow plane as the a′ b′ plane with c′ perpendicular to a′ b′ plane. √ ′ ′ ′ The orthorhombic unit cell has a = a, b = c = 2a, i.e. the volume of the orthorhombic unit cell is doubled, and a′ , b′ , and c′ are not equal, where a is for pseudocubic unite cell in the distorted structure. Ceramics based on BaTiO3 are very important in electronic component of capacitors due to its high dielectric constant and low dielectric loss. BaTiO3 thin film is also one of the most attractive ferroelectric thin films for memory device application. This will be introduced in Chapter 4. Pure SrTiO3 is a quantum paraelectric material where quantum fluctuations of atomic positions suppress a ferroelectric transition, leading to a so-called incipient ferroelectric (Müller and Burkard 1979; Zhong and Vanderbilt 1996; Barrett 1952; Hemberger et al. 1995). It means that SrTiO3 is supposed to be ferroelectric but its Curie temperature is absolute 0 K, which can never be reached. Therefore temperature-dependent dielectric constant of SrTiO3 increases continuously when temperature decreases, but it can never have a peak like a normal ferroelectric material (see Figure 2.12). However, one can see that when a bias voltage is applied on the SrTiO3 crystal, a relaxor-like paraelectric–ferroelectric transition happens above 0 K. This is because of the existence of polar nano regions (PNRs) in SrTiO3 at low temperatures. These PNRs can form ferroelectric nano-domains under electric field, giving rise to relaxor-like ferroelectricity that will be introduced later in this section. SrTiO3 single crystal has been used as many oxide thin films substrate material due to its high chemical stability and good lattice matching with many perovskite oxide materials. When SrTiO3 is doped with BaTiO3 , it becomes (Bax Sr1−x )TiO3 (called BST) and its Curie temperature can be tuned between 0 K to T c of BaTiO3 depending on how much BaTiO3 is added. The study of BST, especially BST thin films, has been a very hot topic due to its large dielectric tunability which is very useful in tunable microwave device. BST is attractive for microwave device application mainly because of two reasons. The first is that the dielectric permittivity of BST strongly depends on the

2.5 Ferroelectric Materials

25 1

1- E = 0 kV/cm 2- E = 2 kV/cm 3- E = 5 kV/cm 4- E = 15 kV/cm

ϵ(T ) × 103

20 15 10 2 5 0

3 4 0

100

200

300

Temperature (K)

Figure 2.12 Dielectric constant of SrTiO3 single crystal as a function of temperature and biasing field. Source: Vendik et al. (1999). Reprinted with permission of Springer Nature.

bias electric field; this property is termed as dielectric tunability. Another reason is that BST possesses very low dielectric loss (tan 𝛿) at microwave frequency. In the following paragraphs, properties and applications of BST are introduced. In Figure 2.13, an experimental result shows the change in lattice parameters when different compositions of Ba and Sr are added into the compound. When the composition of Ba increases, it can be seen that the structure of BST varies, changing from cubic to tetragonal, proofing the structural transition from SrTiO3 (STO) to BaTiO3 (BTO). Therefore, it can be concluded that the tetragonality increases when the composition of Ba increases.

Lattice parameters (nm)

0.405

c

0.400

a

T

a

0.395

C

0.390 0.0 SrTiO3

0.2

0.4

0.6

Composition

0.8

1.0 BaTiO3

Figure 2.13 Lattice constant of (Bax Sr1−x )TiO3 (both bulk and films) as a function of composition x. Source: Adapted from Abe and Komatsu (1995).

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2 Introduction to Ferroelectrics

25 000 x=0

20 000

x = 0.27

x = 0.34 x = 0.47

x = 0.65 x = 0.87

15 000 ϵr(T )

32

x = 1.0

10 000

5000

0

0

100

200

300

400

Temperature (K)

Figure 2.14 The phase transition behavior (the dielectric constant as a function of temperature) of (Bax Sr1−x )TiO3 single crystal for various content of Ba–x. Source: Vendik et al. (1999). Reprinted with permission of Springer Nature.

Figure 2.14 shows a theoretical result that the composition of Ba in BST varies the Curie temperature as well as the dielectric properties of the compound. Owing to the increase of tetragonality, when more Ba is added, the Curie temperature increases. One can also notice that the dielectric constant near T c is a few orders of magnitude greater than the rest, indicating a paraelectric to ferroelectric phase transition. Figure 2.15 shows the BST dielectric constant as a function of bias electric field. The dielectric constant has a huge difference when E-field is applied. In the case of x = 0.24, the dielectric constant is greatly reduced from 𝜖 r > 1400 to 𝜖 r < 500 when a field of E = 10 MV/m is applied. The electric field-dependent dielectric permittivity is commonly described as the dielectric tunability n, which is defined as the ratio of the dielectric permittivity at zero electric field bias to the permittivity under electric field bias E, i.e. n=

𝜖r (0) − 𝜖r (E) 𝜖r (0)

Question to students: Why does dielectric constant decrease when an E-field is applied? 2.5.2

From PbTiO3 to PbZrO3

PbTiO3 (PT) is another model ferroelectric material. As illustrated in Figure 2.16, above its Curie temperature, the paraelectric cubic structure is centrosymmetric; but below Curie temperature, it is energetically favorable for the O2− ions to shift slightly below face centers and Ti4+ ions to shift upward from the unit cell center forming tetragonal phase. The relative shift in positions of the Ti4+ and O2− ions produces a spontaneous polarization Ps .

2.5 Ferroelectric Materials

Figure 2.15 Bias field dependence of (Bax Sr1−x )TiO3 dielectric constant under the condition of f = 100 kHz at room temperature. Source: Adapted from Abe and Komatsu (1995).

1600 1400

Dielectric constant ϵ

1200 1000 800 0.24 600 0.44 400 0.68

0

200 1.0 0 –50 –40 –30 –20 –10 0 10 20 30 40 50 Bias field (MV/m) Pb Ti Ti

(a)

T > Tc

O

Ps

(b)

T < Tc

Figure 2.16 (a) Perovskite structure of PbTiO3 in the cubic form above T c and (b) tetragonal structure of PbTiO3 for T < T c presenting spontaneous polarization.

2.5.3

Antiferroelectric PbZrO3

The notion of antiferroelectric (AFE) dates back to the early 1950s. It describes a state where chains of ions in the crystal are spontaneously polarized, but with neighboring chains polarized in antiparallel directions. As a result, the crystal does not display spontaneous macroscopic polarization or the piezoelectric effect. PbZrO3 was the first compound identified as an antiferroelectric crystal (Sawaguchi et al. 1951; Shirane et al. 1951). At that time, Kittel (1951) proposed a theory of antiparallel ionic displacements in dielectrics using the antiferromagnetism scheme. In parallel, Sawaguchi et al. assigned the perovskite lead zirconate, PbZrO3 , as antiferroelectric because of its dielectric behavior. Thereafter, materials that exhibit a structural phase transition between two nonpolar phases with a strong dielectric anomaly at the high temperature side of

33

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2 Introduction to Ferroelectrics

the transition are named antiferroelectrics, and PbZrO3 remains the prototype for this large group of materials. When an antiferroelectric crystal is subject to electric fields, the antiparallel dipoles can be flipped and forced to be parallel, corresponding to an electric field-induced antiferroelectric-to-ferroelectric phase transition (Cross 1967). This phase transition manifests itself by the signature of double-hysteresis loops as the electric polarization P is monitored as a function of applied field E (see Figure 2.17a). In addition to the abrupt change in polarization, significant changes in other quantities, such as linear dimensions (Berlincourt et al. 1964; Pan et al. 1989a; Brodeur et al. 1994) and optical properties (Chandani et al. 1989), also accompany the phase transition. These changes at the phase transition provide the physical foundation for important engineering applications of antiferroelectric materials in digital displacement transducers (Berlincourt 1966; Uchino 1985; Pan et al. 1989b), energy storage capacitors, electrocaloric cooling devices (Tuttle and Payne 1981; Mischenko et al. 2006; Scott 2011), and flat panel displays (Fukuda et al. 1994; Takezoe et al. 2010). For most device applications, antiferroelectric materials in the thin film form are required. As a result, films with various antiferroelectric compositions have been deposited and characterized (Li et al. 1994; Qu and Tan 2006). As illustrated in Figure 2.17b, with respect to the pseudocubic perovskite primitive unit cell that contains one PbZrO √ 3 molecule, the dimensions of the √ orthorhombic cell are a = 2ac , b = 2 2ac , and c = 2ac , where the parameter ac is the lattice constant for pseudocubic unit cell. The primary structural feature that accounts for the AFE is the antiparallel displacements of Pb2+ along the pseudocubic ⟨110⟩ directions, as schematically shown in Figure 2.17b. At room temperature, the average displacement of Pb2+ is ∼0.025 nm. Energy storage is an important application of antiferroelectric material. As illustrated in Figure 2.18a, the energy density W of dielectric capacitors is P

4 b

E 4

c

(a)

a

(b)

Figure 2.17 (a) Typical P–E loop of antiferroelectric and (b) structure models for PbZrO3 -based antiferroelectric. The orthorhombic unit cell is indicated by the box, while arrows represent the direction of shifts of Pb ions along the c-axis. Source: Tan et al. (2011). Reprinted with permission of John Wiley and Sons.

2.5 Ferroelectric Materials

D

D

E

(a)

D

E

(b)

E

(c)

Figure 2.18 The typical electric displacement verses electric field (D–E) hysteresis loops and energy storage characteristics of the three classes of solid dielectric materials (a) linear, (b) ferroelectric, and (c) antiferroelectric.

represented by the shadow area that can be calculated by the following formula: U=

∮

E ⋅ dD =

Pmax

∫Pr

E dD (upon discharging)

where Pr is remnant polarization and Pmax is the maximum polarization under bias electric field E whose change generates electric displacement dD. We can see that the linear dielectric has relatively low energy density (∼0.01 J/cm3 ) because of the low dielectric constant. The ferroelectric material owns large electric polarization and hence has very high dielectric permittivity. However, when electric fields excess the coercive field of the material, the permittivity decreases very rapidly with the electric fields. Moreover, because of the hysteresis characteristic of ferroelectric materials, the energy stored in ferroelectric during the charging process cannot be fully released (see Figure 2.18b). Instead, some of the energy is converted into heat due to the internal friction of dipole moment reorientation. The shape of the double-hysteresis loop of antiferroelectric, as shown in Figure 2.18c, can benefit to the energy storage in two aspects. First, because of the AFE–FE phase transition, AFE can sustain a high dielectric constant in relatively high electric field and hence absorbing a large amount of energy. On the other hand, ferroelectric structures can turn back to AFE phase at low electric field and release nearly all the energy. 2.5.3.1

Pb(Zrx Ti1−x )O3 (PZT)

When PbZrO3 is doped with PbTiO3 forming Pb(Zrx Ti1−x )O3 (PZT), its ferroelectric and piezoelectric properties reach maximum at their morphotropic phase boundary (MPB); these outstanding properties made PZT dominant in the commercial application in piezoelectric devices until today. What makes PZT so significant in piezoelectric properties? The answer may be found from its phase diagram as shown in Figure 2.19, which was established by Jaffe et al. (1971) and Noheda et al. (2000). One can see from PZT phase diagram, as the temperature T decreases, PZT undergoes a paraelectric-to-ferroelectric phase transition, and the cubic unit cell is distorted depending on the mole fraction of PbTiO3 . In the Zr-rich region, the paraelectric phase changes to orthorhombic phase, and an intermediate monoclinic phase exists between the Zr-rich rhombohedral and Ti-rich tetragonal

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2 Introduction to Ferroelectrics

800 Cubic

700

Temperature (K)

36

600

20 –31

500

Rhombohedral(HT)

–32 –25

MB

400

19

–25 –24 13

Tetragonal MA

30

300 –25 –26 –22

200

–12 10 15 26 31

Rhombohedral(LT)

23

R3c/R3m+Cm R3m+P4mm+Cm

100 Orthorombic

16

19 22 30

0.3

0.4

0 0.0

0.1

0.2

0.5

35

0.6

0.7

0.8

0.9

1.0

Ti concentration

Figure 2.19 PZT phase diagram. The crossover between MB and MA structures is marked by a dashed line. Tetragonal and rhombohedral phase regions are ferroelectric phases. Source: Adapted from Zhang et al. (2014).

phases. That’s the point! One should notice that the best piezoelectric property for PZT actually appears at the composition close to the MPB; one of the main reasons for PZT to possess excellent piezoelectric property is the existence of the MPB at which the monoclinic phase is the result of structural uncertainty between the left side rhombohedra and the right side tetragonal structures. It is because of this “uncertainty,” under a small applied electric field, the structure distortion can easily happen between the tetragonal and rhombohedra phases, i.e. non-180∘ polarization rotation, which is the key factor to induce very large strain in the crystal. Of course, the large strain also comes from domain wall movement that will be further discussed later. 2.5.4

Relaxor Ferroelectrics

Relaxor ferroelectrics were discovered about 50 years ago. The so-called relaxor means a delayed dielectric response to high-frequency electric field, and as illustrated in Figure 2.20, a relaxor ferroelectric has three characteristics: 1. Phase transition is in a wide temperature range with its diffused dielectric constant maximum at temperature T m , which represents the transition temperature from ferroelectric phase to paraelectric phase (Figure 2.20a). 2. The ferroelectric relaxation also exhibits a frequency-dependent relative permittivity, whereas its magnitude decreases with increasing frequency (Figure 2.20a).

2.5 Ferroelectric Materials

25 000

Tm

20 000 Dielectric constant

Figure 2.20 (a) Frequency-dependent relative permittivity and (b) P–E loops as a function of temperature of relaxor ferroelectrics.

15 000 f

1000

5000

0 –60

100 kHz 10 kHz 1 kHz 0.1 kHz

–40

–20

0

20

40

60

80

100

Temperature (°C)

(a)

Electric displacement (μC/cm2)

30 20 10 0

–20 –30 –10 –8 –6

(b)

24 °C 46 °C 72 °C 96 °C 124 °C 152 °C 165 °C

–10

–4

–2

0

2

4

6

8

10

Electric field (kV/cm)

3. Ferroelectricity Pr decreases gradually as temperature increases. Figure 2.20b illustrates gradually decreased Pr when temperature increases. In fact, even when temperature is higher than T m , there is still ferroelectricity because of the existence of so-called nano-polar region (PNR). 2.5.4.1

Relaxor Ferroelectrics: PMN-xPT Single Crystal

Among many relaxor ferroelectrics, Pb(Mg1/3 Nb2/3 )O3 –xPbTiO3 (PMN-xPT or PMN-PT for simplicity) single crystal is the most attractive one for its outstanding ferroelectric/piezoelectric properties and engineering applications. The outstanding properties of PMN-PT single crystal include its high piezoelectric coefficient (d33 ∼ 2000 pC/N), large electromechanical coupling factor (k 33 ∼ 92%), and large potential in application of novel devices including electromechanical sensors, high-performance ultrasound transducers and actuators, and even infrared sensors. The most important advantage of PMN-PT is that it can be grown into large size single crystal (at least 3 in.), making them perform much better than PZT, which can only be used in a polycrystalline ceramic form.

37

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2 Introduction to Ferroelectrics

Lead magnesium niobate, Pb(Mg1/3 Nb2/3 )O3 (PMN), is one of the most interesting relaxor dielectric materials due to its quenched order–disorder complex structure. The ordered and disordered distribution of Mg2+ and Nb5+ cations on the equivalent lattice B sites in PMN is the essential structural characteristic of relaxor dielectrics. While lead titanate, PbTiO3 (PT), is a normal ferroelectric material with long-range order domain structure and has a sharp phase transition taking place at the Curie temperature T c = 490 ∘ C. PMN-PT is the solid solution of Pb(Mg1/3 Nb2/3 )O3 and PbTiO3 , in which the substitution of Ti4+ cations in B sites tends to reduce the random field and enhances the size and formation of long-range ordered ferroelectric domains. PMN-PT has a complex perovskite structure with an ABO3 -type unit cell and exhibits a relaxor ferroelectric property. 2.5.4.2

Polar Nano Regions

One of the unique features of a relaxor ferroelectric is the presence of PNRs in the crystal below the Burns temperature T B , which is a few hundred degrees above the T m . Figure 2.21 illustrates the model of PNRs proposed by Cross (1987). The existence of PNRs below T B ∼ 620 K in PMN is confirmed by inelastic neutron scattering and X-ray diffraction around the reciprocal lattice points (Vakhrushev et al. 1996; Naberezhnov et al. 1999; Hirota et al. 2002). The number of PNRs increases and their dynamics slow down enormously during the sample cooling, but at the freezing temperature T f , which is typically below the T m , the PNRs will be frozen into a polarized state. At this polarized state, the average size of PNRs increases, and a sharp decrease of the number of PNRs occurs due to the merging of smaller PNRs into the larger ones. The static polarization can be achieved at any temperature upon further cooling below T f . It is generally accepted that the dynamic properties of relaxor ferroelectrics are closely related to the degree of order/disorder and the unique nature of PNRs in Ferroelectric

Micro-polar clusters interact and material exhibits a remanent polarization

Superparaelectric

Paraelectric

Micro-polar develop but have random orientation due to thermal excitation

Regions are nonpolar and material exhibits no macroscopic polarization

Increasing temperature

Figure 2.21 Polar mechanisms in relaxor ferroelectric materials as postulated by Cross (1987). In the superparaelectric phase, the materials exhibit a nonzero root mean square (RMS) dipole moment but have zero macroscopic polarization. Source: Cross (1987). Reproduced with permission of Taylor & Francis.

2.5 Ferroelectric Materials

the crystal. Although the information about PNRs can be indirectly measured by neutron scattering, X-ray diffraction, high-resolution electron microscopy, etc., their evolution under different conditions cannot be directly observed and is still unclear. The PNRs with sizes around 1.5 and 18 nm in the Pb(Mg1/3 Nb2/3 )O3 and Pb(Zn1/3 Nb2/3 )O3 crystals, respectively, at room temperature have been reported (Gehring et al. 2001; Stock et al. 2004; Xu et al. 2004). Compared with the size of PNRs in PMN determined by means of high-resolution electron microscopy (Yoshida et al. 1998), PNRs size determined from elastic neutron scattering is an order of magnitude smaller. Figure 2.22 shows piezoresponse force (PFM) images of various compositions (001)-oriented PMN-xPT crystals with different PT content x. It is apparent that the domain size reduces when the content of PT is reduced, and when x is equal to 20%, the nanosized domains dominant the domain structure. The domain structure can also be observed under scanning electron microscope (SEM) if the sample is treated by acid. Figure 2.23 is the laminate domain structures of PMN-0.35PT crystal, where one can see that when the content of PT PMN-0.20PT

PMN-0.35PT

PMN-0.40PT

2 μm

Figure 2.22 Piezoresponse force images of various compositions of (001)-oriented PMN-xPT crystals. Source: Adapted from Bai et al. (2004). Figure 2.23 SEM image of laminate structure of PMN-0.35PT crystal (courtesy of R.Y. Wang and L.L. He).

5 μm

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2 Introduction to Ferroelectrics

increases to more than 30% (x > 0.3), laminate micro-sized domains are formed to make the crystal more like a normal ferroelectric. 2.5.4.3

Morphotropic Phase Boundary (MPB) of PMN-PT Crystal

PMN-PT single crystals have a complex phase diagram with rhombohedral (R), tetragonal (T), orthorhombic (O), and monoclinic (M) phases as shown in Figure 2.24, in which the shadow area with x = 0.28–0.36 of PT is termed as MPB. When x < 0.28, a stable rhombohedral ferroelectric phase dominates in the PMN-PT single crystal below the T m . On the other hand, when x > 0.36, PMN-PT is tetragonal phase and behaves as normal ferroelectrics. By means of synchrotron X-ray diffraction and neutron diffraction studies, temperature and electric field-induced phase transitions from rhombohedral phase to monoclinic phase have been found in the lower boundary of MPB. Fu and Cohen (2000) have reported their result of first principle calculations, revealing that the very large piezoelectric response is due to external electric field-induced phase transition by polarization rotation via monoclinic phase. After the discovery of a monoclinic phase near the MPB in PZT (Noheda et al. 1999), a large number of works of X-ray diffraction and neutron scattering studies on PMN-PT and Pb(Zn1/3 Nb2/3 )O3 -PbTiO3 (PZN-PT) have been reported. Noheda et al. have reported the presence of ferroelectric monoclinic phase in the MPB of PMN-PT system (Ye et al. 2001; Noheda et al. 2002). Three types of monoclinic phases, MA , MB , and MC , have been identified in the MPB of electric field-induced phase diagram. The notations are adopted following Vanderbilt and Cohen (2001). Figure 2.25 shows the polarization vectors in a pseudocubic perovskite unit cell. Pb(Mg1/3Nb2/3)TixO3 500 Cubic

Tm

400

Tetragonal

T (K)

40

O R-O

300

M Rhombohedral

200 0.1

0.2

0.3

0.4

0.5

X

Figure 2.24 Phase diagram of PMN-xPT. Source: Guo et al. (2002). Adapted with permission of AIP Publishing.

2.6 Ferroelectric Domain and Phase Field Calculation

Figure 2.25 The polarization vectors in pseudocubic perovskite unit cell. The polarization of MA phase in a plane between rhombohedral (R) and tetragonal (T) phases, the polarization of MB phase in a plane between R and orthorhombic (O) phases, the polarization of MC phase in a plane between O and T phases.

MA

R

T Mc MB

O

2.6 Ferroelectric Domain and Phase Field Calculation When we investigate the ferroelectric domain structure and domain evolution under electric field or temperature change, a simulation method called phase field calculation is worth to be mentioned. Phase field calculation is a method to simulate phase boundaries of different structures by systematically incorporating effects of coherency strain induced by lattice mismatch and applied stress as well as external electric and magnetic fields. This method has been applied to many material processes including solidification, solid-state phase transition, and various types of complex microstructure changes. Due to the nature of ferroelectric domains with different polarizations, the phase field calculation has also been well used to simulate domain structures and evolutions in ferroelectrics where different oriented ferroelectric domains can be treated as different phases. Reversible ferroelectric domain switching controlled by thermal treatment has been recently discovered by Ren (Ren 2004; Liu et al. 2006) in BaTiO3 -based ferroelectric materials. Figure 2.26 shows the schematic illustration explaining the mechanism of the reversible ferroelectric domain switching, which has been proposed by Ren (details of this mechanism is explained in their report). In ferroelectric phase field calculation, the evolutions of the polarization P (which is a function of time and space), i.e. the structural phase transition

Electric field Electric field

Figure 2.26 Schematic illustration of the mechanism of the reversible domain switching. The light and dark gray regions represent the ferroelectric domains having tetragonal crystal structure with deferent orientations, respectively. The black arrows indicate the direction of the c-axis of the tetragonal phase. The small white rectangles represent the spatial symmetry of the point defects (dopant/impurity ions and oxygen-ion vacancies) in ferroelectrics. E appel is an electric potential energy induced from the external electric field. Adapted from Koyama and Onodera (2009).

41

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2 Introduction to Ferroelectrics

Cubic Structure

Tetragonal Structure

Electric field

50 μm

50 μm

Figure 2.27 Two-dimensional simulation of the structural transformation from cubic to tetragonal phase in BaTiO3 at 298 K, and polarization domain microstructure change induced by the external electric field applied along horizontal direction. Adapted from Koyama and Onodera (2009).

and ferroelectric domain evolution, are described by the time-dependent Ginzburg–Landau (TDGL) equation: 𝜕Gsys 𝜕P = −LP 𝜕t 𝜕Pi where LP is the relaxation kinetic coefficient and Gsys is the total free energy of the 𝜕G

system. The quantity 𝜕Psys is the thermodynamic driving force for the spatial and i time-evolution of polarization. The total free energy Gsys is expressed as (Koyama and Onodera 2009) [ ] 3 1 ∑ 2 Gc (Pi , T) + 𝜅P |∇Pi | dr + Estr + Edipole + Eappel Gsys = ∫r 2 i=1 where Gc (Pi , T) is a bulk free energy density, Estr an elastic strain energy, Edipole a dipole–dipole interaction energy among polarization moments in the microstructure, Eappel an electric potential energy induced from the external electric field, and the second term in the integrant is a ferroelectric domain wall energy, which is expressed by the same form as the gradient energy in phase-field method. 𝜅 P is a gradient energy coefficient. It is assumed that the strain field and electric field are always at equilibrium for a given polarization field distribution. T. Koyama and H. Onodera (2009) used this method to simulate BaTiO3 domain formation and evolution as ferroelectric phase at room temperature that is well below Curie point. The results shown in Figure 2.27 illustrate the domain formation developed into 180∘ and head to tail 90∘ domains. Under electric field, 180∘ domain switching and 90∘ domain rotation are realized by domain wall movement.

References Abe, K. and Komatsu, S. (1995). Ferroelectric properties in epitaxially grown Bax Sr1−x TiO3 thin films. Journal of Applied Physics 77 (1995): 6461. Bai, F., Li, J., and Viehland, D. (2004). Domain hierarchy in annealed (001)-oriented Pb(Mg1/3 Nb2/3 )O3−x %PbTiO3 single crystals. Applied Physics Letters 85 (12): 2313–2315. Available at: https://doi.org/10.1063/1.1793353.

References

Barrett, J.H. (1952). Dielectric constant in perovskite type crystals. Physical Review 86 (1): 118–120. Berlincourt, D. (1966). Transducers using forced transitions between ferroelectric and antiferroelectric states. IEEE Transactions on Sonics and Ultrasonics 13 (4): 116–124. Berlincourt, D., Krueger, H.H.A., and Jaffe, B. (1964). Stability of phases in modified lead zirconate with variation in pressure, electric field, temperature and composition. Journal of Physics and Chemistry of Solids 25 (7): 659–674. Available at: http://www.sciencedirect.com/science/article/pii/0022369764901751. Blundell, S.J. and Blundell, K.M. (2009). Concepts in Thermal Physics, 2e. Oxford: OUP. Brodeur, R.P., Gachigi, K., Pruna, P.M., and Shrout, T.R. (1994). Ultra-high strain ceramics with multiple field-induced phase transitions. Journal of the American Ceramic Society 77 (11): 3042–3044. Available at: https://doi.org/10.1111/j.11512916.1994.tb04546.x. Chandani, A.D.L., Gorecka, E., Ouchi, Y. et al. (1989). Antiferroelectric chiral smectic phases responsible for the tristable switching in MHPOBC. Japanese Journal of Applied Physics 28 (7): L1265–L1268. Choi, T., Horibe, Y., Cheong, S.W. et al. (2010). Insulating interlocked ferroelectric and structural antiphase domain walls in multiferroic YMnO3 . Nature Materials 9: 253. Available at: http://dx.doi.org/10.1038/nmat2632. Cochran, W. (1960). Crystal stability and the theory of ferroelectricity. Advances in Physics 9 (36): 387–423. Available at: https://doi.org/10.1080/ 00018736000101229. Cross, L.E. (1967). Antiferroelectric-ferroelectric switching in a simple ‘Kittel’ antiferroelectric. Journal of the Physical Society of Japan 23 (1): 77–82. Cross, L.E. (1987). Relaxor ferroelectrics. Ferroelectrics 76 (1): 241–267. Available at: https://doi.org/10.1080/00150198708016945. Fu, H. and Cohen, R.E. (2000). Polarization rotation mechanism for ultrahigh electromechanical response in single-crystal piezoelectrics. Nature 403: 281–283. Available at: http://dx.doi.org/10.1038/35002022. Fukuda, A., Takanishi, Y., Isozaki, T. et al. (1994). Antiferroelectric chiral smectic liquid crystals. Journal of Materials Chemistry 4 (7): 997–1016. Available at: http://dx.doi.org/10.1039/JM9940400997. Gehring, P.M., Wakimoto, S., Ye, Z.G., and Shirane, G. (2001). Soft mode dynamics above and below the Burns temperature in the relaxor Pb(Mg1/3 Nb2/3 )O3 . Physical Review Letters 87 (27): 277601. Graz University of Technology (n.d.). Landau theory of a first order phase transition. Available at: http://lampx.tugraz.at/~hadley/ss2/landau/first_order.php (accessed 14 August 2019). Guo, Y., Luo, H., Chen, K. et al. (2002). Effect of composition and poling field on the properties and ferroelectric phase-stability of Pb(Mg1/3 Nb2/3 )O3 –PbTiO3 crystals. Journal of Applied Physics 92 (10): 6134–6138. Hayashi, H., Nakamura, T., and Ebina, T. (2013). In-situ Raman spectroscopy of BaTiO3 particles for tetragonal–cubic transformation. Journal of Physics and Chemistry of Solids 74 (7): 957–962. Available at: http://www.sciencedirect.com/ science/article/pii/S0022369713000668.

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Hemberger, J., Lunkenheimer, P., Böhmer, R. et al. (1995). Electric-field-dependent dielectric constant and nonlinear susceptibility in SrTiO3 . Physical Review B 52 (18): 13159–13162. Available at: http://journals.aps.org/prb/abstract/10.1103/ PhysRevB.52.13159. Hirota, K., Ye, Z.G., Wakimoto, S. et al. (2002). Neutron diffuse scattering from polar nanoregions in the relaxor Pb(Mg1/3 Nb2/3 )O3 . Physical Review B 65 (104105): 1–7. Jaeger, G. (1998). The Ehrenfest classification of phase transitions: introduction and evolution. Archive for History of Exact Sciences 53 (1): 51–81. Available at: https:// doi.org/10.1007/s004070050021. Jaffe, B., Cook, W.R. Jr., and Jaffe, H. (1971). Chapter 7. Solid solutions of Pb(Ti, Zr, Sn, Hf )O3 . In: Piezoelectric Ceramics (eds. J.P. Roberts and P. Popper), 135–184. London/New York: Academic Press. Kittel, C. (1951). Theory of antiferroelectric crystals. Physical Review 82 (5): 729–732. Koyama, T. and Onodera, H. (2009). Phase-field simulation of ferroelectric domain microstructure changes in BaTiO3 . Materials Transactions 50 (5): 970–976. Li, J., Viehland, D.D., Tani, T. et al. (1994). Piezoelectric properties of sol–gel-derived ferroelectric and antiferroelectric thin layers. Journal of Applied Physics 75 (1): 442–448. Available at: https://doi.org/10.1063/1.355872. Liu, W., Chen, W., Ren, X. et al. (2006). Ferroelectric aging effect in hybrid-doped BaTiO3 ceramics and the associated large recoverable electrostrain. Applied Physics Letters 89 (17): 172908. Available at: https://doi.org/10.1063/1.2360933. Luspin, Y., Servoin, J.L., and Gervais, F. (1980). Soft mode spectroscopy in barium titanate. Journal of Physics C: Solid State Physics 13 (19): 3761. Available at: http:// stacks.iop.org/0022-3719/13/i=19/a=018. Maris, H.J. and Kadanoff, L.P. (1978). Teaching the renormalization group. American Journal of Physics 46 (6): 652–657. Available at: http://aapt.scitation .org/doi/10.1119/1.11224. Mischenko, A.S., Zhang, Q., Scott, J.F. et al. (2006). Giant electrocaloric effect in thin-film PbZr0.95 Ti0.05 O3 . Science 311 (5765): 1270–1271. Available at: http:// science.sciencemag.org/content/311/5765/1270. Müller, K.A. and Burkard, H. (1979). SrTiO3 : an intrinsic quantum paraelectric below 4 K. Physical Review B 19 (7): 3593–3602. Naberezhnov, A., Vakhrushev, S., Dorner, B. et al. (1999). Inelastic neutron scattering study of the relaxor ferroelectric PbMg1/3 Nb2/3 O3 at high temperatures. European Physical Journal B: Condensed Matter and Complex Systems 11 (1): 13–20. Available at: http://link.springer.com/10.1007/s100510050912. Noheda, B., Cox, D.E., Shirane, G. et al. (1999). A monoclinic ferroelectric phase in the Pb(Zr1−x Tix )O3 solid solution. Applied Physics Letters 74 (14): 2059–2061. Available at: https://doi.org/10.1063/1.123756. Noheda, B., Gonzalo, J.A., Caballero, A.C. et al. (2000). New features of the morphotropic phase boundary in the Pb(Zr1−x Tix )O3 system. Ferroelectrics 237 (1): 237–244. Available at: https://doi.org/10.1080/00150190008216254. Noheda, B., Cox, D.E., Shirane, G. et al. (2002). Phase diagram of the ferroelectric relaxor (1−x)PbMg1/3 Nb2/3 O3 −xPbTiO3 . Physical Review B 66 (5): 054104. Available at: https://link.aps.org/doi/10.1103/PhysRevB.66.054104.

References

Pan, W., Zhang, Q., Bhalla, A., and Cross, L.E. (1989a). Field-forced antiferroelectric-to-ferroelectric switching in modified lead zirconate titanate stannate ceramics. Journal of the American Ceramic Society 72 (4): 571–578. Available at: https://doi.org/10.1111/j.1151-2916.1989.tb06177.x. Pan, W.Y., Dam, C.Q., Zhang, Q.M., and Cross, L.E. (1989b). Large displacement transducers based on electric field forced phase transitions in the tetragonal (Pb0.97 La0.02 ) (Ti,Zr,Sn)O3 family of ceramics. Journal of Applied Physics 66 (12): 6014–6023. Available at: https://doi.org/10.1063/1.343578. Qu, W. and Tan, X. (2006). Texture control and ferroelectric properties of Pb(Nb,Zr,Sn,Ti)O3 thin films prepared by chemical solution method. Thin Solid Films 496 (2): 383–388. Available at: http://www.sciencedirect.com/science/ article/pii/S0040609005017608. Ren, X. (2004). Large electric-field-induced strain in ferroelectric crystals by point-defect-mediated reversible domain switching. Nature Materials 3 (2): 91–94. Sawaguchi, E., Maniwa, H., and Hoshino, S. (1951). Antiferroelectric structure of lead zirconate. Physical Review 83 (5): 1078. Scott, J.F. (2011). Electrocaloric materials. Annual Review of Materials Research 41 (1): 229–240. Available at: https://doi.org/10.1146/annurev-matsci-062910100341. Shirane, G., Sawaguchi, E., and Takagi, Y. (1951). Dielectric properties of lead zirconate. Physical Review 84 (3): 476–481. Shirane, G., Axe, J.D., Harada, J., and Remeika, J.P. (1970). Soft ferroelectric modes in lead titanate. Physical Review B 2 (1): 155–159. Stock, C., Birgeneau, R.J., Wakimoto, S. et al. (2004). Universal static and dynamic properties of the structural transition in Pb(Zn1/3 Nb2/3 )O3 . Physical Review B 69 (9): 094104. ˇ c, M. (2010). Antiferroelectric liquid crystals: Takezoe, H., Gorecka, E., and Cepiˇ interplay of simplicity and complexity. Reviews of Modern Physics 82 (1): 897–937. Tan, X., Ma, C., Frederick, J. et al. (2011). The antiferroelectric ↔ ferroelectric phase transition in lead-containing and lead-free perovskite ceramics. Journal of the American Ceramic Society 94 (12): 4091–4107. Tuttle, B.A. and Payne, D.A. (1981). The effects of microstructure on the electrocaloric properties of Pb(Zr,Sn,Ti)O3 ceramics. Ferroelectrics 37 (1): 603–606. Available at: https://doi.org/10.1080/00150198108223496. Uchino, K. (1985). Digital displacement transducer using antiferroelectrics. Japanese Journal of Applied Physics 24: 460–462. Uchino, K. (2009). Ferroelectric Devices, 2e. CRC Press. Vakhrushev, S., Nabereznov, A., Sinha, S.K. et al. (1996). Synchrotron X-ray scattering study of lead magnoniobate relaxor ferroelectric crystals. Journal of Physics and Chemistry of Solids 57 (10): 1517–1523. Available at: http://www .sciencedirect.com/science/article/pii/0022369796000224. Vanderbilt, D. and Cohen, M.H. (2001). Monoclinic and triclinic phases in higher-order Devonshire theory. Physical Review B 63 (9): 094108. Vendik, O.G., Hollmann, E.K., Kozyrev, A.B., and Prudan, A.M. (1999). Ferroelectric tuning of planar and bulk microwave devices. Journal of Superconductivity 12 (2): 325–338. Available at: https://doi.org/10.1023/A:1007797131173.

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Wang, C. (2010). Theories and methods of first order ferroelectric phase transitions. In: Ferroelectrics (ed. I. Coondoo). InTech. Available at: https://www.intechopen .com/books/ferroelectrics/theories-and-methods-of-first-order-ferroelectricphase-transitions. Xu, G., Shirane, G., Copley, J.R.D., and Gehring, P.M. (2004). Neutron elastic diffuse scattering study of Pb(Mg1/3 Nb2/3 )O3 . Physical Review B 69: 064112. Ye, Z.G., Noheda, B., and Dong, M. (2001). Monoclinic phase in the relaxor-based piezoelectric/ferroelectric Pb(Mg1/3 Nb2/3 )O3 –PbTiO3 system. Physical Review B 64 (18): 184114. Yoshida, M., Mori, S., and Yamamoto, N. (1998). Transmission electron microscope observation of relaxor ferroelectric Pb(Mg1/3 Nb2/3 )O3 . Journal of the Korean Physical Society 32: S993–S995. Zhang, Q., Tan, G., Yu, R. et al. (2013). Direct observation of multiferroic vortex domains in YMnO3 . Scientific Reports 3: 2741. Available at: http://dx.doi.org/10 .1038/srep02741. Zhang, N., Yokota, H., Glazer, A.M. et al. (2014). The missing boundary in the phase diagram of PbZr1−x Tix O3 . Nature Communications 5 (May): 5231. Zhong, W. and Vanderbilt, D. (1996). Effect of quantum fluctuations on structural phase transitions in SrTiO3 and BaTiO3 . Physical Review B 53 (9): 5047–5050.

47

3 Device Applications of Ferroelectrics Some smart systems are based on ferroelectric materials; among their many applications are actually the implementation of their piezoelectric properties since most excellent piezoelectric materials are also ferroelectrics. The device applications of piezoelectric property in smart systems will be introduced in Chapter 7. In this chapter, device applications based on the change of ferroelectric polarization will be introduced.

3.1 Ferroelectric Random-Access Memory For memory application, owing to its nonlinear relationship between the applied electric field and the stored charge, a ferroelectric layer sandwiched by two metallic electrodes with a capacitor-like structure can be used as a non-volatile memory. When an external electric field is applied to it, the spontaneous polarization can be reversed by an appropriate electric field applied in the opposite direction of polarization. Therefore, binary data “0” and “1” can be written and stored by its polarization directions. Ferroelectric random-access memory (FeRAM), or called FRAM, is a non-volatile memory where the non-volatility is achieved by the ferroelectric polarization in the ferroelectric layer acting as the dielectric layer in either a one transistor–one capacitor (1T1C) structure (see Figure 3.1a) or a one transistor (1T) structure (see Figure 3.1b). It is a growing memory technology alternative to flash memory with advantages such as lower power consumption, faster write speed, and higher durability. With almost 30 years of development of FeRAM, it started in mass production commercially. Fujitsu began its volume production of FeRAM in 1999 and more companies are making multi-megabit density of FeRAM for niche application where its performance is better than flash memory, which is very popularly used in USB and SSD. However, the readout operation in a conventional FeRAM is destructive as the read method involves writing a bit to each cell. Thus, there are still limitations in read/write cycle and access time. In addition, the depolarization field that is opposite to that of polarization within the ferroelectric layer reduces the data retention time significantly. The depolarization field is an electric field established

Ferroic Materials for Smart Systems: From Fundamentals to Device Applications, First Edition. Jiyan Dai. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

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3 Device Applications of Ferroelectrics

Ferroelectric film – – – – – – – – – –

Bit line

Word line Word line Bit line

N

N P

Plate line (a) 1T

VR 1V FBB/ RBB

ID ≈ 100 μ

A

Low-VT state

(b)

Figure 3.1 (a) FeRAM device and structural diagram (1T1C) made by Fujitsu, where the ferroelectric layer’s polarization switching induces large difference in capacitance and therefore the source–drain current representing “0” and “1” states. Source: Redrawn from Fujitsu (n.d.). (b) FeRAM (or FeFET) from the Ferroelectric Memory Company and structural diagram showing 1T structure. Source: Adapted from Ferroelectric Memory Company (2018) and Arimoto and Ishiwara (2004).

inside the polarized layer by the generation of surface charge. This field’s direction is opposite to the field that induces polarization and tends to depole the ferroelectric layer; therefore, it is called depolarization field as illustrated in Figure 3.2. Figure 3.3 shows 1T FeRAM diagram, where the gate dielectric layer is ferroelectric with polarization up and down representing the “0” and “1” states. Figure 3.3a illustrates that the surface charge induced by ferroelectric polarization results in the shift of ON/OFF threshold voltage of field effect transistor (FET) exhibiting a memory window. The earlier studies of FeRAM were focused on PZT, since it has much larger remnant polarization compared with other ferroelectric materials such as BaTiO3 and PbTiO3 . However, PZT film faces serious fatigue problem, i.e. after 108 times of polarization switching, its ferroelectricity decays significantly. This is not acceptable for non-volatile memory application (Al-Shareef et al. 1994). This problem has hindered scientists quite a long time but attracted a lot of interest of study. Later on, an alternative ferroelectric material SrBi2 Ta2 O9 (SBT) has been found to be fatigue-free up to 1011 cycles of polarization switching.

3.1 Ferroelectric Random-Access Memory

Figure 3.2 Illustration of surface charges generated in a ferroelectric layer by polarization P0 and depolarization field E d induced by the generated surface charges. P0

Poling field

Ed

Gate –

Spontaneous polarization of ferroelectric layer Source

–

–

–

Memory window

–––––––––

Drain IDS

– – – – – – – – – –

N

N

–––––––––

P –

–

–

–

VG

(a)

(b)

Figure 3.3 (a) Diagram of the 1T FeRAM based on a field-effect transistor with ferroelectric gate dielectric. (b) The ferroelectric polarization-induced interfacial charge alters the on–off voltage of the FET resulting “0” and “1” states.

SBT also has a relatively smaller coercive field making it able to be operated with lower voltage. BiFeO3 (BFO) came into the focus of many scientists in the ferroelectric field due to the discovery of its very large remnant polarization in thin film form, which makes it very promising for FeRAM. However, it is very difficult to grow high-quality BFO thin films with low leakage current and low dielectric loss. This hinders the mass production of BFO-based FeRAM (Ielmini and Waser 2016). An emerging ferroelectric material for FeRAM application is HfO2 , which has been found to be ferroelectric when doped with ZrO2 or other oxides. Even the ferroelectricity of HfO2 is usually not as good as PZT and BiFeO3 , but since it has been implemented in CMOS technology as gate insulator, FeRAM with HfO2 gate oxide should be very compatible to current COMS process and therefore is very attractive to semiconductor industry. However, the origin of ferroelectricity is still not very clear, even it has been practically demonstrated to be ferroelectric. Details will be introduced in Chapter 5.

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3.2 Ferroelectric Tunneling Non-volatile Memory With the development of FeRAM, in the meantime, Esaki introduced a concept of ferroelectric tunnel junction (FTJ) in 1971. A FTJ is composed of two dissimilar metal electrodes with a ferroelectric ultrathin film in-between as a barrier for tunneling. According to quantum mechanical theory, quantum tunneling occurs when the dielectric thickness in the FTJ is thin enough. It describes the situation that there is a finite probability that the wave function may extend beyond the barrier if the barrier is thin enough, i.e. the carrier can tunnel through the junction even its energy is below the barrier height. Unlike the classical physics law, quantum tunneling depends on the potential barrier height. Carriers may tunnel through the barrier proportional to the tunneling probability, which is a function of barrier height. In particular, the thickness, bandgap, and polarization of the ferroelectric film determine the height of potential barrier in the FTJ. Such a structure takes advantage of ferroelectric polarization that can be switched by an electric field, which also adds a new function to the tunnel junction. However, this novel idea requires a few nanometer-thick ferroelectric films with strong polarization, which is very challenging. Since most people believed that the critical thickness for ferroelectricity to exist was above 10–15 nm, no further development or investigation had been done at that time. However, followed by the rapid improvement of nanotechnology in the 2000s, nanoscale characterization techniques, and the high-quality crystallinity in the growth of ferroelectric thin film, this idea was raised again. Since then, there has been a great attraction to scientists to study ferroelectric tunneling-induced resistive switching in relation to ferroelectric thin films. Apart from differentiating the charge polarity in FeRAM, the binary data of new emerging ferroelectric-based memory depends on the resistance state, i.e. electroresistance (ER). In a FTJ, its ER also depends on ferroelectric polarization. The origin of electroresistance effect is introduced by Zhuravlev et al. (2005) with a simple model of ferroelectric film separating two different metal electrodes (M1 and M2). To explain this mechanism, let us look at the energy band diagram of a FTJ as shown in Figure 3.4. In this junction structure, the ferroelectric film should be sufficiently thin but still maintains its ferroelectric polarization, and the charges induced by polarization at the surface of polarization will repel or attract electrons next to ferroelectric layer. These interfacial charges build an electric field giving rise to an additional electrostatic potential at the interface and therefore induce an asymmetric modulation of the potential barrier profile. By solving the time-independent Schrödinger equation (3.1) for this junction, ℏ2 d 2 𝛹 (x) + V (x)𝛹 (x) = E𝛹 (x) (3.1) 2m dx2 where ℏ is the reduced Planck constant, m the particle mass, x the distance in the direction of motion of the particle, V the potential energy of barrier height, and E the energy of the particle, we can obtain the wave function 𝛹 : −

𝛹 (x) = Ae−ikx + Beikx √ −E) with k = 2m(V ℏ2

(3.2)

3.2 Ferroelectric Tunneling Non-volatile Memory

Φ–

δ1 EF

Φ+

EF δ2

FE

M1

M2

M1

FE P

P –

M2

+

Low barrier height (Φ–)

+

–

Low barrier height (Φ+)

Figure 3.4 Energy band diagram of a ferroelectric tunnel junction. Polarizations in two opposite directions cause two average barrier heights resulting in two resistance states. A rectangular barrier, denoted by dashed lines is assumed when ferroelectric is nonpolarized. Source: Garcia and Bibes (2014). Reprinted with permission of Springer Nature.

As the transmission of wave function is exponentially proportional to the square root of barrier height (see Eq. (3.2)), for lower barrier height of potential barrier, more carriers can pass through when an external voltage is applied; therefore, the resistance of the junction is relatively lower and is called “ON” state. On the other hand, when polarization changes to the opposite site, the barrier becomes higher than normal (shown by dot line); as a result, fewer carriers can pass through the barrier, and resistance of this system is relatively higher (i.e. “OFF” state). Hence, the information can be stored because of this barrier height difference. The ratio of electroresistance (ER) of OFF state to ON state defines the efficiency of electrical resistance effect by Eq. (3.3): ER = 3.2.1

I(high conduction) − I(low conduction) ROFF − RON = I(low conduction state) RON

(3.3)

Tunneling Models

In a simple metal–ferroelectric–metal (MFM) structure, electron transport strongly depends on ferroelectric barrier properties, polarization orientation, and metal properties. Pantel and Alexe (2010) calculated the influence of possible factors on MFM current flowing density across the ferroelectric thin film with three possible transport mechanisms including direct tunneling (DT), Fowler–Nordheim tunneling (FNT), and thermionic emission (TE). Figure 3.5a illustrates the transport mechanisms through an ultrathin ferroelectric layer.

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3 Device Applications of Ferroelectrics

FNT

10–7

DT

(a)

V (V)

(b)

DT 2.0

TE

–1.5 –1.0 –0.5 0.0 0.5 1.0

3.0

FNT

DT

1.0 0.5

FNT

10–1 10–4

TE

4.0 d (nm)

FNT

102

TE

0.0 –0.5 –1.0

1.5

–1 .5 –1 .0 –0 .5 0. 0 0. 5 1.0 1.5

TE

ER

5.0

105 j (A/cm2)

52

(c)

V (V)

Figure 3.5 (a) Illustration and (b) voltage dependence of the current density contributions for 3.2 nm-thick BaTiO3 of three possible transport mechanisms through ultrathin ferroelectrics: direct tunneling (DT), Fowler–Nordheim tunneling (FNT), and thermionic emission (TE). Solid and dashed lines stand for two polarization orientations. (c) Corresponding electroresistance (ER) as a function of voltage and ferroelectric thickness d. The dashed lines illustrate the boundaries where each DT, FNT, and TI mechanism dominates. Source: Pantel and Alexe (2010). Reprinted with permission of American Physical Society.

The direct tunneling (DT) is a quantum-mechanism phenomenon, whereas electron flow can be estimated in a trapezoidal potential barrier using Wentzel–Kramers–Brillouin (WKB) approximation. Current characteristics at low voltage in different temperatures can be fitted into Brinkman’s model and Gruverman’s model. The fitting determines the presence of direct tunneling through the barrier with polarization-induced interfacial effects. Also, mean potential barrier height is calculated using the following equation: { [( )3 ( ) 3 ]} 2 2 exp 𝛼 𝜑2 − eV2 − 𝜑1 + eV2 IDT = SC [( )1 ( )1 ] 𝛼2 {

× sinh

𝜑2 − eV2 − 𝜑1 + eV2 [( )1 ( ) 1 ]} eV 2 eV 2 3eV − 𝜑1 + 𝛼 𝜑2 − 4 2 2 2

2

(3.4)

with C=−

32𝜋em∗ 9h3

(3.5)

and 1

8𝜋d(2m∗ ) 2 𝛼= 3h(𝜑1 + eV − 𝜑2 )

(3.6)

where I DT is the direct tunneling current given by Gruverman et al. (2009), S the area of the interface, d the film thickness, h the Planck constant, m* the effective carrier mass, and Φ1 and Φ2 are potential steps at the two interfaces. Similar to DT, FNT shares the same physical phenomenon but in different voltage regime. It is the process whereby electrons tunnel through a barrier in the presence of a high electric field. Across a triangular-shaped potential barrier, FNT is present. The tunnel probability can be derived from time-independent Schrödinger equation, and the calculation of current yields the following

3.2 Ferroelectric Tunneling Non-volatile Memory

relationship between the current density J FN and the electric field in the barrier E: ( √ 3 ) ∗ (e𝜑 ) 2 2m 4 B JFN = CFN E2 exp − (3.7) 3 eℏ E where 𝜑B is the barrier height at the conductor/barrier interface. TE describes the electron flow as the charge carriers that overcome the potential barrier by thermal energy. Usually, the I–V curves can be fitted by TE model. The forward bias currents are described by the theory of TE as the following equation: ( ) ] ( e𝜑 ) [ eV Iforward = SA∗ T 2 exp − B exp −1 (3.8) kT nkB T where S is the junction area, A* the Richardson constant, T the absolute temperature, e the electron charge, k the Boltzmann constant, n the ideal factor, and 𝜑B the Schottky barrier height. From the linear relationship in TE modeling, we can identify the barrier height in OFF state using Eq. (3.8). The corresponding depletion region width, where electrons are depleted in interface, can be estimated by Eq. (3.9): [ ]1 2 2𝜖s (Vbi − V ) (3.9) WD = eND where W D is the depletion region width, 𝜖 s the dielectric constant of semiconductor, N D the donor concentration, V bi = 𝜑B + 𝜑n , and 𝜑n = EF − Ec . In Figure 3.5b, the current density as a function of voltage across 3.2-nm thick ferroelectric thin film is shown. For the two opposite polarizations (P > 0 and P < 0 with solid and dashed lines, respectively), the current density behaves differently in three mechanisms. DT is prominent in low voltage regime, while FNT is in high voltage regime. With a small range of voltage, DT gives a parallel current branch for both polarization directions showing the independence of applied voltage on electroresistance. On the other hand, FNT is barely observed in low voltage. Because of insufficient band tilting, there is virtually no current recorded in low voltage regime. In the case of TE, electroresistance is obvious as its value is given by the change in barrier height on polarization reversal. However, the total current density decreases compared with the other two. Therefore, it is usually suitable for thicker films that tunneling cannot happen (see Figure 3.5c). Among these three mechanisms, we can understand the possibility in electronic transportation across ferroelectric thin film. Certainly, parameters related to those mechanisms, such as effective mass of oxide film in direct tunneling and FNT as well as temperature in TE, can shift between mechanisms. I–V measurement is the first step to see a ferroelectric junction’s memory characteristics. However, I–V curves alone are not sufficient to clarify the underlying switching mechanism that should be attributed to polarization switching. By performing sequential ferroelectric poling, piezoresponse force microscopy (PFM) imaging, and conductive-atomic force microscope (C-AFM) imaging of the poled domains, the resistance difference measured with the C-AFM or

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3 Device Applications of Ferroelectrics

other electrical methods between positively and negatively poled regions would therefore correspond to the ferroelectric-induced ER effect (Gruverman et al. 2009). Garcia et al. (2009) investigated the local tunneling electroresistance based on this method and demonstrated resistive readout of the polarization state through its influence on the tunnel current using PFM and C-AFM at room temperature even the film is a 1 nm-highly strained BaTiO3 film. The resulting electroresistance effect scales exponentially with ferroelectric film thickness and it can reach 75 000% at 3 nm thick. In fact, only by tunneling at such thin thickness is able to read the polarization state without destroying it, otherwise undesirable leakage current will dominate. Apart from using conductive tips as top electrodes, Chanthbouala et al. (2012b) reported tunnel electroresistance on Au/Co/BaTiO3 (2 nm)/LSMO(30 nm)/NdGaO3 heterostructure with resistance switching occurring at the coercive voltage of ferroelectric switching. They show an alternative way to ensure that the electroresistance is originated from ferroelectric response (Figure 3.6).

Figure 3.6 Out-of-plane PFM (a) phase and (b) amplitude measurements on Au/Co/BaTiO3 /LSMO/NdGaO3 heterostructure showing ferroelectric switching versus resistive switching. (c) Resistance versus writing voltage measured in remanence after applying successive voltage pulses. The open and filled circles represent two different scans to show reproducibility. Source: Adapted from Chanthbouala et al. (2012b).

Phase (°)

90 45 0 –45

Amplitude (a.u.)

–90

(a)

10

1

(b)

107 Resistance (Ω)

54

106

105 –4

(c) –3

–2

–1

0 1 Vwrite (V)

2

3

4

3.2 Ferroelectric Tunneling Non-volatile Memory

3.2.2

Metal–Ferroelectric–Semiconductor Tunnel Junction

Wen et al. (2013) proposed to replace one of the metal electrodes with a heavily doped semiconductor to form a metal–ferroelectric–semiconductor (MFS) heterostructure. Since the transmittance of a tunnel junction depends on both barrier height and width exponentially, the tunneling electroresistance can be enhanced by tuning the depletion region in semiconductor surface by the ferroelectric field effect. Figure 3.7a shows the schematic drawing of MFS structure with corresponding potential barrier. At ON state, the polarization is pointing toward the semiconductor electrode, and the positive bound charges in the ferroelectric thin layer near the ferroelectric/semiconductor interface drive the n-type semiconductor surface into accumulation. In this situation, the semiconductor layer can be treated as a metal electrode and the junction is similar to a MFM FTJ. With the development of depolarization field, the barrier height is lower than that in metal/ferroelectric interface, and thus more electrons can tunnel through the thin barrier layer. While at the OFF state (polarization pointing to the metal electrode), the negative bound charges in ferroelectric have to be screened by immobile ionized donor within the depletion region in the surface of semiconductor. In contrary to the ON state, depolarization field increases the barrier height in ferroelectric/semiconductor interface. Also, due to the presence of depletion region, electrons have to overcome both increased barrier height and barrier width induced by ferroelectric field effect, and therefore, tunneling current is greatly reduced. Figure 3.7b shows the resistance changes in two states based on this phenomenon. With an increased thickness of depletion layer in the semiconductor electrode, a giant electroresistance can be obtained even at room temperature. F

107

S

EF EC EV

M

F

S

EF EC

106 Resistance (Ω)

M

w

OFF

105 104

ON

103

r

EV

(a)

wr

P

ON

w

P

w –5

(b)

OFF

–4

–3

–2

–1 0 1 Voltage (V)

2

3

4

5

Figure 3.7 (a) Schematic drawings of the metal/ferroelectric/semiconductor structures and corresponding potential energy profiles for the low (top) and the high (bottom) resistance states. (b) Resistance hysteresis loops at room temperature. The loop composed of dark squares and circles is measured using the pulse train shown schematically in the bottom-left (top-right) inset. The device is preset to the ON (OFF) state by a positive (negative) pulse. The corresponding domain evolution is shown schematically in the bottom-right, top-left, and middle-left insets for the ON, OFF, and intermediate (ON → OFF) states, respectively. Source: Wen et al. (2013). Reprinted with permission of Springer Nature.

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3 Device Applications of Ferroelectrics

3.2.3

Ferroelectric Tunneling Memristor

Chanthbouala et al. (2012a) introduced a memristive behavior that the ratio of two resistance changes can be controlled by the nucleation and growth of ferroelectric domains. They mentioned an important degree of freedom that has not yet been exploited in FTJs, i.e. the domain structure of the ferroelectric tunnel barrier. As we know that the domain size in ferroelectrics scales down with film thickness, so that nanometer-sized domains are expected for ferroelectric tunnel barriers. By applying different positive writing voltage pulses on the virtually homogeneous up-polarized NiFe/BaTiO3 (5 nm)/LSMO heterostructure, the voltage starts nucleating the down-polarized domain. From our research results as shown in Figure 3.8a, one can see the multiple logic states of resistance when the amplitude of writing voltage pulse increases. This result suggests that the ferroelectric tunneling-based memory behaves like a memristor whose

1.0 V 1.5 V 2.0 V 2.5 V

1000 OFF Resistance (kΩ)

56

100

ON 10

–4 (a)

–3

–2

–1

0 Voltage (V)

2.5 V

0V

–2.5 V

1.0 V

1

2

3

4

0V 2.0 V

(b)

Figure 3.8 (a) Tunneling resistance with varying maximum (positive or negative) writing voltages and (b) configuration extracted from the PFM phase images showing the relative fraction of ferroelectric domains pointing downward with increased voltage. Source: Reprinted with permission from Yau et al. (2015).

3.2 Ferroelectric Tunneling Non-volatile Memory

resistance states can be continuously tuned by writing voltage. The corresponding ferroelectric domain orientation at different levels of resistance is shown in Figure 3.8b, where one can see that the degree of polarization switching depends on writing voltage applied. These results suggest that the multiple resistance states can be modified by controlling the nucleation and growth of ferroelectric domains, i.e. memristive characteristics. A memristor is considered to be the fourth electrical component whose resistance has multiple values that can be altered by writing voltage with different magnitudes or different time sequences. The so-called artificial neural network is built with these memristors that are also called artificial synapses since their electrical characteristics are similar to biological synapses in our neuron system. Therefore, in recent years, scientists are trying to implement these artificial synapses into artificial neural network to realize artificial intelligence (AI) function. 3.2.4

Strain Modulation to Ferroelectric Memory

Recently, we have demonstrated the strain engineering in the Pt/BaTiO3 / Nb:SrTiO3 metal/ferroelectric/semiconductor heterostructured FTJ. As shown in Figure 3.9, with an extra compressive strain induced by mechanical stress that is applied beyond the lattice mismatch, the ON/OFF ratio of FTJ can be increased significantly up to an extremely high value of 107 ; while a mechanical erasing effect can be attained when a tensile stress is applied. This pure strain modulation to ferroelectricity and therefore the memory state within a simple structure avoids other effects, and therefore, verifies the role of ferroelectricity in FTJ as a non-volatile resistive memory. This external strain engineering also provides a way to tune resistive switching including ON/OFF ratio and other performance for future development in non-volatile memory application. 107 Pt BTO

106

In-plane strain ε 4.30%

ON/OFF ratio

105

NSTO

2.18%

104 103 102 101

0.18%

c a

(a)

100

Ba 2+ Ti 4+ O 2–

–4.0

–3.0 –2.0 Strain ε (%)

–1.0

0.0

(b)

Figure 3.9 (a) Illustration of in-plane strain effect and (b) the corresponding ON/OFF ratio of Pt/BaTiO3 /Nb:SrTiO3 metal/ferroelectric/semiconductor heterostructured ferroelectric tunnel junction. Source: Adapted from Yau et al. (2017).

57

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3 Device Applications of Ferroelectrics

3.3 Pyroelectric Effect and Infrared Sensor Application 3.3.1

Pyroelectric Coefficient

Another important application of a ferroelectric material is the implementation of its pyroelectric property in infrared (IR) sensors. Compared with semiconductor infrared sensors, advantages of pyroelectric sensors are their wide wavelength response, uncooled operation, high sensitivity, compact structure, and low cost. These advantages enable pyroelectric-based infrared sensors a variety of applications such as temperature sensing, flame and fire detection, gas monitoring, energy harvesting, night vision, and thermal imaging (Rogalski 2003; Whatmore 1986; Yang et al. 2012; Tien et al. 2009). Infrared (IR) radiation exists in the electromagnetic spectrum at a wavelength longer than visible light. To detect infrared radiation, a transducer is required to convert the infrared radiation to electrically detectable signal with electrical circuits. Traditional ferroelectric materials, including triglycine sulfate (TGS), lithium tantalite (LiTaO3 ), and barium strontium titanate (BST) have been utilized for infrared detector applications (Watton 2011; Buser et al. 1997). Recently, the superior pyroelectric performance of PMN-PT single crystal makes it a promising candidate for high-performance uncooled infrared detectors and thermal imagers. The pyroelectric properties of PMN-PT can be nearly independent of incident beam frequency in the range of 50 Hz to 10 kHz (Yang et al. 2012). The pyroelectric effect can be quantified by the pyroelectric coefficient p, which is defined as dPs (3.10) p= dT With a pyroelectric coefficient p, a small and gradual temperature change ΔT in a pyroelectric crystal will cause a change in the spontaneous polarization vector ΔPS given by ΔPS = pΔT

(3.11)

Given the pyroelectric coefficient, the pyroelectric current induced by temperature change can be derived by ( ) dPs dPs dT dT j=− =− × =p (3.12) dt dT dt dt Ferroelectric materials have negative pyroelectric coefficients, i.e. an increase in temperature leads to a decrease in the spontaneous polarization. The polarization will suddenly fall to zero if the crystal is heated above its Curie temperature. The temperature-dependent spontaneous polarization Ps of a ferroelectric material makes the pyroelectric sensing element generate surface electric charges when exposed to heat in the form of infrared radiation. However, one should know that a pyroelectric infrared sensor can only detect temperature change but has no response to a steady temperature since a pyroelectric sensing element can only sense polarization change when the sensor is subject to temperature change. However, too fast change of radiation makes the pyroelectric effect reduced due

3.3 Pyroelectric Effect and Infrared Sensor Application

Thermometer

Electrometer Thermocouple Sample PC

Driver

Peltier element

Lock-in amplifier Signal-out

Reference Signal-in

Figure 3.10 Schematic setup of dynamic pyroelectric coefficient measurement system.

to slow change of temperature restricted by heat capacity and heat conductivity. So, there is a frequency limit of temperature change (usually up to 1 kHz, depending on the size of sensing element and electrode material, thickness, etc.). The pyroelectric coefficient is usually determined from a dynamic measurement, and its schematic setup is shown in Figure 3.10. This dynamic measurement method is based on the measurement of pyroelectric current induced by a sinusoidal variation of temperature of a heater using a thermoelectric Peltier device. As the pyroelectric current is proportional to the partial derivative of the sample’s temperature, the pyroelectric coefficient p can be obtained: ( ) ( ) 1 ΔQ 1 dQ∕dt I 𝜕P =− (3.13) =− = p=− 𝜕T A ΔT A dT∕dt A(dT∕dt) The pyroelectric current is recorded by an electrometer that acts as a current to voltage converter, and the amplitude of the voltage is measured by a lock-in amplifier. The sample’s temperature is sinusoidal modulated around 30 ∘ C with a frequency in the range of a few mHz. After the signal becomes stable, the reading of the pyroelectric coefficient can be recorded. The pyroelectric coefficients of some ferroelectric materials are presented in Table 3.1. 3.3.2

Pyroelectric Infrared Sensor

A pyroelectric sensor consists of a slice of pyroelectric material with metal electrodes on opposite faces. A ferroelectric material should be poled so that the polarization direction is perpendicular to the electrode faces and the material forms a parallel-plate capacitor. The electrode configuration is shown in Figure 3.11a, and the electric polarization change under thermal radiation is illustrated in Figure 3.11b,c.

59

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3 Device Applications of Ferroelectrics

Table 3.1 Pyroelectric coefficient of some typical pyroelectric materials.

Material

Pyroelectric coefficient (𝛍C/(m2 K))

References

PMN-0.26PT (111)

1530

Tang and Luo (2009)

PMN-0.29PT (111)

1280

Tang et al. (2005)

Mn-PMN-0.26PT

1720

Liu et al. (2009)

Mn-PMN-0.29PT

1620

Tang et al. (2006)

LiTaO3

230

Kulwicki et al. (1992)

PZT

330

Kulwicki et al. (1992)

–

–

+

+ +

P –

–

+

–

–

–

+ +

+

+

+ –

–

–

–

–

+

+

+

+

+

P

+ +

–

– – – –

–

– –

–

–– –

–

+

+

+

+

+

+

+

+

+

dT dt

0

(b)

–

+

–

dT dt

(a)

– +

– –

– – – – – –

+

+

+ +

+ + +

–

– –

–

–

+

P

– –

+

+ + +

– – –

–

+

–

+ + + + + + + + IP

dT dt

0

IP 0

(c)

Figure 3.11 (a) Diagram of pyroelectric infrared sensor, (b) decreased polarization and change of surface charges when thermal radiation is on, and (c) increased polarization and change of surface charges when thermal radiation is off. The ellipsoids are only guidance for electric polarization change.

Since the change of polarization in a solid is accompanied by the change in number of surface charges, it can be detected by measuring induced current in an external circuit. If the pyroelectric material is perfectly electrical insulated from its surroundings, the surface charges are eventually neutralized by charge flow in the circuit. Infrared radiation is radiated at the front surface that is usually “blackened” by black gold (a gold film grown at very poor vacuum is black like a carbon film) or graphite so that the efficiency of heat absorption can be maximized. The heat diffuses through the graphite layer or gold electrode into the pyroelectric material and causes finite temperature change. When the amount of radiation striking the crystal changes, the number of charges associated with the spontaneous polarization also changes and a small pyroelectric current is thus induced. The current can then be amplified and measured with a sensitive FET device built into the sensor. TO5 package is a typical kind of electronic packages that is commonly applicable to pyroelectric sensors. Poled pyroelectric materials are protected and isolated from the surrounding by TO5 package, which can significantly reduce the noise. Figure 3.12 shows the device structure and circuit. 3.3.3

Pyroelectric Figures of Merit

Figure of merit (FOM) is an important parameter to assess the performance of a pyroelectric sensor. In order to select a suitable pyroelectric material as a sensing

3.3 Pyroelectric Effect and Infrared Sensor Application

+5 V Amplifier Pyroelectric elements

(a)

R1

R2

(b)

Figure 3.12 (a) Basic structure and (b) circuit of a pyroelectric detector.

element to fabricate an infrared (IR) sensor, many factors such as the operating temperature, frequency, and pyroelectric response should be considered. It is possible to formulate a group of FOMs to assess the performance of IR device. The current FOM, F i , which describes the influence of the pyroelectric material on the current responsibility of a pyroelectric sensor, is defined as Fi =

p 𝜌Cp

(3.14)

where p is the pyroelectric coefficient, 𝜌 is the density of the sample, and C p is the heat capacity. The voltage FOM, F v , which evaluates the influence of the pyroelectric-material on the voltage responsibility of a pyroelectric sensor, is expressed as Fv =

p 𝜌Cp 𝜖0 𝜖r

(3.15)

where 𝜖 0 and 𝜖 r are the permittivity of free space and relative permittivity of the sample, respectively. The detectivity FOM, F d assesses the influence of the pyroelectric material on the detection sensitivity of a pyroelectric sensor and is given by Fd =

√

p

𝜌Cp 𝜖0 𝜖r tan 𝛿

(3.16)

where tan 𝛿 is the dielectric loss tangent. This equation tells us that to make a pyroelectric sensor with high performance, the pyroelectric coefficient should be as large as possible; other parameters such as heat capacity, dielectric permittivity and dielectric loss, and specific weight should be the smaller the better. PMN-PT single crystal has superior pyroelectric performances to make it a promising candidate for high-performance uncooled infrared detectors and thermal imaging. From Table 3.1, the pyroelectric coefficient of PMN-PT single crystal is proven to be larger than some classical ferroelectric materials such as PZT ceramic. However, the measured pyroelectric coefficients of PMN-PT are

61

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3 Device Applications of Ferroelectrics

Table 3.2 Pyroelectric figures of merit of some typical pyroelectric materials. Material

F i (pm/V)

F v (m2 /C)

F d (𝛍Pa−1/2 )

PMN-0.26PT

610

0.11

153

PMN-0.29PT

525

0.11

98

Mn-PMN-0.26PT

688

0.12

402

Liu et al. (2009)

Mn-PMN-0.29PT

648

0.11

371

Tang et al. (2006)

72

0.17

157

Kulwicki et al. (1992)

122

0.02

12

Kulwicki et al. (1992)

LiTaO3 PZT

References

Tang and Luo (2009) Tang et al. (2005)

far away from another Mn-doped PMN-PT single crystal. Hence, the pyroelectric coefficient is varied from composition and crystal orientation. (Students: Think how to choose the crystal orientation for largest pyroelectric response.) Table 3.2 summarizes PMN-PT materials’ group of FOMs with comparison of the other ferroelectric materials. We can see that PMN-PT relaxor ferroelectric crystal has superior FOM compared with the ever best pyroelectric LiTaO3 (Li et al. 2014; Tang and Luo 2009; Tang et al. 2005, 2006; Liu et al. 2009; Kulwicki et al. 1992). Another method to characterize the pyroelectric effect is to use infrared laser with a chopper and lock-in amplifier. As illustrated in Figure 3.13, an infrared sensor based on pyroelectric effect can be characterized with infrared radiation from an incident of infrared laser light, and current or charges can be measured Pyroelectric materials Absorption layer IR laser A Chopper Lock-in amplifier

Pyroelectric materials

IR laser Absorption layer IR pulse

A

Signal generator Lock-in amplifier

Figure 3.13 Experimental setup for infrared sensor characterization.

3.4 Application in Microwave Device

C/Au

MoS2 e– e– e– e– e– e–

Au

MoS2

Au PMN-PT

S

D C/Au

PMN-PT

–

–

–

–

–

MoS2

Au

ITO

–

e–

e–

e–

–

–

–

Au PMN-PT

IR C/Au (a)

IR

(b) 0.018 6 mW/mm2 IR off

VD = 0.5 V

Current (μA)

0.016 0.014 0.012 0.010 IR on 0.008 (c)

0

200

600 400 Time (s)

800

Figure 3.14 (a) A schematic view of the ferroelectric gate FET, (b) the working mechanism of the ferroelectric field-effect transistor modulated with IR illumination, and (c) the time-resolved photocurrent in response to IR on/off at an irradiance of 6 mW/mm2 with 1064 nm laser. Source: Reprinted with permission from Fang et al. (2015). Copyright 2015, The Optical Society.

by lock-in technique. In the setup, the alternative laser incident can be realized by a laser chopper or by a laser beam excited by sinusoidal or square wave of electrical input that is also used as the reference input to the lock-in amplifier. Figure 3.14 shows our result of an example to integrate pyroelectric effect with a semiconducting 2D material to form an infrared sensor. This device uses pyroelectric-induced field effect to modulate the surface conductivity of 2D MoS2 to enable it to detect infrared radiation, or otherwise it is only sensitive to visible or UV light.

3.4 Application in Microwave Device Compared with BaTiO3 , ferroelectric (Bax Sr1−x )TiO3 (BST) exhibits much larger dielectric constant change with applied dc electric field. This unique property of BST is in favor to the application in the miniaturization of tunable microwave devices. The development of microwave devices in voltage-controlled oscillators, tunable filters, and phase shifters are currently based on this property. Recently,

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3 Device Applications of Ferroelectrics

BST thin films have been employed in many applications, for example, the BST tunable capacitor used for matching networks in antenna and amplifier. By changing its capacitance as bias voltage changes, it can tune antennas, improve the effectiveness of the phone’s antenna, and help the phone to avoid dead spots in order to improve performance especially in regions with low signal. The electronically steerable system saves space and weight in satellite communication and radar system. Not limited to communication technology, BST thin films also have the potential in integration with semiconductor microelectronic circuits, including microwave tunable switches and resonators (Figure 3.15) (Terrence 2013). Although BST is currently competing with several materials used in the market, such as semiconductor GaAs and micro-electrical–mechanical systems (MEMS), the outstanding properties of BST still shows its value in the development. Table 3.3 shows the comparison of the properties of GaAs, MEMS,

BST Gap

Patch Substrate Ground plane

Reflection magnitude (dB)

64

0 –5 –10 BL- Corner BL- Emin BL- Middle No BL

–15 –20 –25 9

9.5

10 10.5 11 11.5 Frequency (GHz)

(a)

12 (b)

Figure 3.15 (a) Antenna devices with tunable BST layer acting as a distributed capacitance and corresponding reflection (Karnati et al. 2015). (b) The wafer-level antenna devices with BST thin films. Source: Adapted from Terrence (2013). Table 3.3 Comparison of the properties of GaAs, MEMS, and BST thin film. GaAs

MEMS

BST

Tunability n

∼2–6: 1

∼1.5–3: 1

∼2–4: 1

Loss

∼20–50 at 10 GHz

Very high

∼20–100 at 10 GHz

Control voltage

0

Figure 3.20 (a) An illustration of the basic principle of the electrocaloric effect in a polar material and (b) a multilayered electrocaloric device where the electrocaloric films are ferroelectric. Source: Adapted from Epstein and Malloy (2010).

where 𝜌 is the density ) C is( the)specific heat. These equations are based on the ( and 𝜕D 𝜕S = 𝜕E , which establishes the relationship between Maxwell equation 𝜕T E T the change of polarization and the change of entropy under changing electric field. By sweeping the electric field from E1 to E2 , heat can be pumped out from the device and refrigeration can be realized. The recent development in this field is to find new materials and new device design with higher efficiency toward commercialization.

References Al-Shareef, H.N., Bellur, K.R., Auciello, O., and Kingon, A.I. (1994). Fatigue and retention of Pb(Zr0.53 Ti0.47 )O3 thin film capacitors with Pt and RuO2 electrodes. Integrated Ferroelectrics 5 (3): 185–196. Available at: https://doi.org/10.1080/ 10584589408017011. Arimoto, Y. and Ishiwara, H. (2004). Current status of ferroelectric random-access memory. MRS Bulletin 29 (11): 823–828. Bao, P., Jackson, T.J., Wang, X., and Lancaster, M.J. (2008). Barium strontium titanate thin film varactors for room-temperature microwave device applications. Journal of Physics D: Applied Physics 41 (063001): 21. Buser, R.G., Tompsett, M.F., Kruse, P.W. et al. (1997). Ch.4 Hybrid Phyroelectric-Ferroelectric Bolometer Arrays. In: Uncooled Infrared Imaging Arrays and Systems, 1e (ed. C.M. Hanson), 154–168. Academic Press. Cao, D., Zheng, F., Shen, M. et al. (2012). High-efficiency ferroelectric-film solar cells with an n-type Cu2 O cathode buffer layer. Nano Letters 12 (6): 2803–2809. Chanthbouala, A., Garcia, V., Barthélémy, A. et al. (2012a). A ferroelectric memristor. Nature Materials 11: 860–864. Available at: http://www.ncbi.nlm.nih .gov/pubmed/22983431. Chanthbouala, A., Crassous, A., Garcia, V. et al. (2012b). Solid-state memories based on ferroelectric tunnel junctions. Nature Nanotechnology 7 (2): 101–104. Available at: http://www.ncbi.nlm.nih.gov/pubmed/22138863.

References

Chen, F.S. (1969). Optically induced change of refractive indices in LiNbO3 and LiTaO3 . Journal of Applied Physics 40 (8): 3389–3396. Available at: https://doi .org/10.1063/1.1658195. Chen, L., Luo, B.C., Wang, D.Y. et al. (2014). Enhancement of photovoltaic properties with Nb modified (Bi,Na)TiO3 –BaTiO3 ferroelectric ceramics. Journal of Alloys and Compounds 587: 339–343. Available at: http://www.sciencedirect .com/science/article/pii/S0925838813026339. Choi, T., Lee, S., Cheong, S.W. et al. (2009). Switchable ferroelectric diode and photovoltaic effect in BiFeO3 . Science 324 (5923): 63–66. Available at: http:// science.sciencemag.org/content/324/5923/63. Epstein, R.I. and Malloy, K.J. (2010). Electrocaloric refrigerator and multilayer pyroelectric energy generator, Pub. No.: US 2010/0037624 A1. pp. 1–10. Available at: https://patents.google.com/patent/US20100037624. Fang, H., Yan, Q., Dai, J.Y. et al. (2015). Infrared light gated MoS2 field effect transistor. Optics Express 23 (25): 31908–31914. Available at: http://www .opticsexpress.org/abstract.cfm?URI=oe-23-25-31908. Ferroelectric Memory Company (2018). One-transistor FeFET memory. Available at: https://ferroelectric-memory.com/technology/one-transistor-fefet-memory/ (accessed 19 August 2019). Fujitsu (n.d.). FRAM structure. Available at: http://www.fujitsu.com/global/ products/devices/semiconductor/memory/fram/overview/structure/ (Accessed 14 March 2018). Garcia, V. and Bibes, M. (2014). Ferroelectric tunnel junctions for information storage and processing. Nature Communications 5: 4289. Available at: http:// www.ncbi.nlm.nih.gov/pubmed/25056141. Garcia, V., Fusil, S., Bibes, M. et al. (2009). Giant tunnel electroresistance for non-destructive readout of ferroelectric states. Nature 460 (7251): 81–84. Available at: http://www.ncbi.nlm.nih.gov/pubmed/19483675. Gruverman, A., Wu, D., Lu, H. et al. (2009). Tunneling electroresistance effect in ferroelectric tunnel junctions at the nanoscale. Nano Letters 9 (10): 3539–3543. Gu, H., Qian, X.S., Ye, H.J., and Zhang, Q.M. (2014). An electrocaloric refrigerator without external regenerator. Applied Physics Letters 105 (16): 162905. Available at: https://doi.org/10.1063/1.4898812. Huang, H. (2010). Solar energy: ferroelectric photovoltaics. Nature Photonics 4: 134–135. Ielmini, D. and Waser, R. (eds.) (2016). Resistive Switching: From Fundamentals of Nanoionic Redox Processes to Memristive Device Applications. Weinheim, Germany: Wiley-VCH Verlag GmbH & Co. Karnati, K.K., Shen, Y., Trampler, M.E. et al. (2015). A BST-integrated capacitively loaded patch for Ka- and X-band beamsteerable reflectarray antennas in satellite communications. IEEE Transactions on Antennas and Propagation 63 (4): 1324–1333. Kulwicki, B.M., Amin, A., Beratan, H.R., and Hanson, C.M. (1992). Pyroelectric imaging. In: ISAF’92: Proceedings of the Eighth IEEE International Symposium on Applications of Ferroelectrics, 1. IEEE https://doi.org/10.1109/ISAF.1992.300607. Li, L., Zhao, X., Luo, H. et al. (2014). Scale effects of low-dimensional relaxor ferroelectric single crystals and their application in novel pyroelectric infrared

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detectors. Advanced Materials 26 (16): 2580–2585. Available at: https://doi.org/ 10.1002/adma.201304546. Liu, L., Li, X., Wu, X. et al. (2009). Dielectric, ferroelectric, and pyroelectric characterization of Mn-doped 0.74Pb(Mg1/3 Nb2/3 )O3 –0.26PbTiO3 crystals for infrared detection applications. Applied Physics Letters 95 (19): 192903. Available at: https://doi.org/10.1063/1.3263139. Neese, B., Chu, B., Zhang, Q.M. et al. (2008). Large electrocaloric effect in ferroelectric polymers near room temperature. Science 321 (5890): 821–823. Available at: http://www.sciencemag.org/cgi/doi/10.1126/science.1159655. Pantel, D. and Alexe, M. (2010). Electroresistance effects in ferroelectric tunnel barriers. Physical Review B 82 (13): 134105. Qin, M., Yao, K., and Liang, Y.C. (2008). High efficient photovoltaics in nanoscaled ferroelectric thin films. Applied Physics Letters 93 (12): 122904. Available at: https://doi.org/10.1063/1.2990754. Rogalski, A. (2003). Infrared detectors: status and trends. Progress in Quantum Electronics 27 (2): 59–210. Available at: http://www.sciencedirect.com/science/ article/pii/S0079672702000241. Tang, Y. and Luo, H. (2009). Investigation of the electrical properties of (1 − x)Pb(Mg1/3 Nb2/3 )O3 –xPbTiO3 single crystals with special reference to pyroelectric detection. Journal of Physics D: Applied Physics 42 (7): 75406. Available at: http://stacks.iop.org/0022-3727/42/i=7/a=075406. Tang, Y., Zhao, X., Feng, X. et al. (2005). Pyroelectric properties of [111]-oriented Pb(Mg1/3 Nb2/3 )O3 –PbTiO3 crystals. Applied Physics Letters 86 (8): 82901. Available at: https://doi.org/10.1063/1.1865337. Tang, Y., Luo, L., Jia, Y. et al. (2006). Mn-doped 0.71Pb(Mg1/3 Nb2/3 )O3 –0.29PbTiO3 pyroelectric crystals for uncooled infrared focal plane arrays applications. Applied Physics Letters 89 (16): 162906. Available at: https://doi.org/10.1063/1.2363149. Terrence, T. (2013). Paratek antenna technology will make an appearance in BlackBerry Z30! BerryReview. Available at: http://www.berryreview.com/2013/ 09/18/paratek-antenna-technology-will-make-appearance-blackberry-z30/ (accessed 12 March 2018). Tien, N.T., Seol, Y.G., Lee, N. et al. (2009). Utilizing highly crystalline pyroelectric material as functional gate dielectric in organic thin-film transistors. Advanced Materials 21 (8): 910–915. Available at: https://doi.org/10.1002/adma.200801831. Watton, R. (2011). Ferroelectric IR bolometers – from ceramic hybrid arrays to direct thin film integration. Ferroelectrics 184 (1): 141–150. Wen, Z., Li, C., Wu, D. et al. (2013). Ferroelectric-field-effect-enhanced electroresistance in metal/ferroelectric/semiconductor tunnel junctions. Nature Materials 12 (7): 617–621. Available at: http://www.ncbi.nlm.nih.gov/pubmed/ 23685861. Whatmore, R.W. (1986). Pyroelectric devices and materials. Reports on Progress in Physics 49 (12): 1335–1386. Yang, S.Y., Seidel, J., Byrnes, S.J. et al. (2010). Above-bandgap voltages from ferroelectric photovoltaic devices. Nature Nanotechnology 5 (2): 143–147. Yang, Y. et al. (2012). Flexible pyroelectric nanogenerators using a composite structure of lead-free KNbO3 nanowires. Advanced Materials 24 (39): 5357–5362. Available at: https://doi.org/10.1002/adma.201201414.

References

Yau, H.M., Yan, Z.B., Dai, J.Y. et al. (2015). Low-field switching four-state nonvolatile memory based on multiferroic tunnel junctions. Scientific Reports 5: 12826. Available at: http://www.nature.com/doifinder/10.1038/srep12826. Yau, H.M., Xi, Z., Dai, J.Y. et al. (2017). Dynamic strain-induced giant electroresistance and erasing effect in ultrathin ferroelectric tunnel-junction memory. Physical Review B 95: 214304. Yuan, Y., Reece, T.J., Huang, J. et al. (2011). Efficiency enhancement in organic solar cells with ferroelectric polymers. Nature Materials 10: 296–302. Zhuravlev, M., Sabirianov, R., Jaswal, S., and Tsymbal, E. (2005). Giant electroresistance in ferroelectric tunnel junctions. Physical Review Letters 94 (24): 246802. Available at: http://link.aps.org/doi/10.1103/PhysRevLett.94.246802.

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4 Ferroelectric Characterizations In recent years, many new emerging materials are being claimed to be ferroelectric, such as two-dimensional (2D) materials MoS2 , SnSe, and topological insulator material CsPbI3 (Liu et al. 2016). Even the solar cell material hybrid perovskite CH3 NH3 PbO3 has been claimed to be ferroelectric, which is also attributed to be one of the main reasons to facilitate the photovoltaic efficiency, but the result is still under debate. To clarify that a material is ferroelectric needs critical experimental evidence such as P–E loop and switchable electric polarization. In this chapter, basic techniques and recent advances of ferroelectric characterization methods are introduced.

4.1 P–E Loop Measurement P–E hysteresis loop is the most direct evidence to prove a material to be ferroelectric, from which the saturation polarization, remnant polarization, and coercive field can be determined. However, the P–E loop cannot be measured directly; it can only be obtained by measuring electric displacement D forming D–E hysteresis loop, providing that 𝜖 0 E is relatively much smaller compared with the polarization P. In fact, to a ferroelectric material, this is true, i.e. D = 𝜖0 E + P ≈ P

(4.1)

The measurement of P–E hysteresis loop of a ferroelectric sample is usually by means of modified Sawyer–Tower circuit that contains the ferroelectric sample as a capacitor (C x ) in series with a linear reference capacitor (C ref ) (see Figure 4.1). There are two prerequisites for this measurement method: (i) assuming the charge density (electric displacement D) of the ferroelectric sample is the same as the amount of charges per unit area on the reference capacitor, and (ii) the capacitance of the reference capacitor is constant and independent of the applied voltage and it should be much larger than the capacitance of the sample (around 1000 times so that the voltage drop across the reference capacitor can be neglected). From the Sawyer–Tower measurement, Vref Vx

=

Cx Cref

(4.2)

Ferroic Materials for Smart Systems: From Fundamentals to Device Applications, First Edition. Jiyan Dai. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

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4 Ferroelectric Characterizations

Sample

Cx

R

Rx Cref

Figure 4.1 A modified Sawyer–Tower circuit for ferroelectric P–E loop measurement.

where V ref is the voltage for the reference capacitor and V x is the voltage for the sample. Given the dimension of the sample area S, thickness d, and dielectric constant 𝜖 r , under electric field E, the sample’s ferroelectric polarization can be found by Vref =

Cx 𝜖𝜖 SV 𝜖𝜖S S V = r 0 x = r 0 E= P Cref x Cref d Cref Cref

(4.3)

Then, we have P=

Vref Cref S

(4.4)

During the hysteresis loop measurement, an ac signal generated by a function generator and amplified by a voltage amplifier is applied to the sample placed in a silicone oil bath. The input and output signals of Sawyer–Tower circuit are observed by a digital oscilloscope, and the data of hysteresis loop at different temperatures are recorded in a computer. As an example, Figure 4.2 shows our experimental result of P–E loop measurement, from which we can determine that the remnant polarization Pr = 0.25 C/m2 or 25 μC/cm2 and saturation polarization Ps = 0.29 C/m2 or 29 μC/cm2 . The coercive field Ec = 1.80 × 105 V/m. To obtain a P–E loop of a typical ferroelectric material is not difficult; however, for many ferroelectric materials with large leakage current (especially thin films), it is not an easy task. Figure 4.3a sketch an ideal linear response of a capacitor in the P–E loop measurement, and Figure 4.3b shows an ideal resistor response. For a leaky ferroelectric thin film, the measured hysteresis loops sometimes only show ellipsoid feature without saturation as illustrated in Figure 4.3c. This kind of loop cannot be called P–E loop since it only reflects a leaking dielectric characteristic, and therefore, this kind of result cannot support the conclusion of ferroelectric claim of the corresponding materials (Scott 2008). A successful P–E loop should present at least saturation in the loop as shown in Figure 4.3d.

4.1 P–E Loop Measurement

Ps = 0.29 C/m2

0.3

Polarization (C/m2)

0.2 PR = 0.25 C/m2

0.1

0.0 Ec = 1.80 × 105 V/m –0.1

–0.2

–0.3 –1.5 × 106

–1.0 × 106

–5.0 × 105 0.0 5.0 × 105 Electric field (V/m)

1.0 × 106

1.5 × 106

Figure 4.2 Typical P–E loop from a PMN-PT relaxor ferroelectric single crystal.

P

P

E

(a)

E

(b) P

P

E

(c)

E

(d)

Figure 4.3 (a) An ideal linear response of capacitor, (b) an ideal resistor response, (c) an unsuccessful “P–E loop” without a saturation, which is a typical dielectric loss induced loop, and (d) a typical P–E loop showing ferroelectricity.

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4 Ferroelectric Characterizations

4.2 Temperature-Dependent Dielectric Permittivity Measurement A very important characteristic for ferroelectrics is the presence of a peak in dielectric permittivity–temperature curve as introduced in the Curie–Weiss law. Therefore, another practical way to illustrate ferroelectricity of a material is to measure its temperature-dependent dielectric permittivity 𝜖 r –T, where the peak indicates ferroelectric to paraelectric phase transition and the corresponding temperature is the Curie temperature. As shown in Figure 4.4a, the 𝜖 r –T ϵr (T) 25 000 x=0 20 000

x = 0.27

x = 0.34 x = 0.47

x = 0.65 x = 0.87

15 000

x = 1.0

10 000

5000

0

0

100

(a)

200 300 Temperature (K)

400

PMN-PT 2

104ϵ

76

1 Frequency increases

0 –100 (b)

–50

0

50

100

Temperature (°C)

Figure 4.4 Dielectric–temperature curve of (a) (Bax Sr1−x )TiO3 where x varies from 0 to 1, i.e. from SrTiO3 to BaTiO3 and (b) PMN-xPT. Source: Adapted from Vendik et al. (1999) and Cheng et al. (1998).

4.3 Piezoresponse Force Microscopy (PFM)

curves of (Bax Sr1−x )TiO3 (BST) show different Curie temperatures for different components of x. The 𝜖 r –T curves at different frequencies also present the frequency dispersion if a ferroelectric is relaxor type such as PMN-PT as shown in Figure 4.4b.

4.3 Piezoresponse Force Microscopy (PFM) Piezoresponse force microscopy (PFM) is a special function in an atomic force microscope (AFM) that is designed to detect piezoelectric response of a sample to an applied electric field. This technique has been widely used in ferroelectric domain observation and was systematically introduced by Gruverman in his book (Gruverman and Kalinin 2007). In this session, details of this technique and results from our group of research are introduced. 4.3.1

Imaging Mechanism of PFM

In PFM imaging, when an ac electric field is applied between a conductive probe and bottom electrode of a sample, local oscillatory deformation with the same frequency as the ac field will be induced on the sample surface due to the converse piezoelectric effect. Depending on the direction of ferroelectric polarization, this oscillatory deformation response has 0∘ or 180∘ phase relationship with the added ac electric field, and this phase angle together with amplitude can be measured by lock-in technique and imaged to reveal the ferroelectric polarization and domain structure. The conductive AFM probe detects the surface oscillation and sends the signal through the reflected laser beam to photodiode detector and be processed by a lock-in amplifier. The amplitude measured by the lock-in amplifier represents the scale of piezoresponse that is proportional to piezoelectric coefficient; and the phase angle represents the polarization direction, where 0∘ corresponds parallel and 180∘ the antiparallel alignment of polarization to the electric field added on the AFM tip. By forming an image with phase angle derived from the lock-in amplifier, the PFM technique can image ferroelectric domains by presenting the bright–dark contrast, and the amplitude tells how strong the piezoelectricity is. In the PFM imaging setup as shown in Figure 4.5, an ac voltage V cos 𝜔t is applied to the bottom electrode to provide an electric field on the sample while the conductive probe is electrically grounded, or vice versa. The signal detected by photodiode from the local polarization, A cos(𝜔t + 𝜑), with the reference signal V cos 𝜔t will be fed into the lock-in amplifier. By passing through a low-pass filter, the lock-in amplifier will output a signal with dc component Ã, which is the time-averaged value of the piezoresponse amplitude and the phase angle 𝜑 of electromechanical response of the sample to the ac electric field. The domain images can be obtained by imaging à and 𝜑 over an observation area. Figure 4.6 is a PFM image of a PZT grain in a thin film structure, where one can see that within one PZT grain there are multiple ferroelectric domains forming 90∘ and 180∘ domain walls. These domain polarizations can be switched by adding a dc field on the tip.

77

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4 Ferroelectric Characterizations

Output electrical signal Photodiode detector Laser source

Low-pass filter

Feedback circuit

Piezo-material Scanner Lock-in amplifier Reference signal Amplitude, A

Topographic

ac modulate signal

Phase, φ

Domain image

Figure 4.5 Schematic diagram of piezoresponse force microscopy imaging mechanism. 120°

(101)

(110) (011)

Figure 4.6 (a) Our PFM phase image of a PZT grain in a PZT film showing ferroelectric domain structure as illustrated in (b). Source: Adapted from Zhao (2005).

[111] (a)

(b)

Not only the domain pattern, when an area is scanned with a conductive APM tip being added with a dc electric field, but also the polarization of the scanned area can be switched. Therefore, the PFM technique can also demonstrate polarization switching, which is the key criterion for a ferroelectric. Furthermore, a so-called butterfly loop (see Figure 4.7) can be achieved by PFM technique when the piezoresponse amplitude is measured as a function of dc electric field added on the sample. The butterfly loop illustrates that the increased dc field results increased polarization leading to larger piezoresponse. It is worthy to note that the nonzero piezoresponse at zero field is due to the remnant polarization, and the corresponding phase angle change from 0∘ to 180∘ is due to the polarization switching. Therefore, this butterfly loop and the corresponding 180∘ phase change can be used as criteria for clarification of ferroelectricity. To demonstrate ferroelectric switching is essential to claim ferroelectricity. In PFM imaging, when a dc voltage is applied on the conductive AFM tip while scanning an area, the subsequent PFM image can reveal the switched polarization as illustrated in Figure 4.8, where (a) is the amplitude image and (b) the phase image.

4.3 Piezoresponse Force Microscopy (PFM)

Figure 4.7 Our PFM out-of-plane phase hysteresis loop and amplitude butterfly loop as a function of applied dc voltage for an epitaxial BaTiO3 thin films.

250

Phase (°)

200

150

100

50

Amplitude (a.u.)

0

–6

–4

–2

0 2 Voltage (V)

a.u.

(a)

6

° 180

1

1 μm

4

1 μm

0

0

(b)

Figure 4.8 Our PFM results from epitaxial BaTiO3 thin films. (a) Amplitude and (b) phase images recorded after writing an area of 3.0 × 3.0 μm2 with −4.5 V and then the central square with +4.5 V using a biased conductive tip. Source: Yau et al. (2017). Reproduced with permission of American Physical Society.

79

80

4 Ferroelectric Characterizations

For the switched areas with polarizations facing up and down, the brightness of amplitude image represents the strength of piezoresponse, while the phase image represents the polarization direction. 4.3.2 PFM

Out-of-plane Polarization (OPP) and In-plane Polarization (IPP)

The real situation is that in a ferroelectric crystal or film, the polarizations of domains are not only facing up and down but also in-plane or inclined with an angle to the surface. Therefore, a PFM should also be able to map domains with different polarization directions. Fortunately, for the PFM technique, out-of-plane polarization (OPP) and in-plane polarization (IPP) PFM can be done simultaneously. In the OPP-PFM, the induced local vertical deformation is measured by the up-and-down deflection of the cantilever; while the induced local shear deformation is measured by the torsional twisting of the cantilever in the IPP-PFM as shown in Figure 4.9. The photo-detector in the AFM can distinguish the vertical up-and-down movement and torsion (twist) movement of the tip by sensing the reflected laser beam. There are usually four sectors in the photo-detector, where (A + B) − (C + D) signal measures the tip’s vertical movement and (A + C) − (B + D) signal measures the tip’s torsional movement. The OPP- and IPP-PFM images can be obtained by collecting the piezoelectric response signals demodulated by two lock-in amplifiers in PFM. Here we demonstrate the OPP- and IPP-PFM techniques with our results in PMN-PT single crystal domain imaging. Figure 4.10 shows the OPP- and IPP-PFM images of the (001)-cut PMN-0.38PT crystal at room temperature, where stripe-like domains with width of around 700 nm can be observed. In the PFM images, the dark contrast in the OPP and IPP amplitude images corresponds to the regions without piezoresponse signal, and the bright regions correspond to large piezoresponse. While in the OPP phase image, the upward Torsion

E

Out-of-plane

A

P

a-domain

c-domain

D

E

(a)

E P

P

B In-plane

C

Deflection

(b)

PP ε5 = d51E1

X3 X2

X1 X1

E

P

X2

X3

ε3 = d33E3 ε1 = d13E3

Figure 4.9 (a) Illustration of four sectors of a photo-detector in AFM. (b) Quantitative explanation of the contrast formation in IPP- and OPP-PFM. The electric field induced from a conductive tip causes surface displacement due to the converse piezoelectric effect. Source: Adapted from Felten et al. (2004).

4.3 Piezoresponse Force Microscopy (PFM)

(a)

(b)

(c)

OPP

IPP

IPP

IPP

IPP

OPP

(d)

(e)

Figure 4.10 Piezoresponse (a) OPP phase, (b) OPP amplitude, (c) IPP amplitude images of the as-grown (001)-cut PMN-0.38PT single crystal with scan size of 5 μm, (d) the schematic diagram of tetragonal unit cell with six polar directions, and (e) illustration of proposed domain boundary configuration revealed from the OPP- and IPP-PFM images. Source: Adapted from Wong et al. (2006).

polarization (pointing out of the paper) appears dark in contrast, and regions with downward polarization appear bright (please notice that the orientation depends on the direction of electric field). One can see that both upward and downward domains appear dark and bright alternatively in the OPP phase image (Figure 4.10a); while they appear bright in OPP amplitude image (Figure 4.10b). It is interesting to notice that in the OPP amplitude image, the dark contrast regions with the width of around 40–90 nm at the domain boundaries, however, appear bright in the IPP amplitude image shown in Figure 4.10c. It indicates that the domain boundaries contain large IPP without OPP. If an IPP phase image is formed, the IPP direction can also be mapped as bright and dark contrasts. According to a schematic diagram of a tetragonal unit cell with six polar directions (Figure 4.10d), it can be seen that the dark and bright stripe-like domains in the OPP phase image represent two opposite OPPs, and four IPPs exist in the domain boundaries. Figure 4.10e shows the proposed domain boundary configuration revealed from the OPP- and IPP-PFM images. In order to understand the nature of PFM imaging, finite element modeling (FEM) has been carried out to illustrate the surface deformation under a finite point electrode induced electric field. In the FEM simulation with commercial software ANSYS, the size of the sample is set as 2 mm × 3 mm × 50 μm, which

81

4 Ferroelectric Characterizations

is same as the real sample in PFM imaging. A metal layer under the bottom of sample as bottom electrode is fixed. Because the size of the conductive tip is very small, the probing electric field is strongly inhomogeneous and decays rapidly from the tip into the crystal. Therefore, we can substitute the electric field with the same magnitude as being used in the PFM experiment into the simulation. The displacements of two samples possessing OPP and IPPs are computed in the ANSYS simulation. In the simulation results (Wong 2007), the light blue and purple curves in Figures 4.11 and 4.12 represent the applied ac electric field across the sample and the calculated displacement in the x, y, and z directions, respectively. From the OPP simulation result, it can be seen that along the direction of OPP, denoted as the z-direction, there is a large displacement as shown in Figure 4.11d, while there is no obvious displacement along the x and y directions in Figure 4.11b,c. By contrast, in the IPP simulation result, large displacement along the x direction can be observed as shown in Figure 4.12b; but there is no obvious displacement along the y and z directions. Therefore, according to the movement of sample surface, the IPP and OPP can be detected by PFM tip.

6.25 5 3.75

ac field

2.5

z

Value

Out-of-plane polarization

x

1.25 0 –1.25 –2.5 –3.75

y

–5 (×10−6)

–6.25 0

(a)

–3

–4

–5

–6

–7

–8

–9

1

Time

6.25

5

5

3.75

3.75 2.5 1.25

Value

2.5 1.25 0 –1.25

0 –1.25

–2.5

–2.5

–3.75

–3.75 –5

–5 (×10−6)

–6.25 0

(c)

–1

–2

(b) 6.25

Value

82

–1

–2

–3

–4

–5

–6

Time

–7

–8

–9

(×10−6)

–6.25 0

1

(d)

–1

–2

–3

–4

–5

–6

–7

–8

–9

1

Time

Figure 4.11 (a) The sample surface movement of out-of-plane polarization ANSYS simulation. (b–d) The graph of x, y, and z displacements, respectively. The applied ac electric field is along the out-of-plane direction. Source: Adapted from Wong (2007).

4.3 Piezoresponse Force Microscopy (PFM) 6.25 5

ac field

3.75 2.5

Value

z

x

In-plane polarization

1.25 0 –1.25 –2.5 –3.75

y

–5 –6.25 0

(a)

–3

(b)

–6

–5

–7

–8

–9

(×10−6) 1

Time

5

5

3.75

3.75 2.5 1.25

Value

2.5 1.25 0 –1.25

0 –1.25

–2.5

–2.5

–3.75

–3.75 –5

–5 (×10−6)

–6.25 0

(c)

–4

6.25

6.25

Value

–1

–2

–1

–2

–3

–4

–5

–6

–7

–8

–9

Time

–6.25 0

1

(d)

–1

–2

–3

–4

–5

–6

–7

–8

–9

(×10−6) 1

Time

Figure 4.12 (a) The sample surface movement of in-plane polarization ANSYS simulation. (b–d) The graph of x, y, and z displacements, respectively. The applied ac electric field is along the out-of-plane direction. Source: Adapted from Wong (2007).

4.3.2.1

Electrostatic Force in PFM

When an ac electric field is applied to a ferroelectric surface through a conductive AFM tip, the displacement D of the AFM tip includes three parts as illustrated by the equation D = ad33 E + b𝛼E2 + cE2

(4.5)

where the first term is from the piezoresponse of the sample surface with piezoelectric constant d33 and the second term is contributed by the strain generated from electrostrictive effect with coefficient 𝛼, which exists in all dielectrics and is proportional to E2 . The last term is the contribution from the interaction between the tip cantilever and sample surface due to the electrostatic force (also called capacitance force since the cantilever and sample surface form a parallel plate capacitor) that can be repulsive or attractive depending on the positive or negative charges of the sample surface and the instant voltage of the cantilever. The constants a, b, and c represent the relative contributions to the tip displacement due to the imaging condition, such as the cantilever’s force constant, and the tip/sample surface condition (for example, when the cantilever is on metal electrode or near the edge of the sample, the electrostatic attraction force is much lower).

83

84

4 Ferroelectric Characterizations

PFM amplitude image without electrostatic force

Amplitude

PFM amplitude image with electrostatic force

Amplitude

Ferroelectric polarization (a)

Figure 4.13 Expected piezoresponse amplitude signals, and accompanying domain images for (a) without and (b) with electrostatic force.

Electrostatic force (b)

During PFM imaging, in order to produce an ac electric field across the sample, an ac voltage is applied to the bottom electrode; while the conductive tip is electrically grounded, or vice versa. Since there is an air gap between cantilever and sample surface, capacitive force between the cantilever and sample surface needs to be considered in PFM imaging. The capacitive force is an attractive force between the cantilever and sample surface that alters the PFM image and shadows out the information of domain boundary structure. Hong et al. (2002) have investigated the capacitive force effect during PFM imaging and discovered that, at the center position, the attractive capacitive force leads to a vertical shift of amplitude signal in the upward and downward domains. Figure 4.13 illustrates the electrostatic capacitance force effect to the PFM amplitude image, where panel (a) is what we suppose to see from a ferroelectric polarization induced contrast and panel (b) is what we see when a strong capacitance force exists. However, it can be reduced by placing the tip near the edge of sample surface due to the small capacitive coupling. In order to avoid the effect coming from attractive force and increase the signal-to-noise ratio, a relatively large force constant (e.g. 40 N/m) tip should be used in PFM imaging. With a “shorter” and “stronger” cantilever, the capacitive force effect can be significantly reduced, and clear domain boundary feature can be obtained in the PFM images. 4.3.2.2

Perspectives of PFM Technique

PFM is becoming one of the key technologies to investigate ferroelectricity and domain structures of new emerging materials such as 2D, topological, and hybrid perovskite materials. For example, the ferroelectricity in halide-perovskite solar cell material has been considered to be important to its photovoltaic efficiency. But the ferroelectric domain structure is still arguable. Switchable polarization has not been demonstrated so far. It has been said “you can almost obtain PFM image in any material.” Sometimes this is correct for careless experiments, i.e. non-ferroelectric materials can produce PFM signals that are similar to those of standard ferroelectrics (Vasudevan et al. 2017). If you look at Eq. (4.5), the contribution to the PFM image, a domain contrast stimulated by an external electric field added on a conductive AMF tip

4.3 Piezoresponse Force Microscopy (PFM)

comes from at least three integration force where only the first term is real piezoelectric response. Electrostrictive and electrostatic responses should also be considered, especially when the piezoresponse is very small in those non-typical ferroelectric materials (such as ultrathin films, 2D materials, and other emerging new materials in which only monolayer or a few layers of some 2D materials present weak ferroelectricity). To deeply understand the mechanism and improve the PFM technology (such as resonance tracing; Xie et al. 2012) are very important in PFM study of very thin films and new materials’ ferroelectricity and domain structure. The piezoresponse and PFM sensitivity can usually be enhanced by driving the ac modulation voltage near the resonance of the cantilever-specimen system as shown in Figure 4.14(a), and thus magnify the piezoresponse amplitude A ∼ d33 V 0 Q by orders of magnitude, where Q is the quality factor of the system Δω Amplitude

V0eiox

ΔA

d33V0Q d33V0

d33V0ei(ox + φ)

ω0

ω′0

Frequency

150

7.3 Fit amplitude phase

100

4.7

50

3.4

0

2.1

–50

0.8

–100

–0.5

ω1 255

280

ω2

305 Frequency (kHz)

330

355

Phase (°)

Amplitude (a.u.)

6

–150

Figure 4.14 Schematics of dual frequency resonance tracking (DFRT) technique: (a) tip-nanofiber harmonic oscillation system, (b) resonance enhancement using single frequency, and (c) schematics of DFRT using actual experimental data of piezoresponse versus ac driving frequency. Source: Xie et al. (2012). Reproduced with permission of The Royal Society of Chemistry.

85

4 Ferroelectric Characterizations

typically in the range of 10–100 (Xie et al. 2012). For the single frequency resonance enhancement technique, the frequency is locked at the resonance in a particular location. As the resonance shifts during scanning due to the change of contact stiffness, the lock-in will actually drive the oscillator away from the resonance at the current location, resulting in loss of enhanced sensitivity, as schematically shown in Figure 4.14b. The dual frequencies adopted, on the other hand, allow us to trace resonance when it varies over the sample surface and minimize the cross-talk with topography, as schematically shown in Figure 4.14c.

4.4 Structural Characterization Structural distortion from a centrosymmetric structure is essential for ferroelectric phase transition, such as from cubic to tetragonal (T), rhombohedral (R), orthorhombic (O), or monoclinic (M) phases. Using XRD characterization to determine the structure distortion is a very important step and is also a convenient method to identify the ferroelectric phase. Figure 4.15a shows that tetragonal phase results in split of (002) peak where the two equal intensity peaks are (002) and (020) since c > a for T structure. While for orthorhombic phase with ⟨110⟩ polarization direction, the (002) peak shows a shoulder peak. As shown in Figure 4.15b from PMN-0.28PT rhombohedral phase, the (111) peak splits into two peaks due to the elongation along one particular (111) direction. The map of symmetric Bragg peak can also be used to separate tilt (off-set) or strain in the structure of a material. Figure 4.16a is a reciprocal space map for pseudocubic {103}c peaks at a representative lower temperature (30 ∘ C), panel (b) is a reciprocal space map for {113}c peaks at 30 ∘ C, and panel (c) illustrates the pseudocubic unit cell of the lower temperature phase with a monoclinic distortion where the c-axis direction tilts up toward [100]c . Figure 4.16(d) is the reciprocal space mapping for pseudocubic {103}c peaks and (e) {113}c peaks performed at 200 ∘ C, respectively, and (f ) presents the pseudocubic unit cell at the kα1

(002) (020)

kα2 (002)

Intensity (a.u.)

20% LBFO Intensity (a.u.)

86

15% LBFO (002) SRO(002)

10% LBFO (002)

BFO 40 (a)

42

[111] poled PMN–0.28PT

50 K

150 K 300 K

44 46 2θ (°)

48

50

37.5 (b)

38.0

38.5

39.0

2θ (°)

Figure 4.15 Typical X-ray 𝜃–2𝜃 scans of (a) BiFeO3 and La:BiFeO3 films with different La concentrations on SrRuO3 /SrTiO3 /Si substrates, and (b) the (111) surface of [111]-poled PMN-0.28PT single-domain crystal. Source: Adapted from Chu et al. (2008) and Li et al. (2016).

4.5 Domain Imaging and Polarization Mapping by Transmission Electron Microscopy

4.05

3.95

50

45

2.

40

2.

35

(–113)c (1–13)c

^ bc

(113)c

3.95

βH

3.90

qII (Å–1)

50

βH

2.

2.

2.

(e)

45

200 °C

20

1.60 1.65 1.70 1.75 qII (Å–1)

2ΔH

40

200 °C

4.00

2.

(103)c (013)c

3.95

(d)

(–1–13)c

4.10 4.05

4.05 4.00

4.15

35

(–103)c (0–13)c

^ cm ^ T = 200 °C (MA) cc

{113}c {203}m

4.20

2.

{103}c {113}m

^ ac

(c)

30

q⊥ (Å–1)

4.10

2.

20

(b)

4.15

βL

30 °C

2.

(a)

^ bc

3.90

2.

30 °C 1.60 1.65 1.70 1.75

2.

3.95

(103)c

25

4.00

ΔL

(1–13)c (113)c

4.00

30

ΔL

2.

q⊥ (Å–1)

4.05

(–1–13)c (–113)c

4.10 (0–13)c (013)c

^ cc T = 30 °C (M ) C

{113}c

2.

(–103)c

4.10

4.20 4.15

25

{103}c

2.

4.15

(f)

^ am

^ ac ^ –bm

Figure 4.16 Reciprocal space mapping of the MA and MC phases in PMN-xPT crystal. Source: Ko et al. (2011). Reprinted with permission of Springer Nature.

higher temperature with the direction of the c-axis tilt toward [110]c. The high temperature structural phase is similar to the normal R-BiFeO3 film in terms of the direction of the distortion; however, it is apparently different in terms of the large c-axis lattice parameter.

4.5 Domain Imaging and Polarization Mapping by Transmission Electron Microscopy Transmission electron microscopy (TEM) is an important technology in material structural characterization including ferroelectric materials. Beyond the general techniques such as electron diffraction, high-resolution imaging, bright–dark field imaging, and in situ observation of ferroelectric domain evolution under electric field, direct mapping of polarization by Cs-corrected high-angle annular dark-field (HAADF) imaging is emerging as a very powerful technique to directly mapping ionic displacement and determine the electric polarization. Bright and dark field images can present domain structures by different contrasts due to their different scattering of incident electron beam. Figure 4.17(a) shows existence of 180∘ and 90∘ domain structures in K0.15 Na0.85 NbO3 single crystal, and the corresponding selected area electron diffraction pattern is shown in Figure 4.17(b) (Chen et al. 2015).

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4 Ferroelectric Characterizations

0.2 μm

(a)

(b)

Figure 4.17 (a) Low magnification TEM images of the KNN single crystal along ⟨100⟩ axes showing typical laminate domain structures and (b) the corresponding selected area electron diffraction pattern. Source: Adapted from Chen et al. (2015).

4.5.1

Selected Area Electron Diffraction (SAED)

Selected area electron diffraction (SAED), in general, is difficult to distinguish a ferroelectric phase of either a tetragonal, rhombohedral, or orthorhombic from a paraelectric cubic phase, since the resolution of electron diffraction pattern is much lower than XRD. However, if the distortion is relatively large, SAED may still show split spots. In the study of antiferroelectrics, SAED can clearly show the structural information by presenting superlattice structure in diffraction pattern. As shown in Figure 4.18, a series of SAED patterns of different zone axes of Bi0.85 Nd0.15 FeO3 ceramics were taken from the regions where the 1/4(00l) superstructure appears. The fundamental spots in Figure 4.20 come from the pseudocubic perovskite (a)

(b)

– 120

(c) 004

004

[210] (d) 040

200

(e) – 044

044

– 204

040

200

[100]

[010] (f)

– 120 – 124

[001]

– 004

––– [211]

040 – 208

[401]

Figure 4.18 SAED patterns of different zone axes of Bi0.85 Nd0.15 FeO3 ceramics taken from the of superstructure regions. The zone axes corresponding to (a–f ) are [210], [100], [010], [001], [211] and [401], respectively. Source: Pu et al. (2015). Reprinted with permission of Elsevier.

4.5 Domain Imaging and Polarization Mapping by Transmission Electron Microscopy

structure. In comparison with the fundamental spots, the emergence of 1/4{hh0} and 1/4(00l) diffraction spots indicates √that the √ superstructure resembles the antiferroelectric Pbam (with unit cell 2a × 2 2a × 4a) structure of PbZrO3 but with quadrupled c-axis instead of the doubled c-axis. It is seen that all the diffraction √ can be well indexed using an orthorhombic unit cell with dimen√ spots sions 2a × 2 2a × 4a (Pu et al. 2015). High-resolution TEM image can present clear crystal lattice image of perovskite structure such as BaTiO3 ; however, the imaging mechanism of wave interference and resolution limitation make it impossible to see the relative movement of ions from B site to A site. Thanks to the development of Cs-corrected STEM technique, now people can use it to observe relative shift of ions from B site to A site by forming real atomic position image. With this progress, the polarization direction can be directly mapped out from the atomic images, and therefore the domain structure can be identified. This is a technique using focused electron beam in the size of ∼0.1 nm to scan sample surface and using HAADF imaging mode to collect scattered electrons forming the position image of A and B sites cations in the perovskite crystal structure, where the heavier atoms appear brighter contrast. A recent research has reported the observation of polarization vertex (circular polarization direction loop compared with normally existed 90∘ and 180∘ domains) (Tang et al. 2015). Wang reported the growth of BiFeO3 (BFO) films on the tetragonal SrRuO3 (SRO)-buffered STO (001) substrate. Using aberration-corrected HAADF Z-contrast STEM imaging, they visualize, at the atomic scale, the out-of-plane atomic displacement in the 2D BFO films as shown in Figure 4.19. BFO is a well-known ionic-displacive ferroelectric material. Both the oxygen octahedra and the positive (Bi) ions have displacements from the center of the

BFO

20 pm

1 2

SRO (b)

STO

[001]

O

[100] (a)

(c)

DFe

Fe

c

a

Bi

Figure 4.19 (a) Atomically resolved HAADF-STEM images of 2 unit-cell-thick BiFeO3 . (b) B-site atomic displacement vector map from the area marked with a white dashed rectangle in (a), where the yellow arrows superimposed on the atomic image represent the relative displacement DFe of the Fe atoms as shown in (c). Source: Adapted from Wang et al. (2018).

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Fe cation tetragonal cell that give rise to the spontaneous polarization. However, due to difficulty in imaging the light element (O), the relative displacement between the Fe cation and the mass center of an elongated rectangle formed by its four Bi neighbors (denoted as DFe in Figure 4.19) can be used to determine the polarizations of BFO unit cells. Another important technique in TEM is the in situ observation with electric field applied to the sample. Under electric field driven, domain evolution can be dynamically observed (see Figure 4.20). No domain structure is observed without electric field, but an applied positive electric field leads to the domain growth

Top electride (Pt/Au)

Pdown Pup (a)

After contact (0 V)

(b)

3V

(c)

10 V

(d)

–4 V

(e)

–10 V

Nb-STO

Figure 4.20 A series of TEM image of in situ observation of ferroelectric domain evolution induced by external biases. Source: Han et al. (2014). Reprinted with permission of Springer Nature.

4.5 Domain Imaging and Polarization Mapping by Transmission Electron Microscopy

toward the top electrode. When the negative field is applied, it is noted that the domain configuration is reversible. 4.5.2 Convergent Beam Electron Diffraction (CBED) for Tetragonality Measurement Convergent beam electron diffraction (CBED) is a technique widely used for determination of structure, symmetry details, and atom positions in a crystal as small as a few nanometers in size in the field of TEM. Conventionally, until the discovery of CBED, electron diffraction from a thin crystal in TEM was obtained using a method called SAED using a parallel electron beam. By replacing the parallel beam with a convergent beam, the diffraction spots are enlarged into CBED discs. Compared with the smallest area of SAED (500 nm), the area for diffraction in CBED is chosen by focusing the incident beam into a very fine spot (a few nanometers) on the region of interest. CBED has been widely used for nanostructure analysis. Detailed analyses of CBED patterns could give good amount of additional and useful information about a crystal. These include (i) three-dimensional information about the reciprocal lattice, (ii) point and space-group symmetry details, (iii) lattice parameters from regions as fine as 1 nm, (iv) atom positions within a unit cell, and (v) defects in crystals. A ferroelectric tetragonal phase can be determined using the CBED method. Figure 4.21a shows a CBED pattern of the tetragonal phase of PbTiO3 taken at room temperature with the [100] incidence. The CBED pattern is seen to have mirror symmetry parallel to the c-axis. Figure 4.21b schematically shows the corresponding [100]-projected structure of the tetragonal phase, where the direction of the ferroelectric polarization is indicated by arrow. Figure 4.21c shows a CBED pattern simulated with the structure in Figure 4.21b using software MBFIT for dynamical diffraction calculations. The features of the CBED pattern of Figure 4.21a are qualitatively reproduced, indicating that the local structure of the tetragonal phase of PbTiO3 has the symmetry of the tetragonal structure. Sim.

Exp. c*

c*

Ba

001 Ti O

000 010

c m (a)

(b)

Ps

b

m (c)

Figure 4.21 (a) CBED pattern of the tetragonal phase of PbTiO3 taken at room temperature with the [100] incidence, where the mirror expected from the tetragonal structure is indicated by a line. (b) Corresponding crystal structure of the tetragonal phase with space group P4mm. (c) CBED pattern simulated with the structure in (b). Source: Tsuda and Tanaka (2013). Reprinted with permission of IOP.

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References Chen, Y., Wong, C.M., Dai, J.Y. et al. (2015). Ferroelectric domain structures in ⟨001⟩-oriented K0.15 Na0.85 NbO3 lead-free single crystal. AIP Advances 5 (3): 037117. Available at: https://aip.scitation.org/doi/pdf/10.1063/1.4914936? class=pdf. Cheng, Z.-Y., Katiyar, R.S., Yao, X., and Bhalla, A.S. (1998). Temperature dependence of the dielectric constant of relaxor ferroelectrics. Physical Review B 57 (14): 8166–8177. Available at: https://link.aps.org/doi/10.1103/PhysRevB.57.8166. Chu, Y.H., Zhan, Q., Yang, C.-H. et al. (2008). Low voltage performance of epitaxial BiFeO3 films on Si substrates through lanthanum substitution. Applied Physics Letters 92 (10): 102909. Available at: https://doi.org/10.1063/1.2897304. Felten, F., Schneider, G.A., Saldaña, J.M., and Kalinin, S.V. (2004). Modeling and measurement of surface displacements in BaTiO3 bulk material in piezoresponse force microscopy. Journal of Applied Physics 96 (1): 563–568. Available at: https:// doi.org/10.1063/1.1758316. Gruverman, A. and Kalinin, S.V. (2007). Piezoresponse force microscopy and recent advances in nanoscale studies of ferroelectrics. In: Frontiers of Ferroelectricity (eds. S.B. Lang and H.L.W. Chan), 107–116. Boston, MA: Springer US. Available at: https://doi.org/10.1007/978-0-387-38039-1_10. Han, M.-G., Marshall, M.S.J., and Zhu, Y. (2014). Interface-induced nonswitchable domains in ferroelectric thin films. Nature Communications 5: 4693. Available at: https://doi.org/10.1038/ncomms5693. Hong, S., Shin, H., Woo, J., and No, K. (2002). Effect of cantilever-sample interaction on piezoelectric force microscopy. Applied Physics Letters 80 (8): 1453–1455. Ko, K.-T., Park, J.H., Yang, C.H. et al. (2011). Concurrent transition of ferroelectric and magnetic ordering near room temperature. Nature Communications 2: 567. Available at: http://www.ncbi.nlm.nih.gov/pubmed/22127063. Li, F., Zhang, S., Chen, L.Q. et al. (2016). The origin of ultrahigh piezoelectricity in relaxor-ferroelectric solid solution crystals. Nature Communications 7: 13807. Available at: http://dx.doi.org/10.1038/ncomms13807. Liu, S., Kim, Y., Tan, L.Z., and Rappe, A.M. (2016). Strain-induced ferroelectric topological insulator. Nano Letters 16 (3): 1663–1668. Available at: https://doi .org/10.1021/acs.nanolett.5b04545. Pu, S., Zheng, H., Zou, H. et al. (2015). The microstructure and ferroelectric property of Nd-doped multiferroic ceramics Bi0.85 Nd0.15 FeO3 . Ceramics International 41: 5498–5504. Scott, J.F. (2008). Journal of Physics: Condensed Matter 20: 021001. Tang, Y.L., Zhu, Y.L., Ma, X.L. et al. (2015). Observation of a periodic array of flux-closure quadrants in strained ferroelectric PbTiO3 films. Science 348 (6234): 547–551. Available at: http://www.sciencemag.org/cgi/doi/10.1126/science .1259869. Tsuda, K. and Tanaka, M. (2013). Convergent-beam electron diffraction study of the local structure of the tetragonal phase of PbTiO3 . Applied Physics Express 6 (10): 101501. Available at: http://stacks.iop.org/1882-0786/6/i=10/a=101501.

References

Vasudevan, R.K., Balke, N., Maksymovych, P. et al. (2017). Ferroelectric or non-ferroelectric: why so many materials exhibit “ferroelectricity” on the nanoscale. Applied Physics Reviews 4 (2): 021302. Vendik, O.G., Hollmann, E.K., Kozyrev, A.B. et al. (1999). Ferroelectric tuning of planar and bulk microwave devices. Journal of Superconductivity 12 (2): 325–338. Available at: https://doi.org/10.1023/A:1007797131173. Wang, H., Tian, H., Chen, J.S. et al. (2018). Direct observation of room-temperature out-of-plane ferroelectricity and tunneling electroresistance at the two-dimensional limit. Nature Communications 9 (1): 3319. Available at: https:// doi.org/10.1038/s41467-018-05662-y. Wong, K., Tian, H., and Chen, J.S. (2007). Study of Domain Structure and Evolution in PMN-30%PT Single Crystals by Means of Piezoresponse Force Microscopy. Hong Kong Polytechnic University. Available at: http://theses.lib.polyu.edu.hk/ handle/200/812. Wong, K.S., Dai, J.Y., Choy, C.L. et al. (2006). Study of domain boundary polarization in (111)-cut [Pb(Mg1/3 Nb2/3 )O3 ]0.7 (PbTiO3 )0.3 single crystal by piezoresponse force microscopy. Applied Physics Letters 89 (9): 092906. Xie, S., Gannepalli, A., Li, J. et al. (2012). High resolution quantitative piezoresponse force microscopy of BiFeO3 nanofibers with dramatically enhanced sensitivity. Nanoscale 4 (2): 408–413. Available at: http://xlink.rsc.org/?DOI=C1NR11099C. Yau, H.M., Xi, Z., Dai, J.Y. et al. (2017). Dynamic strain-induced giant electroresistance and erasing effect in ultrathin ferroelectric tunnel-junction memory. Physical Review B 95: 214304. Zhao, X. (2005). Ferroelectric Domain Study by Piezoresponse Force Microscopy. Hong Kong Polytechnic University.

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5 Recent Advances in Ferroelectric Research There are very broad applications for ferroelectric materials, especially thin films, and the research interest in ferroelectrics is showing accelerated increase in the twenty-first century. One of the main reasons is the successful demonstration of ferroelectric non-volatile memory in the field effect transistor (FET) structure. Other new developments in ferroelectrics are stimulated by nanotechnology, where the ferroelectricity (FE) in nanometer scale and the corresponding advanced characterization techniques for emerging new materials such as two-dimensional (2D) and topological materials, have been extensively studied. This chapter introduces the most recent progress in ferroelectric research in terms of new materials, new phenomena, and advanced characterization techniques and applications.

5.1 Size Limit of Ferroelectricity Due to the nature of crystal symmetry breaking-induced ferroelectricity, it is expected that when a single grain of ferroelectric crystal approaches the regime of a few nanometers, the distorted crystal lattice, tetragonal, for example, will relax into cubic structure and crystal symmetry breaking will disappear, making the ferroelectricity vanished (Shaw et al. 2000). This is true in a freestanding grain without external stress. Several systematic studies of size effects have been carried out on powders processed to achieve small particles (Jiang et al. 2000; Ishikawa et al. 1996). These studies have focused on the prototypical ferroelectric perovskite PbTiO3 , and suppression of ferroelectricity was found for the small particles with diameters from 10 to 20 nm manifested by the increase of T c compared with the unstrained and stoichiometric bulk value. These data were extrapolated to predict a ferroelectric critical size of a few nanometers at room temperature. However, in a thin film form of ferroelectric material, due to the substrate clamping effect and interfacial lattice mismatch, the film could be retained in a tetragonal structure instead of relaxing to a cubic structure. For example, a BaTiO3 film as thin as 4 unit cells still behaves very good ferroelectric properties due to the substrate clamping effect that retains its ferroelectricity. In Garcia’s work (Garcia et al. 2009), BaTiO3 films were grown with thicknesses ranging Ferroic Materials for Smart Systems: From Fundamentals to Device Applications, First Edition. Jiyan Dai. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

5 Recent Advances in Ferroelectric Research 2

180 Phase (°)

Height (nm)

96

0

0 (a)

(b)

(c)

Figure 5.1 Demonstration of ferroelectricity for a 1-nm BaTiO3 film grown on top of a 30 nm LSMO electrode. (a) Topography (636 mm2 ) and (b) PFM out-of-plane (OOP) phase image after writing eight alternating positive (13.5 V) and negative (23.5 V) 430.5-mm2 voltage stripes. (c) A 232-mm2 dark square is written on top of the pattern with 13.5 V, and a 131-mm2 white square (23.5 V) is subsequently written on top of that square. Source: Garcia et al. (2009). Reprinted with permission of Springer Nature.

between 1 and 50 nm on NdGaO3 substrate with a 30-nm-thick La0.67 Sr0.33 MnO3 (LSMO) buffer. It demonstrated the ferroelectricity of BaTiO3 by piezoresponse force microscopy (PFM) even in 1-nm thick (see Figure 5.1). The retained ferroelectricity at a few atomic layer thickness is due to the substrate clamping effect forcing the BaTiO3 film into a tetragonal structure or otherwise should be a cubic structure at this thickness. Fong et al. (2004) have investigated this hypothesis by using X-ray diffraction to record the structure of PbTiO3 thin films as a function of temperature over a range of different film thicknesses. Their method allows them to distinguish between the centrosymmetric structure, in which there is no net polarization, and the off-centered ferroelectric phase. They observed that the ferroelectric state persists down to a thickness of 12 Å, which corresponds to just three unit cells of PbTiO3 . The retaining of ferroelectricity in this size is attributed to the formation of domain structure and interfacial charge to screen the depolarization field. Otherwise, the thickness limit for ferroelectricity to exist may be doubled as predicted by first-principle calculation. Retaining ferroelectricity beyond 5-nm thick is essential to realize ferroelectric tunneling device application. Therefore, the study of size limitation is a guideline for ferroelectric application, especially in thin films and nanodevices.

5.2 Ferroelectricity in Emerging 2D Materials As one of the emerging group of new materials, 2D materials exhibit wide range of symmetry-breaking quantum phenomena such as crystalline order, superconductivity, magnetism, and charge density wave, which persist in the limit of a single-unit cell thickness. However, compared with these well-studied properties, ferroelectricity in 2D materials is less explored. The reason is that, compared with those bulk ferroelectric materials, the ferroelectricity in the ultrathin films is much weaker, and therefore the observation of ferroelectric properties is challenging. One should know that usually only monolayer or a few layers of the 2D

5.2 Ferroelectricity in Emerging 2D Materials

materials are ferroelectric; while their bulk is not, since multilayers average out each layer’s electric polarization. Since the first prediction of 2D Van der Waals ferroelectric (Wu et al. 2013), theoretical studies have proposed that 2D ferroelectrics may appear in distorted monolayer MoS2 (Shirodkar and Waghmare 2014), buckled honeycomb AB monolayer (DiSante et al. 2015), phosphorene analogues (GeS, GeSe, SnS, and SnSe) (Wu and Zeng 2016), hydrogenated carbon nitride (Tu et al. 2017), and chemically functionalized 2D materials (Wu et al. 2016). For instance, first-principle calculations predict that the phosphorene analogue (GeS, GeSe, SnS, and SnSe) monolayers are multiferroic with coupled ferroelectricity and ferroelasticity. The structure of SnS and SnSe monolayers are non-centrosymmetric, and the parallel zigzag chains with dipole moments along the armchair direction may lead to a spontaneous polarization. If the polarization is switchable, these materials can be classified as intrinsic ferroelectrics. Let’s have a look at SnSe or SnS as an example to understand its ferroelectric nature. Figure 5.2 illustrates ferroelectric switching in SnS/SnSe monolayer. One can see that when an electric field is applied to the right, the Sn atoms with positive charge will move toward the right from state I; while the S/Se atoms with negative charge will move toward the left, then reach an intermediate state II with a square lattice and finally get to the final state III with the polarization switched. Therefore, this SnSe (or SnS) is intrinsically ferroelectric. It is interesting to note that SnS/SnSe adopts a layered orthorhombic structure (space group Pnma) at room temperature in state I originally. However, when the field is applied, the structure converts to the nonpolar square structure (Cmcm phase) with the inversion symmetry (intermediate state II) similar to the high temperature condition (Li et al. 2015; Zhao et al. 2014), then back to the Pnma phase. I

ll

Ill P

Sn S/Se P

P P

y x

E-field direction

Figure 5.2 Illustrations of (top) top view and (bottom) side view along y direction of the structure showing the pathway of ferroelectric switching for SnS/SnSe monolayer. Black arrows denote external electric fields in different directions.

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At the same time, the spontaneous in-plane polarization in phosphorene analogues MXs (GeS, GeSe, SnS, and SnSe) was also identified through first-principle calculations in another work (Fei et al. 2016). Furthermore, an effective Hamiltonian and parameters were used to investigate the phase transition via Monte Carlo simulations. The calculated Curie temperatures of the monolayer GeS and GeSe are rather high, while the Curie temperature of SnS and SnSe are low (T c_SnS = 1200 K and T c_SnSe = 326 K). Beyond monolayers, because of the restored inversion symmetry, the polarization of even-number-layered MXs is always zero; while the odd-number-layered MXs show spontaneous polarization. Interestingly, although polarization of the odd-number-layered MXs decays with the increasing thickness, the Ps is still comparable to ferroelectric BaTiO3 nanostructures (see Figure 5.3). Thus, it is possible to observe the ferroelectricity in currently available few-layer chalcogenides. However, until 2016, 2D ferroelectricity was finally realized experimentally in atomic-thick SnTe. The scanning tunneling microscope (STM) method was employed to probe the ferroelectricity of 1-unit cell SnTe film at the liquid helium temperature (Chang et al. 2016). Except of the observed domain formation, lattice distortion, and band-bending, the polarization manipulation by electric field was performed, and the results strongly support the occurrence of ferroelectricity in the 1-unit cell SnTe film (see Figure 5.4).

Polarization (10–10 C/m)

2.0 Armchair direction Eq. (2) fit Zigzag direction

1.5

1.0

0.5

0.0

100

(a)

200 300 400 Temperature (K)

1.0 Polarization (C/m2)

98

GeS GeSe SnS SnSe

0.8 0.6 0.4 0.2 0.0

(b)

500

1

3 5 MX layers

7

Figure 5.3 (a) Temperature dependence of polarization obtained from Monte Carlo simulations of monolayer SnSe and (b) polarization at zero temperature versus the layer number of MXs. Source: Fei et al. (2016). Reprinted with permission of American Physical Society.

5.3 Ferroelectric Vortex

(a)

(b)

Domain wall

Figure 5.4 (a) The stripe domain of a 1 unit cell SnTe film and (b) schematic of the lattice distortion and atom displacement in the ferroelectric phase. Source: Adapted from Chang et al. (2016).

In bulk form, the highest ferroelectric transition temperature T c of SnTe is only 98 K, owing to the screening effect of charge carriers. However, when the thickness of SnTe film decreases, the T c is strongly enhanced, which is near 270 K for 1-unit cell SnTe. The strong ferroelectricity enhancement in SnTe films can be attributed to the lower density of defects and larger energy bandgap, and the in-plane lattice expansion as well.

5.3 Ferroelectric Vortex To answer the question whether ferroelectric phase transitions still occur in low-dimensional structures, Naumov and Fu (Naumov et al. 2004) performed ab initio studies on ferroelectric nanoscale discs and rods of PZT and demonstrated the existence of previously unknown phase transitions in zero-dimensional ferroelectric nanoparticles. It is found that phase transitions do exist in zero-dimensional ferroelectric nanostructures, and the minimum diameter of the discs that display low-temperature structural bistability is determined to be 3.2 nm. The phase transitions are very different from those occurring in bulk material; below the critical temperature, they actually lead to the formation of spontaneous vortices with nonzero local toroid moment rather than spontaneous polarization. Vortex as a topological entity exists in nature, from the shape of a flower to galaxy, for example, as shown in Figure 5.5. In microscopic world, the vortex domain structure, characterized by the continuous rotation of ferroelectric polarization or magnetic spin, was firstly predicted more than 70 years ago. Landau and Lifshitz (1935) and Kittel (1946) showed that formation of circular domains was likely in ferromagnetic nanodots due to the surface boundary conditions. In 1979, their 2D structure in terms of winding numbers was investigated and then extended to three-dimensional vortex domains (Mermin 1979; Williams and Wright 1998). The toroidal domains can arise for two unrelated physical reasons: finite size effects and boundary conditions.

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(a)

(b)

(c)

(d)

Figure 5.5 Vortex structures in nature from flower (a) to snail (b), tornado (c), and galaxy (d).

The vortex domains are well studied in nanoferromagnets, but rarely reported in ferroelectrics experimentally since it was extended to ferroelectrics around 1984 (Ginzburg et al. 1984) and predicted by Naumov and Fu (Naumov et al. 2004) from ab initio studies. Compared with magnetic vortex domain, the ferroelectric vortex is more difficult to be formed due to its metastable state and is also technologically difficult to be observed. Gruverman et al. (2008) reported the direct observation of a doughnut-shaped domain pattern appearing during polarization reversal in micrometer-sized ferroelectric capacitors (see Figure 5.6). Domain imaging of the PZT disc was performed through the top electrode of circular 1-μm diameter and square 2.5-μm edge by applying a reading oscillating bias on a 100 ns scale. For the circular capacitors, an unswitched domain remains in the center forming a vortex domain pattern. However, the vortex structure remains only for >1 second, after that the unswitched central domain collapses to leave a uniformly polarized ground state. Though it is difficult to make unambiguous detection of the in-plane polarization, a 2D map of the in-plane polarization distribution Pxy shows that the vortex structure is consistent with the pattern as predicted by simulation, which is supported by the model of Naumov and Fu. Recently, Lane Martin and coworkers (Damodaran et al. 2017; Yadav et al. 2016) explored the coexistence of polarization vortices with other forms of polar 1 μs

3 μs

Pxy

Figure 5.6 PFM images of instantaneous domain configurations with out-of-plane polarization component Pz developing at different stages, (a) 1 μs and (b) 3 μs, of polarization reversal in 1-μm-diameter circular capacitor. (c) two-dimensional maps of polarization components Pxy corresponding to an instantaneous domain pattern in the 1-μm-diameter circular capacitor during switching. Source: Adapted from Gruverman et al. (2008).

5.3 Ferroelectric Vortex

A single vortex–antivortex pair

SrTiO3 PbTiO3 5 nm

The curl of the polar displacement

[001]pc [100]pc

Cross-sectional high-resolution STEM image

Phase-field simulated polarization vectors

Figure 5.7 Exploring the phase boundary between (SrTiO3 )10 /(PbTiO3 )10 superlattice and vortex–antivortex phases in each PbTiO3 layer. Source: Yadav et al. (2016). Reprinted with permission of Springer Nature.

ordering in finely layered perovskite oxides. They demonstrated how to arrange different regions in an easily controllable and precise fashion throughout a system where different orderings are in very close competition with each other like the PbTiO3 /SrTiO3 superlattice system (see Figure 5.7). Because of the competition between the tendency of the oxygen octahedral to rotate and the off-centering of the ions that leads to polarization in a thin layered superlattice of PbTiO3 and SrTiO3 , a hybrid improper ferroelectricity was produced. The up and down polarization domains tend to have sharp boundary polarization transitions, which are described as stripe domains. However, as the DyScO3 substrates have a larger lattice spacing, when this lattice spacing is imposed on the superlattice, it is easy for the polarization to be in-plane. Under this strain condition, with tuned layer thickness, highly ordered vortex-like domains form. When a field is applied in the same direction with both polarization orders, it is found that it switches the film into a pure vortex state using a conductive atomic force microscope (AFM) tip, while the opposite direction should maintain the vortex state. These unique features make the vortex ferroelectricity a potential for interesting applications in memory technology if one can control these responses in a local and non-volatile way. More recently, skyrmion in ferroelectrics or ferroelectric controlled skyrmion in ferromagnetic systems is becoming another hot topic of research in ferroelectric community. First proposed by Skyrme in the 1960s (Skyrme 1962), the localized, particle-like configurations are topologically protected that they cannot be changed by a continuous deformation of the field configuration unless very strong perturbations to the system. The configurations in this model are called skyrmions, and the topologically protected skyrmions can be stabilized in chiral magnets in the form of stable spin textures, referred to as magnetic skyrmions.

101

102

5 Recent Advances in Ferroelectric Research

(a)

(b)

Figure 5.8 (a) Schematic configurations of a single chiral skyrmion and (b) radial skyrmion with the corresponding cross-section along the dashed line. The arrows indicate the direction of the spins, and their colors represent the normal component to the plane, that is, from up direction (red) to the down direction (blue). Source: Kezsmarki et al. (2015). Reprinted with permission of Springer Nature.

y=0

m=1

y=π

y = –π/2

y = π/2

ny

nx m = –1

Figure 5.9 Skyrmion structures with varying vorticity m and helicity 𝛾. The arrows indicate the direction of the in-plane order parameter component, and the brightness indicates the normal component to the plane, with white denoting the up direction and black the down direction. Source: Nagaosa and Tokura (2013). Reprinted with permission of Springer Nature.

The typical spin textures corresponding to a single skyrmion and radial skyrmion are shown in Figure 5.8a and Figure 5.8b, respectively, each of them shows a vector field that all the vectors point along a given direction at the external surface of the skyrmion. Figure 5.9 shows a few patterns of the different possible tangent vector fields in 2D xy-plane. Skyrmion structure can be indexed by vorticity m and helicity 𝛾, where m stands for vorticity and measure how much the pattern rotates and 𝛾 indicates the initial angle that the vector field makes with the x-axis. With this definition, the skyrmion in Figure 5.8a can be indexed as m = 1 and 𝛾 = 𝜋/2, and the one shown in Figure 5.8b corresponds to m = 1 and 𝛾 = 𝜋. By integrating ferroelectricity to magnetic skyrmion system, ferroelectrically controlled magnetic skyrmions in ultrathin oxide heterostructures may be formed. As reported by Wang et al. (2018), ferroelectric control of skyrmion

5.3 Ferroelectric Vortex

1.0

ρAHE + ρTHE

0.4

0.5

0.0

0.0

–0.2

M (μB/Ru)

ρxy – R0H (μΩ cm)

0.2

–0.5 ρAHE (M)

–0.4

ρTHE

B20S4 –4

–2

0

2

–1.0 4

μ0H (T)

Figure 5.10 Ordinary 𝜌xy , anomalous 𝜌AHE and topological Hall resistivities 𝜌THE versus H for the BTO/SRO bilayer heterostructures. The 𝜌AHE –H curve is derived from the normalized M–H curve. The unit of M is μB/Ru. The H-sweeping directions are marked by solid arrows. The inset is the schematic skyrmion structure. Source: Adapted from Wang et al. (2018).

properties in ultrathin BaTiO3 /SrRuO3 (BTO/SRO) bilayer heterostructures were first observed via PFM and Hall measurements. They first examined the basic skyrmion properties by determining transverse Hall resistivity 𝜌xy . Figure 5.10 shows that the skyrmions are excitations from the ferromagnetic background, and their emergence is assisted by the ferromagnetic domain switching. When magnetic field H is aligned parallel with the core spin orientation of the skyrmion and approaches the coercive field (H C ), topological Hall resistivity (𝜌THE ) increases sharply and peaks at approximately ±1.65 T. Further sweeping H across H C , 𝜌THE starts to decrease gradually and vanishes at the critical field (±3.9 T), at which the M–H hysteresis loop closes. Wang then employed an AFM-based electrical gating approach to realize the FE-mediated controllability. With different fraction of upward-poling, the relationship between polarized area and skyrmion phase is demonstrated in Figure 5.11. By selectively poling the sample at room temperature (Figure 5.11a), the upward-poled Hall bar exhibits a slight enhancement in 𝜌THE compared with that in the pristine FE state (Figure 5.11b). It should be noted that the skyrmions studied in the earlier works can only be observed at low temperatures in epitaxially grown ultrathin films, and this type of skyrmions has not been directly relevant for technological application. However, based on most recent works, the skyrmions can also be found at room temperatures, which may pave the way for the development of the ultimate small achievable size for non-volatile memory application.

103

5 Recent Advances in Ferroelectric Research (a) 200 + –

–8 V

Phase (°)

B20S4

Vtip = –8 V

P

+8 V

+8 V

–8 V

–8 V

+8 V

0 Room-T domain writing

(b) 0.4 10 K

ρxy

ρxy (μΩ cm)

104

0.2 0.0 –0.2

Hsk

ρTHE + ρAHE ρTHE

–0.4 –4 –2

Low-T hall measurement

(A)

(B)

0

2

4 –4 –2

μ0H (T)

0

2

μ0H (T)

4 –4 –2

0

2

μ0H (T)

4 –4 –2

0

2

4

μ0H (T)

Figure 5.11 (A) Schematic diagram of the experimental setup for FE domain switching and Hall measurements. (B) PFM phase images (a), hall resistivity signals (b) of the BTO/SRO bilayer in different FE poling states. The scale bar in (B) corresponds to 10 μm. The critical H as 𝜌THE decreases to zero is marked as Hsk . Source: Adapted from Wang et al. (2018).

5.4 Molecular Ferroelectrics Molecular materials have attracted much attention due to their advantages of easy modification and fabrication, environmental-friendly and less energy-cost processing, transparent, light weight, mechanical flexibility, and possible multifunctionalities, compared with their counterparts of pure inorganic or metallic ones. Many functionalities and properties including ferroelectricity have been discovered and realized in molecule-based materials (Zhou et al. 2012; Long and Yaghi 2009; Cheetham and Rao 2007; Rao et al. 2008). Molecular ferroelectrics have been proven to be applicable in many device applications such as temperature sensing, data storage, and energy harvesting. The first ferroelectrics, Rochelle salt, which was discovered in 1920, is in fact a molecular material, and later on a few other molecular systems, such as the well-known potassium dihydrogen phosphate and tri-glycine sulfate, were discovered and developed. However, compared with pure inorganic ferroelectrics such as BaTiO3 and PZT, molecular ferroelectrics usually have much lower spontaneous polarization (Ps ) and dielectric constant (𝜖 ′ ), but relative larger dielectric loss (𝜖 ′′ or tan 𝛿) (Lines and Glass 2001; Jona and Shirane 1962). However, these shortcomings for molecular ferroelectrics have been gradually overcome (Horiuchi et al. 2007, 2010, 2012; Horiuchi and Tokura 2008; Fu et al. 2011). For example, an organic ferroelectric, croconic acid has been reported to possess a Ps of 21 μC/cm2 at room temperature, comparable to BaTiO3 (Horiuchi et al. 2010). Very recently, a research team led by Ren-Gen Xiong has reported (Fu et al. 2013) that a molecular ferroelectric crystal of diisopropylammonium bromide salt (DIPAB, Figure 5.12) showed a Ps of 23 μC/cm2 , high T c of 426 K,

5.4 Molecular Ferroelectrics

b

α

b a

β

c a

c

Figure 5.12 The structures of the two phases of DIPAB, α phase at 293 K and the β phase at 438 K, and the phase transition routes. Color scheme: C – black, N – cyan, H – white, Br – golden. Green dashed bonds are N—H· · ·Br hydrogen bonds. Source: Adapted from Fu et al. (2013).

large 𝜖 ′ up to 103 , and low tan 𝛿 of ∼0.4%, all close to or beyond the pure inorganic BaTiO3 , representing a breakthrough in the research of molecular ferroelectrics. Figure 5.12 shows the two phases 𝛼 and 𝛽 of DIPAB crystal. Below the Curie temperature (T c = 426 K), the 𝛼 phase is in the monoclinic chiral space group P21 , which belongs to the polar point group C 2 . When heating the crystals to temperatures above T c , it results in the high temperature 𝛽 phase of a centrosymmetric structure belonging to the monoclinic space group P21 /m in the nonpolar point group C 2h . The 𝛼 to 𝛽 phase transition is reversible, and accompanied to the transition, the 𝛼 crystal displays a prominent dielectric anomaly at T c = 426 K, with the peak 𝜖 ′ as high as 1.6 × 103 at the lowest frequency of 400 Hz (Figure 5.13a). The polar 𝛼 crystal displayed typical ferroelectric hysteresis loops (Figure 5.13b, inset). This result opens a new avenue to the applications of molecular ferroelectrics that are much more environmental friendly, easier to fabricate, lighter weight, and so on, compared with the pure inorganic oxide ferroelectrics.

500

380

0

416 K

418 K

420 K

422 K

10 0

–10

–10

–10

0 E (kV/cm)

10

–20

0 (a)

10

P (μC/cm2)

ε′

1000

20

400 Hz 1 kHz 10 kHz 100 kHz 1 MHz

Ps (μC/cm2)

1500

400

420 T (K)

440

380 (b)

350

400

450

T (K)

Figure 5.13 (a) The temperature dependence of the dielectric constant of the 𝛼-DIPAB crystal under several frequencies. (b) The Ps versus T plots measured by a pyroelectric technique, and inset, the ferroelectric hysteresis loops under several temperatures below T c . Source: Adapted from Fu et al. (2013).

105

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5 Recent Advances in Ferroelectric Research

5.5 Ferroelectricity in HfO2 and ZrO2 Fluorite Oxide Thin Films Perovskite-structure ferroelectrics have been intensively studied among the various ferroelectric materials, and the structural origin of the spontaneous polarization in perovskite materials is well defined as the oxygen octahedron distortion, i.e. the relative change in positions of the B-site cation and O2− ions in a tetragonal ABO3 structure. Recently, ferroelectricity has also been found in fluorite-structure oxide, mainly HfO2 thin films. Fluorite oxide consists of four metal and eight oxygen ions in a unit cell. As shown in Figure 5.14, metal ions are located at eight corners and six face centers, while oxygen ions are at the eight tetrahedral sites. Unlike the perovskite-structure, the ferroelectricity of fluorite-structure oxide has been raised by the distortion of oxygen hexahedron. Figure 5.15 shows the origins of ferroelectricity in perovskite and fluorite structures. As reported by Materlik et al. (2015), in fluorite structure, the oxygen hexahedron with body-centered metal ion can be found in two adjacent unit cells for the cubic phase, and a nonpolar phase is formed due to the centrosymmetry. When the cubic unit cell is elongated along a-axis, the orthorhombic polar phase can be formed, in which four oxygen ions are significantly shifted from the stable position in a cubic phase, and each of them has two stable positions. As a result, a ferroelectric phase is formed. One of the well-known examples of fluorite oxide, hafnium dioxide (HfO2 ), exhibits three different crystal phases at normal pressure: a monoclinic phase (P21 /c, m-phase) at room temperature, a tetragonal phase (P42 /nmc, t-phase) above 2050 K, and finally a cubic phase (Fm3m, c-phase) above 2803 K. It is known that, same as ZrO2 , HfO2 exhibits a martensitic transformation to the monoclinic phase when its thin films are cooled down from high temperature Figure 5.14 Illustration of fluorite oxide structure, large and small spheres denote oxygen and metal ions, respectively.

5.5 Ferroelectricity in HfO2 and ZrO2 Fluorite Oxide Thin Films

Nonpolar phase

Polar phase

Oxygen octahedron in perovskite oxide

: Oxygen ion : Metal ion Oxygen hexahedron in fluorite oxide

c

a/b

Figure 5.15 A schematic for the structural origin of the ferroelectricity in perovskite and fluorite ferroelectrics. Source: Adapted from Park et al. (2017).

tetragonal phase. Although t-phase and c-phase can be stabilized at room temperature via doping or nanostructuring, all these structures are centrosymmetric and reported as non-ferroelectric. However, the recent discovery of the possible factor such as doping and crystallization with capping layer, HfO2 structure can be stabilized into non-centrosymmetric orthorhombic phase (Pbc21 ) exhibiting ferroelectricity (see Figure 5.16). In 2011, Böscke et al. firstly observed a non-centrosymmetric orthorhombic phase during the study of silicon doped HfO2 , which exhibited a remnant polarization (Pr ) of ∼10 μC/cm2 at a coercive field of 1 MV/cm (see Figure 5.17). In fact, ferroelectricity (FE) or antiferroelectric (AFE) can be induced in HfO2 by several Figure 5.16 The formation of the orthorhombic phase proceeds by transformation from the tetragonal phase. The right arrow indicates two different polarization states of the ferroelectric phase (large atoms are O and small atoms are Hf ).

P

P T phase, P42/nmc Ferroelectric O phase, Pca21

107

5 Recent Advances in Ferroelectric Research

20 2.6 mol% 3.1 mol% 4.3 mol% 5.6 mol%

15 10 Polarization (μC/cm2)

108

5 0 –5 –10 –15 –20 –2

–1

0

1

2

Field (MV/cm)

Figure 5.17 Polarization measurement of metal–insulator–metal capacitor samples with different SiO2 admixture. A gradual transition from ferroelectric to antiferroelectric polarization curves is observed with increasing silicon content. Source: Adapted from Böscke et al. (2011).

Table 5.1 The ferroelectric properties of HfO2 films with various dopants (O: observable, X: not observable). Dopant (valence)

Theoretical Pr (𝛍C/cm2 )

Experimental Pr (𝛍C/cm2 )

E c (MV/cm)

Doping concentration (%)

AFE

Si (4+)

41

24

0.8–1.0

4

O

Zr (4+)

50

34

5

50

X

Y (3+)

40

24

1.2–1.5

5.2

X

Al (3+)

48

16

1.3

4.8

O

Gd (3+)

53

12

1.75

2

X

Sr (2+)

48

23

2.0

9.9

X

La (3+)

N/A

45

1.2

N/A

N/A

Source: Adapted from Park et al. (2015) and Wei et al. (2018).

dopants (Y, Gd, Al, Sr, La, Mg, Ba, Nd, Sm, Er, Ga, In, Co, N, and Ni), especially for ZrO2 -doped HfO2 . As shown in Table 5.1, the largest reported value of Pr is 45 μC/cm2 , observed in La-doped HfO2 film. On the other hand, researchers have succeeded to suppress the monoclinic phase in sub-10 nm HfO2 by controlling the oxygen fraction in deposition ambient, and a Pr of 13.5 μC/cm2 in a 6.9-nm-thick dopant-free HfO2 film has been observed by Ashish Pal et al. (2017).

5.5 Ferroelectricity in HfO2 and ZrO2 Fluorite Oxide Thin Films

HfO2 has a large bandgap (Eg > 5 eV), a larger remnant polarization (Pr ∼ 45 μC/cm2 ) and coercive field (∼1–2 MV/cm) than conventional ferroelectric films. The recent discovery of ferroelectricity in HfO2 , which only appears at the nanoscale and becomes stronger at smaller dimensions, makes it able to retain ferroelectricity for extremely thin films ( 20 nm), the m-phase crystal structure appears as the predominant phase. Thus, the effective electric field applied to ferroelectric phase is decreased as a consequence of the increase amount of non-ferroelectric phase. It can therefore explain the generally decreasing polarization with the increasing thickness. Since the scaling of the physical thickness of ferroelectric thin film is the fundamental issue for FeRAM fabrication, the thickness of the conventional perovskite-structure ferroelectrics is limited to achieve robust ferroelectricity. The remnant polarization of perovskite oxide is not greater than that of fluorite oxide, and the difficulty of depositing perovskite-structure ferroelectrics with two or more cation components in high-aspect-ratio structure via atomic layer deposition (ALD) is a critical issue. This makes fluorite-structure ferroelectric able to replace conventional ferroelectrics for memory applications. However, in 2013, Zhou et al. discovered a critical problem, the so-called “wake-up effect,” in Si-doped HfO2 ferroelectric film. This effect refers to the increase in remnant polarization with increasing cycling number of switching field. Unlike conventional ferroelectrics such as BaTiO3 and PZT, the ferroelectricity of HfO2 relies on large number of switching cycles, making it difficult for application. Figure 5.19 shows the general switching I–E curves and remnant polarization change of ferroelectric HfO2 -based thin films with three different states of field cycling. From the graph, one can see that the three states, starting Wake-up

Fatigue

Pr

Wake-up

Pr,rel

Fatigue

Current (a.u.)

Polarization (a.u.)

Pr /Pr,relax (a.u.)

110

Pristine

Electric fied (a.u.)

10–1 100 (a)

101

102 103 Cycles

104

105

106

107

–4 (b)

–2

0

2

4

Electric fied (MV/cm)

Figure 5.19 (a) Evolution of the ferroelectric remnant polarization during bipolar cycling (4 V at 10 kHz) measured on Sr:HfO2 -based ferroelectric capacitors. (b) Current–voltage characteristics for three states, pristine, wake-up, and fatigue. Source: Park et al. (2018). Reprinted with permission of Cambridge University Press.

5.5 Ferroelectricity in HfO2 and ZrO2 Fluorite Oxide Thin Films

from pristine, wake-up to fatigue, can be observed during the cycling. The remnant polarization as a function of cycles shows two distinct trends as described by Peši´c et al. (2016): i. “Wake-up” – increases with cycling. This increase of the Pr value corresponds to an opening or de-pinching of the pristine pinches hysteresis loop with field cycling. ii. “Fatigue” – the Pr value decreases resulting in a closure of the difference between positive and negative remnant polarizations after a certain number of cycles due to aging mechanisms. As shown in the current–voltage characteristics curve, the switching current curve has double peaks in both positive and negative bias regions in pristine state and, consequently, forms a pinched hysteresis loop. It is different from the ideal switching current curve of a ferroelectric, where only a single peak is seen at each region. However, the switching current peaks move toward each other with continuous bipolar cycling and finally merge together in the wake-up regime. When cycling further increases, the switching polarization decreases, which corresponds to the typically observed fatigue in many ferroelectrics. According to the discussion of Peši´c et al. (2016) and Schenk et al. (2014), the behavior of splitting of switching current peaks in the pristine state may come from different built-in field within the film. The inhomogeneous internal electric field distribution can originate from polycrystallinity of the film and the unevenly distributed charged defects, such as oxygen vacancies. A high oxygen vacancy concentration is expected to be present near the electrodes due to the reduction of the doped HfO2 layer by the metal nitride electrodes, whose distribution may be mostly asymmetric. Therefore, the resulting asymmetric distribution of the oxygen vacancies can be an origin of the internal field in the pristine material. When it is in cycling process, the oxygen vacancies at the interface may diffuse into the bulk regions of the ferroelectric-HfO2 -based films, and their distribution becomes more homogeneous, in which the wake-up process will occur, accompanied by the merging of the switching current peaks. Another mechanism of the wake-up effect is discussed as the field cyclinginduced phase transition. From the observation of a decrease in the dielectric constant and an increase in remnant polarization with an increasing number of cycling, a phase transition from a non-ferroelectric phase to a ferroelectric phase was suggested in 2015 (Lomenzo et al. 2015). Later on, many experimental results show both direct and indirect evidences suggesting a phase transition to the ferroelectric o-phase from non-ferroelectric phase, either t-phase or m-phase. In Figure 5.20, the phase evolution trend shows the schematic transition to o-phase in pristine, wake-up, and fatigue states, which indicates that oxygen vacancy distribution as well as the phase transition plays a significant role in fluorite-structure ferroelectric wake-up process. Owing to the industrial-compatible fabrication process, most of the ALD-grown HfO2 films are polycrystalline and contain multiple phases. That is the reason why a wake-up effect and further treatment have to be done before obtaining the better ferroelectricity. Recently, highly oriented ferroelectric

111

112

5 Recent Advances in Ferroelectric Research

Mix phase

O-p M-p

has

e

Mixed phase

(a)

has

e

TiN TiOxNy O-

ph

Vo

M-

ph

Sr;HfOx

O-

ph

as

e

M-

ph

as

e

as e

as

e

TiO2 TiN

(b)

(c)

Figure 5.20 Phase evolution model of the three different stages of the ferroelectric capacitance lifetime. (a) Prestine, (b) wake-up, and (c) fatigue. Source: Adapted from Peši´c et al. (2016).

Hf0.5 Zr0.5 O2 thin films are deposited on (001)-oriented LSMO/STO substrate using pulsed-laser deposition (Wei et al. 2018), and without wake-up cycles, the film under epitaxial compressive strain and predominantly (111)-oriented displays large ferroelectric polarization values up to 34 μC/cm2 . Also, it is curious that Wei et al. discovered a new phase with ferroelectricity besides conventional phase, i.e. rhombohedral polymorph instead of orthorhombic phase. Their results proposed that pulsed laser deposition (PLD)-grown thin film at high temperature enables the in situ crystallization of Hf0.5 Zr0.5 O2 (HZO). Due to the favorable epitaxial relationships induced by the STO/LSMO stack, the growing crystallites are subjected to a large epitaxial compressive strain that elongates the cubic unit cell along the out-of-plane [111] direction, inducing rhombohedral symmetry with a polar unit cell (see Figure 5.21). Our group has been working on HfO2 thin films for high-k gate dielectric application for many years, and we have first reported the successful growth of epitaxial HfO2 on Si and have done pioneer work in doping Al into HfO2 forming amorphous HfAlOx film (Dai et al. 2003; Lee et al. 2003). Our recent work also demonstrates the structural characterization and ferroelectric properties of HfO2 film on mica. Mica is emerging as a good candidate for flexible device. Compared with organic polymers, mica holds the advantages of high stability and sustainable for high temperatures, making it very suitable for functional oxide growth that are usually grown at temperatures higher than 500 ∘ C. With a suitable oxide electrode epitaxially grown on mica, high-quality HfO2 film (even epitaxy) can be grown. Our recent work also demonstrates the structural characterization and ferroelectric properties of Al-doped HfO2 film on different oriented SrTiO3 . As a preliminary trial, we have grown HfAlOx film by PLD deposited (La,Sr)MnO3 film on STO substrate. Figure 5.22 shows the HRTEM image of PLD-grown doped HfO2 /LSMO on STO (111) substrate and its corresponding PFM amplitude and phase hysteresis loops. For the ferroelectric property characterization, the piezoresponse (the out-of-plane amplitude and phase hysteresis loops) of 35 nm thick Al doped HfO2 on LSMO/STO (111) can be clearly seen. From this result, coercive voltages of ±4 V can be obtained at the minima of the amplitude signal.

5.5 Ferroelectricity in HfO2 and ZrO2 Fluorite Oxide Thin Films

[111]

[111]

[111]

[11-1]

[11-1] [11-1]

LSMO (φ = 15°)

2 nm (a)

b c b a

[111]

[11-1]

[-111] ~71° 120°

[1-11]

b

a

(b)

c

(c)

Figure 5.21 (a) Cross-sectional high-angle annular dark field-scanning transmission electron microscopy (HAADF-STEM) image of a 4-nm-thick film, observed along the zone [111], (b) sketch of the proposed rhombohedral structure of the HZO film with polarization along the [111] direction, and (c) two views of the R3m phase obtained for epitaxially compressed HZO and HfO2 . Green (cyan) spheres represent Hf/Zr (O) atoms. The arrows show the polar distortion with respect to a reference paraelectric structure with appropriately R3m state. Source: Wei et al. (2018). Reprinted with permission of Springer Nature.

AI:HfO2

(a)

90

250 200 0

150 100 50

LSMO 10 nm

OFF-field

0

[111]

(b)

PFM phase (°)

Column-like

PFM amplitude (a.u.)

300

–90 –20

–10

0 Voltage (V)

10

20

Figure 5.22 (a) TEM image of Al:HfO2 /LSMO and (b) PFM amplitude and phase hysteresis loops from doped HfO2 on STO (111) substrate.

113

114

5 Recent Advances in Ferroelectric Research

5.6 Ferroic Properties in Hybrid Perovskites Recent studies have shown that hybrid perovskites have the property of spontaneous electric polarization like many inorganic perovskites. For an oxide perovskite ABO3 , the spontaneous electric polarization, i.e. ferroelectricity, comes from the broken of crystal inversion symmetry, where the B cation shifts slightly from the geometric center of the BO6 octahedron (Glazer 1972). When the A point is taken by polar organic cations, the central-symmetry breaking is further strengthened because of the organic cations’ intrinsic asymmetric atomic structure. According to the Berry phase calculation within the modern theory of polarization (Resta 1994; King-Smith and Vanderbilt 1993), J.M. Frost et al. (2014a) reported the magnitude of the bulk polarization of four hybrid lead halide perovskite, which is shown in Table 5.2. For CH3 NH3 PbI3 , the most widely used material to fabricate hybrid perovskite solar cells, 38 μC/cm2 of electric polarization has been predicted, which is in the same level of the ferroelectric oxide perovskite like KNbO3 (30 μC/cm2 ) (Dall’Olio and Dovesi 1997). This group also reported a temperature and electric field-dependent rich domain structure of twinned molecular dipoles in the hybrid perovskite by classical Monte Carlo simulation. They propose that the internal electrical fields associated with microscopic polarization domains contribute to hysteretic anomalies in the current–voltage response of hybrid organic–inorganic perovskite solar cells (Frost et al. 2014b). The reason why the hybrid perovskite solar cells have such unexpected good performance is not completely understood by scientists today, but a recently proposed theory claimed that the photovoltaic operation is largely enhanced by the strong polarization in the following three ways. Firstly, the high local polarization-induced electric field at the domain walls makes the exciton separation more efficiently (Yang et al. 2010). Secondly, the long electron and hole diffusion length results in a low possibility of recombination (Stranks et al. 2013). Thirdly, the open circuit voltage of devices is larger than the bandgap of the material because of the built-in potential. As the hybrid perovskite solar cells are being developed rapidly, the idea to fabricate efficient solar cells based on ferroelectric oxides is also demonstrated Table 5.2 Calculated properties of four hybrid lead halide perovskites from density functional theory.

Cation

Pseudo-cubic lattice constant a (Å)

NH4

6.21

8

CH3 NH3

6.29

38

CF3 NH3

6.35

48

NH2 CHNH2

6.34

63

Lattice electronic polarization 𝚫P (𝛍C/cm2 )

Source: Frost et al. (2014a). Reproduced with permission of American Chemical Society.

5.6 Ferroic Properties in Hybrid Perovskites

500

200

–50

150

300 100 200 50

100

200 nm

–100

0

0

(a)

(b)

Phase (°)

0

Amplitude (a.u.)

400 50

–4

–3

–2

–1

0

1

2

3

4

Voltage (V)

Figure 5.23 (a) Surface morphology of CH3 NH3 PbI3 perovskite crystal showing strip-like domain structure and (b) piezoelectric butterfly loop of CH3 NH3 PbI3 perovskite crystal (J.Y. Dai 2013 unpublished work).

by Grinberg et al. (2013). Our group has been studying the existence of ferroelectricity in CH3 NH3 PbI3 single crystal for many years. Figure 5.23 shows the AFM/PFM results of CH3 NH3 PbI3 single crystal where the surface presents domain structure and PFM strain-bias butterfly loop. These results suggest that CH3 NH3 PbI3 single crystal may have ferroelectric property. Our group has tried to combine ferroelectric material into this perovskite solar cell structure as shown in Figure 5.24. We proposed structures utilizing ferroelectric nanodots embedded in the CH3 NH3 PbI3 perovskite or a ferroelectric network structure at the bottom and top electrodes. It is expected that polarized

bl 3

C

G la

O

×10 000

G la ss

IT

P

5.0 kV

ss

IT

O

O

O Zn

3P

H

3N

H

H

Polarized discrete ferroelectric layer

Zn

3N

-O iro

P

C

Sp

P Me -O iro Sp

Au

3P

D TA

H

M

Au

eT

bl 3

AD

Polarized discrete ferroelectric layer Polarized discrete ferroelectric layer

(b)

3P

M

H

-O

O

ss G

la

Ferroelectric nanodots

IT

P

Zn

O

C

H

3N

iro Sp

Au

eT

bl 3

AD

(a)

SEI

(c)

1 μm

(d)

Figure 5.24 (a) A discrete ferroelectric layer in between electron-collection layer and ITO electrode, (b) double ferroelectric layers at cathode and anode, (c) ferroelectric nanodots embedded 0–3 nanocomposite perovskite solar cell structure, and (d) hexagonal nanocells of PZT discrete film on ITO/glass made by sol–gel method followed by nano-patterning technique. The conductivity of the ITO layer underneath has negligible degradation during the PZT synthesis process (J.Y. Dai and H.J. Fang 2014 unpublished).

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5 Recent Advances in Ferroelectric Research

ferroelectrics may further prevent the recombination of electron and holes generated by sunlight in the CH3 NH3 PbI3 perovskite. However, it is difficult to experimentally increase the efficiency benefited from ferroelectric composite. Alternating polar and nonpolar structures in ferroic domains resolves a key puzzle of CH3 NH3 PbI3 that they exhibit strong piezoresponse in polar domains. Yet, the domains cannot be switched by an electric field applied either locally through a SPM tip or globally through external electrodes (Strelcov et al. 2017). However, most recently, Huang’s group believe this is precisely the consequence of alternating polar and nonpolar domains, since polar domains that are normally switchable by an electric field are now locked by nonpolar ones that sandwich them (Huang et al. 2018). Figure 5.25 shows domains structures of CH3 NH3 PbI3 crystals revealed by SEM, AFM topography, polarized optical microscopy, and PFM. It is apparent that they all exhibit characteristic lamellar patterns that have been often seen in ferroic materials. Striking PFM amplitude mapping acquired through single frequency scanning is seen in Figure 5.25d, and the alternating domains exhibit large contrast in piezoresponse up to one order of magnitude difference. This

pm 250

0.0

2 μm

2 μm

–250

(a)

(b) a.u. 545

310

2 μm

4 μm

75.0

(c)

(d)

Figure 5.25 Ferroic domain patterns of CH3 NH3 PbI3 crystals revealed by (a) SEM, (b) AFM topography, (c) polarized optic microscope, and (d) vertical PFM amplitude. Source: Adapted from Huang et al. (2018).

References

suggests that those alternating domains have their different origins. It is found that polar and nonpolar orders of CH3NH3PbI3 can be distinguished from their distinct lateral piezoresponse, energy dissipation, first and second harmonic electromechanical couplings, and temperature variation, even though their difference in crystalline lattice is very subtle. These findings resolve key questions regarding polar nature and ferroelectricity of CH3NH3PbI3 and its implication on photovoltaics.

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6 Piezoelectric Effect: Basic Theory 6.1 General Introduction to Piezoelectric Effect Piezoelectric effect refers to the generation of electric charges or voltage in response to a mechanical stress or the generation of strain under applied electric field. It has been explained in Chapter 2 that all ferroelectrics are also piezoelectric since the electric field-induced polarization switching and domain wall movement will generate strain. However, one should note that even without polarization switching (which is a must to be ferroelectric), changes of polarization strength under electric field can also induce strain. That is why some piezoelectric materials such as ZnO and AlN, which possess electric polarization due to their non-centrosymmetric crystal structure, are not ferroelectric since their polarizations are not switchable. Piezoelectric effect can be classified into direct and converse piezoelectric effects as illustrated in the diagram (Figure 6.1). Direct piezoelectric effect refers to the generation of charges under stress, while the converse piezoelectric effect refers to the generation of strain under electric field. To design a piezoelectric device, the following parameters should be considered when selecting a suitable piezoelectric material along with its shape, size, orientation, etc. Among many parameters, piezoelectric constant (charge constant and voltage constant), dielectric constant, electromechanical coupling factor, and mechanical quality factor are the most important. Table 6.1 lists the piezoelectric properties of some important piezoelectric materials. Before systematic introduction of all parameters, let us have some general view on how to select a material for device applications. In general, for high-power applications, for example, ultrasonic motor, hard PZT is a good choice: here “hard” means high mechanical quality factor. On the other hand, to pursue a large piezoelectric response with large mechanical strain output, soft PZT is a good choice: here, “soft” means large strain that can be easily induced with relatively lower electric field but the output force is relatively small (its mechanical quality factor is smaller compared with hard piezoelectric materials). While for ultrasound sensor application, detectors made of PVDF may be more sensitive, even its piezoelectric charge constant is much smaller compared with PZT. The reason is that the very large piezoelectric voltage constant of PVDF makes it able to output relatively higher voltage under mechanical Ferroic Materials for Smart Systems: From Fundamentals to Device Applications, First Edition. Jiyan Dai. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

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Applied stress

Piezoelectric plate

Induced strain

+

+

+

+

–

–

–

–

P

Electrodes Direct piezoelectric effects Converse piezoelectric effects

Figure 6.1 Direct and converse piezoelectric effects. Table 6.1 Piezoelectric properties of representative piezoelectric materials.

Parameters

d33 (pC/N) −3

g 33 (10 kt

V m/N)

Quartz

BaTiO3

PZT 4 (hard)

PZT 5H (soft)

PVDF–TrFE copolymer

2.3

190

189

593

33

57.8

12.6

26.1

19.7

380

0.09

0.38

0.51

0.5

0.3

0.33

0.58

0.65

1700

1300

3400

6

500

65

3–10

328

193

kp 𝜖r

5

Qm

>105

T c (∘ C)

120

Source: From Uchino (2009) and Chen (2013).

stress. The following sections will give a detailed description of all parameters in piezoelectric materials.

6.2 Piezoelectric Constant Measurement According to the directions of electric polarization and electric field as well as the vibration mode, the piezoelectric constant and electromechanical coupling factor are expressed by d33 , d31 , d15 , k 33 , k 31 , and k 15 , where the indications 1, 3, and 5 are defined in Figure 6.2. Designing different devices needs to consider different parameters such as dielectric constant, mechanical compliance, etc. To get the whole set of parameters, usually five samples with different vibration modes are needed. For an anisotropic piezoelectric material, its physical constants are related to both the direction of the applied mechanical or electrical force and the directions perpendicular to the applied force. Therefore,

z

3 6 P

5

4 1 x

Figure 6.2 Definition of directions for piezoelectric constants.

2 y

6.2 Piezoelectric Constant Measurement

each constant should have two subscripts indicating the directions of the two related quantities, electric field and strain, for piezoelectricity. The direction of positive polarization is usually made to coincide with the z-axis of a rectangular system of x-, y-, and z-axes (Figure 6.2). Directions of x, y, and z are represented by the subscripts 1, 2, and 3, respectively, and shear about one of these axes is represented by the subscripts 4, 5, and 6, respectively. Piezoelectric constant is an important indicator of a material’s suitability for strain-dependent applications. The following give definition of the most frequently used physical constants and their interrelating for piezoelectric materials, such as piezoelectric charge constant d, the piezoelectric voltage constant g, and the dielectric permittivity 𝜖. 6.2.1

Piezoelectric Charge Constant

The piezoelectric charge constant, dij , is the amount of charges generated per unit area (polarization P generated) of mechanical stress (𝜎) applied to a piezoelectric sample or, alternatively, is the mechanical strain (𝜀) generated to a piezoelectric sample per unit of electric field E applied. As illustrated in Eq. (6.1), the first subscript i indicates the direction of the applied field strength or, alternatively, is the direction of induced polarization when the electric field is zero or constant. The second subscript j is the direction of the induced strain generated or the applied stress in the sample, respectively: ( ( ) ) 𝜕𝜀j 𝜎 𝜕Pi E dij = = (6.1) 𝜕𝜎j 𝜕Ei In Eq. (6.1), the first term corresponds to the direct piezoelectric effect and the second term corresponds to the converse piezoelectric effect. The superscript E indicates a zero or constant electric field, and the superscript 𝜎 indicates a zero or constant stress. Based on the definition of indication and orientation as shown in Figures 6.2 and 6.3, the often used d33 , d31 , and d15 are defined as follows: d33 is the induced polarization (electric displacement) in direction 3 per unit stress applied in direction 3 or induced strain in direction 3 per unit electric Piezoelectric plate

(a)

(b)

(c)

Electrodes

Polarization direction Stress direction

Figure 6.3 Diagram showing the definitions of (a) d33 , (b) d31 , and (c) d15 .

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field applied in direction 3. In this case, the applied electric field or stress should be parallel to direction in which piezoelectric plate is polarized. d31 is similar to the definition of d33 , but for d31 , the applied stress or the induce strain should be along the direction 1, and the measured polarization or the applied E-field should be in direction 3. In this case, the applied electric field or stress should also be parallel to the direction in which piezoelectric plate is polarized. d15 is the induced shear strain in direction 5 per unit electric field applied in direction 1 or the sample’s polarization generated along direction 1 by a shear stress applied in direction 5. In this case, the applied electric field or stress should be perpendicular to direction in which piezoelectric sample is polarized. There are two contributions to the generated polarization Pi , i.e. the applied stress and electric field. With the absence of an applied field, the polarization produced by a stress 𝜎 jk , which is a second-rank tensor, is Pi = dijk 𝜎jk

(6.2)

where dijk is the third-rank tensor of piezoelectric charge constant. The second contribution to induced polarization Pi from the applied electric field is instantaneous dielectric response to the electric field. It can be quantified by Pi = 𝜖0 𝜒ij Ej

(6.3)

where 𝜒 ij denotes the dielectric susceptibility measured at constant stress. By combining these two contributions, the generated polarization to a linear piezoelectric material can be expressed as Pi = dijk 𝜎jk + 𝜖0 𝜒ij Ej

(6.4)

It should be noted that this expression is only right at low field regime where the linear relationship is hold. One should also note that dijk is a third-rank tensor, but by invoking elastic symmetries along with electric symmetries due to poling, the tensor can be reduced to second rank, i.e. what has been introduced as dij , typically d33 , d31 , and d15 . In fact, d31 = d32 ; and d15 = d24 , and all others are zero. For the converse piezoelectric effect, the strain is expressed as E 𝜀ij = Sijkl 𝜎kl + dkij Ek

(6.5)

where SE is the fourth-rank compliance tensor measured at constant field.

6.2.2

Piezoelectric Voltage Constant

The piezoelectric voltage constant, g, is the electric field generated by a piezoelectric material per unit of mechanical stress applied or, alternatively, is the mechanical strain experienced by a piezoelectric material per unit of electric displacement applied. g is important for assessing a material’s suitability for sensing (sensor)

6.2 Piezoelectric Constant Measurement

applications, and the very often used g 33 , g 31 , and g 15 are defined in the same way as Figures 6.2 and 6.3, i.e. ) ) ( ( 𝜕𝜀j 𝜎 𝜕Ei E gij = − = (6.6) 𝜕𝜎j 𝜕Di where the “−” sign is due to the fact that the field built from electric polarization reduces under positive compressive stress. Dielectric permittivity is the linkage between the piezoelectric charge and voltage constants. This can be understood by the relation between voltage and charge, whereas their ratio is capacitance, which is proportional to permittivity. Therefore, the relationship between d and g is g=

d 𝜖0 𝜖r

(6.7)

From the relationship of d and g, we can understand why PVDF is a good choice for sensor applications such as hydrophone and pressure sensor. It is the very low dielectric constant that gives rise to a larger g factor, even its d factor is much smaller than PZT; while PZT is excellent for transducer application due to its large d value, making it able to generate large strain under electric field. 6.2.3

Dielectric Permittivity

The dielectric permittivity, 𝜖 = 𝜖 0 𝜖 r (𝜖 r is relative dielectric constant and 𝜖 0 = 8.85 × 10−12 F/m is vacuum dielectric constant), is the dielectric displacement per unit electric field for a piezoelectric material. Usually, 𝜖 𝜎 is the permittivity at constant stress, and 𝜖 𝜀 is the permittivity at constant strain. Dielectric permittivity is second-rank tensor that can be expressed as 𝜖 ij , where the first subscript i to 𝜖 indicates the direction of the dielectric displacement and the second j is the direction of electric field. 6.2.4

Young’s Modulus (Elastic Stiffness)

Young’s modulus Y is an indicator of the stiffness (elasticity) of a material. Y is determined from the value of the stress applied along an axis to the material divided by the value for the resulting strain along that axis in the range of stress in which Hooke’s law holds: 𝜎ij Yijkl = 𝜀kl 6.2.5

Elastic Compliance

Elastic compliance Sijkl is the strain produced in a material per unit of stress applied and is the reciprocal of the modulus of elasticity (Young’s modulus, Y ). SD is the compliance under a constant electric displacement; SE is the compliance under a constant electric field. The first two subscripts indicate the direction of strain, and the last two is the direction of stress 𝜎: 𝜀ij = Sijkl 𝜎kl

127

128

6 Piezoelectric Effect: Basic Theory

The elastic compliance and Young’s modulus follow the equation Sijkl =

1 Yijkl

For the general anisotropic piezoelectric material, the stress–strain relation for a linear elastic material in terms of matrices can be written as ⎡ 𝜀11 ⎤ ⎡ S1111 S1122 S1133 2S1123 2S1131 2S1112 ⎤ ⎡ 𝜎11 ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 𝜀22 ⎥ ⎢ S2211 S2222 S2233 2S2223 2S2231 2S2212 ⎥ ⎢ 𝜎22 ⎥ ⎢ 𝜀 ⎥ ⎢ S S S 2S3323 2S3331 2S3312 ⎥ ⎢ 𝜎33 ⎥ ⎢ 33 ⎥ = ⎢ 3311 3322 3333 ⎥ ⎢ ⎥ ⎢ 2𝜀23 ⎥ ⎢ 2S2311 2S2322 2S2333 4S2323 4S2331 4S2312 ⎥ ⎢ 𝜎23 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ 2𝜀31 ⎥ ⎢ 2S3111 2S3122 2S3133 4S3123 4S3131 4S3112 ⎥ ⎢ 𝜎31 ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ 2𝜀12 ⎦ ⎣ 2S1211 2S1222 2S1233 4S1223 4S1231 4S1212 ⎦ ⎣ 𝜎12 ⎦ To simplify, the tensor notation can be rearranged based on the indices in Figure 6.2, i.e. 11 → 1; 22 → 2; 33 → 3; 23 → 4; 31 → 5; 12 → 6. The utilization of the structural and electric symmetries along with matrix reindexing then yields the system ε1

E E E S12 S13 0 S11 E E E S12 S22 S13 0 E E E 0 S13 S13 S33

ε2 ε3 0 ε4 ε5 = 0 0 ε6 0 D1 0 D2 d31 D3

0 0 0 0 0 d31

0 0 0 0 0 d33

E S44 0 0 0 d15 0

0 0 0

0 0 0

0 0 0

0 E S44 0 d15 0 0

0 0 E S66 0 0 0

0 d15 0 σ ϵ11 0 0

0 d31 0 d31 0 d33 d15 0 0 0 σ ϵ 11 0

0 0 0 0 0 ϵ σ33

σ1 σ2 σ3 σ4 σ5 σ6 E1 E2 E3

It summarizes the converse and direct piezoelectric effects. In matrix form, the components can be written as 𝜀 = SE 𝜎 + d∗ E D = d𝜎 + 𝜖 𝜎 E where d* denotes the transpose of d. Note: With all the previously mentioned definitions, strain and dielectric constant share the same letter 𝜀 as in most literatures, i.e. 𝜀 is used to represent strain and dielectric constant. For friendly reading, in this book, 𝜀 and 𝜖 denote strain and dielectric constant, respectively. 6.2.6

Electromechanical Coupling Factor

The electromechanical coupling factor k ij is defined as the conversion efficiency between the electrical energy and mechanical energy. The ratio of the stored or converted (mechanical or electrical) energy to the input of the second energy

6.2 Piezoelectric Constant Measurement

(electrical or mechanical) is defined as the square of the coupling coefficient. The subscripts denote the relative direction along which the electrical and mechanical quantities and the kind of motion involved: √ √ mechanical energy stored electrical energy stored k= or k = electrical energy applied mechanical energy applied Depending on the vibration modes of shapes and the directions of electric field and polarization, the electromechanical coupling factors can be defined as k 33 , k 31 , k 15 , k t , and k p . Detailed information about the requirement of testing samples to satisfy these vibration modes will be introduced in later sections. 6.2.6.1

How to Measure Electromechanical Coupling Factor?

A high electromechanical coupling factor k is desirable for efficient energy conversion. In a well-designed system, piezoelectric elements can exhibit k factor over 90%. As mentioned earlier, vibration mode-dependent k ij factors can be defined as illustrated in Figure 6.4, where k 33 , k 31 , and k 15 are defined same as the way of d33 , d31 , and d15 . While k t is for a thin round disc vibrating in thickness direction with electrodes on two faces, k p is also for a thin round disc but vibrating along the radial direction (perpendicular to direction in which piezoelectric material is polarized). The electric field and polarization direction are the same as the case of k p in direction 3 (parallel to the thickness in which piezoelectric material is polarized). There are two different methods to measure the k factors. One is from the measurement of strain under electric field at static or low frequencies where the d factor, mechanical compliance, and dielectric permittivity are involved. The other one is the high-frequency impedance measurement at resonance state.

3

Poling direction k15

k33

kp, kt 2

1

k31

Electrodes

Figure 6.4 Diagram showing the definition (the relationship between the poling and electric field direction, where the electric field direction is parallel to the two electrodes) of k33 , k31 , k15 , kt , and kp.

129

130

6 Piezoelectric Effect: Basic Theory

Under Static or Low Frequency For a rod/bar sample with 33 vibration mode, 2 d33

2 = k33

E 𝜎 S33 𝜖33

For a plate sample with 31 vibration mode, 2 d31

2 = k31

E 𝜎 S11 𝜖33

=1−

D S11 E S11

.

For a plate sample with 15 vibration mode, 2 d15

2 k15 =

E 𝜎 S15 𝜖11

For a disc sample with thickness vibration mode, kt2 =

2 𝜀 d33 𝜖33 D C33

=

2 d33 E 𝜎 S33 𝜖33

For a disc sample with radial vibration mode (Meitzler et al. 1973), kp2 =

2 2 2k31 2d31 2(k P )2 = = , 𝜎 E E 1 − υp 1 + υp + 2(k P )2 𝜖33 (S11 + S12 ) SE

where υp = − S12E as a planar Poisson’s ratio, and a planar radial piezoelectric cou11

pling coefficient k P is given by (k P )2 = p

C11 = p

e31 =

p

(e31 )2 p p 𝜖33 C11

𝜎 ,where 𝜖33 = 𝜖33 − p

2 2d31 E E S11 +S12

,

E S11 E 2 E 2 (S11 ) − (S12 ) d31 E E S11 + S12

High-Frequency Resonance Measurement A more popular way to determine k

factor is through measuring frequency-dependent impedance where the resonance f r (or f m ) and anti-resonance f a (or f n ) states can be identified as shown in Figure 6.5. Here r denotes resonance, and m means minimum; a denotes anti-resonance, and n means maximum. Why is the impedance minimum at its resonance state and maximum at anti-resonance state? This will be explained by equivalent circuit that is introduced in the following section. It should be noticed that the sample must be poled before resonance measurement; otherwise, a piezoelectric material with random polarizations microscopically and zero polarization macroscopically will not generate strain and will not present any resonance peak in impedance spectrum. At higher frequencies, the different k factors at different resonance modes are given: For a rod sample with 33 vibration mode, √ [ ] 𝜋 (fa − fr ) 𝜋 fa tan k33 = 2 fr 2 fa

6.2 Piezoelectric Constant Measurement

For a plate sample with 31 vibration mode, √ ] [ √ 𝜋f 𝜋 (fa −fr ) √ a √ 2 fr tan 2 fr k31 = √ ] [ √ f (f −f ) 1 + 𝜋2 fa tan 𝜋2 a f r r

r

For a plate sample with shear vibration mode, √ [ ] 𝜋 (fa − fr ) 𝜋 fr k15 = tan 2 fa 2 fa For a disc sample with thickness vibration mode, √ [ ] 𝜋 (fa − fr ) 𝜋 fr tan kt = 2 fa 2 fa For a disc sample with radial vibration mode, kp2 =

2(k P )2 1 + υp + 2(k P )2

where 𝜐p = −

E S12 E S11

is the planar Poisson ratio. Nevertheless, in an ultrasonic transducer design, very often, the vibration mode is not any of the previously mentioned situations due to the restriction of dimension, so the k factor is called k eff that is quantified as 2 = keff

fa2 − fr2 fa2

Figure 6.5 Frequency-dependent dielectric permittivity showing resonance and anti-resonance.

Impedance Z

fa (fn)

fr (fm) Frequency f

131

132

6 Piezoelectric Effect: Basic Theory

Cs

Rs

Ls

Figure 6.6 Equivalent electrical circuit of a piezoelectric resonator.

Cp

6.3 Equivalent Circuit One unique characteristic of a piezoelectric material is the presence of electrical resonances and anti-resonances in frequency-dependent impedance spectrum. At frequencies below f r and above f a , the piezoelectric resonator behaves capacitively; however, between these two frequencies, it behaves inductively. At resonance state, large strain and capacitance changes are induced, and the current flow into the sample easily. While at the anti-resonance state, the strains induced in the sample compensate completely and the current cannot easily flow into the sample. For each mechanical resonance in the piezo-element, a resonance/anti-resonance pair will exist in the impedance spectrum. This characteristic piezoelectric impedance’s behavior can be modeled by the following equivalent electrical circuit (see Figure 6.6) representing the piezoelectric sample’s electrical response as an electrical device. This equivalent circuit is commonly referred to as Van Dyke’s model and is recommended by the IEEE Standard on Piezoelectricity. In this circuit, inductance Ls represents the mass; capacitance C s represents compliance; resistance Rs represents the internal friction (or dielectric loss) of the piezoelectric element; and C p is used to represent the capacitance that is formed because of sample’s mechanical molding. This model is only valid near the resonance. This circuit diagram consists of series resonance and parallel resonance, i.e. two resonance frequencies. If the reactance X C produced by capacitance C s is equal (in amplitude) and opposite (in phase) to the reactance X L produced by inductance Ls , the series resonance occurs. Thus, the impedance is the lowest and approximately equal to the resistance Rs during this condition. If the reactance of the series wing becomes inductive and equal to the reactance caused due to capacitance C p , parallel resonance occurs. At this condition, the circuit results in very high impedance. This can be understood by the following equivalent circuit analyses. Now we use a real sample’s impedance spectrum to explain the important meaning of the equivalent circuit. An example impedance-frequency spectrum from a relaxor ferroelectric crystal of PMN-0.3PT is shown in Figure 6.7, where one can see the resonance and anti-resonance peaks as well as 180∘ phase difference between these two states. Let us analyze the equivalent circuits by classifying the impedance into A–E five regimes with their corresponding equivalent circuits shown in Figure 6.8.

90

fa

B 100

E A

C

0

10 fr 1

3.0

3.5

D 4.0

Phase ϕ (º)

Impedance Z (Ω)

6.3 Equivalent Circuit

–90

4.5

5.0

5.5

6.0

Frequency f(MHz)

Figure 6.7 Impedance-frequency spectrum of a poled PMN–PT single crystal with f r and f a .

A and E

C

XL

XL R

R Cs

Rs

Rs

Ls Xc

Xc XL

B

Ls

Rs Cs

D

Rp

YL 1 Rp

R Lp Xc

Cp

Yc where YL = 1 = 1 XL jωL and YC = 1 = jωC XC

Figure 6.8 Simplified equivalent circuit corresponding to the regions A–E in Figure 6.7.

In the equivalent circuit, impedance Z and its phase angle 𝜑 depend upon the reactance values of the components; and we know that reactance X is zero when the circuit element is resistive, positive when the circuit element is inductive, and negative when it is capacitive. Their impedances are quantified: ZR = R ZC =

1 j𝜔C

ZL = j𝜔L where j is the imaginary notation. In the previously mentioned equations, it is apparent that the impedance of an inductor is directly proportional to the frequency, while that of capacitor is inversely proportional to frequency. As a result,

133

134

6 Piezoelectric Effect: Basic Theory

in the series LCR circuit, X C is larger than X L when it is at lower frequencies. Thus, the circuit becomes capacitive that the phase angle becomes negative (i.e. regime A). However, in regime B, the inductive reactance magnitude X L increases as frequency increases, while capacitive reactance magnitude X C decreases with the increase in frequency. At a certain frequency, these two reactances are equal but opposite in sign, so they will cancel each other and only resistance is left. This frequency is called resonance frequency f r at which the total impedance reaches its minimum with Ztotal = R and phase angle 𝜑 = 0. When Z is minimum, i.e. series resonance occurs, j(XLs + XCs ) = 0 1 =0 2𝜋f Cs 1 f = fr = √ 2𝜋 Ls Cs 2𝜋f Ls −

When the frequency further increases, the circuit element becomes inductive that X L is larger than X C . Thus, it becomes inductive and phase angle is positive (i.e. region C). In region D, the piezoelectric material behaves like a parallel LC circuit. As the frequency increases, the parallel circuit will produce a parallel resonance (also called anti-resonance) circuit when the resultant current through the parallel combination is in phase with the applied voltage. At anti-resonance, there is a large circulating current between the inductor and the capacitor due to the energy of the oscillations, and then parallel circuit produces current resonance. Dissimilar to region B, the parallel LC circuit acts like an open circuit rather than short circuit, so the current flow into the circuit is zero at anti-resonance. For a parallel equivalent circuit, admittance Y can better illustrate the situation, which is given as Y =

1 1 − j𝜔Cp + RP j𝜔LP

where at anti-resonance state, the admittances of L and C are the same values but antiphase resulting in a minimum value of total admittance. Thus, the total impedance of this resonance circuit at anti-resonance is at its maximum value creating a circuit condition of high resistance and low current. At f = f a , the impedance of the circuit represents the so-called maximum dynamic impedance, i.e. Zd = Z max . When parallel resonance occurs, i.e. Z is maximum, and the imaginary parts of Y become zero, Im(Y ) = 0 so 1 − 2𝜋f LP = 0 2𝜋f Cp

6.4 Characterization of Piezoelectric Resonator Based on a Resonance Technique

The solution of this equation is f = fa =

1 √ 2𝜋 LP CP

At region E, frequency further increases, the circuit becomes capacitive again, which is similar to that of region A.

6.4 Characterization of Piezoelectric Resonator Based on a Resonance Technique Most parameters of the piezoelectric samples can be evaluated using the frequency spectra of impedance and phase, which are measured by an impedance analyzer (for example, Agilent 4294A). When an ac signal is applied on a poled piezoelectric sample, different vibrational modes are excited depending on the shape of the sample. With the measured resonance and anti-resonance frequencies of specified vibrational modes, most properties of the sample can be evaluated. Five samples with specified shapes and polarization directions, including circular disc, long bar, thin rod, thin square plate, and thin shear plate are usually used in these measurements (Berlincourt et al. 1964). The piezoelectric parameters related to these geometries are listed in Table 6.2. Specific resonant modes are excited in samples with specific shapes. In order to avoid mode coupling and to obtain a pure vibration mode in the impedance and phase-frequency spectra, thickness and lateral dimensions of the samples should have substantial difference. In order to use the equations in the IEEE and IRE standards, assumptions are set for the specific sample listed in the Table 6.3. The following shows the details of the calculation of the material parameters, including the elastic compliance constant SE and SD , elastic stiffness constant Y E and Y D , piezoelectric voltage constant g, piezoelectric charge constant d, electromechanical coupling factor k, and frequency constant N. The parameters with superscript D and E represent conditions of the constant electric displacement and constant electric field, respectively. The constant electric displacement condition corresponds to open circuit (zero current across the electrodes), while the constant electric field corresponds to the short circuit condition (zero voltage across the electrodes). 6.4.1

Length Extensional Mode of a Rod

As illustrated in Figure 6.9, a bar of piezoelectric PZT ceramic with its length along direction 3 (the poling direction), and its end faces electrodes E D , S33 , and k 33 . From the (normal to the direction 3), is used to determine S33 impedance-frequency spectrum, the resonance and anti-resonance frequencies f r and f a can be determined, from which its piezoelectric, mechanical, and electrical parameters can be calculated using [ ] 𝜋 (fa − fr ) 𝜋 fr 2 k33 = tan 2 fa 2 fa

135

Table 6.2 Resonance modes of the five geometries and their related piezoelectric parameters. Piezoelectric modes of vibration and resonance

Shape

Dimensions l – length, w – width, t – thickness, d – diameter

Poling direction

Vibration mode

Resonant mode

Related parameters Piezoelectric

Mechanical

t > 10d

Along t

Thickness extension mode

Thickness resonance

𝜎 kt , 𝜖33

D E E D C33 , C33 , S33 , S33 , Qt

l > 5(t, w)

Along t

Transverse length mode

Length extensional resonance

𝜀 k31 , d31 , g31 , 𝜖33

D E S11 , S11 , Q31

l > 3.5(t, w)

Along l

Thickness shear mode

Thickness shear resonance

𝜎 𝜀 k15 , d15 , g15 , 𝜖11 , 𝜖11

D D E C55 , S55 , S55 , Q15

d > 10t

Along t

Thickness extension mode

Thickness resonance

𝜎 kt , 𝜖33

D E E C33 , C33 , S33 , Qt

d > 10t

Along t

Radial mode

Planar resonance

𝜀 𝜎 kp , 𝜖33 , 𝜖33

E 𝜎 E , S11 , Qp

6.4 Characterization of Piezoelectric Resonator Based on a Resonance Technique

Table 6.3 The assumption for specific samples (l – length, w – width, t – thickness, and D – diameter). Sample

Assumption

Circular disc

d≫t

Square plate

l, w ≫ t

Long bar

l ≫ w, t or t ≫ l, w

Shear plate

l, w ≫ t

Figure 6.9 A long rod with the electric field parallel to its length.

3 Electrodes

Poling direction

t

∼ d

2

1 D S33 =

1 = 2 𝜌(2tfr )2 1 − k33 √ E 𝜎 = k33 S33 𝜖33

E S33 =

d33

1 𝜌(2tfa )2 D S33

g33 =

d33 𝜎 𝜖33

where the dielectric constant 𝜖ij𝜎 can be determined from the capacitance C0𝜎 measured at 1 kHz, i.e. 𝜖 LF (a frequency substantially lower than the lowest resonance frequency of the crystal plate), in which case the measurements yield the dielectric permittivity as constant stress. The equation is given by C0𝜎

=

𝜖ij𝜎 A

t where A is the area of the electrode. Usually, the constant strain or “clamped” relative permittivity 𝜖ij𝜀 can be measured at high frequency (frequencies that are high compared with the principal natural frequencies of the plate but well below any ionic resonances), i.e. 𝜖HF = 𝜖ij𝜀 . C0𝜎 =

𝜖ij𝜀 A t

137

6 Piezoelectric Effect: Basic Theory

In case of length extensional mode, 𝜀 = 𝜖HF = 𝜖33 𝜎 = 𝜖LF = 𝜖33

6.4.2

C0𝜎 t A 𝜀 𝜖33 2 1 − k33

Extensional Vibration Mode of a Long Plate

E To determine the transverse mode constants S11 and k 31 , a piezoelectric long plate sample is used (see Figure 6.11). The lateral extensional mode is excited, and similar to Figure 6.10, the corresponding resonance and anti-resonance frequencies f r and f a can be found, from which electromechanical coupling factor k 31 can be calculated as ] [ 𝜋 fa 𝜋 (fa −fr ) tan 2 fr 2 fr 2 k31 = ] [ (f −f ) 𝜋 fa 1 + 2 f tan 𝜋2 a f r r

r

E can be determined by And its dielectric permittivity and elastic compliance S11 𝜎 𝜖33 = 𝜖LF =

𝜖HF = E S11

C0𝜎 t wl

𝜎 𝜖33

2 1 − k31 1 = 𝜌(2lfr )2

100 1000 000

80 60 40 20

100 000

0 –20

Phase ϕ (º)

Impedance Z (Ω)

138

–40

10 000

–60 –80 –100

1000 100

200 Frequency f(kHz)

Figure 6.10 Electrical impedance and phase angle versus frequency spectra of a PZT piezoelectric ceramic rod.

6.4 Characterization of Piezoelectric Resonator Based on a Resonance Technique

Figure 6.11 A length-expander plate with the electric field perpendicular to its length.

3

Poling direction Electrodes t

w

∼

l 2 1

The piezoelectric charge and voltage constants of 31 mode are √ E 𝜎 𝜖33 d31 = k31 S11 g31 =

6.4.3

d31 𝜎 𝜖33

Thickness Shear Mode of a Thin Plate

Figure 6.12 illustrates the relationship of the electrode surface and polarization direction of a thin shear piezoelectric plate with electric field applied along direction 1. A shear mode resonance can be exited and its resonance and anti-resonance frequencies f r and f a can be found, from which electromechanical coupling factor k 15 can be calculated as [ ] 𝜋 (fa − fr ) 𝜋 fr 2 k15 = tan 2 fa 2 fa Other elastic, piezoelectric, and dielectric constants at this shear mode are given by D C44 = 𝜌(2tfa )2

Figure 6.12 A shear plate with the electric field parallel to its thickness.

1 Electrodes

t

w l 3

2

Poling direction

∼

139

140

6 Piezoelectric Effect: Basic Theory E D 2 C44 = C44 (1 − k15 ) E S44 =

1 E C44

𝜀 𝜖11 = 𝜖HF =

𝜖LF = d15

wl

𝜀 𝜖11

2 1 − k15 √ E 𝜎 = k15 S44 𝜖11

g15 =

6.4.4

C0𝜎 t

d15 𝜎 𝜖11

Thickness Mode of a Thin Disc/Plate

A schematic diagram of the piezoelectric thin plate is shown in Figure 6.13. A pure thickness resonance mode will be excited by electric field. By finding f r and f a , electromechanical coupling factor k t can be calculated as [ ] 𝜋 (fa − fr ) 𝜋 fr 2 kt = tan 2 fa 2 fa E D Elastic stiffness C33 and C33 can be determined by D C33 = 𝜌(2tfa )2 E D C33 = 4𝜌fa2 t 2 (1 − kt2 ) = C33 (1 − kt2 ) 𝜀 The free relative permittivity 𝜖33 can be calculated from the capacitance C 0 with the equation given by 𝜀 = 𝜖HF = 𝜖33

𝜖LF =

C0𝜎 t A

𝜀 𝜖33

1 − kt2 Figure 6.13 A thin plate with the electric field parallel to its thickness.

3

Poling direction

Electrodes t d 2

1

∼

References

Figure 6.14 A thin disc with the electric field parallel to its thickness.

3 Electrodes r

Poling direction

t

∼

r

6.4.5

Radial Mode in a Thin Disc

By using a thin disc, the planar electromechanical coupling coefficient k p and planar Poisson’s ratio 𝜐p can be deduced. The major surfaces of the disc are fully covered by electrodes, and it is required that its radius r should be large compared with its thickness t (r/t > 20). The disc is poled in the thickness direction as illustrated in Figure 6.14. The radial mode is a special case that it requires three critical frequencies to analyze. By measuring the fundamental resonance frequency f r1 , anti-resonance frequencies f a , and the second series resonance frequency f r2 and of a disc, two constants k p and 𝜐p can be obtained (Meitzler et al. 1973). More parameters calculation can be found elsewhere (Anon 1988). 6.4.6

Mechanical Quality Factor

Mechanical quality factor QM is the ratio of the reactance to the resistance in the series equivalent circuit. Therefore, it can be determined by the equivalent circuit analysis function built-in to the Agilent 4294 impedance/gain phase analyzer. The mechanical quality factor QM can be calculated by QM =

𝜔Ls fa2 1 = Rs 2𝜋fr Rs (Cp + Cs ) fa2 − fr2

Electrical quality factor QE is the inverse of the dissipation factor tan 𝛿, where tan 𝛿 can be measured by the impedance analyzer at 1 kHz: QE =

1 tan 𝛿

References Anon (1988). IEEE Standard on Piezoelectricity. In: ANSI/IEEE Std 176-1987. New York, NY: IEEE Standards Board. Berlincourt, D.A., Curran, D.R., and Jaffe, H. (1964). Piezoelectric and piezomagnetic materials and their function in transducers. In: Physical Acoustics: Principles and Methods (ed. W.P. Mason), 169–270. New York/London: Academic Press. Available at: https://doi.org/10.1016/B978-1-4832-2857-0.50001-0. Chen, Y. (2013). High-Frequency and Endoscopic Ultrasonic Transducers Based on PMN–PT and PIN–PMN–PT Single Crystals. The Hong Kong Polytechnic University. Available at: http://theses.lib.polyu.edu.hk/handle/200/6985.

141

142

6 Piezoelectric Effect: Basic Theory

Meitzler, A.H., Obryan, H.M., and Tiersten, H.F. (1973). Definition and measurement of radial mode-coupling factors in piezoelectric ceramic materials with large variations in Poisson’s ratio. IEEE Transactions on Sonics and Ultrasonics 20 (3): 233–239. Uchino, K. (2009). Ferroelectric Devices, 2e. CRC Press.

143

7 Piezoelectric Devices Piezoelectric materials ha.ve very broad applications in smart systems as transducers, including sensors and actuators. It is very easy to list a few dozens of electronic devices utilizing piezoelectric properties. For example, piezoelectric materials are active elements in many acoustic devices such as ultrasonic transducers for medical imaging and sonar system and hydrophone for underwater detections. Actuators utilizing piezoelectric materials include micro-positioners, active damping systems, and piezoelectric transformers etc. With piezoelectric thin films, surface acoustic wave (SAW) devices and micro-electro-mechanical systems (MEMS) devices have been widely used in telecommunications. More recently, piezoelectric nanostructures such as ZnO nanowires also found application in nano-generators for energy harvesting. In this chapter, device applications of piezoelectric materials in smart systems in terms of device design, fabrication, characterization, and modeling will be introduced with examples from our own research work.

7.1 Piezoelectric Ultrasonic Transducers Acoustic transducers are devices to convert electrical energy to acoustic energy and vice versa, and those transducers with operating frequency within the range of ultrasound are called ultrasonic or ultrasound transducers. Most of the ultrasound transducers are made of active elements by piezoelectric materials, so they are called piezoelectric ultrasound transducers. Applications of ultrasound transducers are very broad, from ultrasound cleaner to ultrasound bonding and from non-destructive test (NDT) to medical ultrasound (including therapeutics and diagnostics). In this session we use medical imaging ultrasonic transducers as examples to introduce the transducers design, fabrication, characterization, and modeling. 7.1.1

Structure of Ultrasonic Transducers

The basic structure of an ultrasonic transducer is shown in Figure 7.1, where the main components include matching layer, backing layer, and piezoelectric (or active) element. The matching layer between the piezoelectric element and Ferroic Materials for Smart Systems: From Fundamentals to Device Applications, First Edition. Jiyan Dai. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

144

7 Piezoelectric Devices

Piezoelectric element Backing layer

Matching layer

Ultrasonic beam Input pulse

Focusing lens

Figure 7.1 The basic geometry of a single-element ultrasonic transducer while focusing lens can be convex or concave lens depending on the acoustic velocity between lens and front-loading medium.

medium allows maximum transmission of the ultrasonic wave, and the backing layer is to damp the back-radiated acoustic wave to reduce the ring down. Piezoelectric materials, such as PZT ceramics or relaxor ferroelectric single crystals, are usually used as the active element to generate and receive ultrasonic waves. Other piezoelectric materials such as PVDF, LiNbO3 , and lead-free piezoelectric materials have also been used as active element of ultrasonic transducers. Transducers with broad bandwidth (BW) and high sensitivity are important for ultrasonic imaging applications. Usually one or two acoustic matching layers are used to increase sound transmission into tissue to compensate the acoustic impedance (Z) mismatch between piezoelectric material and human tissue. Acoustic impedance Z is a parameter used to evaluate the acoustic energy transfer between two materials. Acoustic impedance of a medium is defined as Z=

pressure volume velocity

(7.1)

For materials, Z = 𝜌c

(7.2)

where 𝜌 and c are the density and the sound velocity, respectively, of the material. The unit of the acoustic impedance is kg/(m2 s) or Rayl. Table 7.1 lists the acoustic impedance for some nonbiological and biological materials (Zagzebski 1996). After applying electrical pulses on the piezoelectric active element, the pressure waves are generated in both back and front directions. The transmission coefficient (𝛼 T ) evaluates the pressure transmits into the front-loading medium: 𝛼T = where Zij =

4Zpf (1 + Zpf )2 cos2 (km ⋅ L) + (Zmf + Zpm )2 sin2 (km ⋅ L) Zi , Zj

(7.3)

and Zp , Zf , and Zm are acoustic impedances of the piezoelec-

tric element, front-loading medium, and matching layer, respectively (refer to

7.1 Piezoelectric Ultrasonic Transducers

Table 7.1 Acoustic impedances of materials. Tissue

Impedance (MRayl)

Air

0.0004

Fat

1.34

Water

1.48

Blood

1.65

Muscle

1.71

Skull bone

7.8

PZT

∼30

PMN-PT

∼28

Epoxy

∼2.0

Note: The acoustic impedances of PZT, PMN-PT, and epoxy depend on the composition. The values given in the table are only correct for some range of compositions. Source: Adapted from Zagzebski (1996).

Figure 7.1). k m denotes the wave numbers in matching layer, and L the thickness of matching layer. The transmission coefficient is a function of the acoustic impedance, frequency, and the matching layer thickness. Without the matching layers, the transmission coefficient 𝛼 T is calculated to be only about 10% for the values Zf ∼ 1.5 MRayl for human tissue and Zp ∼ 30 MRayl for PZT ceramic element. Therefore, it is necessary to insert a matching layer between the loading medium and piezoelectric element to improve the transmission coefficient of the ultrasonic wave at the interface (Zhou et al. 2011; Zhu 2008). When the matching thickness is equal to quarter wavelength, i.e. L = 𝜆4 , the transmission coefficient of Eq. (7.3) can be simplified to be 𝛼T =

4Zpf (Zmf + Zpm )2

(7.4)

For example, if a matching layer has an acoustic impedance of 15 MRayl with a quarter wavelength thickness, for the previously mentioned PZT-based transducer, 𝛼 T can reach more than 55%. 7.1.2

Theoretical Models of Ultrasonic Transducer (KLM Model)

There are many sophisticated one-dimensional equivalent circuit models that are used to simulate the behavior of ultrasonic transducer, such as the Mason model, the Redwood model, and the Krimholtz–Leedom–Matthae (KLM) model (Krimholtz et al. 1970). Among them, the KLM model is the most popular one. The commercial transducer simulation software PiezoCAD (version 3.03 for Windows, Sonic Concepts, Woodinville, WA), which has been extensively used in transducer design, is based on the KLM model. The effects of matching layers and backing material can be readily included as sections of transmission lines.

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7 Piezoelectric Devices

L/2

L/2 + Back acoustic V1 port –

I1

Z1ʹ v

I2

Z2ʹ v

+ Front acoustic V2 port –

Acoustic transmission line + I3 C 0 Electrical V3 port –

I= L

Cʹ

1: Φ

+V3–

Figure 7.2 KLM electrical equivalent circuit model for a piezoelectric transducer. Source: Adapted from Kirk Shung (2015).

This model divides a piezoelectric element into two halves, each is represented by an acoustic transmission line. The acoustic transmission line serves as a secondary circuit that is linked with an electrical primary circuit by an ideal transformer as shown in Figure 7.2 (Kirk Shung 2015). With the KLM model, the electrical impedance of the transducer is given as Zin =

Z1 Z2 1 1 + + j𝜔C0 j𝜔C ′ 𝛷2 (Z1 + Z2 )

(7.5)

where Z1 and Z2 are the input impedances of the acoustic transmission line looking toward the front acoustic port and back acoustic port, respectively, and C 0 and C′ are the clamped and series capacitance due to the acoustic transmission line, respectively. 𝛷 is the electromechanical turns ratio. Desilets et al. also determined the optimum impedance of the matching layer(s) based on the KLM model (Desilets et al. 1978). For one matching layer, the acoustic impedance of the matching material Z m satisfies Eq. (7.6) Zm = (Zp Zf2 )1∕3

(7.6)

In order to enhance front matching, two or more matching layers with quarter wavelength are often used for transducer fabrication. The acoustic impedances of two matching layers should be as follows (Desilets et al. 1978): Zm1 = (Zp4 Zf3 )1∕7

(7.7)

Zm2 = (Zp Zf6 )1∕7

(7.8)

It should be pointed out that usually two matching layers with quarter wavelength thickness of each layer are used in commercial product as a tradeoff of price and performance. More matching layers and even gradient matching with the impedance ranging from the piezoelectric element to human tissue is more favorable in performance, but the fabrication process is complicated and cost is very high.

7.1 Piezoelectric Ultrasonic Transducers

7.1.3

Characterization of Ultrasonic Transducers

Ultrasonic transducers are usually characterized by pulse-echo method in water tank, from which the basic parameters of transducers such as bandwidth, insertion loss (IL), and resolution can be determined. The generated ultrasound field distribution can also be measured by hydrophone or other techniques. Figure 7.3 shows the setup of pulse-echo measurement, where the ultrasound beam is generated by the piezoelectric element that also acts as a sensor to detect the reflected acoustic wave from the target. 7.1.3.1

Bandwidth (BW)

From the measured echo signal and its Fourier transform as shown in Figure 7.3, the frequency range emitted by a transducer is termed the characteristic bandwidth. The bandwidth of a transducer refers to the frequency range at its −6 dB points (at which the magnitude of the amplitude in the spectrum is 50% of the

Ultrasound analyzer Ultrasound absorb tank Transducer Designed focus length Bubble-free DI water

Planar reflecting target (a) Frequency (MHz)

10.0 M

15.0 M –10

1.0

–20

0.5

–30 –40

0.0

–50

–0.5

–60

–1.0 –1.5 (b)

Amplitude (dB)

Voltage (V)

1.5

5.0 M

–70 13.0 μ

14.0 μ

15.0 μ

16.0 μ

–80 17.0 μ

Time (s)

Figure 7.3 (a) Setup of pulse-echo measurement and (b) typical measurement results showing echo signal and its Fourier transform in frequency domain.

147

148

7 Piezoelectric Devices

maximum) of the spectrum. The center frequency f c can be determined as fl + fu (7.9) 2 where the f l and f u are lower and upper −6 dB frequencies and the −6 dB bandwidth is f −f BW = u l × 100% (7.10) fc fc =

7.1.3.2

Sensitivity of the Transducer

A standard method to characterize a transducer’s sensitivity is using the IL, which is the ratio of the input power Pi delivered to the transducer from the driving source to the transducer output power Po . By assuming that the input load resistance Ri is equal to the output load resistance Ro , the IL is simplified as the ratio of the pulse excitation voltage V i to the transducer and the echo output voltage V o . ) ( ( ) ( ) Vi2 ∕Ri Vi Pi = 20 log IL = 10 log = 10 log (7.11) 2 Po Vo Vo ∕Ro To obtain the IL of a transducer by Eq. (7.11), practically, a standard method is to generate a tone burst using an input sinusoidal wave of 20 cycles with amplitude V i at the resonance frequency of the transducer and received echo wave with amplitude V o . 7.1.3.3

Resolution

The axial and lateral resolutions are the distinguishable minimum distance between two reflectors along the axis and perpendicular to the ultrasound beam direction, respectively. Axial resolution (in depth) is determined by the ultrasound pulse duration, while the lateral resolution is related to the beam width in the lateral direction. For a circular transducer working at center frequency f c , the axial resolution, Raxial , and the lateral resolution, Rlateral at the focus point are c (7.12) Raxial = 2BW ⋅ fc 𝜆d Rlateral = (7.13) a where c is the sound velocity in the loading medium, BW the bandwidth of the transducer, 𝜆 the wavelength at the loading medium, d the focal length, and a is the diameter of the aperture (see Figure 7.4). Therefore, increasing the transducer’s bandwidth and center frequency will improve the axial and lateral resolutions of the imaging. Practically, the resolution can be determined by imaging metal wire phantom and find the resolution as shown in the figure. As shown in Figure 7.4, the resolution can be determined by measuring the width of the −6 dB points in real space from the image signal intensity.

7.1 Piezoelectric Ultrasonic Transducers

Figure 7.4 Illustration of resolution definition ultrasound imaging, where −6 dB is commonly used as the cut-off attenuation.

Well resolved

Barely resolved

Not resolved

a

Resolution d 6 dB Rlateral

7.1.4 7.1.4.1

Types of Ultrasonic Transducers Medical Application

Applications of the medical ultrasound transducers include diagnostic and therapeutic. Therapeutic ultrasounds usually require focused and high power in order to generate enough heat to burn cancer cells, for example, or break stones in the kidney. For diagnostic imaging purpose, there are many different types of transducers such as phase array, linear array, annular array, and radial array. In the past 10 years, our group has been developing these transducers with relaxor ferroelectric PMN-PT single crystals. Figure 7.5 shows different types of medical ultrasound transducers for diagnostic imaging. 7.1.5

Piezoelectric Film Application in Ultrasound Transducers

In order to increase the ultrasound imaging resolution, people have been trying to further increase the transducers frequency beyond 100 MHz and even to GHz by implementing thick or thin piezoelectric films such as PZT, ZnO, and AlN (Zhu et al. 2008). These very high frequency (f c > 100 MHz) transducers not only have potential applications as clinical tools for the examination of the anterior segment of the eye, skin, and intravascular imaging but may also be used in bimolecular imaging and research. Sol–gel coating of piezoelectric film with thickness in the order of a few micrometers and vacuum coating of the piezoelectric films with thickness of 100-nm scale have gain success in fabrication of very high-frequency transducers beyond 100 MHz. One interesting application of such piezoelectric-miniature ultrasonic transducer (PMUT) is piezoelectric-based fingerprint ID system in mobile phone as shown in Figure 1.8. Ultrasound fingerprint identification system based on piezoelectric ultrasonic transducer and imaging system can obtain a three-dimensional (3D) image of fingerprint with a certain depth. To realize this 3D fingerprint image, piezoelectric thick film forming a 2D array is needed to

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7 Piezoelectric Devices

(a)

(c)

(b)

(d)

(e)

Figure 7.5 Different types of medical ultrasound transducers developed from our group. (a) Single-element focused transducer, (b) single-element rotation transducer for intravascular imaging, (c) 128-element radial array transducer for endoscopic ultrasound imaging, (d) annular array for dynamic focusing eye imaging, and (e) an ultrasound image of rabbit eye by our 40-MHz focus transducer.

form a 2D phase-array transducer that can control the ultrasound beam to be focused and scanned electronically. 7.1.6 Challenges and Trend of Developing New Advanced Ultrasound Transducers Driven by the progress in new piezoelectric materials and new applications, there are a few fields of further development of ultrasound transducers: 1. Relaxor piezoelectric-based transducer for high sensitivity and larger bandwidth. 2. Lead-free piezoelectric-based transducer for environmental-friendly requirement. 3. 2D array transducers for 3D imaging. 4. High-frequency transducers up to GHz for cell imaging.

7.2 Ultrasonic Motor Suggested by the name, an ultrasonic motor (USM) is driven by a traveling ultrasound wave transmitted inside the medium of an USM. A traveling wave is a mechanical wave that is created by a vibrating object and subsequently travels through a medium where its propagation involves particle interaction causing

7.2 Ultrasonic Motor

Figure 7.6 Simulation of motion on stator surface. MX

Y

displacements of neighboring particles by sequence. Compared with electromagnetic motor, a USM possesses the advantages of large torque, low speed, high angular resolution and self-lock when stopped. USM can find its unique applications in some special situations where electromagnetic motor cannot work properly or satisfy the job. Like an earthquake, a mechanical traveling wave in a solid material can be classified into two general types – Rayleigh wave and flexural wave. The Rayleigh wave creates large amplitude at the surface, but the wavelength is too large and not suitable for use in USM. However, the flexural wave propagates with a snake-like motion (the elliptical motion as shown in Figure 7.6) provides the drive in traveling wave motors. It is obvious that an elliptical motion is created on the surface of the stator. There are two stages for the energy conversion in USM. The first stage is to convert electrical energy into mechanical energy through piezoelectric conversion by a piezoelectric material in which a transverse mode of vibration induces a flexural wave. In the second stage of energy conversion, the high-frequency oscillation of stator drives rotor into motion by frictional coupling between rotor and stator surfaces. Similar to the mechanism introduced by Sashida and Kenjo (1991), Figure 7.7 illustrates this basic principle of USM driven by flexural traveling wave and our fabricated linear ultrasonic motor. 7.2.1

Terminologies

There are several terminologies to examine the performance of an USM. Usually, a larger diameter and larger number of waves result in a larger torque and better performance of the motor. As indicated, the crests of the waves “push” the rotor forward for linear or circular motion. It can be predicted that more crests would provide larger “pushing” force on the rotor. For an USM, the most important characteristic is the torque – velocity dependence. This is also the most significant factor that describes the performance of an electromagnetic motor. For USM, another important factor is the admittance Y . As mentioned, Y can be expressed as G + jB. The admittance helps to understand the resonance behavior and to find the mechanical quality factor. By plotting the imaginary part of Y against the real part, a typical Nyquist diagram as shown in Figure 7.8 is obtained, from which resonance and anti-resonance frequencies,

151

152

7 Piezoelectric Devices

Pressure

Direction of motion

Traveling wave

Elliptical motion

Stator metal P

Electrode (a)

P

P

~

P

~

A

A sin ωt

Bulk piezoelectric material

B

B sin ωt

(b)

Figure 7.7 (a) Schematic diagram of motion between stator and rotor and (b) our fabricated linear motor based on PZT ceramic generated flexural traveling wave.

Im(Y) f fh fs

j0Cd f1

fp fa

1/Rd

fr 1/Rd+1/R

Re(Y)

Figure 7.8 Nyquist diagram around its fundamental resonance and anti-resonance frequencies. Source: Adapted from El and Helbo 2000.

7.2 Ultrasonic Motor

blocking capacitance and resistance, and other parameters can be found. The maximum and the minimum of the motional admittance occur at the mechanical resonance (series) f s and anti-resonance (parallel) f p , respectively. The resonance and the anti-resonance of the whole system are given by f r and f a , respectively, whereas the maximum and the minimum of the total admittance are given by f h and f l . The characteristic frequencies that are commonly used in the evaluation of the equivalent circuit parameters are f s and f p . When the quality factor is sufficiently high (Qm > 100), the pair (f s , f p ) is obtained from the average of the two other pairs: (fh , fl ) + (fr , fa ) (7.14) 2 If the G–B diagram obtained from impedance measurement is larger and more rounded, the resonant characteristics of the motor is better. The mechanical quality factor Qm can be determined from either the Nyquist diagram or the Bode diagram (Figure 7.9). In the Bode diagram, the sharpness of the admittance around the resonance frequency is used to determine Qm from the pass band at −3 dB of the frequency response. (fs , fp ) =

Qm =

7.2.2

fs Δf−3 dB

(7.15)

Design of USM

In the ultrasonic motor, the stator–ceramic–rotor assembly is made to resonate to create the flexural traveling waves in order to drive the rotor. As mentioned before the output torque and speed are directly related to the dimensions of the stator and the adhesion between the stator and the rotor. Depending on the application of individual ultrasonic motor, for rotors requiring smaller dimensions, the torque output would be smaller since the torque depends on the radius, and it is difficult to fabricate ceramics with large number of waves in small radius. Figure 7.10 shows diagram of a USM stator structure and a piezoelectric ceramic disc used in ultrasonic motor. Sectors labeled with dots are polarized up and the rest sectors are poled down. In theory, the 34 𝜆 region should be connected to ground in order to allow the flexural wave to transmit through. Admittance 0 dB

Y

–3 dB

fr

Frequency (Hz)

Figure 7.9 Bode diagram around its fundamental resonance and anti-resonance frequencies.

153

154

7 Piezoelectric Devices + 280 – 265 +246 + 258 – 296 – 280 + 293

(a)

– 293

End plane

+ 297

(b)

Stator

Case Assembly

Rotor (c)

(d)

Figure 7.10 (a) Diagram of stator and (b) the PZT ring. The numbers indicate d33 measurement results. (c) the parts and (d) the assembly of the motor.

But in some occasions, to make full use of piezoelectric ceramics, the 34 𝜆 region would be divided into two sectors and also be poled. In this configuration, the flexural traveling wave will be produced by exciting both the inputs concurrently. When an electric field is applied, the positively and negatively poled region will contract and expand correspondingly. With a 90∘ phase difference between the two waves, and an unpoled 𝜆/4 region, the flexural waves will travel on the ceramic surface, and make the teeth-shaped stator vibrate in the same way driving the rotor to rotate through friction force.

7.3 Surface Acoustics Wave Devices The SAW device is an electronic component widely used as a frequency filter or resonator in communication systems such as a mobile phone, optical communication systems, and so on. It is one of the key devices in modern communication systems due to its compactness, superior stability, and high performance of frequency characteristics. For a conventional SAW device as shown in Figure 7.11, a pair of interdigital transducers (IDTs) electrodes are placed on top of a piezoelectric crystal to generate and detect the SAW signal. The piezoelectric crystal is periodically poled with the same pattern of the IDTs. The input IDT applies an electric field to the substrate and launches a SAW by the converse piezoelectric effect, and the receiver reconverts the acoustic wave into an electrical signal by a direct piezoelectric effect. Acoustic waves propagating along the

7.3 Surface Acoustics Wave Devices

λ Rs RL Vin

s w Input IDTs

Surface acoustic wave

Output IDTs

Figure 7.11 Schematic diagram of SAW filters.

surface of a piezoelectric material provide a means of implementing a variety of signal-processing devices at frequencies ranging from several MHz to a few GHz. 7.3.1

Interdigital Electrode in SAW Device

A SAW device consists of two comb-shaped metal electrodes, called IDT electrode, placed on a piezoelectric material. An electric field created by applying a voltage to the electrodes induces dynamic strains in the piezoelectric material, thus creates the elastic waves. When an ac voltage is applied to the electrodes, the stress wave induced by finger pairs travels along the surface of crystal in both directions (Gardner et al. 2013). To ensure constructive interference and in-phase strain, the sum of spacing (s) and width (w) between the adjacent fingers should be equal to the half of the wavelength (𝜆). i.e. 𝜆 (7.16) 2 The frequency of the signal, which can be effectively converted to SAW and vice versa, is given by v = 𝜆f (where v is the phase velocity of SAW). Only this signal with this particular frequency can be generated with this designed SAW device with the particular spacing s and w. s+w=

7.3.2

Acoustic Wave

The type of acoustic wave generated in a piezoelectric material depends mainly on the substrate material properties, the crystal cut, and the structure of the electrodes used to transform the electrical energy into mechanical energy. The acoustic waves are mainly divided into two types: bulk acoustic wave (BAW) and surface acoustic wave (SAW). The BAW is an elastic wave propagates in the solid, while the SAW is a transverse wave propagating along the surface. The acoustic wave devices usually contain the Rayleigh wave (“true” SAW) and pseudo-SAW wave propagation, such as leaky-surface acoustic wave (LSAW), surface skimming bulk wave (SSBW), and surface transverse waves (STWs) (Campbell 1998a). In a conventional SAW device, a piezoelectric substrate is used in the device fabrication, and the elastic Rayleigh wave is usually used as the electrical–mechanical–electrical energy conversion through piezoelectric effect. As shown in Figure 7.12, the Rayleigh wave is a transverse wave that travels

155

156

7 Piezoelectric Devices

Elliptical displacement of atoms

Figure 7.12 Schematic diagram of Rayleigh wave. Source: Adapted from Gardner et al. (2013).

along the surface and a classical example is the ripples created on the surface of water by a water droplet. The elastic Rayleigh wave has both a surface-normal component and a surface-parallel component that is parallel to the direction of propagation. The energy of the SAW is confined to a zone close to the surface with a few wavelengths thick. However, due to the relatively low acoustic velocity in the piezoelectric material, the operation frequency is limited in a range of a few hundred MHz. This cannot satisfy the requirement of high frequency and high bit-rate communication systems. The increasing demand of high-frequency SAW devices accelerates the development of SAW devices in the GHz range. The high-frequency SAW devices may be fabricated by either implementing high-resolution photolithography or by using high acoustic velocity materials as the substrate with a piezoelectric layer on top (Nakahata et al. 2003), since the operating frequency is proportional to the acoustic velocity and inversely proportional to the sum of line width and space of the IDT (see Figure 7.11). However, the use of high-resolution photolithography results in dramatical increase in cost and poorer reliability. Therefore, using a substrate with high acoustic velocity is preferred, and layer-based SAW devices have been introduced to achieve this goal. Most substrates possessing high acoustic velocity are not piezoelectric and thus must be combined with a piezoelectric layer such as zinc oxide (ZnO), aluminum nitride (AlN), lead zirconate titanate (PZT), lithium niobate (LiNbO3 ), and lithium tantalate (LiTaO3 ) for the generation of the SAW. Consequently, in the fabrication of high-frequency SAW devices, many combinations of materials are currently under investigation as potential candidates, such as ZnO/diamond, LiNbO3 /sapphire, LiNbO3 /diamond, and LiTaO3 /sapphire (Nakahata et al. 2003; Uchino 2009; Gardner et al. 2013; Campbell 1998a). In the selection of materials, a systematic study of the required properties of piezoelectric materials and substrates is needed. Some important parameters for selecting SAW device structures are (i) acoustic velocity, (ii) electromechanical coupling coefficient, (iii) film morphology, and (iv) structural and interfacial qualities. However, none of the material combinations can satisfy all these requirements. The lattice mismatch, thermal expansion coefficient, and resistance to oxidation of the materials would directly affect the structure of films and finally adversely affect the performance of the devices. A careful investigation and selection of materials are essential in the device fabrication. Tables 7.2 and 7.3 compare the relevant properties for some piezoelectric materials and substrates achieved

7.3 Surface Acoustics Wave Devices

Table 7.2 Comparison of some piezoelectric materials properties.

Materials

Shear wave acoustic velocity (ms−1 )

Electromechanical coupling coefficient (%)

Curie temperature (∘ C)

LiNbO3

4000

5.5

1210

LiTaO3

3300

0.8

603

AlN

6700

0.7

/

ZnO

2600

1.9

/

PZT

2400

2.3

370

Table 7.3 Comparison of shear acoustic velocity of substrate materials.

Materials

Shear wave acoustic velocity (ms−1 )

Diamond

12 000

Sapphire

6 000

Quartz

3 500

SiC

6 830

from SAW devices, respectively (Nakahata et al. 2003; Gardner et al. 2013; Campbell 1998b). 7.3.3

Piezoelectric Property Considerations for SAW Devices

Started in the 1970s, SAW devices have become attractive as components incorporated in electronic systems (Morgan 1973). At that time the substrate (bulk) materials used to fabricate SAW components are limited. Due to the relatively large electromechanical coupling coefficient and high acoustic velocity, LiNbO3 has been widely investigated for the SAW device fabrication. The y-cut z-propagating LiNbO3 is one of the most common materials and orientations used in fabricating SAW transducers (Reilly et al. 1973). This orientation has advantages of having strong electromechanical coupling, low attenuation, and low scattering of waves. The disadvantage is that bulk wave can also be generated, and those bulk waves traveling close to the surface can excite the output transducer and produce unwanted output signals (Wong 2002). In general, the choice of cut or orientation of the substrate materials depends on the application. LiNbO3 is one of the most important materials for both fundamental and applied research in optics and materials science, owing to its electrical, electro-optical, nonlinear optical, and crystallographic properties. As a ferroelectric crystal, LiNbO3 presents a spontaneous polarization at temperatures below the Curie temperature T c (1210 ∘ C). In the microscopic viewpoint,

157

158

7 Piezoelectric Devices

lithium niobate consists of a grid of oxygen ion (O2− ) layers in which the lithium ions (Li+ ) and niobium ions (Nb5+ ) are positioned, as shown in Figure 7.13a. The displacement of the lithium ions and niobium ions relative to the neutral configuration (paraelectric phase) gives rise to a dipole moment and thus to the spontaneous polarization along the c-axis. In the ferroelectric phase as shown in Figure 7.13b, lithium ions and niobium ions are displaced above or below the oxygen ion layers. The dipole moment of LiNbO3 is pointing upward so that the direction of spontaneous polarization is upward. Figure 7.13c shows the schematic diagram of the crystal structure of lithium niobate in paraelectric phase where the niobium ions are placed at the center of the oxygen ions octahedron and the lithium ions are distributed within the next two octahedra. One possible approach to improve the device performance is to widen the choice of substrate materials by using piezoelectric films with IDTs to generate and detect surface waves on non-piezoelectric substrates. LiNbO3 has been deposited on different substrates or combined with another piezoelectric layer to enhance the performance of SAW devices. Film and substrate combinations can be chosen to control the coupling efficiency, to achieve low propagation loss, to regulate temperature coefficient of delay, and to increase fundamental frequency and reduce spurious signals. Thus, not many combinations are available. LiNbO3 has also been used as a substrate material incorporating with AlN or ZnO piezoelectric layer in SAW device application (Wu et al. 2002; Kao et al. 2004; Kao et al. 2003). For the LiNbO3 film deposition, it is usually done on the sapphire substrate. Since these materials are in hexagonal structure and their lattice mismatch are not so large, high quality films can be obtained by deposition. Shibata et al. claimed that their grown LiNbO3 films have enough piezoelectricity for SAW device fabrication, and it was also shown that different directions of propagation affect the signal transmission and different modes of acoustic wave transmission were Li+

Nb5+ P

P

O2– c a (a)

T < TC (b)

T > TC (c)

Figure 7.13 (a) Stereoscopic view of the ideal crystal stacking of LiNbO3 along the crystallographic c axis, (b) side view of ferroelectric phase LiNbO3 , and (c) side view of paraelectric phase LiNbO3 . Source: Kang et al. (2016). Adapted with permission of Royal Society of Chemistry.

7.3 Surface Acoustics Wave Devices

obtained (Shibata et al. 1995). The center frequency of these devices is much higher than that of the conventional type of SAW devices with similar size of IDT. 7.3.4

Characterization of SAW Devices

The network analyzer is usually used to characterize a SAW device, where the characteristic admittance (Y ) or impedance (Z) and scattering parameters (S11 , S21 , S12 , and S22 ) can be measured over a set of discrete frequency data points (Gardner et al. 2001). The scattering parameters (S-parameters) are the ratio of reflection and transmission coefficients between the incident and reflected waves and can be characterized by a magnitude in decibel (dB) and expressed as Sij , where i, j = 1 or 2. The S-parameters are the important parameters in the microwave measurement, since they can well describe the performance of a device over a range of frequency. These parameters can also be expressed in different formats, such as rectangular or polar coordinate systems and in Smith chart. Figure 7.14 shows the details of S-parameters. In characterizing a SAW device, S21 and S11 are plotted as a function of frequency. As an example, a SAW device based on the LiNbO3 /diamond contains two aluminum IDTs with 200 couples of fingers for each pair and an aperture length of 1000 μm. The IDTs are fabricated by depositing 100-nm-thick aluminum

a1 Forward b1

Z0 load

b1

S21 Transmitted

Incident S11 Reflected

a2 = 0 S22 Reflected

a1 = 0

Transmitted

S12

Incident

b2 Z0 load

b2 Reverse a2

S11 =

reflected b = 1 a1 incident a2 = 0

S22 =

reflected b = 2 a2 incident a1 = 0

S21 =

transmitted b2 = a1 incident a2 = 0

S12 =

transmitted b1 = a2 incident a1 = 0

b1 = S11 a1 + S12 a2 b2 = S21a1 + S22a2

Figure 7.14 Detail information of S-parameters.

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7 Piezoelectric Devices

film on the surface of the LiNbO3 film followed by photolithography and wet etching process. The designed wavelength of the IDTs was 16 μm (both the line width and space of IDTs are 4 μm). Figure 7.15 shows the schematic diagram of LiNbO3 /diamond SAW device. The performance of the devices is shown in Figure 7.16, where one can see that the IL is 30 dB at a center frequency of 940 MHz. The calculated phase velocity (v = 𝜆f ) from the center frequency is 15 000 ms−1 , which are well above the shear wave sound velocity for diamond and lower than that of the longitudinal wave.

AI IDTs Δ μm

LiNbO3 piezoelectric film 4 μm

Al2O3 buffer layer

Diamond

Figure 7.15 Schematic diagram of SAW device structure (LiNbO3 /Al2 O3 /diamond). Inset is the aluminum IDTs image.

1 –30 0

–1

–40

–2 –50

S21 (dB)

S11 (dB)

160

–3 –60 –4

–5

600

800

1000

1200

1400

–70

Frequency (MHz)

Figure 7.16 Scattering parameters S11 response of the SAW device fabricated on LiNbO3 films deposited diamond substrate. Source: Adapted from Lam et al. (2004).

7.3 Surface Acoustics Wave Devices

Beyond communication application, SAW devices can also be used as sensors by detecting the SAW that affected by physical stimuli such as weight from sensing elements. This sensing mechanism can be used as very sensitive sensors for biomaterials and chemical reactions where the weight of biomolecules and reactant can be detected by the wavelength shift. 7.3.5

Lead-Free Piezoelectric Materials

It is well known that, as the most important applications of ferroelectric materials, almost all commercial piezoelectric actuation and sensing applications are based on lead zirconate titanate ( PZT) (Jo et al. 2012; Rödel et al. 2009). However, Pb makes both the manufacturing and disposal of PZT hazardous to people and the environment (Rödel et al. 2015) in the form of Pb vapor during sintering and the Pb leaching in soil (Kosec et al. 1998). These issues have led many countries to regulate and strictly limit the use of Pb in electrical and electronic systems and devices. Since the early twenty-first century, there has been significant progress in the development of lead-free ferroelectrics. Up to date, under certain conditions, the electromechanical properties of some compositions have shown successful match or even surpassed commercially available PZT over a wide temperature range, making them potentially attractive for applications. However, there is still no single system to be comparable to PZT in satisfying all criteria in application. The large strains and piezoelectric responses from Pb-free ferroelectrics could be markedly different to classical ferroelectrics such as PbTiO3 and BaTiO3 . Lead-free ferroelectrics that have potential for device application can be broadly classified into two primary groups: (i) ferroelectric and (ii) relaxor ferroelectric. The first group (i) is primarily based on (Kx ,Na1 − x )NbO3 (KNN) (Saito et al. 2004), and the second group (ii) is primarily based on (Na1/2 ,Bi1/2 )TiO3 (NBT), a pseudo-rhombohedral ferroelectric. Solid solutions containing MPBs such as KNN-BT and NBT-BT-KNN show promise, particularly when coupled NBT-BT single crystal

Dice the plate for the first time

Epoxy Dice direction

Fill the kerfs for the second time

Fill the kerfs with epoxy for the first time Dice the plate for the second time

Dice the plate for the third time Fill the kerfs for the third time

Ground to remove the top and bottom part

Figure 7.17 Schematic procedures of the modified dice-and-fill method used for fabricating 1–3 composites based on very fragile piezoelectric materials. Zhou et al. (2012). Reprinted with permission of Elsevier.

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7 Piezoelectric Devices

with substitutional doping. Ultimately, more complex solid solutions and doping schemes will have to be explored because these provide an increasing number of degrees of freedom for identifying extraordinary properties (Aksel and Jones 2010). Our group has demonstrated successful application of Pb-free ferroelectric materials in medical ultrasound transducers. Figure 7.17 shows the diagram of fabricating NBT-BT/epoxy 1–3 composite for transducer fabrication. This 1–3 composite structure can effectively increase the piezoelectric strain due to the 33 vibration mode of each NBT-BT single crystal element and can also reduce the acoustic impedance due the low acoustic impedance of epoxy matrix. With such composite, we have fabricated single-element and array ultrasound transducers, and the transducers show great performance in both sensitivity and bandwidth as shown in Figures 7.18 and 7.19. Matching layer NBT-BT / epoxy 1–3 composite Conductive adhesive Backing material Coaxial cable

Figure 7.18 Construction of the NBT–BT/epoxy 1–3 composite single-element and linear array transducers. Source: Zhou et al. (2012). Reprinted with permission of Elsevier. 0.6 0 Magnitude (dB)

0.4 Amplitude (V)

162

0.2 0.0 –0.2 –0.4 –0.6

(a)

BW (–6dB) = 104%

–10 –20 –30

9

11 10 Time (µs)

0

12

1

4 5 2 3 Frequency (MHz)

6

7

(b)

Figure 7.19 (a) Pulse-echo waveform and (b) frequency spectrum of a single array element of the fabricated NBT–BT/epoxy 1–3 composite linear array ultrasonic transducer. Source: Zhou et al. (2012). Reprinted with permission of Elsevier.

References

References Aksel, E. and Jones, J.L. (2010). Advances in lead-free piezoelectric materials for sensors and actuators. Sensors 10 (3): 1935–1954. Campbell, C.K. (1998a). Basics of piezoelectricity and acoustic waves. In: Surface Acoustic Wave Devices for Mobile and Wireless Communications, 19–62. San Diego: Academic Press. Campbell, C.K. (1998b). Principles of linear-phase SAW filter design. In: Surface Acoustic Wave Devices for Mobile and Wireless Communications, 67–96. San Diego: Academic Press. Desilets, C.S., Fraser, J.D., and Kino, G.S. (1978). The design of efficient broad-band piezoelectric transducers. IEEE Transactions on Sonics and Ultrasonics 25 (3): 115–125. El, G.N. and Helbo, J. (2000). Equivalent circuit modeling of a rotary piezoelectric motor. IASTED International Conference on Modelling and Simulation (MS’2000), Pittsburgh, Pennsylvania, USA (May 15–17, 2000). Gardner, J.W., Varadan, V.K., and Awadelkarim, O.O. (2001). IDT microsensor parameter measurement. In: Microsensors, MEMS, and Smart Devices, 337–339. Available at: https://doi.org/10.1002/9780470846087.ch11. Gardner, J.W., Varadan, V.K., and Awadelkarim, O.O. (2013). Introduction to SAW devices. In: Microsensors, MEMS, and Smart Devices, 303–317. Wiley-Blackwell. Available at: https://onlinelibrary.wiley.com/doi/abs/10.1002/9780470846087 .ch9. Jo, W., Dittmer, R., Acosta, M. et al. (2012). Giant electric-field-induced strains in lead-free ceramics for actuator applications – status and perspective. Journal of Electroceramics 29 (1): 71–93. Available at: https://doi.org/10.1007/s10832-0129742-3. Kang, X., Sang, Y., Liu, H. et al. (2016). Formation mechanism and elimination methods for anti-site defects in LiNbO3 /LiTaO3 crystals. CrystEngComm 18 (42): 8136–8146. Available at: http://dx.doi.org/10.1039/C6CE01306F. Kao, K.S., Cheng, C.C., Chen, Y.C., and Lee, Y.H. (2003). The characteristics of surface acoustic waves on AlN/LiNbO3 substrates. Applied Physics A 76 (7): 1125–1127. Available at: https://doi.org/10.1007/s00339-002-2022-3. Kao, K.S., Cheng, C.C., Chen, Y.C., and Chen, C.H. (2004). The dispersion properties of surface acoustic wave devices on AlN/LiNbO3 film/substrate structure. Applied Surface Science 230 (1–4): 334–339. Available at: http://www .sciencedirect.com/science/article/pii/S0169433204001850. Kirk Shung, K. (2015). Diagnostic Ultrasound: Imaging and Blood Flow Measurements, 2e. Boca Raton, London, New York: CRC Press. Kosec, M., Malic, B., Wolny, W. et al. (1998). Effect of chemically aggressive environment on the electromechanical behaviour of modified lead titanate ceramics. Journal of the Korean Physical Society 32: S1163–S1166. Krimholtz, R., Leedom, D.A., and Matthaei, G.L. (1970). New equivalent circuits for elementary piezoelectric transducers. Electronics Letters 6 (13): 398–399. Lam, H.K., Dai, J.Y., and Chan, H.L.-W. (2004). Highly-oriented LiNbO3 films on polycrystalline diamond substrate for high frequency surface acoustic wave devices. Japanese Journal of Applied Physics 43 (6A): L706–L708. Available at: http://dx.doi.org/10.1143/JJAP.43.L706.

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Morgan, D.R. (1973). Surface acoustic wave devices and applications. Ultrasonics 11 (3): 121. Available at: http://www.sciencedirect.com/science/article/pii/ 0041624X73906082. Nakahata, H., Fujii, S., Higaki, K. et al. (2003). Diamond-based surface acoustic wave devices. Semiconductor Science and Technology 18 (3): S96–S104. Available at: http://stacks.iop.org/0268-1242/18/i=3/a=314. Reilly, N.H.C., Milsom, R.F., and Redwood, M. (1973). Generation of Rayleigh and bulk waves by interdigital transducers on y-cut z-propagating lithium niobate. Electronics Letters 9 (18): 419–420. Rödel, J., Jo, W., Seifert, K.T.P. et al. (2009). Perspective on the development of lead-free piezoceramics. Journal of the American Ceramic Society 92 (6): 1153–1177. Available at: http://dx.doi.org/10.1111/j.1551-2916.2009.03061.x. Rödel, J., Webber, K.G., Dittmer, R. et al. (2015). Transferring lead-free piezoelectric ceramics into application. Journal of the European Ceramic Society 35 (6): 1659–1681. Available at: http://www.sciencedirect.com/science/article/pii/ S0955221914006700. Saito, Y., Takao, H., Tani, T. et al. (2004). Lead-free piezoceramics. Nature 432: 84. Available at: http://dx.doi.org/10.1038/nature03028. Sashida, T. and Kenjo, T. (1991). An Introduction to Ultrasonic Motors. Clarendon Press. Shibata, Y., Kaya, K., Akashi, K. et al. (1995). Epitaxial growth and surface acoustic wave properties of lithium niobate films grown by pulsed laser deposition. Journal of Applied Physics 77 (4): 1498–1503. Available at: https://doi.org/10 .1063/1.358900. Uchino, K. (2009). Ferroelectric Devices, 2e. CRC Press. Wong, K.-K. (2002). Acoustic wave propagation and properties. In: Properties of Lithium Niobate. London: INSPEC, The Institution of Electrical Engineers. Wu, S., Chen, Y.-C., and Chang, Y.S. (2002). Characterization of AlN films on Y-128∘ LiNbO3 by surface acoustic wave measurement. Japanese Journal of Applied Physics 41 (Part 1, No. 7A): 4605–4608. Available at: http://dx.doi.org/10.1143/ JJAP.41.4605. Zagzebski, J.A. (1996). Essentials of Ultrasound Physics. USA: Mosby: Maryland Heights. Available at: https://books.google.com.hk/books?id=XahsQgAACAAJ. Zhou, Q., Lam, K.H., Dai, J.Y. et al. (2011). Piezoelectric films for high frequency ultrasonic transducers in biomedical applications. Progress in Materials Science 56 (2): 139–174. Available at: http://www.sciencedirect.com/science/article/pii/ S0079642510000563. Zhou, D., Lau, S., Wu, D., and Kirk Shung, K. (2012). Lead-free piezoelectric single crystal based 1–3 composites for ultrasonic transducer applications. Sensors and Actuators A: Physical 182: 95–100. Available at: http://www.sciencedirect.com/ science/article/pii/S0924424712003299. Zhu, J. (2008). Optimization of Matching Layer Design for Medical Ultrasonic Transducer. The Pennsylvania State University. Zhu, B.P., Wu, D.W., Zhou, Q.F. et al. (2008). Lead zirconate titanate thick film with enhanced electrical properties for high frequency transducer applications. Applied Physics Letters 93: 012905. Available at: https://doi.org/10.1063/1 .2956408.

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8 Ferromagnetics: From Material to Device Ferromagnetic materials exhibit spontaneous magnetization (magnetic moment per unit volume) at temperatures below the Curie point T c . Magnetic moments are originated from the orbital and spin of electrons, as shown in Figure 8.1a,b. Same as ferroelectrics, ferromagnetic materials also exhibit hysteresis loop in the curve of magnetization–magnetic field as shown in Figure 8.1c. A ferromagnetic material also tends to form domains when the grain size is larger than a critical value that is normally a few nanometers. Since the first discovery of Fe3 O4 (the so-called lodestones) more than 3000 years ago, many ferromagnetic materials have been found and applied in different fields from magnetic compass to electrical power generation and telecommunications. In recent decades, magnetic materials have been very successfully used in data storage from magnetic tape to hard disc. In this book, we focus on device applications of magnetic materials in smart systems such as transducers utilizing magnetostrictive effect and thin film-based ferromagnetic sensors and memory devices.

8.1 General Introduction to Ferromagnetics The origin of ferromagnetism (FM) is time reversal symmetry breaking due to the existence of unpaired spin of electrons and the associated current. This statement can be understood by considering that if time is reversed, the spin or current flow direction will thus be reversed and this results in magnetization direction reversion, i.e. time reversal symmetry breaking. Figure 8.2 illustrates time reversal symmetry breaking. Any atoms or ions with existing unpaired electrons exhibit ferromagnetism such as Fe, Co, and Ni. Magnetization can exist, even very weak, in non-ferromagnetic materials such as graphene as long as their defects or edge structures induce dangling bonds with unpaired electrons (Ma et al. 2012; Liu et al. 2013). However, those ferromagnetic materials with practical application potentials are usually compounds containing Fe, Co, or Ni. In terms of magnetization, materials can be classified into paramagnetic, diamagnetic, ferrimagnetic, ferromagnetic, antiferromagnetic (AFM), and superparamagnetism. A ferromagnetic material usually forms permanent magnet or Ferroic Materials for Smart Systems: From Fundamentals to Device Applications, First Edition. Jiyan Dai. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

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8 Ferromagnetics: From Material to Device

M Magnetic moment m

Ms

Magnetic moment m

Electron

–Hc

Atomic nucleus

Mr

+Hc

H

Direction of spin

(a)

(c)

(b)

Figure 8.1 Diagram showing the magnetic moment associated with (a) orbital motion and (b) spin motion of an electron; (c) ferromagnetic M–H loop where Ms and Mr are the saturation and remnant magnetization and Hc is the coercive magnetic field. Figure 8.2 Illustration of time reversion symmetry breaking. Source: Adapted from Eerenstein et al. (2006).

m

m Time reversal m switched

m Spatial inversion m invariant

is attracted to magnets, and it undergoes phase change to paramagnetic above Curie temperature through a second-order phase transition. Ferromagnetism (including ferrimagnetism) is the strongest type; it is the only type that creates forces strong enough to be felt and is responsible for the common phenomena of magnetism encountered in everyday life. An AFM has magnetization microscopically but has no macroscopic magnetization due to the cancelation of antiparallel spins of alternative layers. The temperature from an AFM to paramagnetic phase transition is called Néel temperature. For example, BiFeO3 is an AFM material with Néel temperature of 635 K (Kiselev et al. 1963). Ferrimagnetic is a relatively weak ferromagnetic, such as CoFe2 O4 spinel. A diamagnetic generates weak but negative magnetization when a magnetic field is applied. At this point, you may have already seen analogies of ferromagnetics to ferroelectrics by comparing these terms: ferromagnetic to ferroelectric, paramagnetic to paraelectric, and AFM to antiferroelectric. The fact is that the theory of ferromagnetics was established much earlier than that of ferroelectrics, and scientists in field of ferroelectrics translated the ferromagnetic theory to ferroelectrics. If you treat a spin electron as a magneton and a polarized unite cell as polariton,

8.1 General Introduction to Ferromagnetics

Table 8.1 Fundamental properties showing analogies between ferromagnetism and ferroelectricity. Ferromagnetism

Ferroelectricity

Magnetic field H

Electric field E

Magnetic induction B (or magnetic flux density)

Electric displacement D

B = 𝜇0 H + 𝜇0 M

D = 𝜖0 E + P

Vacuum permeability 𝜇0

Vacuum permittivity 𝜖 0

Magnetization M

Polarization P M = 𝜒 mH

Magnetic susceptibility 𝜒 m B = 𝜇H

P = 𝜖0 𝜒 e E Electric susceptibility 𝜒 e D = 𝜖 0 (1 + 𝜒 e )E = 𝜖E

Permeability 𝜇 = 𝜇0 𝜇r

Permittivity 𝜖 = 𝜖 0 𝜖 r

Relative permeability 𝜇r = 1 + 𝜒 m

Relative permittivity 𝜖 r = 1 + 𝜒 e

they similarly have properties of being switchable and vanished above Curie temperature. There are phenomenological similarities between ferromagnetic and ferroelectric, where ferromagnetic is due to time-reversal symmetry breaking and ferroelectric originates from spatial inversion symmetry breaking. More analogies can be seen from Table 8.1 comparing their fundamental properties. Electrons have an intrinsic property called spin that contributes to their magnetic moment. The classical model of electron spin is the electron spinning on its axis, i.e. either spin-up or spin-down. Then the value of magnetic moment can be calculated with respect to the spin of an electron. The total magnetic moment of an atom is the vector sum of the orbital and spin magnetic moments. A magnetic solid, made up of atoms with magnetic moment, has quantum exchange interactions that tend to align the magnetic moments at low temperature. When T < T c , the macroscopic magnetization arises and retains even in the absence of a magnetic field. The magnetic moments tend to align in the same direction without the aid of an external magnetic field. This is known as the ferromagnetic phase. In the ferromagnetic category, materials are divided into strong and weak ferromagnets. By introducing the total particle number N = N ↑ + N ↓ and the spin polarization s = N ↑ − N ↓ , a strong ferromagnet has almost s = 100% at the Fermi energy as shown in Figure 8.3a, while a weak ferromagnet gets a smaller spin polarization and paramagnet has no net spin polarization (Figure 8.3b,c). Meanwhile, there is another type of magnets that possess magnetization microscopically, but have no macroscopic magnetization due to the cancelation of antiparallel spins of neighboring pairs – this is known as AFM phase. A magnet that exhibits no macroscopic magnetization at high temperature (when H = 0) is called as the paramagnetic phase, in which the magnetic moment induced by the applied H-field is rather weak. Figure 8.4 shows the relationship between magnetization and temperature of a ferromagnet, where one can see that the magnetization starts decreasing close to T c , and when T > T c , magnetic

167

168

8 Ferromagnetics: From Material to Device

E

E

E

EF

EF

DOS (a)

DOS

DOS

DOS

(b)

DOS

DOS

(c)

Figure 8.3 Schematic densities of states (DOSs) for ferromagnetism: (a) strong ferromagnet, (b) weak ferromagnet, and (c) paramagnet.

M

Mr

Tc

T

Figure 8.4 Illustration of magnetization versus temperature. Arrows inside the boxes denote the magnetic moments alignment.

moments align randomly resulting in zero macroscopic magnetization. This behavior is similar to polarization versus temperature characteristics of a ferroelectric under second-order phase transition.

8.2 Ferromagnetic Phase Transition: Landau Free-Energy Theory Landau has given a general description of ferromagnetic to paramagnetic phase transitions with free energy approach by writing out a series expansion for the free energy close to Curie temperature. The magnetic free energy is defined as a power series in the order of magnetization M. By only considering the symmetries, the series must only contain terms that respect the symmetry of the

8.3 Domain and Domain Wall

T > Tc

Energy

T = Tc

T < Tc Magnetization

Figure 8.5 Sketch of Landau free energy F(M) = 𝛼M2 + 𝛽M4 for T > T c , T = T c , and T < Tc .

order parameter: F(M) = F0 + 𝛼M2 + 𝛽M4 + …

(8.1)

F is minimized with respect to M: 𝜕F (8.2) = 2𝛼M + 4𝛽M3 = 0 𝜕M 𝛼 M = 0 or M2 = − (8.3) 2𝛽 In Figure 8.5, the symmetry changes precisely at T = T c and the transition occurs when 𝛼 changes its sign. When T < T c , there are two energy minima M = ± Ms that indicate 𝛼 < 0 and 𝛽 > 0. When T > T c , a single energy minimum M = 0 indicates 𝛼 > 0 and 𝛽 > 0. The temperature dependence of 𝛼 can be identified as ( ) T − Tc 𝛼(T) = 𝛼0 Tc With the presence of an external magnetic field H, a linear term is added as F(M) = 𝛼M2 + 𝛽M4 + … − 𝜇0 HM

(8.4)

As shown in Figure 8.6, when an external magnetic field breaks the symmetry, the global minimum of F(M) changes from two minima to a single minimum. Therefore, we can see that even when T > T c , a paramagnetic phase of material, which has no macroscopic magnetization, can also be attracted with the presence of an externally applied magnetic field.

8.3 Domain and Domain Wall Ferromagnetic materials also exhibit domain structure where each domain has its own magnetization direction that can be switched by external magnetic field.

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8 Ferromagnetics: From Material to Device

T > Tc T = Tc

Energy

170

T < Tc

Magnetization

Figure 8.6 Sketch of Landau free energy F(M) = 𝛼M2 + 𝛽M4 − 𝜇0 HM for T > T c , T = T c , and T < T c under a positive magnetic field.

There are easy axes in ferromagnetic materials that are the directions magnetic moments should follow. For example, for a cubic structured FM material such as Fe, the ⟨100⟩ directions are usually the easy axes, while for a hexagonal-structured FM material such as Co, the ⟨0001⟩ are the easy axes. Due to this difference, in a cubic structure, both 90∘ and 180∘ magnetic domains can be formed (Figure 8.7a), while in a hexagonal structured ferromagnetic material, the domains are usually aligned with an angle of 180∘ (Figure 8.7b). The easy axes of cubic-structured Ni are ⟨111⟩, so it can form 180∘ , 71∘ , and 109∘ magnetic domains; these are possible angles between all ⟨111⟩ easy axes. The magnetization switching can be realized by not only external magnetic field but also mechanical stress. This mechanical stress-induced magnetization switching is similar to ferroelectric material whose polarization can be switched by both electric field and mechanical stress.

(a)

(b)

Figure 8.7 Domain structures in cubic (a) and hexagonal (b) structured ferromagnetic materials.

8.4 Magnetoresistance Effect and Device

Bloch wall

Néel wall

(a)

(b)

Ising wall (c)

Figure 8.8 Domain wall structures of (a) Bloch-type ferromagnetic domain walls, (b) Néel-type ferromagnetic domain walls, and (c) Ising-type ferroelectric domain walls.

Ferromagnetic domain wall structure is somewhat different from ferroelectric domains in terms of domain wall thickness. In fact, a ferromagnetic domain wall is much thicker than ferroelectric domain wall thickness, which is only one to a few atomic layers. Figure 8.8a illustrates the 180∘ domain wall structure in ferromagnetic materials, where the more common one is the Bloch wall, but in thinner films a Néel wall is often favored (Figure 8.8b). By contrast, Figure 8.8c demonstrates a narrow domain wall (Ising type) structure usually as the case of a ferroelectric domain structure.

8.4 Magnetoresistance Effect and Device Among many applications of magnetic thin films, magnetoresistance effect has achieved great success in sensors and memory devices. By changing the value of its electrical resistance in an externally applied magnetic field, anisotropic magnetoresistance (AMR), giant magnetoresistance (GMR), tunneling magnetoresistance (TMR), and colossal magnetoresistance (CMR) can be observed in magnetic thin films. The discovery of GMR in 1988 provided scientists with a new perspective for understanding polarized carriers in ferromagnetic metals and possibilities to apply this new technologies in non-destructive testing (García-Martín et al. 2011; Poon et al. 2013) and magnetic storage (Daughton 1992). 8.4.1

Anisotropic Magnetoresistance (AMR)

AMR is a property of a ferromagnetic conductor reported in 1857 by British physicist Lord Kelvin. AMR is a phenomenon in which a dependence of electrical resistance on the angle between the directions of electrical current and ferromagnetic magnetization. In general, the longitudinal resistance reaches maximum when the current is parallel to magnetization, and the transverse resistance reaches minimum when the current is perpendicular to magnetization (Nickel 1995). The AMR ratio is defined as AMR =

R∥ − R⟂ R∥

(8.5)

The AMR is originated from the spin–orbit coupling that leads to spin-dependent scattering of conduction electrons (Inoue 2013). When the magnetization of

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8 Ferromagnetics: From Material to Device

H-field direction

H-field

M

172

e–

direction

M

e–

(a)

θ

(b)

Figure 8.9 Schematic diagram of origin of the AMR effect, (a) high and (b) low resistance states raised by the current parallel and perpendicular to the magnetization, respectively. Ovals represent the scattering cross-section of the bound electronic orbits.

a material rotates, the electron cloud of each nucleus deforms and it changes the amount of conduction electron scattering, i.e. the magnetization direction rotates the closed orbit orientation with respect to the current direction. As demonstrated in Figure 8.9a, if the field and magnetization are oriented parallel to the current, the electronic orbits are oriented perpendicular to the current, so the cross-section for scattering increases, resulting in a high resistance state. On the contrary, if the field is transverse to the current, the electronic orbits are in the plane with current; therefore, less scattering occurs due to the small cross-section, resulting in a low resistance state (Figure 8.9b). The relation of electrical resistance R on the angle 𝜃 between the directions of electrical current and magnetization can be obtained by R = R⟂ + (R∥ − R⟂ )cos2 𝜃

(8.6)

Owing to the angular dependence, AMR effect is popularly used in magneto-resistor and sensors. For example, electronic compass for the measurement of Earth’s magnetic field, traffic detection, and linear position and angle sensing. A typical AMR magnetoresistive sensor is constructed of a combination of magneto-resistors, which can be used to measure strength and direction of magnetic field. Besides sensors, AMR effect was used as the basis of reading in magnetoresistive random-access memory (MRAM). However, its amplitude in ferromagnetic materials is weak (about 2% variations in resistance on changing the relative orientation of magnetization and current). The application in reading head using AMR effect was therefore replaced after the GMR was discovered. However, AMR effect has attracted great interest again because of investigation on AFM materials (Kriegner et al. 2016; Lu et al. 2018; Hu et al. 2012). It represents a major step in the field of AFM spintronics in which the antiferromagnet (with a zero net moment) governs the transport instead of just playing a passive supporting role in the traditional ferromagnetic (FM) spintronics. 8.4.2

Giant Magnetoresistance (GMR)

GMR is a quantum mechanical magnetoresistance effect observed in multilayers composed of the super lattice of alternating ferromagnetic and non-magnetic

8.4 Magnetoresistance Effect and Device

conductive layers (Baibich et al. 1988). Unlike AMR, GMR requires nanostructured composites and each layer is only a few nanometers thick. The magnetoresistance was reported as the largest so far observed for magnetic metal films (about 40% for Fe/Cr multilayers), so it was called as GMR. The 2007 Nobel Prize in Physics was awarded to Albert Fert and Peter Grünberg for the discovery of GMR. Let us look at the details of the work being awarded 2007 Nobel Prize in Physics. As reported in 1987–1988 (Grünberg et al. 1987; Baibich et al. 1988), they grew Cr/Fe superlattice structure and measured the in-plane resistance in the presence of magnetic field. Due to the exchange coupling between the FM Fe layer and Cr layer, this superlattice structure can have parallel and antiparallel magnetization layers forming a GMR effect as illustrated in Figure 8.10. One can see that the maximum magnetoresistance reaches 80% from the superlattice structure with Cr layer thickness of 0.5 nm. This GMR effect has been successfully implemented in hard disc drive date storage. Not only ferromagnetic metals, GMR was also found to occur in some oxide thin films of manganates of the formula La1−x Ax MnO3 (A = alkaline earth metal). One of the well-known examples is La1−x Srx MnO3 . The parent manganate LaMnO3 , AFM insulator, is an orthorhombic perovskite with distorted MnO6 octahedra. When La is replaced by a divalent ion, Mn4+ ions are created, and when Mn4+ is sufficiently large, the material changes from AFM to ferromagnetic, exhibiting an insulator-to-metal transition because of double exchange mechanism. The magnetization and electrical resistivity behavior of

R/R(H = 0)

1.0 0 Fe(3 nm)/Cr(1.8 nm) 0.9 .9 ~+80% 8

0.8 0 .8 3 nm)/Cr(1.2 nm) Fe(3

0.7 0.6

(0.9 nm) Fe(3 nm)/Cr(0.9 0.5 –40

–20

0 H (kG)

20

40

Figure 8.10 Magnetoresistance of Fe/Cr superlattice showing the change in the resistance of Fe/Cr superlattice at 4.2 K in external magnetic field H. The arrow indicates maximum resistance change. Source: Adapted from Baibich et al (1988).

173

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8 Ferromagnetics: From Material to Device

La1−x Ax MnO3 offers the oxide thin films an opportunity for ferromagnetism and GMR effect applications (Rao and Mahesh 1997). The nature of the GMR effect is different from that of AMR – that the resistance in the plane of the film is isotropic for the GMR effect but depends on current direction for the AMR. The GMR effect shows a magnetization orientation-dependent component of resistance that varies between magnetizations in alternate layers. Figure 8.11a shows the schematic superlattice structure of ferromagnetic Fe layers alternatively with Cr layers reported by Baibich et al. In this structure, layers of Fe, sandwiching an ultrathin Cr layer, was found to be ferromagnetically ordered with a magnetic moment oriental antiparallel to that

Fe Cr Fe Cr Fe Cr Fe GaAs

External H-field direction

(a) When H ≠ 0,

When H = 0, Fe

Fe Cr

Cr

Fe

Fe

Rparallel (b)

Rantiparallel (c)

Figure 8.11 (a) Diagram showing the superlattice structure of ferromagnetic Fe layers alternatively with Cr layers, white arrows denote magnetization direction, and (b) and (c) are the corresponding low and high resistance states for parallel and antiparallel magnetizations among neighboring Fe layers.

8.4 Magnetoresistance Effect and Device

GMR sensor

AC generator

Figure 8.12 Hybrid probe: eddy current testing (ECT) coil with GMR sensor. Source: Adapted ˇ from Cápová et al. (2008).

of the adjacent Fe layer. Without applied magnetic field, the coupling of Fe films across Cr interlayers with proper thickness is found to be AFM (Saurenbach et al. 1988; Grünberg et al. 1987), and it brings the magnetization to zero. GMR effect is thus achieved in antiferromagnetically coupled Fe/Cr superlattice by aligning the magnetization of adjacent Fe layers with an external field. By well controlling the AFM coupling, the ferromagnetic layers can form a state with antiparallel magnetization. Electrons that transport through this structure will be highly scattered due to the averaged spin-up and spin-down currents, resulting in high resistance state (Figure 8.11c). When a magnetic field is applied in parallel to the film, AFM magnetization can be switched to the magnetic field direction, making the whole film’s magnetization direction into the field direction as shown in Figure 8.11b; electrons transport through this layer experience weak interface scattering and a high transmission for the spin-down electrons, resulting in a relatively much lower resistance. When the thickness of the Cr layer or the temperature increases, more defect and steps may be introduced, which results in weakened AFM coupling within the Cr layer; this makes the GMR effect relatively smaller. This model explains the GMR and explains the thickness-dependence of GMR shown in the diagram. A GMR sensor has large bandwidth making it applicable to different frequencies of magnetic field. GMR sensors also have high sensitivity and low power consumption. Figure 8.12 shows an eddy current probe and a GMR magnetic field sensor being used in power line current stability monitoring by eddy current detection. GMR sensors have also been used in smart parking system where the approaching and leaving of vehicles can be sensed by measuring the MR due to the Earth’s magnetic field change induced by the vehicles’ movement. 8.4.3

Colossal Magnetoresistance (CMR)

CMR is a phenomenon of dramatically changed electrical resistance in the presence of a magnetic field. Most CMR materials are manganese-based perovskite oxides such as Lal−x Ax MnO3+𝛿 (A = Ca, Sr). The magnetoresistance

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of conventional materials enables changes in resistance of up to 5%, but CMR materials may exhibit resistance changes by orders of magnitude. The term colossal has arisen from the huge effects observed, in the order of ΔR/R(H) = 127 000%, as the resistivity dropped from 11.6 Ω/cm at H = 0 to 9.1 mΩ/cm at H = 6 T (Jin et al. 1994). Because of very large MR effect, and to distinguish it from the GMR effect, it is called CMR. The understanding and application of CMR offers tremendous opportunities for the development of new technologies such as read/write heads for high-capacity magnetic storage and spintronics. 8.4.4

Tunneling Magnetoresistance (TMR)

Electron tunneling is a quantum-mechanical effect, where electrons can traverse the potential barrier that exceeds their kinetic energy. When an electric field is applied to the magnetic tunnel junction (MTJ, consisting of two ferromagnetic electrodes and an ultrathin insulating barrier), there is a probability for electron tunneling. As the tunneling is spin dependent (a disproportion in the number of electrons parallel and antiparallel to the magnetization of a ferromagnet, usually referred to as majority- and minority-spin electrons), the imbalance leads to the measurable difference in the tunneling current carried by majority- and minority-spin electrons. This phenomenon is known as TMR. TMR effect is originated from spin-conserving tunneling processes between electrodes with spin-polarized density of state (DOS) as illustrated in Figure 8.13. From the simple heterostructure, we can see that an oxide tunnel barrier is sandwiched by two dissimilar ferromagnetic materials (a free layer and a pin layer). It should be noted that the free layer has a smaller H c than that of pin layer, thus the free layer acts as a soft magnet that its magnetization direction can be easily tuned and the pin layer acts as a hard magnet. In spin and energy conservation during tunneling, spin-up or spin-down electrons can only tunnel from the initial spin-up or spin-down state to an unoccupied spin-up or spin-down final state, respectively. In other words, the magnitude of TMR including current and resistance depends on the alignment of magnetization. Therefore, the TMR can be maximized with highly spin-polarized materials such as half-metallic manganites and Heusler alloys. To obtain the relative resistance change, TMR can be expressed by using Julliere’s model (Julliere 1975): Rap − Rp 2P1 P2 = (8.7) TMR = Rp 1 − P1 P2 where Rap is the resistance in antiparallel magnetization state, Rp in parallel magnetization state, and P1 and P2 spin polarizations of the two electrodes. The following equation describes the spin polarization P calculated from the spin-dependent DOS at the Fermi energy, where N ↑ is spin-up DOS and N ↓ is the spin-down DOS (Mazin 1999): P1 and P2 =

N↑ − N↓ N↑ + N↓

(8.8)

8.4 Magnetoresistance Effect and Device

Free layer (ferromagnetic) Tunnel barrier (oxide) Pin layer (ferromagnetic)

Iwrite

Iread E

E

Iwrite E

E

eU

eU

DOS of pinned layer

Iread

DOS of free layer

DOS of pinned layer

DOS of free layer

Figure 8.13 Diagram of spin-dependent TMR effect in parallel (left) and antiparallel (right) magnetization of a MTJ. The solid arrow represents larger tunneling probability.

8.4.4.1

Spin-Transfer Torque Random-Access Memory (STT-RAM)

In order to fulfill the demand of data storage, a memory based on MRAM combining with spin-transfer torque technology was developed recently, which is called STT-RAM as shown in Figure 8.14. Similar to a hard disk drive using materials with magnetic properties, STT-RAM also controls the orientation of magnetization except that it is operated with electrical current; the writing process involves an electrical current aligning the spin direction of electrons flowing through a MTJ element. The resultant resistance difference of the MTJ element is then used for information readout. The memory mainly consists of three layers same as MTJ, i.e. an ultrathin oxide barrier is sandwiched by two dissimilar ferromagnetic layers, one of them has a larger coercive field H c so that its magnetization alignment is difficult to be reoriented (it is called pin layer or hard layer). On the other hand, the free layer has a smaller H c , so its magnetization is much easier to be reoriented (it is called soft layer). Usually, the data bit is stored in this so-called soft layer according to its magnetization orientation.

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Bit line MTJ

Write word line

Bit line Bypass line

Cladding

MTJ Gate

Source

Silicon substrate (a)

Gate Drain

Source

Drain

Silicon substrate (b)

Figure 8.14 Diagram of (a) a conventional MRAM cell and (b) STT-RAM cell memory structures.

Similar to a ferroelectric tunnel junction (FTJ), wave function tunnels through this ferromagnetic layers sandwiched barrier and the memory state depends on the tunneling resistance. However, unlike FTJ, this effect is spin-polarized electron tunneling. As illustrated in Figure 8.14, in spin and energy conservation during tunneling, spin-up or spin-down electrons can only tunnel from the initial spin-up or spin-down state to an unoccupied spin-up or spin-down final state, respectively. It means that the magnitude of TMR depends on the alignment of magnetization. For TMR, if both magnetization alignments are parallel to each other, electrons will be able to tunnel, and the memory cell is in the low resistance state. However, if they are in antiparallel, the magnetoresistance is high. Most MRAMs that are now being developed write data by applying the magnetic field generated by a current running through a wire near a TMR element to change the magnetization. That enables fast operation switching but consumes more power. In the case of STT technology, a large and directional (spin-polarized) current changes the magnetic orientation of the free layer when it is writing. When it is reading, data can be identified by the resultant resistance difference of MTJ. This STT-RAM consumes less power because the spin valve or MTJ can be modified using spin-polarized current. The smaller memory cell that enhances scalability is also in favor of commercial applications. Starting from 2005, Renesas Technology Corp. and Hynix Semiconductor Inc. licensed Grandis’ intellectual property (IP) and began developing STT-RAM for embedded and standalone applications. In 2010, Grandis has developed a 90 nm, 265 Kbit MTJ devices, and they demonstrated STT-RAM write current of less than 200 μA, a write/read speed of 20 ns, and an endurance of 1013 cycles.

8.5 Magnetostrictive Effect and Device Applications Magnetostrictive effect refers to mechanical strains induced by magnetic field. This effect results from magnetization switching under applied magnetic field

8.5 Magnetostrictive Effect and Device Applications

and only happens below Curie temperature when a magnetic material is in the ferromagnetic phase. This effect exists in all magnetic materials either smaller or larger and can be positive and negative. The mechanical strain induced by magnetic field is called direct magnetostrictive effect, while the magnetization change resulted from applied mechanical strain is called inverse magnetostrictive effect. You may have experience to hear the low-frequency noise “electric hum” near transformers and high-power electrical devices. That sound comes from the magnetostrictive vibration of magnetic core material in the transformer at a frequency of power line that is usually 50 Hz. For device applications, in order to have large response to a magnetic field making it applicable for sensors or actuators, the magnetostrictive effect should be significant. Table 8.2 lists properties of magnetostrictive effect of some typical magnetic materials. In this table, 𝜀max = 32 𝜆s is the maximum strain that can be generated under a prestress condition, where 𝜆s is the saturation magnetostriction that quantifies strains generated from a relaxed ferromagnetic state to the state with all domains aligned along the magnetic field direction. Ms is the saturation magnetization in response to an applied magnetic field, which should be large for strong transduction; and T c is the Curie temperature at which a ferromagnetic to paramagnetic phase transition occurs. The magnetomechanical coupling coefficient k quantifies the efficiency to convert magnetic energy to mechanical energy, or vice versa, and elastic modulus Y is its mechanical properties quantifying the strain that can be generated under mechanical stress. These parameters are important for magnetostrictive material-based transducer design. This table provides general guidelines to select a suitable magnetostrictive (MS) material for device application. Fe, Ni, Co, and CoFe2 O4 possess negative and much smaller value of maximum strain and usually small magnetomechanical coupling factor k. These parameters make them less attractive for device application. Tb and Dy have very large magnetostriction, but their T c is too low to make them meaningful for device application. The most significant one is Terfenol-D (terbium: Ter, iron: Fe, Naval Ordnance Laboratory: NOL, dysprosium: D), which can produce strains up to 1600 μl/l at Table 8.2 Properties of magnetostrictive effect of some ferromagnetic materials. Material

𝜺max (×10−6 )

Ms (T)

T c (∘ C)

Y (GPa)

k

Fe

−14

2.15

770

285

—

Ni

−50

0.61

358

210

0.31

Co

−93

1.79

Tb

3000

— —

−184

61.4

—

380

110

0.77

Dy

6000

—

1620

1.0

CoFe2 O4

210 55.7

−48

Terfenol-D Metglas 2605SC

1120

—

60

1.65

370

25–200

0.92

−885

0.53

520

142

—

All measurements are at room temperature except where specified Source: Tsai et al. (2013). Reproduced with permission of AIP Publishing.

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room temperature and therefore can greatly expand the role of magnetostrictive smart material for transducer applications. Metglas is also practically very popular since it has the largest magnetomechanical coupling factor k, making it more efficient and having broader bandwidth of response to magnetic field. 8.5.1

Magnetostrictive Properties of Terfenol-D

Terfenol-D crystals are normally grown along [112] orientation, while their magnetization easy axes are along ⟨111⟩ directions. As what has been illustrated in piezoelectric that non-180∘ polarization rotation is essential to generate large strain, it is the same for MS. A Terfenol-D crystal bar along [112] at room temperature has randomly distributed magnetization domains along 8 equiv. ⟨111⟩ directions. To realize non-180∘ switching of magnetization moment, the crystal should be prestressed by compression in the bar direction, i.e. [112] orientation. This will make those ⟨111⟩ moments that are not perpendicular to the [112] direction to be switched to the [111] and [111] directions that are perpendicular to [112]. This can be demonstrated in Figure 8.15. You can imagine that by applying a magnetic field along [112] direction, these [111] and [111] moments tend to be switched back to their original ⟨111⟩ directions through non-180∘ rotation, therefore maximum strain along [112] can be generated. The general term magnetostriction refers to strains generated during the paramagnetic–ferromagnetic phase transition or in response to an applied field that constitutes magnetoelastic coupling. Usually there are three parameters of interest: the spontaneous magnetostriction 𝜆0 , saturation magnetostriction 𝜆s, and Joule magnetostriction 𝜆 below saturation. The relationship between these parameters and the total strain 𝜖 is illustrated in Figure 8.16. It has been calculated that if the total strain from the prestressed state to the final state is 𝜀, the prestress makes the crystal shrink 1/3 𝜀, which is the same as

[112] [111] [001] [112] Prestress [110] [111]

2a

a

Figure 8.15 Terfenol-D crystal orientation where [112] is the rod direction and ⟨111⟩ are magnetization easy axes.

8.5 Magnetostrictive Effect and Device Applications

(a) When T > Tc’ ε

(b) When T < Tc’ and field is absent,

λ0 = 1 ε 3

λs = 2 ε 3 (c) When T < Tc’ and H-field is applied along ↑ direction or under stress along →← direction,

(d) When T < Tc’ and H-field is applied along → direction

Figure 8.16 Illustration of the relationship among the spontaneous magnetostriction 𝜆0 , saturation magnetostriction 𝜆s , and the total strain 𝜀. Spheres represent the isotropic domains of paramagnetic phase above Curie temperature (a) and the ellipsoids represent the elongated domain shape at ferromagnetic phase below Curie temperature (b)–(d). (b) Without magnetic field, (c) magnetic field or mechanical stress forces domains along perpendicular direction, and (d) magnetic field along parallel direction.

the spontaneous magnetostriction 𝜆0 when the crystal passes from paramagnetic phase to ferromagnetic phase. Without prestress, the crystal can generate 2/3 𝜀 under magnetic field that is called saturation magnetostriction 𝜆s . Detailed calculation can be found at “Theory of the magnetomechanical effect” (Jiles 1995). As introduced in Jiles’s book (Jiles 2015), the spontaneous magnetostriction of a magnetic material when being cooled down to T c below, if a single domain with magnetization generate maximum strain 𝜀, the spontaneous strain along any direction 𝜆0 , can be expressed as 𝜀(𝜃) = 𝜀 cos2 𝜃 This equation is only suitable for the case when an isotropic paramagnetic (represented by the round circle in the diagram) changes to isotropic FM (represented by ellipsoids). Without external magnetic field, these domains, each of them is represented by one ellipsoid, are randomly orientated, and the spontaneous strain

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generated can be calculated by 𝜆0 =

𝜋∕2

∫ −𝜋∕2

𝜀 cos2 𝜃 sin 𝜃d𝜃 =

𝜀 3

where one can see that the spontaneous strain is 1/3 of the total strain. When these domains align their magnetization along external magnetic field, the extra strain (called saturated strain 𝜆s ), is then equal to 2/3 of the total strain, i.e. 𝜆s = 𝜀 − 𝜆 0 =

2 𝜀 3

Figure 8.16d illustrates the maximum strain that can be achieved from such isotropic magnetostrictive material through prestress to force all domains into alignment to the direction perpendicular to the external magnetic field, subsequently applied to rotate the magnetization moment by 90∘ . As illustrated in Figure 8.16, it also suggests the practical way to measure the total strain by measuring the difference of strains when the magnetic field is, respectively, perpendicular and parallel to the rod, i.e. 𝜆s|| − 𝜆s⟂ =

3 𝜆 =𝜀 2 s

The state 𝜆s⟂ can also be obtained by applying prestress forcing the domains into a state with magnetic moment along the perpendicular direction as illustrated by the ellipsoids in Figure 8.16d. One can imagine that when the parallel magnetic field and the prestress along the rod direction reach a metastable sate, minimum perturbation of ac magnetic field will cause the magnetic momentum to rotate from perpendicular to parallel or from parallel to perpendicular. In fact, this is the way to achieve maximum strain in real application that will be further demonstrated in next chapter. Another important relationship can be derived from 𝜀(θ) = 𝜀 cos2 𝜃 By substituting with M = Ms cos 𝜃 and then, 𝜆(𝜃) =

3 𝜆 cos2 𝜃 2 s

This results in a quadratic relationship between the magnetostriction and the 𝜆 magnetization, i.e. 𝜆(M) = 32 Ms M2 . s One should notice that electrostrictive effect is also the quadratic relation. Compared with P–E loop and 𝜀 − E butterfly loop of ferroelectric materials, M–H and 𝜀 – H exhibit similar hysteresis and butterfly loops as shown in the Figure 8.17.

8.5 Magnetostrictive Effect and Device Applications

1.0 0.8 0.6 0.4

M/Ms

0.2 0 –0.2 –0.4

f = 1 Hz f = 500 Hz f = 1000 Hz f = 2000 Hz

–0.6 –0.8 –1.0 –1.5

–1

–0.5

0

0.5

1

H (A/m)

(a)

1.5 ×105

1800 f = 1 Hz f = 500 Hz f = 1000 Hz f = 2000 Hz

Magnetostrictive strain (10–6)

1600 1400 1200 1000 800 600 400 200 0 –1.5 (b)

–1

–0.5

0

0.5

H (A/m)

1

1.5 ×105

Figure 8.17 The magnetic (a) hysteresis loops and (b) magnetostrictive strain hysteresis loops of Terfenol-D under the sinusoidal driving magnetic field of same amplitude and different frequencies. Source: Xiao et al. (2017). Reprinted with permission of Elsevier.

8.5.2

Magnetostrictive Ultrasonic Transducer

Magnetostrictive materials have very broad application in magnetic field-driven transducers, such as micromachining, spray nozzle in gas injection, ultrasound wave generation by magnetic field, etc. Figure 8.18 shows a diagram of ultrasonic transducer that can generate ultrasound wave driven by magnetic field.

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Figure 8.18 Schematic diagram of a magnetic field-driven ultrasonic transducer utilizing magnetostrictive effect.

Solenoid with electrical ac input

Magnetostrictive laminations Magnetization and strain induced by magnetic field

A magnetostrictive ultrasonic transducer consists of laminations of a FM material to be ultrasonically activated when an electrical coil produces an ac magnetic field. A magnetostrictive element extends when current passes through the solenoid and shrinks when the current is off; this causes the ferromagnetic laminates to vibrate at their resonant frequency. Although piezoelectric transducers dominate in the ultrasonic world today, magnetostrictive technology is still available from a limited number of ultrasonic manufacturers and, in fact, is the technology of choice of some users in some low frequency applications up to 30 kHz. 8.5.3

Magnetoelastic Effect

The inverse magnetostrictive effect (also known as magnetoelastic effect or Villari effect) refers to the change of the magnetic susceptibility of a material when a mechanical stress is applied on it. The magnetostriction characterizes the shape change of a FM material during magnetization, whereas the inverse magnetostrictive effect characterizes the change of sample’s magnetization (for a given magnetizing field strength) when a mechanical stress is applied to the sample. This inverse magnetostriction can be understood by considering the strain-induced crystal symmetry change, which causes the change of easy axis leading to magnetization switching. Therefore, under a given uniaxial mechanical stress, the flux density for a given magnetization field may increase or decrease. Magnetoelastic effect can be used in development of force sensors applied in engineering, such as civil engineering and biomedical engineering. The force can be measured by detecting the sample’s magnetization change under a

8.5 Magnetostrictive Effect and Device Applications

given magnetic field. Maximum sensitivity can be realized if an appropriate dc magnetic field and prestress are applied. Analogous to their ferroelectric counterpart, in ferromagnetic materials, magnetic field and stress induce strain and magnetization that can be linearly modeled under low field drive. In engineering application such as transducer design, the linear approximation in modeling can be reasonably accurate; however, one should notice that in the regime of high field, either electric or magnetic, the piezoelectric and ferromagnetic responses are nonlinear, presenting hysteresis characteristics. The physics behind this nonlinearity is the domain evolution including domain wall movement. Analogous to piezoelectric linear relation as shown in Chapter 6, the direct and inverse magnetostrictive effect can be summarized by the linear constitutive matrix relations 𝜀 = SH 𝜎 + d∗ H M = d𝜎 + 𝜒 𝜎 H where 𝜀 is strain, SH compliance at constant field H, 𝜎 stress, M the magnetization, 𝜒 𝜎 magnetic susceptibility at constant stress, and d* the transpose of piezomagnetic coefficient d. 8.5.3.1

Magnetomechanical Strain Gauge

Apart from Terfenol-D, Metglas is another very common magnetostrictive composite that is the amorphous alloy Fe81 Si3.5 B13.5 C2 with its trade name Metglas 2605SC. Originated from its amorphous structure, Metglas has no anisotropic properties and there are no crystalline grain boundaries to prevent motion of magnetic domain walls, so it shows excellent magnetic properties such as high permeability and low loss while having a high-saturation magnetic flux density. Though the maximum strain is much smaller, about 60 microstrains, compared with Terfenol-D, Metglas 2605SC exhibits both magnetostriction and magnetoelastic effect. The very good mechanical elastic properties, a very large magnetomechanical coupling factor, and the energy conversion efficiency make Metglas attractive for magnetic micro-electro-mechanical systems (MEMS) devices. A strain gauge using Metglas 2605SC was predicted to have 103 to 105 more sensitivity than a conventional strain gauge (Savage and Spano 1982). Figure 8.19

Force Metglas ribbon

V Detection coil Excitation coil

Figure 8.19 Schematic diagram of force sensor using Metglas core.

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shows a diagram of the force sensors using magnetoelastic effect. When a force is added, the magnetoelastic material deforms, and as a result, the strain can be detected by measuring the flux density change.

8.6 Characterizations of Ferromagnetism Magnetization and the corresponding properties can be measured directly or indirectly by employing different instruments and techniques that can be categorized into inductive, force, magneto-optic, and electrical methods. In the following sections, some commonly used instruments and techniques for magnetic property characterizations are briefly introduced. 8.6.1

Vibrating Sample Magnetometer (VSM)

Measuring M–H loop is the most fundamental characterization technique to prove a material to be FM, with which the saturation magnetization, remnant magnetization, and coercive field can be determined. The M–H loop can be obtained by vibrating sample magnetometer (VSM) that uses a technique of induction to measure magnetic moment of a sample. The change of magnetization can be determined by measuring the change of magnetic flux density inside the sample if an external magnetic field is known. During measurement, a sample is placed at one end of the rod and starts vibrating along vertical direction at a fixed frequency as illustrated in Figure 8.20a. In the meantime, an external magnetic field, generated by a pair of electromagnets, magnetizes the sample along the horizontal parallel direction. The sample’s magnetic moment then alters with the strength of the applied magnetic field and results in a M–H loop for ferromagnetism. The basic principle of manipulation is to detect the induced voltage in pick-up coil when a magnetic flux of the sample is changing, and the induced voltage e.m.f . is obtained synchronously using ( ) 𝜕 𝜕𝛷 = B ⋅ dS e.m.f . = − 𝜕t 𝜕t ∫ where 𝛷 is the flux linked by the coils and dS is an element of vector area. Referencing to a well-defined sample, the absolute values of magnetic moments of sample can be attained. VSM is also applicable to low-temperature characterization by combining with cryostat equipment, for example, magnetic moment can be detected from 77 to 350 K by using liquid nitrogen in a VSM system. For further cooling, 2 K can also be achieved if liquid helium is used. Figure 8.20b,c shows the magnetic moment of La0.7 Sr0.3 MnO3 and Ni0.81 Fe0.19 thin films in response to the external magnetic field. Except for FM classification, VSM can also be used to observe the strength of magnetization in in-plane and out-of-plane directions. It is apparent that both thin films have a stronger magnetization signal parallel to H-field than the perpendicular one. It describes that the magnetic dipoles are pointing parallel to the film instead of pointing perpendicular to the film. In the meantime, an extra high magnitude of field is required to switch the moment within the same sample for

8.6 Characterizations of Ferromagnetism

Sample rod with oscillation

Sample

Pick-up coils

Electromagnets

Magnetic field (a) 150 LSMO NiFe

Moment (μemu)

Moment (μemu)

400 200 0 –200

LSMO NiFe

100 50 0 –50 –100

–400 –200

(b)

0

Magnetic field (Oe)

–150

200

(c)

–200

0

200

Magnetic field (Oe)

Figure 8.20 (a) Schematic diagram of VSM system and M–H loops of two magnetic materials LSMO and NiFe in in-plane (b) and (c) out-of-plane magnetization directions at 78 K.

the out-of-plane measurement. Based on the results, VSM can also be used for specifying the easy axis of magnetization for materials. Using inductive method, for instance, VSM, FM properties including the direction of easy axis, saturation magnetization, and coercivity can be obtained by analyzing M–H loops of materials. Other common inductive methods include extraction magnetometry, AC susceptometry, and superconducting quantum interference device (SQUID) magnetometry. 8.6.2

Superconducting Quantum Interference Device (SQUID)

Although SQUID and VSM are both inductive methods, VSM is not suitable for materials that are magnetically diluted or that possess very small sample volumes, i.e. nanoscale magnet (Spinu et al. 2013). The sensitivity of VSM may be insufficient to properly characterize such materials, while SQUID is a more sensitive magnetometer used to measure extremely subtle magnetic fields, based on superconducting loops containing Josephson junctions that is a tunnel junction between two superconducting layers sandwiched with an ultrathin insulating layer.

187

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Josephson junctions

Ia

Is

I

I

Φ

Is

Ib

Superconducting loop

Figure 8.21 Diagram of a DC SQUID.

Quantum effect in conjunction with superconducting detection coil circuitry are used in SQUID-based magnetometers to measure the magnetic properties of materials. In a DC SQUID, it has two Josephson junctions in parallel in a superconducting loop as shown in Figure 8.21. An input biasing current I enters the superconducting loop and splits into path I a and path I b equally. If an external magnetic field is applied to the loop, a current I s begins circulating within the loop that generates an opposite magnetic field against the external flux 𝛷. In the two branches, the total current in each branch is changed because of the induced current, i.e. one is I/2 + I s and the another one is I/2 − I s . When the total current in either branch exceeds the critical current I c of the Josephson junction, a voltage appears across the junction. The voltage is therefore a function of the applied magnetic field and the period of the magnetic flux quantum 𝛷0 . R ⋅ Δ𝛷 L Similar to flux to voltage converter, Δ𝛷 can be estimated as the function of ΔV using this equation, where R is a shunt resistance connected across the junction to eliminate the hysteresis and L is the self-inductance of the superconducting ring. Theoretically, SQUIDs are capable of achieving sensitivities of 10−12 emu, but practically they are limited to 10−8 emu, because SQUIDs also pick up environmental noise similar to VSM. SQUIDs may also be used to perform measurements from low to high temperatures; so temperature-dependent magnetization and other magnetic properties can also be characterized by SQUIDs. ΔV = R ⋅ ΔI =

8.6.3

Magnetic Force Microscopy (MFM)

Besides the macroscopic magnetization determination, ferromagnetic domains can be observed. Again, similar to the manipulation of piezoelectric materials using piezoresponse force microscopy (PFM), the magnetic structure and kinds of magnetic interactions of sample surface can be reconstructed by magnetic force microscopy (MFM).

8.6 Characterizations of Ferromagnetism

Tip with ferromagnet coating 30 25

μm

20

Path of cantilever

15 10 5 0

Magnetic sample with domains (a)

(b)

0

5

10

15

μm

20

25

30

Figure 8.22 (a) Schematic diagram of MFM mechanism and (b) MFM image of a magnetic recording disc displaying a multi-scale domain structure.

As illustrated in Figure 8.22a, In MFM measurement, when a sharp tip coated with a ferromagnetic thin film scans a magnetic sample, local oscillatory deformation as the tip-sample force will be induced on the sample surface due to the magnetic interactions. The frequency and phase change of oscillatory deformation response possess a kind of relationship with the strength of the magnetic field gradient, and this phase angle together with amplitude can be measured by lock-in technique and imaged to reveal the ferromagnetic magnetization and domain structure. The magnetic AFM probe detects the surface oscillation and sends the signal through the reflected laser beam to photodiode detector and be processed by a lock-in amplifier. The amplitude measured by the lock-in amplifier represents the strength of magnetization (if well calibrated), and the phase angle represents the magnetic moment direction; under ideal conditions, 0∘ and 180∘ phase angles represent the up and down magnetization directions, respectively. By forming an image with phase angle, the MFM technique can image ferromagnetic domains by presenting the bright-dark contrast. With this method, magnetic characteristics of such domains and magnetic defects can be revealed on a variety of materials, especially for their surface properties. Figure 8.22b is a MFM image of a magnetic recording disc showing magnetic domains on magnetic film representing memory states of 0 and 1. 8.6.4

Magneto-Optical Kerr Effect (MOKE)

Magneto-optical Kerr effect (MOKE) is originated from the optical anisotropy of the materials, and it describes the change to light reflected by a magnetized surface. MOKE imaging is simply based on the rotation of the polarization plane of linearly polarized light when reflected from the surface of a magnetic material. By integrating domain contrast relating to the magnitude and direction of the magnetization in the materials, their electronic and magnetic properties such as the magnetic domain structure, spin DOSs, and even magnetic phase transition dynamics can be revealed. MOKE technique has been widely used for

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Polarized incident light

Reflected light

Magnetic sample with domains

(c) (b) (a)

Figure 8.23 Schematic illustration showing the optical experiment for observing the (a) polar, (b) longitudinal, and (c) transverse Kerr effects.

recent progress on high-quality nanostructures and 2D materials (e.g. graphene, topological insulators, and transition metal dichalcogenides) for enhanced control of light at nanoscale for integrated photonic and spintronic devices. In MOKE effect, both polarization and reflected intensity of the reflected light from a magnetized surface can be changed. As shown in Figure 8.23, when a polarized light incidents to the magnetic sample and reflects, due to the different magnetization orientations with respect to the reflecting surface and the plane of incidence, MOKE can be divided into three types. First, in the polar Kerr effect (Figure 8.23a), the magnetization is perpendicular to the sample plane and is parallel to the plane of incidence. In this case, the MOKE detection is out-of-plane that the Kerr effect generally alters the polarization of the incident light from plane to elliptically polarized with the major axis rotated. Nearly normal incidence is usually employed when doing experiments in the polar geometry. Second, in the longitudinal Kerr effect as shown in Figure 8.23b, incident light with angle (typically 60∘ (Celotta et al. 2002)) is employed. The in-plane magnetization vector is parallel to both the reflection surface and the incident plane, and the linearly polarized light incidents on the surface is also elliptically polarized same as the polar Kerr effect. Third, in the transverse Kerr effect, the magnetization is also in the plane of the sample but is perpendicular to the incident plane (Figure 8.23c). In this case, the incident light is not normal to the reflection surface, but there is no change in the polarization of the incident light. Indeed, only the intensity of light will be changed, and the change in reflectivity is proportional to the magnetization orientation. The longitudinal and transverse Kerr effects are generally used to study the in-plane magnetic anisotropy, whereas the polar effect is used to study thin films that exhibit perpendicular anisotropy. Typically, magneto-optic signal and images can be generated by using MOKE magnetometer and MOKE microscopy, respectively. For microscopy, magneto-optic images can be obtained either by

8.7 Hall Effect

using conventional imaging optics or a finely focused laser spot across the sample surface. Therefore, the technique is moderately surface sensitive and can be used to image magnetic domains in thin films that are only a few monolayers thick.

8.7 Hall Effect 8.7.1

Ordinary Hall Effect

As shown in Figure 8.24a, Hall effect describes the production of a potential difference (the Hall voltage V H ) across a current-carrying conductor in the presence of magnetic field, which is perpendicular to both current and the magnetic field. The Lorentz force F on a charge carrier can be expressed as F = q(E + (v × B)) where E is the Hall electric field, v the drift velocity of the current, q the charge of a particle, and B the magnetic induction of the conductor. At a steady state with F = 0, the Hall voltage can be obtained as Ix Bz nte where I is the current, n the charge carrier density, t the thickness of conductor, and e the charge of each electron. The Hall coefficient is defined as VH =

RH =

Ey jx B

=

VH t 1 =− IB ne

where j is the current density of the carrier electrons and Ey is the induced electric field. As a result, the carrier density or the magnetic field can be measured with the aim of Hall effect. 8.7.2

Anomalous Hall Effect

In addition to the ordinary Hall effect presented in semiconductors and metals, an effect called anomalous Hall effect (AHE) has been discovered claiming that part of Hall voltage is proportional to the magnetization at fields up to magnetic saturation (Pugh and Rostoker 1953). AHE is usually much larger than the ordinary Hall effect. Therefore, for ferromagnetic elements Fe, Co, and Ni, the Hall voltage is determined mainly by the magnetization, resulting in an unusual nonlinear relationship between Hall electric field and magnetic induction coming from AHE. The AHE has been recognized as a useful tool for measuring M–H hysteresis loops of perpendicular magnetic recording media, spintronic devices, and diluted magnetic semiconductors. The relationship between Hall electric field per current density EH , magnetic field Hz , and magnetization Mz in most ferromagnetic materials is established as EH = R0 Hz + R1 Mz

191

192

8 Ferromagnetics: From Material to Device

B

M

–F

–

– –

– –

–

– t

–

–

–

E

w

F

– t

I

E

w

B=0

I

(a)

(b)

Figure 8.24 (a) Ordinary Hall effect and (b) anomalous Hall effect (AHE). B is the magnetic induction rise by external magnetic field; M is the intrinsic spontaneous magnetization.

where the first term represents the ordinary Hall effect mentioned, while the second term represents the AHE contribution due to the spontaneous magnetization as illustrated in Figure 8.24b. Unlike R0 that only depends on the carrier density, R1 depends on the effects of the spin–orbit interaction of spin electrons (Karplus and Luttinger 1954). As a result, specific magnetic parameters of FM materials, such as magnetization processes and types of materials, can be determined using AHE technique. 8.7.3

Spin Hall Effect

There is another transport phenomenon in magnetic materials analogous to the classical Hall effect – the spin Hall effect (SHE), which consists of the appearance of spin accumulation on the lateral surfaces of a current-carrying sample. From Figure 8.25a, we can see that SHE causes electrons carrying different signs of spins to drift in opposite boundaries, thus generating a transverse pure spin current. When the current direction is reversed, the directions of spin orientation are also reversed. Spin Hall voltage is originated from two possible mechanisms, where an electrical current with moving charges transforms into a spin current with moving spins without charge flow. The first possible mechanism is spin-dependent Mott scattering, where carriers with opposite spins diffuse in opposite direction when Spin current Is

σ

Electromotive force EISHE +

σ

– Spin curr

ent I

s

(b) (a)

Charge current Ic

Figure 8.25 (a) Spin Hall effect and (b) inverse spin Hall effect, 𝝈 denotes the spin polarization vector of the spin current.

References

colliding with the impurities within the material. The second possible mechanism is caused by the intrinsic properties of the material that the trajectories of carriers are distorted due to spin–orbit interaction as a consequence of the asymmetries in the material. Owing to the accumulation of spins of both lateral boundaries, the current and spin Hall voltage can be detected. By the inverse spin Hall effect (ISHE, Figure 8.25b) in a strong spin–orbit coupling material, the spin–orbit interaction bends trajectories of these two electrons in the same direction. The spin current can then be converted into an electric field EISHE with a charge current again in the transverse direction, resulting in charge accumulation at the sample edges. For the SHE, no magnetic field is required as it is a purely spin-based phenomenon dissimilar to the ordinary Hall effect, but it behaves similar to AHE, known for a long time in ferromagnet, which also originates from spin–orbit interaction. SHE can be used to manipulate electron spins electrically, and a pure spin current can even be generated by non-magnetic materials.

References Baibich, M.N., Broto, J.M., Fert, A. et al. (1988). Giant magnetoresistance of (001)Fe/(001)Cr magnetic superlattices. Physical Review Letters 61 (21): 2472–2475. ˇ ˇ Cápová, K., Cáp, I., Janoušek, L., and Smetana, M. (2008). Recent trends in electromagnetic non-destructive sensing. Advances in Electrical and Electronic Engineering 7 (1): 322–325. Available at: http://hdl.handle.net/10084/84200. Celotta, R.J., Unguris, J., Kelley, M.H., and Pierce, D.T. (2002). Characterization of Materials. In: Techniques to measure magnetic domain structures (ed. E.N. Kaufmann), 1. https://doi.org/10.1002/0471266965.com046. Available at: http:// doi.wiley.com/10.1002/0471266965.com046. Daughton, J.M. (1992). Magnetoresistive memory technology. Thin Solid Films 216: 162–168. Eerenstein, W., Mathur, N.D., and Scott, J.F. (2006). Multiferroic and magnetoelectric materials. Nature 442 (7104): 759–765. Available at: http://www .ncbi.nlm.nih.gov/pubmed/16915279. García-Martín, J., Gómez-Gil, J., and Vázquez-Sánchez, E. (2011). Non-destructive techniques based on eddy current testing. Sensors 11 (3): 2525–2565. Available at: http://www.mdpi.com/1424-8220/11/3/2525. Grünberg, P., Schreiber, R., Pang, Y. et al. (1987). Layered magnetic structures: evidence for antiferromagnetic coupling of Fe layers across Cr interlayers. Journal of Applied Physics 61 (8): 3750–3752. Hu, J., Li, Z., Chen, L., and Nan, C. (2012). Design of a voltage-controlled magnetic random access memory based on anisotropic magnetoresistance in a single magnetic layer. Advanced Materials 24 (21): 2869–2873. Available at: https:// onlinelibrary.wiley.com/doi/abs/10.1002/adma.201201004. Inoue, J.I. (2013). GMR, TMR, BMR, and related phenomena. In: Nanomagnetism and Spintronics, 2e, 15–106. Elsevier B.V. Available at: http://dx.doi.org/10.1016/ B978-0-444-63279-1.00002-2.

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Jiles, D. (2015). Introduction to Magnetism and Magnetic Materials, 3e. CRC Press. Jiles, D.C. (1995). Theory of the magnetomechanical effect. Journal of Physics D: Applied Physics 28: 1537–1546. Jin, S., Tiefel, T.H., McCormack, M. et al. (1994). Thousandfold change in resistivity in magnetoresistive La-Ca-Mn-O films. Science 264 (5157): 413–415. Available at: http://www.sciencemag.org/cgi/doi/10.1126/science.264.5157.413. Julliere, M. (1975). Tunneling between ferromagnetic films. Physics Letters A 54 (3): 225–226. Karplus, R. and Luttinger, J.M. (1954). Hall effect in ferromagnetics. Physical Review 95 (5): 1154–1160. Kiselev, S.V., Ozerov, R.P., and Zhdanov, G.S. (1963). Detection of magnetic order in ferroelectric BiFeO3 by neutron diffraction. Soviet Physics – Doklady 7 (8): 742–744. Kriegner, D., Výborný, K., Olejník, K. et al. (2016). Multiple-stable anisotropic magnetoresistance memory in antiferromagnetic MnTe. Nature Communications 7: 11623. Available at: http://dx.doi.org/10.1038/ncomms11623. Liu, Y., Tang, N., Wan, X. et al. (2013). Realization of ferromagnetic graphene oxide with high magnetization by doping graphene oxide with nitrogen. Scientific Reports 3: 2566. Available at: http://dx.doi.org/10.1038/srep02566. Lu, C., Gao, B., Dong, S. et al. (2018). Revealing controllable anisotropic magnetoresistance in spin–orbit coupled antiferromagnet Sr2 IrO4 . Advanced Functional Materials 28 (17): 1706589. Available at: https://onlinelibrary.wiley .com/doi/abs/10.1002/adfm.201706589. Ma, Y.W., Lu, Y.H., Ding, J. et al. (2012). Room temperature ferromagnetism in Teflon due to carbon dangling bonds. Nature Communications 3: 727. Available at: http://dx.doi.org/10.1038/ncomms1689. Mazin, I.I. (1999). How to define and calculate the degree of spin polarization in ferromagnets. Physical Review Letters 83 (7): 1427–1430. Nickel, J. (1995). Magnetoresistance Overview: HP Laboratories Technical Report: HPL-95-60. Poon, T.Y., Tse, N.C.F., and Lau, R.W.H. (2013). Extending the GMR current measurement range with a counteracting magnetic field. Sensors 13 (6): 8042–8059. Available at: http://www.mdpi.com/1424-8220/13/6/8042. Pugh, E.M. and Rostoker, N. (1953). Hall effect in ferromagnetic materials. Reviews of Modern Physics 25 (1): 151. Rao, C.N.R. and Mahesh, R. (1997). Giant magnetoresistance in manganese oxides. Current Opinion in Solid State and Materials Science 2 (1): 32–39. Available at: http://www.sciencedirect.com/science/article/pii/S1359028697801027. Saurenbach, F., Walz, U., Hinchey, L. et al. (1988). Static and dynamic magnetic properties of Fe-Cr-layered structures with antiferromagnetic interlayer exchange. Journal of Applied Physics 63 (8): 3473–3475. Savage, H.T. and Spano, M.L. (1982). Theory and application of highly magnetoelastic Metglas 2605SC. Journal of Applied Physics 53: 8092. Spinu, L., Dodrill, B.C., and Radu, C. (2013). Magnetometry measurements. In: , 1–3. Magnetics Technology International. Tsai, C.Y., Chen, H.R., Hsieh, W.F. et al. (2013). Stress-mediated magnetic anisotropy and magnetoelastic coupling in epitaxial multiferroic

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PbTiO3 -CoFe2 O4 nanostructures. Applied Physics Letters 102 (13): 132905. Available at: https://doi.org/10.1063/1.4800069. Xiao, Y., Gou, X.-F., and Zhang, D.-G. (2017). A one-dimension nonlinear hysteretic constitutive model with elasto-thermo-magnetic coupling for giant magnetostrictive materials. Journal of Magnetism and Magnetic Materials 441: 642–649.

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9 Multiferroics: Single Phase and Composites 9.1 Introduction on Multiferroic Multiferroics is defined as a physical phenomenon of simultaneous exist of more than one ferroic order parameters in a material. The basic primary ferroic order parameters are ferroelectricity, ferromagnetism, and ferroelasticity. Non-primary order parameters such as antiferroelectricity, antiferromagnetism, and ferrimagnetism can also be included into the definition of multiferroics. More than a century ago, Pierre Curie predicted that there should be materials whose magnetism is induced by electric field E and its polarization by magnetic field H (Curie 1894). This conjecture can be illustrated by the following equations: M𝛼 = G𝛼𝛽 E𝛽

(9.1)

P𝛼 = G𝛼𝛽 H𝛽

(9.2)

where M and P are ferromagnetism and ferroelectric polarization, respectively, and G is the corresponding permittivity reflecting the coupling of magnetism and electric polarization. Subscripts 𝛼 and 𝛽 are, respectively, the directions of field applied and response. These two equations illustrate the electrical control of magnetism and magnetic control of electricity. The merit of multiferroic is not only the existence of more than two ferroic properties in one material but also the interplay and mutual control of each other through magnetoelectric (ME) coupling effect. This will broaden the horizon of design and application of electromagnetic devices, and it would be amazing if M–E loop and P–H loops can be easily achieved beyond the well-known P–E and M–H loops as illustrated in Figure 9.1. However, until 1959, this conjecture, ferroelectric magnetism or magnetic ferroelectricity, was first experimentally observed in Cr2 O3 , and in 1966 Schmid reported another multiferroic Ni3 B7 O13 (Ascher et al. 1966). The reason that this conjecture has not been observed for half century since predicted is because of the fact that it is very rare for a single phase material exhibiting detectable level of multiferroics (Dzyaloshinskii 1960; Astrov 1960). The physics behind is introduced in the following. Multiferroic properties are closely linked to crystal structural symmetries. The primary ferroic properties can be characterized by their symmetry breaking Ferroic Materials for Smart Systems: From Fundamentals to Device Applications, First Edition. Jiyan Dai. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

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9 Multiferroics: Single Phase and Composites

M

P

H

E

Figure 9.1 Illustration of multiferroics with expected P–H and M–E hysteresis loops.

under space and time inversions. Space inversion will reverse the direction of polarization P, while leaving the magnetization M invariant. Time reversal, in turn, will change the sign of M, while the sign of P remains invariant. Magnetoelectric multiferroics require simultaneous violation of space and time inversion symmetries. In BiFeO3 , for example, off-centering of ions gives rise to an electric polarization, while additional magnetic ordering breaks time-reversal symmetry. Therefore, BiFeO3 has been extensively studied as a model multiferroic material that will be particularly introduced in the following text. Although scientists have already started experimental observation and theoretical prediction of multiferroics in single phase materials many decades ago, the term multiferroics was firstly used by H. Schmid in 1994 (Schmid 1994). Since then, the concept of multiferroics has been expanded to materials that perform any type of long-range magnetic ordering, a spontaneous electric polarization, and/or ferroelasticity, and some researchers argue that the term multiferroics can only be used in materials that exhibit coupling of the ferroic order parameters. The overlap of these properties in ferroic materials is illustrated in Figure 9.2 (Eerenstein et al. 2006).

Multiferroic

Ferroelectric

Ferromagnetism

Electrically polarizable

Magnetically polarizable

Magnetoelectric

Figure 9.2 Relationship between ferroelectrics, ferromagnetics, multiferroics, and even the magnetoelectrics. Source: Adapted from (Eerenstein et al. (2006)).

9.3 Why Are There so Few Magnetic Ferroelectrics?

9.2 Magnetoelectric Effect The magnetoelectric effect is defined as the generation of electric polarization in a material under an applied external magnetic field, or conversely, the production of an induced magnetization by an external electric field. For device application, a detectable ME coupling coefficient is essential. In ME composites, the ME coupling is associated with mechanical interaction between the magnetostrictive and piezoelectric phases. The induced polarization P is related to the magnetic field H by the expression P = 𝛼H

(9.3)

where 𝛼 is the second-rank ME-susceptibility tensor. A parameter of importance is the ME voltage coefficient 𝛼 E that is defined as 𝜕E (9.4) 𝛼E = 𝜕H The relationship between 𝛼 and 𝛼 E is 𝛼 = 𝜖 0 𝜖 r 𝛼 E and 𝜖 r is the relative permittivity of the dielectric materials. While in single phase materials, the ME effect is via the coupling between magnetic and ferroelectric orders; this is extensively described in Landau’s theory by expressing the free energy F of the system in terms of an applied magnetic field H and an applied electric field E (Landau and Lifshitz 1960). From the free energy term, the ME coupling coefficient can be derived by second-order differentiation. Expansion of the free energy in isotropic system following Siratori’s expression (Siratori et al. 1992) F(E, H) = F0 − P 𝟎 ⋅ E − M 𝟎 ⋅ H − ET ⋅ 𝜒e ⋅ E − H T ⋅ 𝜒m ⋅ H − ET ⋅ 𝛼 ⋅ H − … where F 0 is the free energy independent of E and H, P0 and M 0 denote the spontaneous polarization and magnetization, and tensors 𝜒 e and 𝜒 m are the dielectric and magnetic susceptibilities. The tensor 𝛼 denotes ME coupling coefficients, corresponding to induction of polarization by a magnetic field or of magnetization by an electric field that is designated as the linear ME effect. Derivative of F in E leads to the electric polarization: 𝜕F = P 𝟎 + 2𝜒e ⋅ E + 𝛼 ⋅ H + … P=− 𝜕E Derivative of F in H leads to the magnetization: 𝜕F M=− = M 𝟎 + 2𝜒m ⋅ H + 𝛼 ⋅ E + … 𝜕H ME coupling factor can be found by 𝛼=

𝜕2F 𝜕M 𝜕P = = 𝜕H𝜕E 𝜕E 𝜕H

9.3 Why Are There so Few Magnetic Ferroelectrics? The reason behind the fact that it is rare for a material to be magneticferroelectrics is the mutual exclusion of magnetism and ferroelectricity. In

199

200

9 Multiferroics: Single Phase and Composites

fact, a ferroelectric material must be an insulator, while ferromagnets are often metals. For example, the driving force for ferromagnetism in the elemental ferromagnets Fe, Co, and Ni and their alloys is the high density of states at the Fermi level, resulting in metallicity. The magnetism requires partially occupied d or f orbitals, while ferroelectricity usually requires that d orbitals should be empty (Matthias rule requires diamagnetic ions, i.e. d0 ), such as Ti4+ and Ta5+ . Therefore, multiferroism involves multiple competing factors, among which d-electron occupancy in the transition metal is critical. To be a multiferroic with partially occupied d orbital and also distortion of lattice forming ferroelectric polarization, Jahn–Teller effect plays a crucial role. The Jahn–Teller effect (JT effect or JTE) is an important mechanism of spontaneous symmetry breaking in molecular and solid-state systems; this effect is named after Hermann Arthur Jahn and Edward Teller, who first reported studies about it in 1937 (Jahn and Teller 1937). The Jahn–Teller effect, sometimes also known as Jahn–Teller distortion, describes the distortion of molecules and ions to a lower energy conformation that is associated with certain electron configurations. The Jahn–Teller theorem essentially states that any nonlinear molecule with an orbital and spin motion degenerate in electronic state will undergo a geometrical distortion to compensate that degeneracy, because the distortion lowers the overall energy of the species. Not only rare in nature, ME voltage response in single phase compounds is also generally weak. However, relatively strong ME voltage output may be achieved by artificially integrating magnetostrictive and piezoelectric phases in a composite material in which the mechanical deformation from magnetostriction results in measurable voltage generated from piezoelectric effect. In 1970s, Van den Boomgaard firstly synthesized bulk composites of cobalt ferrite and nickel ferrite with BaTiO3 by a solid-state reaction technique, yielding a ME voltage coefficient 𝛼 E of 130 mV/cm⋅Oe (vanSuchtelen 1972). In the recent decades, new single phase multiferroics and ME composites in a variety of forms, such as piezoelectric/magnetostrictive stacks and polymer–ceramic 0–3 composites (particles in matrix), have attracted considerable interest due to their large ME coupling effect and potential application in novel electronic devices. In the following sessions, recent results on single phase and composite multiferroic materials will be introduced.

9.4 Single Phase Multiferroic Materials Typical single phase multiferroics belong to the group of perovskite transition metal oxides. The most typical single phase multiferroics with relatively large and measurable ME effect is BiFeO3 . Other examples include rare-earth manganites and ferrites such as TbMnO3 , BiMnO3 , HoMn2 O5 , and LuFe2 O4 . In addition, some non-oxide materials such as BaNiF4 and spinel chalcogenides, e.g. ZnCr2 Se4 , are also included in the multiferroic family. The history of ME multiferroics can be traced back to the 1960s. After the initial burst of interest of research on antiferromagnetic Cr2 O3 , more and more

9.4 Single Phase Multiferroic Materials

materials either artificially prepared or existed in nature are synthesized and discovered in the form of ceramic, single crystal, and even thin film. Although the progress of research in multiferroic materials remained static until 2000, the discovery of strong magnetic and electric coupling in orthorhombic TbMnO3 (Kimura et al. 2003) and TbMn2 O5 (Hur et al. 2004) stimulated people’s interest again in the field of multiferroic complex oxides. A spontaneous electric polarization in MnWO4 below ∼13 K in the magnetically ordered phase (Heyer et al. 2006) reveals that MnWO4 is a novel multiferroic material. Giant magnetoelastic coupling in multiferroic hexagonal manganites has been observed in single crystal YMnO3 (Lee et al. 2008). This extremely large magnetoelastic coupling, larger by 2 orders of magnitude than any other magnetic materials, is the primary origin of their multiferroic phenomenon. Most importantly, the discovery of large ferroelectric polarization (Wang et al. 2003) and the realization of electric-field-controlled magnetic state (Zhao et al. 2006) in epitaxial thin films of antiferromagnetic BiFeO3 (BFO) create a new degree of freedom to manipulate the ME coupling in nanoscale and pave the way for using multiferroic ME materials to on-chip integration with new generations of electronic devices. Currently as the most promising multiferroic materials, BFO has already shown its potential application in spintronics (Béa et al. 2008) and ferroelectric random access memory (FeRAM) (Mathur 2008). In fact, the theoretical calculation and experimental investigation on BFO started from 1960s (Roginskaya et al. 1966; Krainik et al. 1966; Kiselev et al. 1963), but it did not show any practical potentials for applications until the observation of large room temperature ferroelectric polarization combining interesting magnetic properties in strongly strained epitaxial BFO films (but is always absent in its bulk counterparts). Among the few reported room temperature single phase ferroelectric antiferromagnetism (Kimura et al. 2005), BFO thin film shows the highest ferroelectric polarization of 90 μC/cm2 (Figure 9.3) along its ferroelectric-preferred pseudocubic ⟨111⟩c direction (Chu et al. 2007). The structure and properties of the bulk single crystal form BFO have been extensively studied, and it has been shown to possess a rhombohedrally distorted perovskite structure at ground state. Unlike other ferroelectric antiferromagnetics, BFO possesses a very high ferroelectric Curie temperature (T c ) of ∼1100 K and a high antiferromagnetic Néel temperature (T N ) of ∼640 K. The structure of BFO is illustrated by distorted perovskite unit cells connected along their body diagonal, denoted by the pseudocubic ⟨111⟩c direction, to form a rhombohedral unit cell with ac = 3.96 Å, as shown in Figure 9.4. The two oxygen octahedrals in the connected perovskite units are rotated clockwise and counterclockwise around this axis by 13.8∘ , with the Fe cation shifted by 0.135 Å along the same axis away from the center of an oxygen octahedron (Zavaliche et al. 2006). The ferroelectric state is achieved by a large displacement of the Bi ions relative to the FeO6 octahedral. The domain structure in BFO is illustrated in Figure 9.5, where the eight ⟨111⟩c oriented polarizations forming 71∘ , 109∘ , and 180∘ domain walls are shown. This BFO domain structure indicates the formation of eight possible polarization variants corresponding to four structural variants lying along the pseudocubic ⟨111⟩c directions. It’s magnetic ordering also classify BFO as

201

9 Multiferroics: Single Phase and Composites

120 80 Polarization (μC/cm2)

202

BFO(111) BFO(110) BFO(100)

40 0 –40 –80 –120

0 5 –5 Applied voltage (V)

–10

10

Figure 9.3 Ferroelectric hysteresis loops of epitaxial BFO thin films along different crystalline directions. Source: Chu et al. (2007). Reprinted with permission of Elsevier.

Pi+ Pi+/–

(c)

Figure 9.4 (a) Crystal structure of rhombohedral BFO and its ferroelectric polarized direction. (b) Projection along ⟨111⟩c direction. (c) Projection along ⟨110⟩c direction. Source: Zavaliche et al. (2006). Reprinted with permission of Taylor & Francis.

Pi+ (b)

(a)

Bi Fe O

G-type antiferromagnetic as shown in Figure 9.6, in which the Fe magnetic moments are aligned ferromagnetically within (111)c and antiferromagnetically between adjacent (111)c . Antiferromagnetic ordering is of three types particularly in perovskite-type oxides. They are the following. A-type: the intra-plane coupling is ferromagnetic, while inter-plane coupling is antiferromagnetic. C-type: the intra-plane coupling is antiferromagnetic, while inter-plane coupling is ferromagnetic. G-type: both intra-plane and inter-plane coupling are antiferromagnetic. Figure 9.6a shows that BFO belongs to G-type (antiferromagnetic

9.4 Single Phase Multiferroic Materials

(a)

(b)

(c)

Figure 9.5 Domain structures in BFO with (a) 71∘ , (b) 109∘ , and (c) 180∘ domain walls. Source: Adapted from Baek et al. (2010).

[111] MFe1 M MFe2 (a)

(b)

Figure 9.6 (a) G-type antiferromagnetic plane of BFO is perpendicular to the ferroelectric polarized [111]c direction, where antiferromagnetic exist in both intra-plane and inter-plane. (b) Magnetic structure including the Dzyaloshinskii–Moriya interaction. The two Fe3+ magnetic moment (MFe1 and MFe2 ) are canted in the (111)c plane so that there is a resulting macroscopic remnant magnetization M. Source: Wu et al. (2016). Reprinted with permission of Elsevier.

in both intra-plane and inter-plane). Consequently, the antiferromagnetic and ferroelectric orders can be coupled together, namely, the switching of ferroelectric dipoles coupled with the switching of antiferromagnetic plane, indicating great scientific and technical values in the spintronic, multiferroic tunneling devices, etc. The perovskite multiferroic BFO simultaneously shows ferroelectric and antiferromagnetic orders (Smolenskii and Chupis 1982), exhibiting weak magnetism at room temperature due to a canted spin structure. Spin canting is a phenomenon through which spins are tilted by a small angle about their axis rather

203

9 Multiferroics: Single Phase and Composites

than being exactly parallel or anti-parallel. The antisymmetric exchange is a contribution to the total magnetic exchange interaction between two neighboring magnetic spins; this term is also called the Dzyaloshinskii–Moriya interaction. A magnetically ordered system favors a spin canting of otherwise (anti)parallel aligned magnetic moments and, thus, is a source of weak ferromagnetic behavior in an antiferromagnet.

9.4.1

Switching Mechanism in BFO Films

The coupling between ferroelectric and antiferromagnetic orders of BFO has been predicted (Ederer and Spaldin 2005) and experimentally observed (Apostolova and Wesselinowa 2008). Therefore, in order to control the antiferromagnetic order by electric field in a high-degree of freedom, it becomes significant to investigate the intrinsic ferroelectric domain configuration and the switching path as well as the physics behind the domain dynamics under a bias field. The potential application of BFO in ferroelectric non-volatile memory also depends on deep understanding on its domain structure and switching mechanism. Therefore, ferroelectric switching in monodomain BFO epitaxial thin film will provide a basic insight into the distributions of electric, magnetic, and stress fields within domain structures, allowing us to manipulate and control the domains and domain walls (Seidel et al. 2009). The ferroelectric domain structure of BFO is related to its crystallographic direction. As indicated in Figure 9.7, starting from Figure. 9.7(a), the c

Bi Fe

71 °

[001] 0 (b)

0°

(a)

° 109

(c)

18

204

(d)

Figure 9.7 (a) original polarization, (b) after 71∘ rotation, (c) after 109∘ rotation, and (c) after 180∘ rotation. Source: Adapted from Zhao et al. (2006).

9.5 ME Composite Materials

polarization can be switched by 71∘ to the direction shown in (b) and 109∘ switching to (c) and 180∘ switching to (d).

9.5 ME Composite Materials In order to implement multiferroic properties in device application, reasonably large ME coupling factor is essential. Therefore, people have started to make composites and heterostructures exhibiting the coexistence of piezoelectric and magnetostrictive properties. The ME composite can be in the forms of solid-state reaction-based composite ceramics, piezoelectric and magnetostrictive-layered stacks, and even in thin film multilayers and novel nanostructures. The ME coupling effect in composites originates from the mechanical deformation of magnetostrictive materials under magnetic field, which induces the accumulated charge in piezoelectric phase due to piezoelectric effect. For example, piezoelectric oxide and magnetostrictive ferrite oxide BaTiO3 /CoFe2 O4 composite can be sintered together using their respective oxide powder, where phase separation can be realized under high-temperature sintering. The maximum ME coefficient once reached a value of 130 mV/cm⋅Oe (Van den Boomgaard et al. 1974). However, the phase and element diffusion may happen in the composite during the sintering, leading to enhanced leakage current and failure of electrical poling. Although a lot of researchers made their effort in reducing the leakage current and dielectric loss in the solid-state composite by doping elements in the respective components (Devan and Chougule 2007; Kadam et al. 2005), the ME coefficient was not enhanced significantly. Based on the previously mentioned consideration, people started to put their interests on bulk ME materials with 2–2 structure (multilayer structure), combining the piezoelectric and magnetostrictive components together by a conductive epoxy. Srinivasan et al. reported a ME coefficient of 270 mV/cm⋅Oe in PZT (Pb(Zrx, Ti1−x )O3 )/ferrite laminated composite (Srinivasan et al. 2003). The multilayered composite was also prepared as a 2–2 structure with an alternating piezoelectric and magnetostrictive phases, and a giant ME coefficient of 1500 mV/cm⋅Oe was achieved in the NiFe2 O4 /PZT multilayered laminate (Srinivasan et al. 2001). The ME coefficient could be further increased by improving the mechanical coupling in the piezoelectric and magnetostrictive laminates. In recent years, polymer materials such as PVDF or P(VDF-TrFE) have been chosen as promising candidates for the piezoelectric phase in the ME composite because they have relatively good voltage sensitivity, high electromechanical coupling, low dielectric constant and low dielectric loss. Moreover, polymeric piezoelectric materials can stand freely to avoid substrate clamping effect, which is the main issue for other thin film structures, and their shapes and sizes can also be easily modified by conventional polymeric processing. Nan et al. developed a new polymer-based ME composite, in which piezoelectric and magnetostrictive powders were dispersed into the piezoelectric copolymer, forming a double layer, sandwich, or even a multilayered structure by bonding the two components together (shown in Figure 9.8) (Lin et al. 2005). The maximum ME sensitivity of such composites is about 6 V/cm⋅Oe at about 90 kHz. At the same time,

205

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9 Multiferroics: Single Phase and Composites

Figure 9.8 (a) Schematic diagram of the composite (tp is the thickness of the piezoelectric phase and L is the total thickness) and (b) a typical cross-sectional SEM images of the fractured surface of the composite. TFD denotes Terfenol-D. Source: Adapted from Lin et al. (2005).

Polarization TFD-PVDF layer PZT-PVDF layer

tp

L

TFD-PVDF layer (a)

(b)

theoretical calculation derived by Green’s function technique also supports the experimental data (Nan et al. 2001). The record of ME coefficient was broken again after the appearance of Metglas/PVDF laminate, developed by Dong et al. which has a ∼300 V/cm⋅Oe of ME coefficient at their resonance frequency (Figure 9.9) (Zhai et al. 2006). The demand of miniaturization and high-density storage in next-generation electronic devices drives people’s interest to develop and design thin films and nanostructured ME nanocomposites and devices. The coexistence of strong ferromagnetic and ferroelectric behaviors and realization of ME coupling effect are necessary in the nanocomposites. The observation of ME coupling effect in the self-assembled multiferroic BaTiO3 –CoFe2 O4 (BTO–CFO) nanostructures reveals that the interconversion of energies stored in electric and magnetic fields can be realized in micro- or even nanodevices (Zheng et al. 2004; Zavaliche et al. 2007). As shown in Figure 9.10, the self-assembled ME nanocomposite, consisting of perovskite BaTiO3 (BTO) and spinel CoFe2 O4 (CFO), exhibits a large magnetization change around the ferroelectric Curie temperature of BTO, indicating a strong ME coupling response happened in the mesoscopic order. This effect, for comparison, was not observed in the BTO/CFO 2–2 heterostructure, which means that the large substrate clamping was not released as much as the 1–3 self-assembled nanostructure composite. The theoretical calculation, shown in Figure 9.11, also predicts the large magnetic-field-induced electric polarization in nanostructured multiferroic Figure 9.9 (a) Photograph of a Metglas/PVDF laminate, (b) the unimorph, and (c) three-layer configuration. Source: Adapted from Zhai et al. (2006). (a) M P

(b)

Unimorph PVDF

Metglas

M P M

(c)

T–L

9.6 Modeling the Interfacial Coupling in Multilayered ME Thin Film

ΔM = 16 emu/cm3

Magnetization (emu/cm3)

200

150

Spinel CoFe2O4 nanopollars

BaTiO3 matrix 100 1–3 nanostructure

Substrate

50 300

320

340

360 380 Temperature (K)

400

420

440

Figure 9.10 Temperature-dependent magnetization in self-assembled nanostructure. Inset: Schematic diagram of phase separation of spinel and perovskite nanocomposite on perovskite substrate. Source: Adapted from Zheng et al. (2004).

composite films (Nan et al. 2005), which further proves that the substrate clamping effect will destroy the mechanical coupling between piezoelectric and magnetostrictive phase in the 2–2 heterostructured thin film.

9.6 Modeling the Interfacial Coupling in Multilayered ME Thin Film In this section, epitaxial PZT/CFO multilayers with perovskite/spinel interface and residual strain are used as an example to demonstrate the modeling of interfacial coupling in multilayered ME thin film. The comparison of ferroelectric and ferromagnetic properties between the multilayered films and the single phase is introduced. This interfacial engineering in perovskite/spinel heterostructure provides an approach to control the ferroelectric, magnetic, and ME interfacial coupling in epitaxial multilayers and superlattice multiferroics. 9.6.1

PZT/CFO Multilayered Heterostructures

PZT and CFO as well as their epitaxial multilayered nanocomposite thin films were deposited on Nb:STO single crystal substrates by pulsed laser deposition (PLD). For comparison purpose, the film thickness is maintained as a constant for single phase and multilayered films with 3, 5, and 11 layers. Figure 9.12a is

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9 Multiferroics: Single Phase and Composites

500

400

ΔP3(μC/m2)

208

300 Nano system 200

Bulk system

1–3 2–2 1–3 2–2

100

0 0 (a)

100

200 H3(kA/m)

300

400

Composite film with 1–3 nanostructure

Substrate

Composite film with 2–2 structure

Substrate (b)

Figure 9.11 (a) Theoretical ME coupling effect in nano and bulk systems with 1–3 and 2–2 structures, and (b) the corresponding schematic diagrams of 1–3 and 2–2 nanostructures. Source: Adapted from Nan et al. (2005).

the low-magnification cross-sectional bright-field image of the PZT/CFO multilayered film. The relatively dark area corresponds to the PZT phase due to its relatively heavy elements. Selected area electron diffraction pattern (SAED) is shown in Figure 9.12b. The cubic-on-cubic epitaxial relationship between PZT, CFO, and STO substrate with fourfold symmetry is confirmed.

9.6 Modeling the Interfacial Coupling in Multilayered ME Thin Film

STO022 PZT022 CFO022

500 nm (a)

(b)

Figure 9.12 (a) Low-magnification TEM images of 11-layered PZT/CFO nanocomposite films, (b) the selected area electron diffraction pattern including PZT, CFO, and STO substrate. Source: Adapted from Zhang et al. (2010).

9.6.2

Ferroelectric Properties of PZT/CFO Multilayers

P–E loop measurement is essential for ferroelectric characterization. With the increase of number of layers, remnant polarizations of the PZT film and composition films of 3, 5 and 11 layers are 31.8, 17.9, 15.3, and 8.5 μC/cm2 , respectively. The coercive field increases from 80 kV/cm for the PZT film to ∼200 kV/cm for the multilayered films. Since the coercive field of each PZT layer in the composite film should be a function of external electric field, volume fraction, and dielectric constant of PZT and CFO, if we neglect the conductivity of the multilayers, the actual electric field applied on each PZT layer can be described by EP =

Em

𝜖

𝜙 P + 𝜙C 𝜖 P

(9.5)

C

where Em and Ep are the electric field applied on the multilayers and the actual field applied on each PZT layer, respectively; 𝜙P , 𝜙C , 𝜖 P , and 𝜖 C are the volume fractions and dielectric constants of the PZT and CFO layers in the films. Therefore, we have to increase the external field in the multilayers in order to maintain the switching electric field in each PZT layer. It is easy to understand the significant enhancement of the coercive field in the films compared with that in the single phase PZT film. To modeling the number of layer-dependent polarization in the multilayered thin films, the average residual stress suffered by the PZT phase in the epitaxial multilayers can be estimated based on the XRD result. The in-plane a-axis lattice parameters of PZT increase from 4.027 to 4.032, 4.035, and 4.041 Å, which correspond to single PZT, 3-layered, 5-layered, and 11-layered films. The Young’s modulus of the epitaxial PZT film is 92.6 GPa as reported elsewhere (Kanno et al. 2004). Therefore the residual in-plane compressive stress suffered by the corresponding epitaxial PZT films should be −520, −410, −330, and −204 MPa; while for the cubic PZT bulk materials without residual stress, the lattice constant a = 4.05 Å (Xu 1991). We adopt Landau–Devonshire’s phenomenological thermodynamic formalism method to calculate the spontaneous polarization

209

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9 Multiferroics: Single Phase and Composites

(Hoon Oh and Jang 1998): ΔG = 𝛼1 (P12 + P22 + P32 ) + 𝛼11 (P14 + P24 + P34 ) + 𝛼12 (P12 P22 + P22 P32 + P32 P12 ) + 𝛼111 (P16 + P26 + P36 ) + 𝛼112 [P14 (P22 + P32 ) + P24 (P32 + P12 ) + P34 (P12 + P22 )] 1 + 𝛼123 P12 P22 P32 − S11 (𝜎12 + 𝜎22 + 𝜎32 ) − S12 (𝜎1 𝜎2 + 𝜎2 𝜎3 + 𝜎3 𝜎1 ) 2 1 2 2 − S44 (𝜎4 + 𝜎5 + 𝜎62 ) − Q11 (𝜎1 P12 + 𝜎2 P22 + 𝜎3 P32 ) 2 − Q12 [𝜎1 (P22 + P32 ) + 𝜎2 (P32 + P12 ) + 𝜎3 (P12 + P22 )] − Q44 (𝜎4 P2 P3 + 𝜎5 P3 P1 + 𝜎6 P1 P2 )

(9.6)

where Pi is the magnitude of the polarization vector along the direction i. 𝛼 i is the dielectric stiffness, 𝛼 ij and 𝛼 ijk are the high-order stiffness coefficients at constant stress, S11 , S12 , and S44 are the elastic compliances measured at constant polarization, and Q11 , Q12 , and Q44 are the electrostrictive coefficients written in polarization notation. 𝜎 i is tensile stresses along direction i. We are now considering the elastic Gibbs function in the presence of a two-dimensional compressive stress in tetragonal phase. The following relations hold under this condition: 𝜎1 = 𝜎2 = −𝜎 𝜎3 = 𝜎4 = 𝜎5 = 𝜎6 = 0 P1 = P2 = 0

(9.7)

Therefore, G = (𝛼1 + 2Q12 𝜎)P32 + 𝛼11 P34 + 𝛼111 P36 − (S11 + S12 )𝜎 2 From the previously mentioned derivation and the condition obtain √ −𝛼11 ± 𝛼11 2 − 3𝛼111 (𝛼1 + 2Q12 𝜎) 2 2 PP = P3 = 3𝛼111

(9.8) dG dP3

= 0, we can

(9.9)

Since the “-” in the equation is meaningless, it should be deleted. In addition, according to the discussion of the “dilution” of the effect of ferroelectric materials in a dielectric matrix in the BTO–STO–CTO superlattice (Lee et al. 2005), when we consider the influence of the non-ferroelectric layer, the electric displacement should be expressed by ( ) 𝜖P 𝜖C 𝜙C 𝜖P D= E+ 1− PP 𝜙 P 𝜖 C + 𝜙C 𝜖 P 𝜙P 𝜎C + 𝜙C 𝜎P where D and P are the electric displacement and polarization of the multilayers. Then the average polarization in the electrostatic model can be derived Pm =

PP 1+

𝜙C 𝜙P

⋅

𝜖P 𝜖C

(9.10)

9.6 Modeling the Interfacial Coupling in Multilayered ME Thin Film

Therefore, we obtain the polarization of the multilayers √ √ −𝛼11 + 𝛼11 2 −3𝛼111 (𝛼1 +2Q12 𝜎)

Pm =

3𝛼111

1+

𝜙C 𝜙P

⋅

(9.11)

𝜖P 𝜖C

where Pm is the average polarization of the PZT/CFO multilayered films and PP indicates the polarization in each PZT layer. The theoretical prediction describes the polarization decrease with the release of in-plane compressive strain, thus qualitatively interprets the experimental data. We consider the influence of the dielectric matrix in the multilayers according to Eq. (9.11). The addition of non-ferroelectric phase reduces the average polarization, compared with the pure PZT films suffered from different in-plane stresses. It is easy to understand the dramatic decrease of the polarization from single PZT film to the multilayered films as what we observed. In addition, the size effect discussed by other reports in ferroelectrics may be another factor to decrease polarization in the multilayers (Wong and Shin 2007). The remnant polarization, if we consider the size effect in PZT phase, is expressed by √ 1 𝛼0 (Tc − T) Pr = (9.12) R 𝛽 where 𝛼 0 and 𝛽 are the modified and original Landau parameters, respectively, 3(t + 4𝛿) 2(t + 6𝛿) 32𝜅 (t + 3𝛿) Tc = T0 − 3𝛼0 t (t + 4𝛿)2

(9.13)

R=

(9.14)

where t denotes the thickness of PZT layer and 𝛿 is the so-called extrapolation length of the PZT/CFO interface; T 0 is the paraelectric to ferroelectric phase transition temperature of bulk PZT, and 𝜅 is the positive constant representing the contribution from the spatial variation of polarization. When we take all of the previously mentioned parameters (Wong and Shin 2007; Zhong et al. 1994; Liu et al. 2005; Kretschmer and Binder 1979) as shown in Table 9.1, the theoretical fitting considering the size effect is in agreement with our experimental observation. We may conclude that the size effect should exists in our experimental observation and theoretical calculations. Table 9.1 Parameters used in calculating the polarization of the PZT film and PZT/CFO multilayered heterostructures. 𝜶 1 (m/F)

𝜶 11 (m5 /C2 F)

𝜶 111 (m9 /C4 F)

−8 × 106

7.12 × 107

2.35 × 106

Q12 (m4 /C2 )

t

𝜹

𝜶0

𝜷

T0

𝜿

−1.54 × 10−2

1

4

1

1

1

1

211

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9 Multiferroics: Single Phase and Composites

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10 Device Application of Multiferroics Multiferroic materials exhibit coupling between electric polarization and electron spin, making them applicable in devices where magnetic switching is being controlled by electric field or vice versa. Composite multiferroic materials based on the mechanical coupling between piezoelectric and magnetostrictive effects have found engineering application in devices of highly sensitive magnetic field sensors. Multistate and multifunctional memories that can be magnetically written and electrically read or vice versa are also very promising in data storage, computing, and AI technology. These will be particularly introduced in this chapter.

10.1 ME Composite Devices Magnetoelectric (ME) effect is the response of polarization P to a magnetic field H applied across a ME device. Due to the limitation of single phase ME material and very low ME coupling factor, scientists have spent a lot of effort trying to realize device-applicable composite materials, utilizing the mechanical coupling between the magnetostrictive and piezoelectric effects. The conversion from magnetic field to electrical signal in ME device is realized by magneto–elasto–electric coupling through stress interaction between the magnetostrictive phase and the piezoelectric phase. Figure 10.1 illustrates different modes of ME composite device, where L represents longitudinal vibration mode, T represents transverse vibration mode, and C represents circular vibration mode. During designing of the device, the piezoelectric coefficient, electromechanical coupling factor, and magnetostrictive effect to the corresponding variation modes should be considered. The integrated device resonance frequency is related to the size of each component and their coupling as well as the preload and therefore is a complicated problem that could be calculated by finite element modeling. Similar to a piezoelectric resonance whose performance can be described by electrical equivalent circuit, an ME device can also be electrically described by equivalent circuit. Figure 10.2 shows equivalent circuit of such ME device under resonance state. In this circuit, the 𝜑m is the magnetoelastic coupling factor, and 𝜑p is the elastic–electric coupling factor. With the input magnetic field H and the induced electrical voltage V , the ME coupling factor can be derived. Ferroic Materials for Smart Systems: From Fundamentals to Device Applications, First Edition. Jiyan Dai. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

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10 Device Application of Multiferroics

Figure 10.1 Various laminates operated in longitudinal vibration mode: L-longitudinal, T-transverse, and C-circular. Source: Adapted from Nan et al. (2008).

Piezoelectric layer M

M P

P

M

M

L–L

T–L

Magnetostrictive layer M

M P

P M

M

L–T

T–T

M P

M

P

M

Push–pull

L–T hybrid M P

M

Multi-L–T

C–C

–C0

Mechanical current Applied magnetic field H

Electric current

C0 Induced voltage V

φ mH

Zm Magnetoelastic coupling φm

Lm

Cm

φp : 1

Elastic–electric coupling φp

Figure 10.2 Magnetoelastic–electric equivalent circuit at resonance. Source: Adapted from Nan et al. (2008) and Dong et al. (2006).

We can design a ME device based on the working principle of L–T mode laminated composite as shown in Figure 10.3. In this device, Terfenol-D fiber mainly vibrates along the length direction when the magnetic field is applied longitudinally. The piezoelectric material vibrates along the length direction due to mechanical coupling, and electric charges are generated across the thickness direction via the d31 piezoelectric effect. To enhance the ME sensitivity, epoxy-bonded Terfenol-D continuous fiber composite is used as the magnetostrictive phase instead of using monolithic Terfenol-D. Composite form possesses several advantages in comparison with monolithic form, for example, higher operation bandwidth, lower material cost, and less brittle. Superior ME coupling generates strong electrical response by external alternating magnetic field especially when the composites are at resonance giving rise to high magnetic field sensitivity. Besides, the composite structure can greatly reduce the eddy current loss that generally occurs in monolithic Terfenol-D.

10.1 ME Composite Devices

Terfenol-D continuous fibers

Epoxy

M

Piezoelectric plate

P

M

Figure 10.3 Schematic diagram of the ME laminate composite device with Terfenol-D continuous fiber composite and piezoelectric plate. The arrows represent the magnetization and polarization directions, respectively.

Multi-purpose FFT analyzer Swept sinusoidal voltage

Signal-out

Channel 2 Channel 1

Induced ME voltage (VME)

Measuring Hac ME laminate composite

DC-coupled AC amplifier Signal-in

Signal-out

Driving signal for Hac Helmholtz coil Electromagnets with driving signal for Hbias

Figure 10.4 Schematic diagram of an automated ME measurement system.

Standard ME properties using an automated measurement system are shown in Figure 10.4. The ME voltages (V ME ) generated from the composite can be measured by applying ac magnetic fields (H ac ) on the composite under a bias magnetic field (H bias ). The sensitivity of the ME laminated composite/magnetoelectric voltage coefficient (MEv ) can be measured by applying H ac of 1 Oe on the composite at frequency f . The MEv at specific H ac and f is usually calculated as follows: MEv =

VME Hac

(10.1)

where the H ac along the longitudinal direction is provided by a Helmholtz coil. H ac and H bias were measured by a pick-up coil and a Gauss/Tesla meter. To work at resonance frequency is important to reach maximum sensitivity. Figure 10.5a shows the spectrum of Z and 𝜃 for the ME laminated composite where the resonance frequency (f r ) of the composite device is 89 kHz, which can

219

10 Device Application of Multiferroics 5

80 Impedance Phase angle

40 0

10

−40

3

θ (°)

4

Z (Ω)

10

−80

(a)

80.0k 100.0k 120.0k 140.0k

Frequency (Hz)

30 20 10 0

2

10 20.0k 40.0k 60.0k

Hbias = 20 Oe Hbias = 100 Oe Hbias = 300 Oe Hbias = 500 Oe Hbias = 700 Oe

40

(ME)v (V/cmOe)

10

0

20

(b)

40

60

80

100

Frequency (Hz)

Figure 10.5 (a) Electrical impedance (Z) and phase angle (𝜃) spectra for a ME laminated composite. (b) Magnetoelectric voltage coefficient (ME v ) as a function of frequency f for a ME laminated composite tested under different dc bias magnetic field. Source: Adapted from Cheung (2009). Hbias = 0 Oe Hbias = 200 Oe

2000

Hbias = 500 Oe

1600 V3 (mV)

220

Hbias = 800 Oe Hbias = 1200 Oe

1200

Hbias = 1600 Oe Hbias = 2000 Oe

800 400 0 0

2

4

6

8

10

H3 (Oe)

Figure 10.6 V ME (V 3 is the voltage across the piezoelectric plate) as a function of Hac with f = 1 kHz for the ME laminated composite tested under different dc bias magnetic fields. Source: Adapted from Cheung (2009).

also be identified in Figure 10.5b, but showing right shift under increased bias field. Figure 10.6 shows the V ME as a function of H ac with f = 1 kHz for the ME laminated composite tested under different dc bias magnetic fields. It is apparent that V ME shows excellent linear relationship with the measured range of H ac under H bias . The linear response to H ac makes the ME laminated composite very suitable for the application in ac magnetic field measurement. During the operation of ME device, the magnetic bias field plays a very important role that affects the sensitivity of the device. Figure 10.7 shows the optimum H bias for the laminated composite device, where one can see that the MEv increases with increasing H bias , but it finally reaches a maximum value at H bias = 400 Oe and then starts to decrease from the maximum. This can be understood by considering the fact that without bias field, the ac magnetic field

10.1 ME Composite Devices

ME coefficient (αE) (V/cmOe)

3.5

@ 1 kHz

3.0 2.5 2.0 1.5 1.0 0.5 0.0 0

200

400

600

800

1000

1200

Hbias (Oe)

Figure 10.7 Magnetoelectric voltage coefficient ME V as a function of dc bias magnetic field (Hbias ) for a ME laminated composite tested under an ac magnetic field (Hac ) of 1 Oe peak with frequency f of 1 kHz. Source: Wang et al. (2008). Reprinted with permission of AIP Publishing.

is hard to induce magnetization switching, and therefore the strain generated by magnetostrictive effect is very small. Under optimized bias field, a small ac magnetic field can easily induce non-180∘ magnetization rotation and therefore can generate largest strain in the magnetostrictive laminate. However, a too large bias field makes the magnetization switching induced by ac field hard, and therefore, the output strain becomes smaller. 10.1.1

Effect of Preload Stress

In Chapter 7, it is explained that non-180∘ domain rotation is essential to generate large strain in Terfenol-D, and this can be achieved by adding H bias and prestress to the Terfenol-D rod. Figure 10.8 shows a ME device illustrating how to make a ME device with presence of dc magnetic field and preloaded mechanical stress resulting in optimized performance. In this device, a dc magnetic field is applied by a permanent magnet with the field parallel to the [112] direction of the Terfenol-D rod. A preloaded mechanical stress is applied by a screw to make the Terfenol-D and Pb(Mg1/3 Nb2/3 )O3 –xPbTiO3 (PMN-PT) crystal being clamped. The measured ME response to an ac magnetic field as a function of frequency and under different level of preload mechanical stress is shown in Figure 10.9, where two general trends can be found. One is that the resonance frequency increases as the stress increases; this can be understood by considering that the stress results in an increase of mechanical compliance. The other trend is that the ME response increases as the stress increases; this is due to the fact that the stress applied on Terfenol-D makes the magnetization direction perpendicular to the rod, and non-180∘ magnetization switching is realized and induces the maximum ME response. However, overstressed rod will result in fully clamped state so that

221

10 Device Application of Multiferroics

22 mm

Stainless frame

Permanent PMN-PT magnet single crystal

Terfenol-D

PMN-PT single crystal

Bolt

Figure 10.8 Schematic diagram of a stress-biased PMN-PT single crystal/Terfenol-D ME sensor. Source: Adapted from Lam et al. (2011). 0.25

0

20

40

60

80

No pressure 0.5 MPa 1.0 MPa 1.5 MPa 2.0 MPa 2.5 MPa 3.0 MPa

0.20 ME voltage coefficient (V/Oe)

222

100 0.25

0.20

0.15

0.15

0.10

0.10

0.05

0.05

0.00

0

20

40

60

80

0.00 100

Frequency (kHz)

Figure 10.9 Frequency dependence of ME v as a function of preloading stress for a PMN-PT single crystal/Terfenol-D ME sensor. Source: Lam et al. (2011). Reprinted with permission of AIP Publishing.

10.2 Memory Devices Based on Multiferroic Thin Films

no domain switching can happen; therefore the ME factor decreases significantly after the peak value. This type of device can find application in magnetic field sensor. Luo’s group has increased the sensitivity of magnetic field measurement to 10−10 T at room temperature.

10.2 Memory Devices Based on Multiferroic Thin Films

Resistance

Multiferroic thin films can also be used for multifunction and/or multistate memory devices. For example, as shown in Figure 10.10, a memory device can be formed based on the coupling between the BiFeO3 (BFO)’s ferroelectricity and anti-ferromagnetism (FE-AFM) and exchange-bias coupling between the BFO film and magnetic electrode. BFO is one of the most attractive multiferroic materials with excellent ferroelectric property and also anti-ferromagnetic and weak ferromagnetic properties. These properties provide BFO thin film great potential for memory device applications. Beyond the well-known application in ferroelectric random-access memory (FeRAM) as ferroelectric gate dielectric layer in a field effect transistor (FET) structure, BFO can also be used as magneto-electric random access memory (MERAM) based on a cross-bar structure. As shown in Figure 10.10, the coupling between the BFO’s FE-AFM makes BFO magnetization moment switchable by electric field. Through exchange bias with a thin ferromagnetic electrode, a spin valve structure with a tunnel barrier layer between the two top ferromagnetic layers provides the two resistive states. Interestingly, BFO can

Rap V+ P

FE – AFM

Electrode

Rp V– P

FE – AFM

Electrode

Voltage

Figure 10.10 MERAM based on exchange-bias coupling between a multiferroic that is ferroelectric and antiferromagnetic (FE-AFM) and a thin ferromagnetic electrode. Source: Bibes and Barthélémy (2008). Reprinted with permission of Springer Nature.

223

δ V (μV)

60 nm

10 Device Application of Multiferroics

Output signal

1 0

S

N S N S S N N S S N NS S N N SN medium S

N

Recording

Read head ME sensor

δ H (Oe)

Substrate FE layer FM layer

BTO

CFO

–1 3

STO

224

0

–3

δH

dM dt

Iw bK aK

0 0 1 1 0 1 1 1 1 1 0 1 1 0 1 0 0 0 1 1 1 0 1 0 0 0 1 0 1 1 0 0

Figure 10.11 Schematic, TEM image (inset) and voltage output of an ME read head sensor based on a bilayer CFO/BTO heterostructure grown on a single-crystal SrTiO3 (STO) substrate. Source: Zhang et al. (2008). Reprinted with permission of AIP Publishing.

act not only as the magnetoelectric active layer but also a ferroelectric tunneling barrier. Another method, which has already been implemented in recording read head application, is a junction working with magnetoelectric (ME) coupling. By using strain as a media, in-plane magnetic field-induced strain in the magnetic component due to magnetostrictive effect can be mechanically transferred to the ferroelectric component and finally can induce an electric polarization change, and thus voltage change, through the piezoelectric effect. Figure 10.11 demonstrates the ME coupling between the magnetostrictive CoFe2 O4 (CFO) and ferroelectric BaTiO3 (BTO) (Zhang et al. 2008).

10.3 Memory Devices Based on Multiferroic Tunneling In Chapter 3, ferroelectric tunneling device as a non-volatile memory has been introduced. Beyond this ferroelectric junction, if ferromagnetic electrodes are implemented, memory device based on multiferroic tunnel junction with multistates and multifunctions can be realized (Gajek et al. 2007; Eerenstein et al. 2006). A multiferroic tunnel junction is made by combining the structure of ultrathin ferroelectric tunnel layer and magnetic tunnel junction into one junction with an ultrathin ferroelectric film sandwiched by two ferromagnetic conductive electrodes. With this artificial multiferroic tunnel junction (MFTJ), four resistance states can be retrieved by the result of tunneling magnetoresistance (TMR) and tunneling electroresistance (TER). As a result, multifunctional (magnetically write and electrically read) four-state memory device can be achieved. Figure 10.12 is a schematic illustration of an artificial multiferroic tunnel junction and its working principle. One can see that the combinations of two ferroelectric polarizations and two ferromagnetic states result in four resistance states.

10.3 Memory Devices Based on Multiferroic Tunneling

FM I

NM FE

FM FE

FM

NM

FM

MTJ R

FTJ

MFTJ

R

Hc

H-field

R

Ec

E-field

E

H Ec

Hc

Figure 10.12 Schematic illustration of artificial multiferroic tunnel junctions and its working principle.

Figure 10.13 Schematic diagram of low-field MFTJ.

Ferromagnetic alloy NiFe

M P Ferromagnetic oxide LSMO

M

Ferroelectric barrier BTO

Substrate STO

Aiming at constructing a MFTJ that can fulfill the constraint of less power consumption, low fields in both ferroelectric and ferromagnetic switching is preferred. As an example of our work, Figure 10.13 shows a ferroelectric BTO tunnel barrier sandwiched by two electrodes composed of different ferromagnetic materials La0.7 Sr0.3 MnO3 (LSMO) and Permalloy Ni0.81 Fe0.19 (NiFe). NiFe is a well-known alloy that allows strong spin electron injection and has relatively much lower coercive field, while LSMO has been predicted to have close to 100% spin polarization. This structure is expected to form a low-switching-field spin valve memory. Figure 10.14 shows resistance–voltage (R–V ) characteristics of non-volatile resistance switching from this MFTJ device. Unlike RRAM-resistive random access memory (ReRAM), no electroforming at higher voltage is needed before the observation of hysteretic resistance changes. During the measurements, pulse train writing voltages are applied in the sequence of 0 V → 1.5 V → 0 V → −1.5 V → 0 V with a step of 0.2 V, while reading voltage is remained constant as 0.1 V. With the top electrode grounded, a positive pulse of

225

10 Device Application of Multiferroics

80 70 60 Resistance (kΩ)

226

50 40 30 20 10 0 –1.5

–1.0

–0.5

0.0 0.5 Voltage (V)

1.0

1.5

Figure 10.14 Electroresistance change of MFTJ at 40 K for 10 cycles. The corresponding ferroelectric polarization direction are shown in the bottom right and top right insets for the ON and OFF states, respectively. Source: Adapted from Yau et al. (2015).

voltage (+V ) is applied on the bottom LSMO electrode, which drives the BaTiO3 ferroelectric polarization pointing to the NiFe electrode and sets the junction to ON state (low resistance state). In contrast to the ON state, a negative voltage pulse (−V ) drives the polarization to the opposite direction and sets the junction to OFF state (high resistance state). Magnetic characteristics of the NiFe and LSMO electrodes are shown in Figure 10.15, where two distinct M–H hysteresis loops present in both layers. NiFe has almost the same magnetization as LSMO; however, the coercive field of LSMO is 20 Oe, while it is only 5.0 Oe for NiFe in the film plane. The different coercive fields for these two dissimilar electrodes determine that LSMO acts as a hard magnet and NiFe electrode acts as a soft magnet in the spin valve. This result suggests that two resistance states can be achieved by interchanging the magnetization orientations using external magnetic field. Figure 10.16 demonstrates the coexistence of TMR and TER in an artificial MFTJ. By placing the sample in the middle of two electromagnets, alternative magnetic field is applied to the junction parallel to the film plane. Then, the junction (ferroelectric barrier) was poled by 1.5 V and −1.5 V for poling upward and downward, respectively. Among the four stages, two resistance states are formed due to the electric polarization switching and the other two are due to the parallel and antiparallel magnetization configurations. By sweeping the external magnetic field from −100 Oe to 100 Oe and returning to −100 Oe, two distinct resistance states can be clearly observed for the two ferroelectric polarization states of up and down, i.e. there are four distinct

10.3 Memory Devices Based on Multiferroic Tunneling

800 LSMO NiFe

Magnetization (μemu/cm2)

600 400 200 0 –200 –400 –600 –800 –100

–50

0

50

100

Magnetic field (Oe)

Figure 10.15 Magnetism of NiFe and LSMO showing the hysteresis loops. Source: Adapted from Yau et al. (2015).

resistance states. When −1.5 V is added, ferroelectric polarization first orients toward LSMO so that the OFF state is reached. While when a −100 Oe magnetic field is applied, the magnetic moments of both NiFe and LSMO are aligned in parallel, and the resultant resistance is obtained as indicated by Stage A in Figure 10.16. Resistance keeps constant when magnetic field changes from −100 Oe to 5.0 Oe because of the unchanged alignment in two ferromagnetic layers. Until the field is larger than 5.0 Oe, the soft magnet NiFe starts to switch, thus, resistance changes due to antiparallel alignment of magnetic moment between NiFe and LSMO (Stage B). When the field is further increased to a certain value that is large enough to switch the hard magnet LSMO (i.e. 20 Oe), orientations of magnetization for two ferromagnetic layers will be aligned in parallel again. As a result, resistance returns to its initial value as Stage C. Stage D is obtained when an opposite sweeping field is applied and NiFe is oppositely switched. From the flow chart illustrated in Figure 10.16b, the configurations of magnetization alignment correlated to magnetic field are explained. It is interesting to notice that when the electric polarization is facing down, the negative TMR exists for the antiparallel magnetization alignment. This can be explained by the fact that the ferroelectric polarization switching may invert the spin polarization at the interface (DeTeresa et al. 1999). Since LSMO is p-type semiconducting, the facing up electric polarization inverts the interfacial layer of LSMO to n-type. This will change the sign of the spin polarization at the interface, and then the sign of TMR of the junction may be changed.

227

10 Device Application of Multiferroics

175.4 D

175.2 175.0 Resistance (kΩ)

228

B C

A

174.8 174.6 174.4 10.54

C′

A′ D′

10.53

B′

10.52 –100

–50

(a)

0 Magnetic field (Oe)

50

100

A

B

C

D

A′

B′

C′

D′

(b)

Figure 10.16 (a) Resistance changes with different magnetic fields at 8 K showing four-state memory tested on the NiFe/BaTiO3 /LSMO structure. (b) The corresponding schematic diagram for magnetization and ferroelectric polarization orientations. Source: Adapted from Yau et al. (2015).

To specify, TMR is expressed in Eq. (10.1) by using Julliere’s model (Julliere 1975). 2PLSMO PNiFe RAP − RP = RP 1 − PLSMO PNiFe

(10.2)

where RAP is the resistance in antiparallel magnetization state, RP in parallel magnetization state, and PLSMO and PNiFe are the spin polarizations of the two electrodes, where the spin polarization P can be calculated from the spin-dependent

References

density of states N at the Fermi energy (refer to Figure 8.3): P=

[N ↑ (EF ) − N ↓ (EF )] [N ↑ (EF ) + N ↓ (EF )]

LSMO has been tested to be almost 100% positive spin polarization, while the spin polarizations of most ferromagnetic metals such as Co and Ni are negative. Therefore, the TMR of the junction built with NiFe and LSMO should be negative, i.e. the tunneling resistance decreases when the spin polarizations are antiparallel in NiFe and LSMO. Thus, negative TMR can be observed (B′ and D′ in the Figure 10.16), and it is reversible when the ferroelectric polarization is reversed (B and D in the Figure 10.16).

References Bibes, M. and Barthélémy, A. (2008). Multiferroics: towards a magnetoelectric memory. Nature Materials 7 (6): 425–426. Available at: http://www.ncbi.nlm.nih .gov/pubmed/18497843%5Cnhttp://www.nature.com/doifinder/10.1038/ nmat2189. Cheung, K. (2009). Ultrasonic Transducer Equipped with a magnetoelectric sensor for weld quality monitoring. Hong Kong Polytechnic University. Available at: https://theses.lib.polyu.edu.hk/handle/200/3843. DeTeresa, J.M., Barthélémy, A., Fert, A. et al. (1999). Role of metal-oxide interface in determining the spin polarization of magnetic tunnel junctions. Science 286 (5439): 507–509. Available at: http://www.sciencemag.org/cgi/doi/10.1126/ science.286.5439.507. Dong, S., Li, J.-F., and Viehland, D. (2006). Magnetoelectric coupling, efficiency, and voltage gain effect in piezoelectric-piezomagnetic laminate composites. Journal of Materials Science 41 (1): 97–106. Available at: https://doi.org/10.1007/s10853005-5930-8. Eerenstein, W., Mathur, N.D., and Scott, J.F. (2006). Multiferroic and magnetoelectric materials. Nature 442 (7104): 759–765. Available at: http://www .ncbi.nlm.nih.gov/pubmed/16915279. Gajek, M., Bibes, M., Fusil, S. et al. (2007). Tunnel junctions with multiferroic barriers. Nature Materials 6 (4): 296–302. Available at: http://www.ncbi.nlm.nih .gov/pubmed/17351615. Julliere, M. (1975). Tunneling between ferromagnetic films. Physics Letters A 54 (3): 225–226. Lam, K.H., Lo, C.Y., Dai, J.Y. et al. (2011). Enhanced magnetoelectric effect in a stress-biased lead magnesium niobate-lead titanate single crystal/Terfenol-D alloy magnetoelectric sensor. Journal of Applied Physics 109 (2): 024505. Available at: https://doi.org/10.1063/1.3536636. Nan, C.-W., Bichurin, M.I., Dong, S. et al. (2008). Multiferroic magnetoelectric composites: historical perspective, status, and future directions. Journal of Applied Physics 103 (3): 031101. Available at: https://doi.org/10.1063/1.2836410. Wang, Y., Or, S.W., Chan, H.L.W. et al. (2008). Enhanced magnetoelectric effect in longitudinal-transverse mode Terfenol-D/Pb(Mg1/3 Nb2/3 )O3 –PbTiO3 laminate

229

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composites with optimal crystal cut. Journal of Applied Physics 103 (12): 124511. Available at: https://doi.org/10.1063/1.2943267. Yau, H.M., Yan, Z.B., Dai, J.Y. et al. (2015). Low-field switching four-state nonvolatile memory based on multiferroic tunnel junctions. Scientific Reports 5: 12826. Available at: http://www.nature.com/doifinder/10.1038/srep12826. Zhang, Y., Li, Z., Nan, C.W. et al. (2008). Demonstration of magnetoelectric read head of multiferroic heterostructures. Applied Physics Letters 92 (15): 152510. Available at: https://doi.org/10.1063/1.2912032.

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11 Ferroelasticity and Shape Memory Alloy Shape memory alloys (SMAs), or generally being called shape memory materials (SMMs) since some polymers also have shape memory characteristics, can be defined as a class of materials having the ability to “memorize” or retain their previous shape in response to stimulus such as thermal and mechanical or even magnetic variations. To be a member in the family of “ferroic,” SMA is characterized as ferroelastic exhibiting hysteresis loop in the curves of strain–stress, strain–temperature, or strain–magnetic field. Analogous to the other two family members of ferromagnetic and ferroelectric, ferroelastic materials also present domain structures, but instead of electrical and magnetic orderings, the domains are segregated regions of martensite variants. Not only SMA and magnetic SMAs but also many ferroelectric and ferromagnetic materials are ferroelastic; however, the desired devices as introduced in previous chapters are mainly related to ferroelectric and ferromagnetic switching. Many metal oxide materials such as SrTiO3 and LaAlO3 also exhibit ferroelasticity, but the range is relative too small for engineering application. One of the first full ferroelastic hysteresis loops (Figure 11.1) was seen in 1976 for the prototypic material Pb3 (PO4 )2 (Salje 1993; Salje and Hoppmann 1976) and was introduced in the review of ferroelastics (Salje 2012).

11.1 Shape Memory Alloy SMAs have drawn significant attention and interest since the end of last century in a broad range of engineering applications such as automotive, aerospace, robotic, and biomedical devices as actuators. The shape memory (SM) process belongs to solid-state displacive phase transition between austenite and martensite phases and can generate strains up to 10%, making it a fascinating candidate for actuators or even sensors in a smart system. There are a number of materials, including alloys composited of nickel–titanium (NiTi), copper–aluminum–nickel (CuAlNi), copper–zinc– aluminum (CuZnAl), and iron–manganese–silicon (FeMnSi), exhibiting shape memory effect. This chapter will introduce the fundamentals, device applications, and modeling of SMA as well as summary of recent advances and new application opportunities.

Ferroic Materials for Smart Systems: From Fundamentals to Device Applications, First Edition. Jiyan Dai. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

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11 Ferroelasticity and Shape Memory Alloy

Pb

P

b

Strain (10–4)

c Stress (bars) –4

–3

–2

–1

1

2

3

4

Figure 11.1 Ferroelastic hysteresis and atomic switching in Pb3 (PO4 )2 . The structural mechanism for ferroelasticity originates from the shift of the Pb atom away from the center of its oxygen cage toward one pair of oxygen ions. This leads to a monoclinic distortion of the crystal structure that can be inverted or rotated (in the direction of the arrows) under external stress. Source: Reprinted with permission from Salje (2012).

11.1.1

SMA Phase Change Mechanism

The ferroelastic characteristic of a SMA is the consequence of phase transition from a high-symmetry austenite phase (cubic structure) to a low-symmetry martensite phase (a distorted cubic structure). This phase transformation can be induced by heat, stress, or electromagnetic fields (for example magneto SMA). Nitinol (NiTi alloy) is the most typical and commercially used SMA that has superior structure and memory capabilities originated from its austenite phase (Figure 11.2a) and martensite phase (Figure 11.2b) at high and low temperatures, respectively. It should be noticed that the volume of a primitive cell (unit cell) of the martensite structure is double of the austenite structural unit cell; to specify, the lattice constants b and c in the martensite structure are close to √ 2a, where a is the lattice constant of the pseudocubic unit cell of austenite structure. To illustrate the nature of phase transformations, we consider the austenitic NiTi lattice cell depicted in Figure 11.3. In this cubic unit cell (where Ni ions occupy the A site and Ti the B site), we can identify six {110} atomic planes, and in response to an applied stress, each of the six face-diagonal planes can shear or shift into four variants. This yields 24 possible variants making the low temperature martensite phase easy to change shape under mechanical stress. All these martensite variants will change back to austenite cubic structure when temperature increases, and as a consequence, the shape in high-temperature phase is resumed. This is the working mechanism of SMAs.

11.1 Shape Memory Alloy

Ni

Ti (a)

(b)

Figure 11.2 (a) Austenite phase (B2) and (b) martensite lattice structure (B19’) of nickel–titanium, where the plane outlined by dash line represents the {100} planes in the primitive cell of the martensite structure.

Figure 11.3 Lattice cell of NiTi and four martensite variants developed from one face-diagonal plane.

The phase transformation between austenite and martensite phases is a first-order transition that can be dictated by the differential scanning calorimeter (DSC) measurement as shown in Figure 11.4. DSC is a thermoanalytical technique in which the difference in the amount of heat flow required to increase the temperature of a sample and a reference is measured as the function of temperature. Both the sample and reference are maintained at nearly the same temperature throughout the experiment. This DSC method shows the amount of heat released or absorbed by a sample of the alloy as it is cooled or heated through its phase transformations. The extra amount of heat during the phase transformation is called latent heat, and the phase change involving latent heat is classified as first-order phase transition. 11.1.2

Nonlinearity in SMA

The shape memory strain response of nitinol is highly nonlinear as illustrated in Figure 11.5, where the length is the spontaneous length change during phase transition. The control of mechanical properties is through the manipulation of phase transition at temperatures of Ms , Mf , As , and Af , where Ms and Mf are the martensite start and finish temperatures and As and Af are the austenite start and finish temperatures.

233

11 Ferroelasticity and Shape Memory Alloy

Heat flow

Cooling

Mf

Ms As

Af

Heating

Temperature

Figure 11.4 A schematic diagram of a typical differential scanning calorimeter (DSC) curve for NiTi shape memory alloy showing presence of latent heat during phase transformation.

100

Mf

As

ΔL

0

Ms

Length

Martensite (%)

234

Af

Temperature

Figure 11.5 Typical transformation–temperature curve of NiTi SMA under constant stress as it is cooled and heated showing hysteresis.

The complete shape memory process showing strain responses to temperature and stress is illustrated in Figure 11.6. The combination of temperature and stress-induced phase transformations provides SMAs with their memory capabilities. One can see that at low-temperature martensite phase, large strain is developed under stress and there is a residual strain from detwinned structure when stress is released. When the detwinned structure is heated above Af , the residual strain is recovered and the alloy returns to its original shape, i.e. the shape memory effect.

11.1 Shape Memory Alloy

Stress Stress-induced strain

Mf

Strain

Cooling Af Heating Temperature

Figure 11.6 Shape memory effect in a uniaxial SMA.

11.1.3

One-Way and Two-Way Shape Memory Effect

The shape memory effect illustrated in Figure 11.6 is actually a one-way shape memory effect since there are only two different shapes (the one at higher temperature can be memorized) and the shape change is realized only by stress (or called loading) at lower temperature, while the original shape at higher temperature will be resumed upon heating. This can be demonstrated more clearly in Figure 11.7a. One can notice that for the one-way shape memory effect, cooling from high temperatures does not cause a macroscopic shape change, even there is an austenite to martensite phase transition. The so-called two-way shape memory effect is the effect that the material remembers two different shapes: one at low temperatures and the other one at high temperatures, i.e. without involving any mechanical stress; these two different shapes can change from one to the other only by heating and cooling. As illustrated in Figure 11.7b, the reason to cause this two-way shape memory effect is due to the mechanical stress-induced plastic deformation (crystal defect) during the training process. The two-way shape memory effect should be considered during device application. A SMA should be trained before usage, during which overloading results in accumulated plastic deformation in the martensite structure and even in the parent austenite structure due to the generation of crystal defect such as staking dislocations. This property gives rise to a spontaneous shape change of SMA in both heating and cooling cycles. Therefore, one can see that the two-way shape memory effect refers to the reversible and spontaneous shape change of SMA during thermal cycling. 11.1.4

Superelastic Effect (SE)

Besides the shape memory effect, the SMAs also have a property known as superelastic effect (SE), making SMA different from the other metal alloys. The

235

11 Ferroelasticity and Shape Memory Alloy

Stress/load

He

ati

ng

Loading

Cooling Heating

(a)

Temperature

He

Stress/load

236

ati

ng

Loading

Cooling Heating

(b)

Temperature

Figure 11.7 Illustration of (a) one-way and (b) two-way memory effects.

SE of Ni–Ti alloys is associated with the recovery of the deformation during unloading. Figure 11.8 illustrates the superelastic behavior during loading and unloading above Af associated with stress-induced martensitic transformation and reversal to austenite upon unloading. 11.1.5

Application Examples of SMAs

SMA can be used for braces and dental arch wires, where the SMA maintains its shape at a constant temperature; and because of the superelasticity of the memory metal, the wires retain their original shape after stress has been applied and removed. The superelastic property of SMAs has also been applied in eyeglass frames that can be bent back and forth with retained shape, demonstrating its superelasticity as shown in Figure 11.9.

11.2 Ferromagnetic Shape Memory Alloys

Figure 11.8 Stress–strain curve illustrating superelastic behavior of SMA when T > Af .

Stress

Loading

Unloading Strain

Figure 11.9 Glasses made of shape memory alloy. Source: Adapted from Burchett (2019).

SMA has also been used in fire security and protection systems, where the SMA should have an immediate action to shut down or open the valve in the presence of increased heat from fire. This can greatly decrease leakage-induced problems in industries that involve petrochemicals, semiconductors, pharmaceuticals, and large oil and gas boilers. Many new applications of SMA are being developed, and even polymer-based SMMs are emerging. These are beyond the scope of this book.

11.2 Ferromagnetic Shape Memory Alloys Ferromagnetic shape memory alloys (FSMAs) are an attractive new class of magneto-active materials that combine the properties of ferromagnetism with shape memory effect. These materials exhibit giant magnetic field-induced strain (MFIS) in the martensitic phase, i.e. when a magnetic field is applied, the shape of the material will change significantly. The mechanism of MFIS originates

237

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11 Ferroelasticity and Shape Memory Alloy

from the redistribution or reorientation of martensitic twin variants under an external magnetic field. The thermally driven SMA has a very slow response of less than 1 Hz. This is attributed to the slow process of heating and cooling. However, the combination of ferromagnetic properties and martensitic transformation in FSMA materials allows realization of magnetically driven shape memory effect in a frequency up to kHz regime. Not only different in mechanisms compared with those of magnetostrictive, piezoelectric, and electrostrictive, the magneto SMA materials also have greater obtainable strains. This presents FSMAs with great potential in various smart structural applications, such as sensors and actuators, which require a large strain and/or fast response. Ullakko et al. presented the idea of redistribution or reorientation of martensitic twin variants under an external magnetic field and also experimentally confirmed the effect of MFIS in 1996 (Ullakko et al. 1996). They reported MFIS of nearly 0.2% observed on unstressed Ni–Mn–Ga single crystals in ∼640 kA/m (79.6 A/m = 1 Oe) magnetic field along the [001] direction at 265 K. This discovery was a major step for the development of FSMA materials, especially in the MFIS field. Since then, interest in this phenomenon as well as these alloys has gained increased attention. Also, several additional material systems, such as Ni–Mn–Fe–Ga, Fe–Pt, Fe–Pd, Co–Ni–Ga, and Ni–Mn–Al, have been tested. After the report of 0.2% MFIS, extensive works on Ni–Mn–Ga intermetallic compounds have been carried out. James et al. obtained a 0.5% reversible strain in an off-stoichiometric Ni–Mn–Ga single crystal (James et al. 1998). Tickle et al. reported separately a reversible strain of 1.3% and maximal extensional strain of 4.3% in Ni51.3 Mn24.0 Ga24.7 (numbers indicate at.%) single crystals (Tickle et al. 1999; Tickle and James 1999). In the next few years, there are a lot of further development and nearly 10% strain has been obtained in modulated martensitic single crystal (Sozinov et al. 2002). For dynamical actuation in these alloys, Henry et al. demonstrated ac-MFIS of 2.6% up to 2 Hz under a bias stress of 1.1–1.5 MPa (Henry et al. 2002). The cyclic MFIS of Ni–Mn–Ga element under a pulse magnetic field was studied by Marioni et al. and Suorsa et al (Suorsa et al. 2004). One key advantage of Ni–Mn–Ga FSMAs is its giant MFIS under an external magnetic field. Different from traditional magnetostrictive materials (e.g. Terfenol-D) that generate strain through magnetic domain rotation, the mechanism of MFIS is attributed to the redistribution or reorientation of martensitic twin variants, as suggested initially by Ullakko in 1996. Thus, the formation of martensitic twin variants and motion of twin boundaries under an external magnetic field are significantly important to understand the mechanism of MFIS in FSMA materials. In this section, the origin and nature of martensitic twin variants as well as the mechanism of twin variants motion under an external magnetic field are presented. 11.2.1

Formation of Twin Variants

Ni–Mn–Ga alloys exhibit a structural (martensitic) phase transformation from a high-symmetry cubic phase to a low-symmetry martensitic phase (e.g. tetragonal structure). Figure 11.10a shows atomic structure of Ni2 MnGa in

11.2 Ferromagnetic Shape Memory Alloys

Ni

Mn

Ga

[110]

[110]

[001]

[001]

– [11 0] – [11 0]

(a)

– [11 0]

– [11 0]

(b)

(c)

Figure 11.10 Illustrations of Ni2 MnGa structured models: (a) 3D cubic structure in the high-temperature austenitic phase with two kinds of stacking (110) plane in cubic phase. Five-layered martensite resulted from shear on these {110}c planes in the (b) cubic phase and (c) modulation. Source: Adapted from Ge (2007).

high-temperature austenite phase. Figure 11.10b,c illustrate how a martensitic phase is developed from a cubic austenite phase. The martensitic transformation is illustrated in Figure 11.11a in two dimensions. In the cubic structure, denoted as a gray block, the two crystallographic axes (three axes in three dimensions) are equivalent. However, a tetragonal martensite has a long axis (a axis) and a short axis (c axis), which are referred to as the hard axis and easy axis (the magnetization is oriented along the easy axis), respectively. Thus, the martensitic structure has two (3 × 3 dimensions) possible tetragonal variants, i.e. two different deformation directions, during the tetragonal deformation. In current case, the easy axis is aligned along c crystallographic axis in the tetragonal unit cell. The arrows “←” and “↑” in the two variant states denote the orientation of easy axes, which are referred to as variant I and variant II, respectively. During martensitic transformation by a shear stress on the closed packed {110} cubic planes, usually a twinned structure can be formed during the tetragonal deformation as illustrated in Figure 11.11b. The twined structure is actually a pair of unit cells in a lattice that the twin variants are oriented as mirror images. The mirror symmetry line of the two variants is the twin boundary. In this case, twin variants have two preferred orientations and can change their directions under an external magnetic field or external mechanical load, and it is the reorientation of twin variants that causes a macro-level deformed strain in a sample. From the microstructure viewpoint, the martensitic twin variants and twin boundaries are correspondence to magnetic domains and domain walls.

239

240

11 Ferroelasticity and Shape Memory Alloy

a

a

a

c

Variant I

a c Variant II (a)

(b)

Figure 11.11 Schematic illustrations of (a) martensitic transformation and (b) twinned structure in two dimensions. Arrows indicate the orientations of the variants (or easy axis). Source: Adapted from Enkovaara et al. (2004). Figure 11.12 A typical twin variants with the 90∘ and the 180∘ magnetic domain structures. Source: Ge et al. (2004). Reprinted with permission of AIP Publishing.

Variant I

Variant II

[100] [001]

[001]

20 μm [100]

A scanning electron microcopy image presenting this microstructural feature is shown in Figure 11.12. This is a typical pattern of the two variant, where the magnetic domains are oriented along the easy axes of the two variants forming a staircase-like pattern. The twin boundaries form 90∘ domain walls, and the 180∘ domain walls are formed in each variant manifested by the black and white contrast (Ge et al. 2004). (A question, why different variants show different contrast in scanning electron microscope (SEM) image?) In a Ni–Mn–Ga single crystal, the motion of martensitic twin boundaries results in magnetic domains evolution under an external magnetic field. As shown in Figure 11.13, initially the single crystal is preset to be a single variant configuration that can be achieved by applying an external magnetic field or an external compressive mechanical load along the sample length direction. Figure 11.13a depicts the 180∘ magnetic domains in a scenario that the overall

11.2 Ferromagnetic Shape Memory Alloys

Micro-scale Large strain

Single variant I H=0

(a) Applying magnetic field

Twin variants Hc < H < Hs Increasing magnetic field

(b) Single variant II H ≥ Hs

(c)

Figure 11.13 Schematic illustrations of a Ni–Mn–Ga single crystal under increasing applied magnetic field levels. Schematics (a)–(c) depict the respective microstructural arrangements of martensitic variants and magnetic domains. Source: Adapted from Kiefer and Lagoudas (2005).

magnetization is zero without external magnetic field applied. When an external magnetic field is applied vertically to the sample length direction (horizontal direction), the magnetic domains tend to align in the direction of the applied field. When a magnetic field is applied but below a critical field H c , the applied field only induces a rotation of magnetization away from the original direction. But when the applied field increases beyond H c , i.e. the magneto-crystalline anisotropic energy (MAE) is higher than the energy required moving a twin, the twin variants are movable and variant II will grow accompanied by reducing the other twin variants. This process generates an axial reorientation strain, and the magnetization in variant II is eventually oriented fully along the applied field direction. Once beyond the saturation magnetic field H s , variant II becomes dominant and completely replaces variant I. Consequently, as illustrated in Figure 11.13c, the magnetic domains align themselves completely along the direction of the applied field. The substitution of variant I with variant II results in an elongation in that direction and contraction in the perpendicular directions. Essentially, the motion of the martensitic twin variants can be seen as the change of volume fraction of the two variants. Thus, twin variants reorientation strain, i.e. MFIS, is quantified by considering the changes in crystallographic axes (a and c axes).

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11 Ferroelasticity and Shape Memory Alloy

7 6 5 Strain (%)

242

4 3

c

c

2

a

a

1 0 0

100

200

400 300 H (kA/m)

500

600

700

Figure 11.14 MFIS dependence of applied magnetic field illustrates the change of twin variants. Source: Adapted from Sozinov et al. (2003).

Figure 11.14 shows the relationship between MFIS and the corresponding change of twin variants; where the coercive field H c and saturation field H s are about 250 and 580 kA/m, respectively, and the maximum MFIS is about 5.5%. In addition, the maximum strain is predictable from the theoretical lattice constant with the equation 𝜀0 = (1 − c∕a) where c and a are the short (or easy axis) and long (or hard axis) lattice constants, respectively. 11.2.2

Challenges for Ni–Mn–Ga SMA

Up to date, ferromagnetic shape memory Ni–Mn–Ga single crystals have shown giant dc-MFIS in the order of 6–10%, making them suitable for some applications (e.g. magnetic field sensors, actuators, and converters). However, there are a few challenges that must be addressed before this FSMA can be largely used in engineering applications. First, large MFIS is obtainable only in single crystals; however, the growth of device-level single crystals is of high cost. Besides, the Ni–Mn–Ga single crystals present poor formability and brittleness, making it difficult to shape the material into required form. Alternatively, some low cost magnetic shape memory alloy (MSMA) such as NiAl etc. may be a good candidate for applications, but of course, their MFIS is still much lower than Ni–Mn–Ga single crystals. Second, even a significant irreversible strain is exhibited under quasi-static actuations, only a very small reversible strain is available. From the practical applications point of view, large cyclic strain in ac actuation is desired; therefore, this disadvantage of significant irreversible strain makes Ni–Mn–Ga single crystals not suitable for application of high cyclic strain.

11.2 Ferromagnetic Shape Memory Alloys

Third, data presented in MFIS curves indicate that a relatively large critical field is needed to initiate shape transformation. Consequently, this relatively large critical field will increase the size of the drive coil/core, which significantly limits the application in realizing miniature active systems. Fourth, the output mechanical stress by Ni–Mn–Ga single crystals driven by MFIS is actually very small (6–10 MPa). This means that the actuation stress in practical application is small, which is difficult in meeting the requirement for large actuation force. 11.2.3

Device Application of MSMA

Similar to the composite device of piezoelectric/magnetostrictive, a heterostructure can also be formed by one layer of piezoelectric PMN-PT crystal sandwiched between two layers of ferromagnetic shape memory Ni–Mn–Ga crystal (Zeng et al. 2010). This device can be a magnetic field sensor utilizing its magnetoelectric (ME) effect or an electric field sensor utilizing its converse ME effect. In Zeng’s work, coefficient of electric field-induced magnetization 𝛼 B , called converse magnetoelectric (CME) effect, of the heterostructure attains its maximum value in the martensitic–austenitic phase transformation temperature 28–39 ∘ C. Giant resonance 𝛼 B of 18.6 G/V under a very low bias magnetic field of 150 Oe has been obtained. Figure 11.15 shows the schematic (a)

Search coil Ni–Mn–Ga Vac PMN–PT

Vind 0.9 MP

MAPT

(b)

AP

Ms

αB (G/V)

0.6

Af 0.3 Heating Mf 0.0 20

25

Cooling

As 30

35

40

45

50

55

60

T (°C)

Figure 11.15 (a) Schematic diagram of the proposed heterostructure. (b) CME coefficient (𝛼 B ) dependence on temperature at an applied ac electric voltage V ac of 20 V peak, a frequency f of 1 kHz, and a magnetic bias field Hbias of 150 Oe upon heating and cooling. Source: Zeng et al. (2010). Reprinted with permission of AIP Publishing.

243

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11 Ferroelasticity and Shape Memory Alloy

diagram of the proposed heterostructure fabricated by bonding one layer of piezoelectric PMN-PT crystal plate between two layers of ferromagnetic shape memory Ni–Mn–Ga crystal plates. The two Ni–Mn–Ga plates have a chemical composition of Ni49.2 Mn29.6 Ga21.2 , dimensions of 11 × 5 × 0.5 mm3 , and the two major surfaces are parallel to the (011) plane.

References Burchett, S. (2019). Global Nitinol Shape Memory Alloy Market 2019 Top Company Profiles, Competition Status, Trends, Industry Growth, And Forecast 2019–2025. themarketfact.com. Available at: https://themarketfact.com/2019/05/02/globalnitinol-shape-memory-alloy-market-2019/ (accessed 21 August 2019). Enkovaara, J., Ayuela, A., Zayak, A.T. et al. (2004). Magnetically driven shape memory alloys. Materials Science and Engineering: A 378 (1): 52–60. Available at: http://www.sciencedirect.com/science/article/pii/S0921509303015181. Ge, Y. (2007). The Crystal and Magnetic Microstructure of Ni–Mn–Ga Alloys. Helsinki University of Technology. Ge, Y., Heczko, O., Söderberg, O., and Lindroos, V.K. (2004). Various magnetic domain structures in a Ni–Mn–Ga martensite exhibiting magnetic shape memory effect. Journal of Applied Physics 96 (4): 2159–2163. Available at: https:// doi.org/10.1063/1.1773381. Henry, C.P., Bono, D., Feuchtwanger, J. et al. (2002). AC field-induced actuation of single crystal Ni–Mn–Ga. Journal of Applied Physics 91 (10): 7810–7811. Available at: https://aip.scitation.org/doi/abs/10.1063/1.1449441. James, R.D., Tickle, R., and Wuttig, M. (1998). Large field-induced strains in ferromagnetic shape memory materials. Materials Science and Engineering. A, Structural Materials: Properties, Microstructure and Processing 273–275: 320–325. Kiefer, B. and Lagoudas, D.C. (2005). Magnetic field-induced martensitic variant reorientation in magnetic shape memory alloys. Philosophical Magazine 85 (33–35): 4289–4329. Available at: https://doi.org/10.1080/14786430500363858. Salje, E.K. (1993). Phase Transitions in Ferroelastic and Co-elastic Crystals. Cambridge, UK: Cambridge University Press. Salje, E.K.H. (2012). Ferroelastic materials. Annual Review of Materials Research 42 (1): 265–283. Available at: https://doi.org/10.1146/annurev-matsci-070511155022. Salje, E.K. and Hoppmann, G. (1976). Direct observation of ferroelasticity in Pb3 (PO4 )2 -Pb3 (VO4 )2 . Materials Research Bulletin 11 (12): 1545–1549. Available at: http://www.sciencedirect.com/science/article/pii/0025540876901070. Sozinov, A., Likhachev, A.A., Lanska, N. et al. (2002). Giant magnetic-field-induced strain in NiMnGa seven-layered martensitic phase. Applied Physics Letters 80 (10): 1746–1748. Available at: https://doi.org/10.1063/1.1458075. Sozinov, A., Likhachev, A.A., Lanska, N. et al. (2003). Effect of crystal structure on magnetic-field-induced strain in Ni–Mn–Ga. Proc. SPIE 5053, Smart Structures and Materials 2003: Active Materials: Behavior and Mechanics 5053: 586–594.

References

Suorsa, I., Pagounis, E., and Ullakko, K. (2004). Magnetic shape memory actuator performance. Journal of Magnetism and Magnetic Materials 272–276 (3): 2029–2030. Available at: http://www.sciencedirect.com/science/article/pii/ S0304885303024922. Tickle, R. and James, R.D. (1999). Magnetic and magnetomechanical properties of Ni2MnGa. Journal of Magnetism and Magnetic Materials 195 (3): 627–638. Available at: http://www.sciencedirect.com/science/article/pii/ S0304885399002929. Tickle, R., James, R.D., Shield, T. et al. (1999). Ferromagnetic shape memory in the NiMnGa system. IEEE Transactions on Magnetics 35 (5): 4301–4310. Ullakko, K., Huang, J.K., Kantner, C. et al. (1996). Large magnetic-field-induced strains in Ni2MnGa single crystals. Applied Physics Letters 69 (13): 1966–1968. Zeng, M., Or, S.W., and Chan, H.L.W. (2010). Effect of phase transformation on the converse magnetoelectric properties of a heterostructure of Ni49.2 Mn29.6 Ga21.2 and 0.7PbMg1/3 Nb2/3 O3-0.3 PbTiO3 crystals. Applied Physics Letters 96 (18): 182503. Available at: https://doi.org/10.1063/1.3427388.

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247

Index a acoustic wave 3, 144, 147, 154, 155, 158 anisotropic magnetoresistance 171 anomalous Hall effect (AHE) 191–192 antiferroelectricity (AFE) 19, 33, 197 antiferroelectric material 34 antiferromagnetic coupling 175 antiferromagnetic ordering 202 artificial intelligence (AI) 1, 57 artificial neural networks (ANN) 1, 2, 57

b bandwidth (BW) 144, 147–148, 150, 162, 175, 180, 218 BaTiO3 phase transition 20, 21, 27 BiFeO3 (BFO) 49, 89, 223 ferroelectric layer 1 switching mechanism 204 Bloch type ferromagnetic domain walls 171 Bloch wall 171 Bode diagram 153 Brinkman’s model 52 bulk acoustic wave (BAW) 155 butterfly loop 78, 79, 115

c colossal magnetoresistance (CMR) 171, 175–176 conventional ferroelectrics 109, 110 convergent beam electron diffraction (CBED) 91

converse magnetoelectric (CME) effect 243 Curie–Weiss law 20–23, 76

d depolarization field 47–49, 55, 96 device application, of ferroelectrics electrocaloric effect 67–68 ferroelectric random access memory (FeRAM) 47–49 ferroelectric tunnel junction (FTJ) 50 microwave device 63–65 photovoltaics 65–67 dielectric constant 8, 16, 17, 21–23, 25, 26, 29–33, 35, 36, 63, 74 dielectric tunability 30–32 differential scanning calorimeter (DSC) 233, 234 direct tunneling (DT) 51–53 domains, and phase field calculation 41–42 domain structure 15, 16, 28, 38, 39, 41, 56, 77, 78, 84, 85, 87–89, 96, 99, 169–171, 189, 201, 203, 231 dual frequency resonance tracking (DFRT) technique 85 Dzyaloshinskii–Moriya interaction 203, 204

e Ehrenfest classification 27 elastic compliance 127–128, 135, 138, 210

Ferroic Materials for Smart Systems: From Fundamentals to Device Applications, First Edition. Jiyan Dai. © 2020 Wiley-VCH Verlag GmbH & Co. KGaA. Published 2020 by Wiley-VCH Verlag GmbH & Co. KGaA.

248

Index

electric polarization 4, 15–18, 20, 59, 60, 67, 73, 87, 97, 114, 123, 124, 127, 197–199, 201, 206, 217, 226 electrocaloric effect 67–68 electromechanical coupling factor 37, 123, 124, 128–132, 138–140, 217 electromechanical sensors 37 electroresistance (ER) 50–55 electrostatic force, in piezoresponse force microscopy 83–84

f feedback control system 1, 3 ferroelectric and anti-ferromagnetic (FE-AFM) 223 ferroelectric-based memory 1, 50 ferroelectricity HfO2 and ZrO2 fluorite-oxide thin films 106–113 size limit 95–96 two-dimensional (2D) materials 96–99 ferroelectric materials 1 antiferroelectric PbZrO3 33–36 BaTiO3 to SrTiO3 29–32 and device application 10–12 ferroelectric RAM (FeRAM) 8–9 infrared sensor 7 magnetoelectric (ME) effect 10, 11 Pb(Zrx Ti1–x )O3 (PZT) 35–36 PbTiO3 to PbZrO3 32–33 piezoelectric device applications 6–7 spin-transfer torque memory 9 ferroelectric phase transition 86 Landau free energy and Curie–Weiss law 21–23 Landau theory of first order phase transition 23–26 ferroelectric polarization 1, 5, 17, 21, 47–50, 74, 77, 84, 91, 99, 108, 112, 197, 200, 201, 224–229 ferroelectric random-access memory (FeRAM) 1, 47–49, 109, 201, 223 ferroelectrics definition 15

dielectric constant 16, 17 P–E loop 15–16 piezoelectric 17–18 pyroelectricity 18 soft phonon mode 19–21 structure-induced phase change 18–19 ferroelectric switching 28–29, 54, 78, 97, 204 ferroelectric tunnel junction (FTJ) direct tunneling 52 electroresistance (ER) 50 ferroelectric memory 57 memristive behavior 56 metal-ferroelectric-semiconductor (MFS) heterostructure 55 nanoscale characterization techniques 50 polarization switching 53 thermionic emission 52 ferroelectric tunnelling-based resistive random-access memory 1 ferromagnetic materials, domain and domain wall 169–171 ferromagnetic shape memory alloys (FSMAs) 237–244 ferromagnetics, paramagnetic phase transitions 168 field effect transistor (FET) structure 1, 49, 50, 95, 223 figure of merit (FOMs) 60 finite element modeling (FEM) 3, 81, 217 Fowler–Nordheim tunneling (FNT) 51, 52

g giant magnetoelastic coupling 201 giant magnetoresistance (GMR) 171–175 Gibbs free energy 27 Google’s AlphaGo 1 Gruverman’s model 52

h hafnium dioxide (HfO2 ) 106 half-metallic manganites 176

Index

Hall effect anomalous Hall effect (AHE) 191–192 ordinary Hall effect 191 spin Hall effect (SHE) 192–193 Heusler alloys 176 high-angle annular dark-field (HAADF) 89 high-frequency resonance measurement 130–132 hybrid perovskites 73, 84, 114–117

i impedance-frequency spectrum 132, 133, 135 industrial-compatible fabrication process, 111 infrared radiation 7, 58, 60, 62, 63 in-plane polarization (IPP) 80–86, 98, 100 interdigital electrode 155 interdigital transducers (IDTs) 154, 159 Ising type ferroelectric domain walls 171

j Jahn–Teller effect 200 Julliere’s model 176, 228

k Krimholtz–Leedom–Matthae (KLM) model 145

l Landau free energy and Curie–Weiss law 21–23 Landau theory 199 of first order phase transition 23–25 of second order phase transition 26–28 lead-free ferroelectrics 161 LiNbO3 film deposition 158 longitudinal vibration mode 217, 218

m magnetic ferroelectrics

197, 199–200

magnetic field-induced strain (MFIS) 224, 237 magnetic field sensor 9–10, 175, 217, 223, 242, 243 magnetic force microscopy (MFM) 188–189 magnetic shape memory alloy (MSMA), device application 243–244 magnetic tunnel junction (MTJ) element 176, 224 magneto-crystalline anisotropic energy (MAE) 241 magneto-elastic coupling 201 magnetoelectric (ME) 10, 11, 199 automated measurement system 219 composite materials 205–207 L–T mode 218 magnetic bias field 220 magnetoelastic coupling factor 217 magneto-elasto-electric coupling 219 preload stress 221–223 PZT/CFO multilayered heterostructures 207–209 Terfenol-D fiber 218 magneto-optical Kerr effect (MOKE) 189–191 magnetoresistance effect anisotropic magnetoresistance 171–172 colossal magnetoresistance (CMR) 175–176 giant magnetoresistance (GMR) 172–175 spin-transfer torque random access memory (STT-RAM) 177–178 tunnel magnetoresistance (TMR) 176–178 magnetoresistive random-access memory (MRAM) 9, 172, 177 magnetostrictive effect 184 properties of Terfenol-D 180–183 ultrasonic transducer 183–184 Mason model 145 mechanical quality factor 123, 141, 151, 153

249

250

Index

medical ultrasound imaging system 6 memory device multiferroic thin films 223–224 multiferroic tunneling 224–229 metal-ferroelectric-semiconductor (MFS) heterostructure 55 Metglas 2605SC 179, 185 microwave device 30, 63–65 molecular materials 104 multiferroic thin films 223–224 multiferroic tunneling 224–229

n nanoferromagnets 100 Néel temperature 166, 201 Néel type ferromagnetic domain walls 171 Néel wall 171 Ni-Mn-Ga single crystals 238, 240, 242, 243 non-destructive test (NDT) 143, 171 Nyquist diagram 151–153

o one transistor-one capacitor (1T1C) structure 47 ordinary Hall effect 191–193 out-of-plane polarization (OPP) 80–86

p paramagnetic phase transitions 166, 168, 179 P–E hysteresis loop ideal linear response of capacitor 74, 75 modified Sawyer–Tower circuit 73 photovoltaics 65–67, 84, 114 piezoelectric effect constants measurement charge constant 125 dielectric permittivity 127 elastic compliance 127–128 electromechanical coupling factor 128–132 voltage constant 126–127 Young’s modulus 127

direct and converse piezoelectric effects 123 equivalent circuit 132–135 extensional vibration mode of long plate 138–139 length extensional mode of rod 135–138 mechanical quality factor 141 properties of 123, 124 radial mode in thin disc 141 thickness mode of thin disc/plate 140–141 thin shear piezoelectric plate 139 piezoelectric-miniature ultrasonic transducer (PMUT) 149 piezoresponse force microscopy (PFM) 188 imaging mechanism 77–80 in-plane polarization (IPP) 80–86 out-of-plane polarization (OPP) 80–86 perspectives of techniques 84–86 polar nano regions 30, 38–40 polymer materials 205 preload stress 221–223 proportional-integrate-derivative (PID) feed-back control system, 3 pulsed laser deposition (PLD) 112, 207 pyroelectric coefficient 58–62, 67 pyroelectric infrared sensor 58–60 pyroelectricity 5, 18 PZT/CFO multilayers, ferroelectric properties, 209–211

q quantum effect 188 quantum tunneling 50

r Rayleigh wave, 151, 155, 156 Redwood model 145 relaxor ferroelectrics 29, 36–41 characteristics 36–37 morphotropic phase boundary (MPB) of PMN-PT crystal 40–41 PMN-xPT single crystal 37–38 polar nano regions 38–40

Index

resonance enhancement Rochelle salt 29, 104

85, 86

piezoelectric property considerations 157–159 symmetric Bragg peak 86

s selected area electron diffraction (SAED) 87–91 shape memory alloys (SMA) 10 ferromagnetic shape memory alloys (FSMAs) 237–244 nonlinearity 233–235 one-way and two-way shape memory effect 235 phase change mechanism 232–233 superelastic effect 235–236 twin variants, formation of 238–242 single phase multiferroic materials antiferromagnetic BiFeO3 (BFO) 201 antiferromagnetic ordering 202 Dzyaloshinskii–Moriya interaction 204 epitaxial BFO films 201 giant magneto-elastic coupling 201 oxygen octahedron 201 rare-earth manganites and ferrites 200 spin canting 203 smart system actuators 3 component 2 proportional-integrate-derivative (PID) feed-back control system 3 sensors 3 transducers 3–4 S-parameters 159 spin canting 203, 204 spin Hall effect (SHE) 192–193 spin-transfer torque random access memory 177–178 structural distortion 86 superelastic effect (SE) 235–236 surface acoustic wave (SAW) device acoustic wave 155–157 characterization 159–161 interdigital electrode 155

t temperature-dependent dielectric permittivity measurement 76–77 Terfenol-D 179–183, 185, 206, 218, 219, 221, 222 thermionic emission (TE) 51, 52 tip-nanofiber harmonic oscillation system 87 tunnel magnetoresistance (TMR) 171, 176–178, 228 two-dimensional (2D) materials atomic-thick SnTe 98 Hamiltonian and parameters 98 in-plane polarization 98 quantum phenomena 96 SnS/SnSe monolayer 97 two-way shape-memory effect 235

u ultrasonic transducers bandwidth (BW) 147, 148 medical application 149 resolution 148, 149 sensitivity of transducer 148 structure of 143–145 theoretical models of 145–146 ultrasound motor (USM) design of 153–154 energy conversion 151 Rayleigh wave 151 terminologies 151–153

v Van Dyke’s Model 132 vibrating sample magnetometer (VSM) 186–187 vortex formation of circular domains 99 in-plane polarization 100 nanoferromagnets 100 skyrmion in ferroelectrics 101–103

251

252

Index

toroidal domains 99 up and down polarization domains 101

x X-ray diffraction

38–40, 96, 110

y w wake-up effect

Young’s modulus 110, 111

127, 128, 209